Application of non-equilibrium statistical mechanics to the analysis of problems in financial markets and economy Andrey Sokolov Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy School of Physics The University of Melbourne September, 2014 Produced on archival quality paper
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Application of non-equilibrium statisticalmechanics to the analysis of problems in
financial markets and economy
Andrey Sokolov
Submitted in total fulfilment of the requirements
of the degree of Doctor of Philosophy
School of Physics
The University of Melbourne
September, 2014
Produced on archival quality paper
Abstract
This thesis contributes to a growing body of work in the emerging inter-
disciplinary field of econophysics, where tools and techniques of statistical
mechanics and other branches of physics are applied to problems in economics
and finance. The thesis examines the following four topics: 1) money flows in
the interbank networks, 2) wealth distributions in multi-agent exchange sys-
tems, 3) variability and dynamics of the foreign exchange markets via lattice
gauge theories, 4) memory loss and patterns of change in Abelian sandpiles
via hidden Markov models.
In economic and financial systems, interactions between the agents oc-
cur on a network, whose properties depend on the system in question. Such
networks are not static but rather dynamic in character, since the links com-
prising the network frequently depend on the actions of the agents as they
interact with one another. Moreover, the links of the network typically repre-
sent flows, e.g. money flows in financial networks, which further complicates
their study. These issues are addressed in the thesis in the case of the financial
networks of money flows between Australian banks.
In most advanced economies, including Australia, high-value transactions
in the banking system are settled in real time via the so called real-time gross
settlement systems, which are controlled by the central banks. Such systems
have been introduced in many countries in the last ten to twenty years in
order to diminish the liquidity risk in the banking system. As such systems
are computerized, the information pertaining to all transactions, including
their source, destination, and value, is recorded by the central banks and can
be used to investigate the properties and dynamics of the interbank flows.
The flows of payments in the interbank networks are not homogeneous but
possess an intricate structure that reflects the nature of various payments. In-
deed, some of the payments recorded by the central bank represent overnight
loans extended from one bank to another and the return payments made on
the following day. The work undertaken in the thesis examines the flows of
overnight loans and the flows of other payments separately. It is shown that
these two kinds of flows are not entirely independent. The flows of overnight
loans appear to counteract the imbalances in the bank’s reserve accounts cre-
ated by the flows of other payments. This is in accord with the dynamics
existing in the interbank money market, where the bank’s with the surplus of
reserves lend the surplus to the banks with depleted reserves.
Agent-based models are gaining popularity in the efforts to understand
economic and financial systems and their dynamics. The strength of these
models lies in the fact that they require the formulation of local rules of interac-
tion only without specifying the global constraints on the system’s behaviour,
which are often poorly understood. The simulations that use agent-based
models reveal the emergent behaviour of the economic and financial systems
that arises as a result of the collective actions of the agents following individ-
ual rules. In particular, agent-based models have been used to address the
problem of the wealth and income distribution in the economies. The work
conducted in the thesis examines one such model, referred to as the giver
scheme, where the rule of exchange stipulates that the transfer amount from
the giver to the receiver is equal to a fixed fraction of the giver’s wealth.
The giver model of asset exchanges is examined in the thesis by means
of multi-agent simulations. The system rapidly evolves to a steady state, in
which the distribution of wealth does not vary with time, even though agents
continue to exchange wealth. This model is amenable to the analysis based on
the master equation, which balances the influx and outflow of agents at every
wealth value. The thesis presents an investigation of the master equation by
means of the Laplace transform. A novel technique for calculating the wealth
distribution in the steady state is introduced and used to investigate wealth
inequality.
The giver model is shown to exhibit two distinct regimes in the steady
state. One of them is reminiscent of exchanges that occur in the economy
where the agents typically exchange a small fraction of their wealth. The other
regime is more akin to gambling and is characterised by exchanges where most
of the giver’s wealth is lost, so that fortunes are made and lost frequently. The
first regime is characterised by relatively low inequality, whereas the second
one is prone to exhibit very large inequality.
In addition to the study of inequality, the thesis investigates applicability
of the Boltzmann entropy as a measure of disorder in the giver model. The
giver model represent a closed system with no sources or sinks of agents or
wealth, i.e. it is conservative and in many respects is similar to an ideal gas.
However, numerical simulations reveal that the Boltzmann entropy does not
evolve monotonically in the giver model and, therefore, is not a faithful mea-
sure of disorder. This paradox is resolved by observing that the exchange rules
of the giver model are not time reversible, i.e. in order to reverse the dynamics
of exchanges a quantitatively different rule of exchange is required.
The foreign exchange market is a complex dynamical system that provides
prodigious amount of financial data, which makes it an attractive subject of
research. One of the approaches to modeling it relies on the lattice gauge
theory, where the lattice is constructed by considering two or more currencies
and discretising time and the gauge refers to the arbitrage on the lattice. A
brief investigation of one such model is undertaken in the thesis. It is shown
that the model is unstable in most realistic situations and thus cannot be used
to model the market behaviour.
Despite wealth of data, the foreign exchange market is difficult to study,
since the internal structure of the market is not well known. The Abelian
sandpile model, a cellular automaton that exhibits self-organised criticality,
is used in the thesis as a toy model that captures some of the features of the
foreign exchange market. The model is used to study temporal correlations in
the observed behaviour and their relation to the underlying internal structure.
A technique based on the site occupancy numbers is proposed in the thesis.
It reveals that the loss of memory in the sandpile model occurs in two distinct
stages, the fast stage characterised by rapid loss of memory and the subsequent
slow stage, during which memory of the initial state is lost at a much reduced
pace. Both stages are shown to be roughly exponential and the scaling of the
time decay is investigated.
The temporal correlations in Abelian sandpiles are also investigated in-
dependently by means of hidden Markov models, which show exceptional ca-
pabilities in detecting patterns in sequences of data. Hidden Markov models
have not been applied to Abelian sandpiles despite their popularity in a broad
range of applications ranging from bioinformatics to speech recognition. It is
demonstrated in the thesis that hidden Markov models do detect patterns in
the temporal variability of avalanche size, consistent with the results based on
the occupancy numbers. However, the connection between these patterns and
the internal structure of the sandpile has not been established. A number of
promising directions to address this problem are proposed.
Declaration
This is to certify that:
1. This thesis comprises only my original work, except where indicated in
the preface.
2. Due acknowledgement has been made in the text to all other material
used.
3. The thesis is less than 100,000 words in length, exclusive of tables, bib-
liographies, and appendices.
Andrey Sokolov
Preface
The majority of the work presented in this thesis is my own. The details of
specific contributions are listed below.
• Chapter 1 is an original review of the key research topics in econophysics,
based on a number of publications quoted in the text.
• Chapter 2 is based very closely on the following paper, co-authored with
Rachel Webster, AndrewMelatos, and Tien Kieu. It uses the data kindly
provided by the Reserve Bank of Australia. The work is original and my
own, including the code I wrote for data analysis and visualisation. The
idea to use the Anderson-Darling test was suggested by an anonymous
referee. Pip Pattison and Andre Gygax provided advice on the network
analysis.
Sokolov, Webster, Melatos, and Kieu (2012)
Loan and nonloan flows in the Australian interbank network.
• Chapter 3 is based very closely on the following publication, co-authored
with Andrew Melatos and Tien Kieu. The work is a result of a close
collaboration with Andrew Melatos who suggested the idea to use the
master equation.
Sokolov, Melatos, and Kieu (2010b)
Laplace transform analysis of a multiplicative asset transfer model.
• Chapter 4 is based very closely on the following publication, co-authored
with Tien Kieu and Andrew Melatos. Tien Kieu provided the gauge
theory expertise for this work.
Sokolov, Kieu, and Melatos (2010a)
A note on the theory of fast money flow dynamics.
• Chapter 5 is based on a paper submitted to Physica A and co-authored
with Andrew Melatos, Tien Kieu, and Rachel Webster. It uses an
Abelian sandpile simulator that I wrote. The idea to use hidden Markov
models can be traced to Hyam Rubinstein.
• Chapter 6 is original work.
Acknowledgements
In 2009, The School of Physics at the University of Melbourne offered a Ph.D.
scholarship in econophysics generously funded by the Portland House Foun-
dation. Tien Kieu, who was Director of Research at the The Portland House
Research Group at the time, played an instrumental role in making the fund-
ing available. The scholarship in econophysics was enthusiastically supported
by Rachel Webster and Andrew Melatos, who along with Tien Kieu became
my supervisors and collaborators. I am grateful to Tien, Rachel, and Andrew
for their support during my studies and research. I am eager to acknowledge
Andrew’s boundless enthusiasm and readiness to get involved, Rachel’s wise
advice, and Tien’s generosity with his time despite his pressing commitments
at The Portland House.
I thank the Portland House Foundation for funding this work.
The work on the interbank network would not be possible without Beth
Webster, who helped us get in touch with Chris Kent and Peter Gallagher
of the Reserve Bank of Australia. I thank the RBA and Peter Gallagher
in particular for making the interbank data available. During this project I
received valuable assistance from Andre Gygax and Pip Pattison.
I thank SIRCA for their timely assistance in obtaining financial data. I
gratefully acknowledge Hyam Rubenstein, Mark Joshi, Omar Foda, and Edda
Claus who advised me in the early stages of my studies. Thanks to David
Jamieson for his continued interest in my work. Many thanks to Sean and the
other members of the IT support group for dealing with my numerous requests.
And thanks to the students and faculty in the Astro group for having a lively
Table 2.1: The number of transactions (volume) and their total value (in unitsof A$109) for each day.
transactions originating from SWIFT1 and Austraclear for executing foreign
exchange and securities transactions respectively. The member banks can also
enter transactions directly into RITS. The switch to real-time settlement in
1998 was an important reform which protects the payment system against
systemic risk, since transactions can only be settled if the paying banks pos-
sess sufficient funds in their exchange settlement accounts. At present, about
3.2 × 104 transactions are settled per day, with total value around A$168
billion.
The data comprise all interbank transfers processed on an RTGS basis by
the RBA during the week of 19 February 2007. During this period, 55 banks
participated in the RITS including the RBA. The dataset includes transfers
between the banks and the RBA, such as RBA’s intra-day repurchase agree-
ments and money market operations. The real bank names are obfuscated
(replaced with labels from A to BP) for privacy reasons, but the obfuscated
labels are consistent over the week. The transactions are grouped into separate
days, but the time stamp of each transaction is removed.
During the week in question, around 2.5 × 104 transactions were settled
per day, with the total value of all transactions rising above A$2 × 1011 on
Tuesday and Thursday. The number of transactions (volume2) and the total
value (the combined dollar amount of all transactions) for each day are given
in Table 2.1. Figure 2.1 shows the distribution of transaction values on a
logarithmic scale. Local peaks in the distribution correspond to round values.
The most pronounced peak occurs at A$106.
In terms of the number of transactions, the distribution consists of two
1Society for Worldwide Interbank Financial Telecommunication2The term “volume” is sometimes used to refer to the combined dollar amount of trans-
actions. In this paper, we only use the term “volume” to refer to the number of transactionsand “total value” to refer to the combined dollar amount. This usage follows the one adoptedby the RBA Gallagher et al. (2010).
32
2. Loan and nonloan flows in the Australian interbank network
Table 2.3: Statistics of the overnight loans identified by our algorithm: thenumber of loans (volume), the total value of the first leg of the loans (in unitsof A$109), and the fraction of the total value of the loans (first legs only) withrespect to the total value of all transactions on a given date.
similar procedure was pioneered by Furfine Furfine (2003); see also Ashcraft
and Duffie (2007).
The application of the above algorithm results in the scatter diagram
shown in Figure 2.2. There is a clearly visible concentration of the revers-
ing transaction pairs in the region v > 2 × 105 and |rt − rh| < 0.5% (red
box). We identify these pairs as overnight loans. Contamination from non-
loan transaction pairs that accidentally give a hypothetical rate close to the
target rate is insignificant. By examining the adjacent regions of the diagram,
i.e. v > 2× 105 and rh outside of the red box, we estimate the contamination
to be less than 2% (corresponding to ≤ 5 erroneous identifications per day).
It is also possible that some genuine loans fall outside our selection criteria.
However, it is unlikely that overnight interest rates are very different from the
target rate; and the lower-value transactions (below A$104), even if they are
real loans, contribute negligibly to the total value.
We identify 897 overnight loans over the four days. A daily breakdown is
given in Table 2.3. Here and below, we refer to the first leg of the overnight
loans as simply loans and to all other transactions as nonloans. The loans
constitute less than 1% of all transactions by number and up to 9% by value
(cf. Tables 2.1 and 2.3). The distribution of loan values and interest rates is
shown in Figures 2.3a and 2.3b. The interest rate distribution peaks at the
target rate 6.25%. The mean rate is within one basis point (0.01%) of the
target rate, while the standard deviation is about 0.07%. The average interest
rate increases slightly with increasing value of the loan; a least-squares fit
yields rh = 6.248 + 0.010 log10(v/A$106).
The same technique can be used to extract two-day and longer-term loans
(up to four-day loans for our sample of five consecutive days). Using the
same selection criteria as for the overnight loans, our algorithm detects 27,
34
2. Loan and nonloan flows in the Australian interbank network
log10v
23-02-2007
log10v
22-02-2007
21-02-2007
20-02-2007
19-02-2007
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
0
500
1000
15000
500
1000
15000
500
1000
1500
Figure
2.1:
Thedistribution
oftran
sactionvaluesv(inAustralian
dollars)on
alogarithmic
scale,
withbin
size
∆log10v=
0.1;
thevertical
axisisthenumber
oftran
sactionsper
bin.Com
pon
ents
oftheGau
ssianmixture
model
areindicated
bythedashed
curves;thesolidcu
rveisthesum
ofthetw
ocompon
ents.Thedottedhistogram
show
stherelative
contribution
oftran
sactions
atdifferentvalues
tothetotalvalue(tocompute
thedottedhistogram
wemultiply
thenumber
oftran
sactionsin
abin
bytheir
value).
35
2. Loan and nonloan flows in the Australian interbank network
r h
log10 v
2 3 4 5 6 7 8 9
4.5
5
5.5
6
6.5
7
7.5
8
Figure 2.2: Hypothetical interest rate rh versus value of the first leg of thetransaction pairs detected by our algorithm, with no restrictions on value orinterest rate. The dotted rectangle contains the transactions that we identifyas overnight loans. The least-squares fit is shown with a solid red line.
67, and 24 two-day loans, with total values A$1.3, A$2.2, and A$1.4 billion,
on Monday, Tuesday, and Wednesday, respectively. The total value of the
two-day loans is 1.5%, 1.0%, and 0.9% of the total transaction values on these
days respectively.
2.4 Nonloans
We display the distributions of the incoming and outgoing nonloan transac-
tions, for which the bank is the destination and the source respectively, for
the six largest banks in Figure 2.4. The distributions are similar to the to-
tal distribution shown in Figure 2.1, with the notable exception of BA (see
below). There is also an unusually large number of A$106 and A$400 transac-
tions from W to T on Monday. Note that the daily imbalance for each bank
is mostly determined by the highest value transactions; large discrepancies
between incoming and outgoing transactions at lower values are less relevant.
The distribution for BA is clearly bimodal; it contains an unusually high
proportion of transactions greater than A$106. Moreover, below A$106, in-
coming transactions typically outnumber outgoing ones by a large amount.
BA is also involved in many high value transactions that reverse on the same
day. These transactions probably correspond to the central bank’s repurchase
agreements, which facilitate intra-day liquidity of the banks (Campbell, 1998).
The banks shown in Figure 2.4 are also the largest in term of the number
36
2. Loan and nonloan flows in the Australian interbank network
log10v
22-02-2007
21-02-2007
20-02-2007
19-02-2007
45
67
89
10
45
67
89
10
45
67
89
10
45
67
89
10
0
10
20
30
400
10
20
30
400
10
20
30
400
10
20
30
40
(a)
r h
66.25
6.5
66.25
6.5
66.25
6.5
66.25
6.5
0
20
40
600
20
40
600
20
40
600
20
40
60
(b)
Figure
2.3:
(a)Thedistribution
ofloan
valuesvon
alogarithmic
scale.
Thevertical
axisisthenumber
ofloan
sper
bin
forbin
size
∆log10v=
0.25.Thedottedlineis
thesamedistribution
multiplied
bythevaluecorrespon
dingto
each
bin
(inarbitrary
units).Thedateof
thefirstlegof
theloan
sis
indicated
.(b)Thedistribution
ofloan
interest
ratesr h.Thevertical
axis
isthe
number
ofloan
sper
bin
forbin
size
∆r h
=0.01.Thedateof
thefirstlegof
theloan
sis
indicated
.Themeanan
dstan
dard
deviation
are6.25%
and0.08%
onMon
day
(19-02-2007),an
d6.26%
and0.07%
ontheother
days.
37
2. Loan and nonloan flows in the Australian interbank network
of transactions, with the exception of BA. The rank order by the number of
transactions matches that by value. For D, which is the largest, the number
of nonloan transactions reaches 48043 over the week. By the number of trans-
actions, the order of the top twelve banks is D, BP, AV, T, W, AH, AF, U,
AP, BI, BA, P. By value, the order is D, BP, AV, BA, T, W, BG, U, A, AH,
AB, BM. The situation is similar when considering the overnight loans. By
value, AV, D, BP, and T dominate. For these four banks, weekly total loans
range from A$11.5 to A$18 billion and number from 254 to 399. For the other
banks the total loan value is less than A$3 billion.
In view of the discussion above, it is noteworthy that Australia’s retail
banking system is dominated by four big banks (ANZ, CBA, NAB, andWBC)3
that in February 2007 accounted for 65% of total resident assets, according to
statistics published by Australian Prudential Regulation Authority (APRA);
see http://www.apra.gov.au for details. The resident assets of the big four
exceeded A$225 billion each, well above the next largest retail bank, St George
Bank Limited4 (A$93 billion). The distinction between the big four and the
rest of the banks in terms of cash and liquid assets at the time was less clear,
with Macquarie Bank Limited in third position with A$8 billion. According
to APRA, cash and liquid assets of the big four and Macquarie Bank Limited
accounted for 56% of the total.
2.5 Loan and nonloan imbalances
In order to maintain liquidity in their exchange settlement accounts, banks
ensure that incoming and outgoing transactions roughly balance. However,
they do not control most routine transfers, which are initiated by account
holders. Therefore, the imbalances arise. On any given day, the nonloan
imbalance of bank i is given by
∆vi = −∑
j
∑
k
vk(i, j) +∑
j
∑
k
vk(j, i), (2.1)
where {vk(i, j)}k is a list of values of individual nonloan transaction from bank
i to bank j, settled on the day. The nonloan imbalances are subsequently
compensated by overnight loans traded on the interbank money market. The
3Australia and New Zealand Banking Group, Commonwealth Bank of Australia, Na-tional Australia Bank, and Westpac Banking Corporation.
4In December 2008, St George Bank became a subsidiary of Westpac Banking Corpora-tion.
38
2. Loan and nonloan flows in the Australian interbank network
Table 2.4: Loan and nonloan imbalances for the six largest banks (in units ofA$109).
loan imbalances are defined in the same way using transactions corresponding
to the first leg of the overnight loans. Note that we do not distinguish between
the loans initiated by the banks themselves and those initiated by various
institutional and corporate customers. For instance, if the funds of a corporate
customer are depleted, this customer may borrow overnight to replenish the
funds. In this case, the overnight loan is initiated by an account holder,
who generally has no knowledge of the bank’s net position. Nevertheless, the
actions of this account holder in acquiring a loan reduce the bank’s imbalance,
provided that the customer deposits the loan in an account with the same
bank.
The loan and nonloan imbalances for the six largest banks are given in
Table 2.4. The data generally comply with our assumption that the overnight
loans compensate the daily imbalances of the nonloan transactions. The most
obvious exception is for BA on Thursday (22-02-2007), where a large negative
nonloan imbalance is accompanied by a sizable loan imbalance that is also
negative. Taking all the banks together, there is a strong anti-correlation
between loan and nonloan imbalances on most days. We see this clearly in
Figure 2.5. The Pearson correlation coefficients for Monday through Thursday
are −0.93, −0.88, −0.95, −0.36. It is striking to observe that many points
fall close to the perfect anti-correlation line. The anti-correlation is weaker on
Thursday (crosses in Figure 2.5), mostly due to BA and AV.
A correlation also exists between the absolute values of loan imbalances
and the nonloan total values (incoming plus outgoing nonloan transactions);
the Pearson coefficients are 0.74, 0.75, 0.66, 0.77 for Monday through Thurs-
day. This confirms the intuitive expectation that larger banks tolerate larger
loan imbalances.
39
2. Loan and nonloan flows in the Australian interbank network
W/34/+
0.3
6W
/20/-0
.36
W/27/+
0.0
8W
/7/-0
.09
T/30/-0
.21
T/21/-0
.62
T/29/-0
.75
T/15/-0
.76
BA/30/-1
.53
BA/30/-0
.32
BA/33/-0
.10
BA/23/+
0.0
3
AV/51/+
1.0
8AV/37/+
0.5
5AV/51/+
1.3
9AV/20/-0
.32
BP/57/+
0.1
6BP/41/+
1.3
8BP/57/+
0.8
0BP/17/+
2.0
8
D/66/-1
.25
D/51/-0
.76
D/60/-0
.28
D/15/-0
.51
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
0
50
100
150 0
100
200 0
20
40 0
100
200 0
100
200 0
100
200
300
Figu
re2.4:
Thedistrib
ution
ofnon
loantran
sactionvalu
esof
thesix
largestban
ksfor
Mon
day
throu
ghThursd
ay(from
leftto
right);
theban
ksare
selectedbythecom
bined
valueof
incom
ingan
dou
tgoingtran
sactionsover
theentire
week
.Black
andred
histogram
scorresp
ondto
incom
ing(ban
kisthedestin
ation)an
dou
tgoing(ban
kisthesou
rce)tran
sactions;red
histogram
sare
filled
into
improve
visib
ility.Theban
ks’an
onymou
slab
els,thecom
bined
daily
valueof
theincom
ingan
dou
tgoingtran
sactions,
andthedaily
imbalan
ce(in
comingminusou
tgoing)
arequoted
atthetop
leftof
eachpan
el(in
units
ofA
$109).
Thehorizon
talax
isis
thelogarith
mof
valuein
A$.
40
2. Loan and nonloan flows in the Australian interbank network
log10|∆
l|
log10 v
∆l
∆v
6 7 8 9 10 11−2 −1 0 1 25.5
6
6.5
7
7.5
8
8.5
9
9.5
−2
−1
0
1
2
Figure 2.5: Left: loan imbalance ∆l vs nonloan imbalance ∆v for individualbanks and days of the week (in units of A$109). Right: the absolute value ofloan imbalance |∆l| vs nonloan total value (incoming plus outgoing transac-tions) for individual banks and days of the week. Thursday data are markedwith crosses.
2.6 Flow variability
For each individual source and destination, we define the nonloan flow as
the totality of all nonloan transactions from the given source to the given
destination on any given day. The value of the flow is the sum of the nonloan
transaction values and the direction is from the source to the destination. On
any given day, the value of the flow from bank i to bank j is defined by
vflow(i, j) =∑
k
vk(i, j), (2.2)
where {vk(i, j)}k is a list of values of individual nonloan transaction from i to
j on the day. For example, all nonloan transactions from D to AV on Monday
form a nonloan flow from D to AV on that day. The nonloan transactions
in the opposite direction, from AV to D, form another flow. A flow has zero
value if the number of transactions is zero. Typically, for any two large banks
there are two nonloan flows between them. The loan flows are computed in a
similar fashion.
Nonloan flows
There are 55 banks in the network, resulting in Nflow = 2970 possible flows.
The actual number of flows is much smaller. The typical number of nonloan
flows is ∼ 800 on each day (the actual numbers are 804, 791, 784, 797). Even
though the number of nonloan flows does not change significantly from day to
day, we find that only about 80% of these flows persist for two days or more.
The other 20% are replaced by different flows, i.e. with a different source
41
2. Loan and nonloan flows in the Australian interbank network
and/or destination, on the following day. Structurally speaking, the network
of nonloan flows changes by 20% from day to day. However, persistent flows
carry more than 96% of the total value.
Even when the flow is present on both days, its value is rarely the same.
Given that 80% of the network is structurally stable from day to day, we as-
sess variability of the network by considering persistent flows and their values
on consecutive days. Figure 2.6 shows the pairs of persistent flow values for
Monday and Tuesday, Tuesday and Wednesday, and Wednesday and Thurs-
day. If the flow values were the same, the points in Figure 2.6 would lie on
the diagonals. We observe that the values of some flows vary significantly, es-
pecially when comparing Monday and Tuesday. Moreover, there is a notable
systematic increase in value of the flows from Monday to Tuesday by a factor
of several, which is not observed on the other days. For each pair of days
shown in Figure 2.6, we compute the Pearson correlation coefficient, which
gives 0.53 for Monday and Tuesday, 0.70 for Tuesday and Wednesday, and
0.68 for Wednesday and Thursday.
To characterize the difference between the flows on different days more
precisely, we compute the Euclidean distance between normalised flows on
consecutive days. We reorder the adjacency matrix {vflow(i, j)}ij of the flow
network on day d as an Nflow-dimensional vector vd representing a list of all
flows on day d (d = 1, 2, . . . , 5). For each pair of consecutive days we compute
the Euclidean distance between normalized vectors vd/|vd| and vd+1/|vd+1|,which gives 0.62, 0.50, 0.50 for all flows and 0.61, 0.49, 0.49 for persistent
flows (the latter are computed by setting non-persistent flows to zero on both
days). Since the flow vectors are normalized, these quantities measure random
flow discrepancies while systematic deviation such as between the flows on
Monday and Tuesday are ignored. For two vectors of random values uniformly
distributed in interval (0, 1), the expected Euclidean distance is 0.71 and the
standard deviation is 0.02 for the estimated number of persistent nonloan flows
of 640. So the observed variability of the nonloan flows is smaller than what
one might expect if the flow values were random.
Loan flows
Variability of the loan flows is equally strong. The number of loan flows varies
from 69 to 83 (actual numbers are 69, 75, 77, 83). Only about 50% of these
flows are common for any two consecutive days. Moreover, persistent flows
42
2. Loan and nonloan flows in the Australian interbank network
log10von22-02-2007
log10von21-02-2007
log10von21-02-2007log10von20-02-2007
log10von20-02-2007
log10von19-02-2007
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
0123456789
10
0123456789
10
0123456789
10
Figure
2.6:
Non
loan
flow
valuepairs
onon
eday
(horizon
talax
is)an
dthenext(verticalax
is).
Only
flow
spresenton
bothdays
areconsidered
.Flowsthat
donot
chan
gelieon
thediagonal
(red
dottedline).Thesolidlineis
theweigh
tedorthogon
alleast
squares
fitto
thescatterdiagram
;theweigh
tshavebeendefi
ned
toem
phasizepoints
correspon
dingto
largeflow
s.
log10von22-02-2007
log10von21-02-2007
log10von21-02-2007
log10von20-02-2007
log10von20-02-2007
log10von19-02-2007
56
78
910
56
78
910
56
78
910
56789
10
56789
10
56789
10
Figure
2.7:
AsforFigure
2.6butforloan
flow
s.
43
2. Loan and nonloan flows in the Australian interbank network
carry only about 65% of the total value of the loan flows on any given day,
cf. 80% of nonloan flows. For persistent loan flows, the Pearson correlation
coefficients are 0.63, 0.90, and 0.76 for the consecutive pairs of days starting
with Monday and Tuesday. The correlation is generally similar to that of the
nonloan flows, with the notable exception of the loan flows on Tuesday and
Wednesday, when the sub-network of persistent loan flows appears to be more
stable.
The Euclidean distances between the normalized loan flows for each pair
of consecutive days are 0.85, 0.68, 0.73 for all flows and 0.63, 0.44, and 0.44
for persistent flows. For two vectors of random values uniformly distributed
in interval (0, 1), the expected Euclidean distance is 0.7 and the standard
deviation is 0.1 for the estimated number of persistent loan flows of 40. So
the observed variability of the persistent loan flows is much smaller than what
one might expect if the flow values were random.
Relation between nonloan and loan flows
Some loan flows do not have corresponding nonloan flows between the same
nodes on the same day. These flows carry about 14% of loan value on Mon-
day, and about 7% on Tuesday through Thursday. Nonloan flows that have
corresponding loan flows account for 35% to 48% of all nonloan flows by value,
even though the number of these flows is less than 10% of the total.
To improve the statistics, we aggregate the flows on all four days. Fig-
ure 2.8 shows nonloan and corresponding loan flow values. We fail to find any
significant correlation between loan and nonloan flows (Pearson coefficient is
0.3). The correlation improves if we restrict the loan flows to those consisting
of three transactions or more; such flows mostly correspond to large persis-
tent flows. In this case the Pearson coefficient increases to 0.6; banks that
sustain large nonloan flows can also sustain large loan flows, even though the
loan flows on average are an order of magnitude lower than the corresponding
nonloan flows. The lack of correlation when all loans are aggregated is due to
the presence of many large loans that are not accompanied by large nonloan
transactions, and vice versa.
44
2. Loan and nonloan flows in the Australian interbank network
2.7 Net flows
The net flow between any two banks is defined as the difference of the opposing
flows between these banks. The value of the net flow equals the absolute value
of the difference between the values of the opposing flows. The direction of the
net flow is determined by the sign of the difference. If vflow(i, j) > vflow(j, i),
the net flow value from i to j is given by
vnet(i, j) = vflow(i, j) − vflow(j, i). (2.3)
For instance, if the flow from D to AV is larger than the flow in the opposite
direction, then the net flow is from D to AV.
General properties
The distributions of net loan and nonloan flow values are presented in Fig-
ure 2.9. The parameters of the associated Gaussian mixture models are quoted
in Table 2.5. The distribution of net nonloan flow values has the same general
features as the distribution of the individual transactions. However, unlike in-
dividual transactions, net flow values below A$104 are rare; net flows around
A$108 are more prominent.
There are on average around 470 net nonloan flows each day. Among these,
roughly 110 consist of a single transaction and 50 consist of two transactions,
mostly between small banks. At the other extreme, net flows between the
largest four banks (D, BP, AV, T) typically have more than 103 transactions
per day each. Overall, the distribution of the number of transactions per net
flow is approximated well by a power law with exponent α = −1.0± 0.2:
Nnet(n) ∝ nα, (2.4)
whereNnet(n) is the number of net nonloan flows that consist of n transactions
(n ranges from 1 to more than 1000). This is consistent with the findings for
Fedwire reported in Bech and Atalay (2010) (see right panel of Fig. 14 in Bech
and Atalay (2010)).
There are roughly 60 net loan flows each day. As many as 40 consist of
only one transaction. On the other hand, a single net loan flow between two
large banks may comprise more than 30 individual loans. The distribution
of the number of transactions per net loan flow is difficult to infer due to
poor statistics, but it is consistent with a power law with a steeper exponent,
45
2. Loan and nonloan flows in the Australian interbank network
log10l
log10 v
5 6 7 8 9 105
6
7
8
9
10
Figure 2.8: Loan flow values versus nonloan flow values combined over fourdays. Triangles correspond to loan flows with three or more transactions perflow. The solid line is the orthogonal least squares fit to the scatter diagram;the weighting is the same as in Figure 2.6.
Table 2.5: Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussianmixture components appearing in Figure 2.9 (u = log10 v).
−1.4± 0.2, than that of the nonloan distribution. There are no net loan flows
below A$105 or above A$109. Comparing net loan and nonloan flows, it is
obvious that net loan flows cannot compensate each and every net nonloan
flow. Not only are there fewer net loan flows than nonloan flows, but the total
value of the former is much less than the total value of the latter.
Net loan and net nonloan flows are not correlated; the correlation coeffi-
cient is 0.3. Restricting net loan flows to those that have three transactions or
more does not improve the correlation. If a net loan flow between two banks
46
2. Loan and nonloan flows in the Australian interbank network
log10v
22-02-2007
log10v
21-02-2007
20-02-2007
19-02-2007
01
23
45
67
89
10
01
23
45
67
89
10
05
10
15
20
25
3005
10
15
20
25
30
Figure
2.9:
Thedistribution
ofvalues
ofnet
non
loan
flow
s(black
histogram
)on
alogarithmic
scalewithbin
size
∆log10v=
0.1.
Thecompon
ents
oftheGau
ssianmixture
model
areindicated
withthedashed
curves;thesolidcu
rveis
thesum
ofthetw
ocompon
ents.Net
loan
flow
sareoverplotted
inred.Thevertical
axis
counts
thenumber
ofnet
flow
sper
bin.
47
2. Loan and nonloan flows in the Australian interbank network
was triggered to a significant degree by the magnitude and the direction of
net nonloan flow between these bank, one expects a correlation between net
loan and nonloan flows. Our examination shows that in this respect loan flows
are decoupled from nonloan flows. The connection between them is indirect.
Namely, nonloan flows cause an imbalance in the account of each bank, which
is subsequently compensated by loan flows, which are largely unrelated to the
nonloan flows that caused the imbalance.
Degree distribution and assortativity
We define the in-degree of node i as the number of net flows that terminate
at i, i.e. the number of net flows with destination i, and the out-degree as
the number of net flows that originate from i, i.e. the number of net flows
with source i. The degree distribution of the nonloan networks is shown in
Figure 2.10a. Node BA has the highest in-degree of 37 on Monday, but on the
other days it drops to 15 on average, while the out-degree is 11.75 on average
for this node. The highest in-degrees are usually found among the four largest
banks (D, BP, AV, T); the only exception is Monday, when AF’s in-degree of
22 is greater than AV’s 21, and BA has the highest in-degree. The highest
out-degrees are usually achieved by D, BP, AV, T, W, and AH; the exceptions
are Monday, when D’s out-degree of 17 is less than AR’s and AP’s 18, and
Thursday, when AV’s out-degree of 16 is less than P’s 18.
It is difficult to infer the shape of the degree distribution for individual
days due to poor statistics. The two-sample Kolmogorov-Smirnov (KS) test
does not distinguish between the distributions on different days at the 5%
significance level. With this in mind, we combine the in- and out-degree data
for all four days and graph the resulting distributions in Figure 2.10b. We
find that a power law distribution does not provides a good fit for either in- or
out-degrees. Visually, the distribution is closer to an exponential. However,
the exponential distribution is rejected by the Anderson-Darling test.
The degree distribution conceals the fact that flows originating or termi-
nating in nodes of various degrees have different values and therefore provide
different contributions to the total value of the net flows. Nodes with lower
degrees are numerous, but the flows they sustain are typically smaller than
those carried by a few high-degree nodes. In particular, for the nonloan flows,
nodes with in-degree d ≤ 10 are numerous, ranging from 35 to 37, but their
outgoing net flows carry about 20% of the value on average. On the other
48
2. Loan and nonloan flows in the Australian interbank network
d
22-02-2007
21-02-2007
20-02-2007
19-02-2007
05
10
15
20
25
30
35
40
95059950599505995059
−9
−5059
−9
−5059
−9
−5059
−9
−5059
(a)
d
100
101
100
101
(b)
Figure
2.10:(a)Degreedistribution
ofthenet
non
loan
flow
networks(for
convenience,in-degrees
arepositivean
dou
t-degrees
arenegative).Thetotalvalueof
thenet
flow
scorrespon
dingto
thespecificdegrees
isshow
nwithreddots(thelogof
valuein
A$10
9isindicated
ontherigh
tvertical
axis).
(b)Degreedistribution
ofthenet
non
loan
flow
swhen
thedegreedataforallfour
daysareaggregated
(in-degrees
arecircles;
out-degrees
aretriangles).
49
2. Loan and nonloan flows in the Australian interbank network
d
05
10
15
9 5 0 5 9 9 5 0 5 9 9 5 0 5 9 9 5 0 5 9
−9
−5 0 5 9
−9
−5 0 5 9
−9
−5 0 5 9
−9
−5 0 5 9
(a)
d
100
101
100
101
(b)
Figu
re2.11:
(a)Sam
eas
Figu
re2.10a,
butfor
thenet
loanflow
netw
orks.
(b)Sam
eas
Figu
re2.10b
,butfor
thenet
loanflow
s.
50
2. Loan and nonloan flows in the Australian interbank network
hand, nodes with d ≥ 17 are rare, but their flows carry 50% of the value. The
same effect is observed for the out-degrees.
The degree distribution of the network of net loan flows is shown in Fig-
ure 2.11a (we ignore the nodes that have zero in- and out- degrees over four
days). Similarly to nonloan flows, the KS test does not distinguish between
the distributions on different days at the 5% significance level. The combined
distribution is shown in Figure 2.11b.
To probe assortativity of the net flow networks, we compute the in-assortativity
defined in Piraveenan et al. (2010) as the Pearson correlation coefficient be-
tween the in-degrees of sources and destinations of the net flows (out-assortativity
is computed similarly using the out-degrees). The net nonloan flow network
is disassortative, with in-assortativity of −0.39, −0.37, −0.38, −0.37 and out-
assortativity of −0.35, −0.38, −0.39, −0.37 on Monday, Tuesday, Wednesday,
and Thursday, respectively. The net loan flow network is less disassortative;
the in-assortativity is −0.16, −0.26, −0.18, −0.19 and the out-assortativity
is −0.03, −0.10, 0.02, −0.20 for the same sequence of days. In biological
networks, the tendency of out-assortativity to be more assortative than in-
assortativity has been noted in Piraveenan et al. (2010).
Topology of the net flows
Given the source and destination of each net flow, we can construct a network
representation of the net flows. An example of the network of net nonloan flows
is shown in Figure 2.12. The size of the nodes and the thickness of the edges
are proportional to the net imbalances and net flow values respectively (on a
logarithmic scale). We use the Fruchterman-Reingold algorithm to position
the nodes Fruchterman and Reingold (1991); the most connected nodes are
placed in the centre, and the least connected nodes are moved to the periphery.
The core of the network is dominated by the four banks with the largest total
value and the largest number of transactions: D, BP, AV, and T. The other
big banks, such as AF, AH, and W, also sit near the core. It is interesting to
note the presence of several poorly connected nodes (Q, V, BF, and especially
X) that participate in large incoming and outgoing flows, which produce only
negligible imbalances in the banks themselves.
The sub-network consisting of D, BP, AV, BA, T, W, U, A, AH, AF, AP,
and P is fully connected on all five days, i.e. every node is connected to every
other node. The sub-network of D, AV, and BP is fully connected, even if we
51
2. Loan and nonloan flows in the Australian interbank network
Figure 2.12: Network of net nonloan flows on Tuesday, 20-02-2007. White(grey) nodes represent negative (positive) imbalances. The bank labels areindicated for each node. The size of the nodes and the thickness of the edgesare proportional to the logarithm of value of the imbalances and the net flowsrespectively.
restrict the net flows to values above A$108.
In Figure 2.12, the flows between the largest nodes are difficult to discern
visually, because the nodes are placed too close to each other in the image.
We therefore employ the following procedure to simplify the network. We
consider the fully connected sub-network of twelve nodes, plus node BG, and
combine all other nodes into a new node called “others” in such a way that
the net flows are preserved (BG is included because it usually participates
in large flows and is connected to almost every node in the complete sub-
52
2. Loan and nonloan flows in the Australian interbank network
network). The result of this procedure applied to the daily nonloan networks
is presented in Figures 2.13a–2.13d. For these plots, we employ the weighted
Fruchterman-Reingold algorithm, which positions the nodes with large flows
between them close to each other. The imbalances shown in Figure 2.13b are
the same as those of the full network in Figure 2.12. The daily networks of
net loan flows for the same nodes are shown in Figures 2.14a–2.14d.
We observe that the largest flows on Monday (19-02-2007) were signifi-
cantly lower than the flows on the subsequent days. The largest nodes (D,
BP, AV, T, W) are always placed close to the center of the network, because
they participate in the largest flows. The topology of the flows is complex and
difficult to disentangle, even if one concentrates on the largest flows (above
A$5 × 108). For instance, on Monday, probably the simplest day, the flow of
nonloans is generally from BG to “others” to D to BP. There are also siz-
able flows from T to AV and from AV to “others” and BP. However, lower
value flows (below A$5 × 108) cannot be neglected completely because they
are numerous and may contribute significantly to the imbalance of a given
node.
Nodes D, T, BP, AV, and W form a complete sub-network of net loan flows
on Monday, Tuesday, and Wednesday. This sub-network is almost complete on
Thursday too, except for the missing link between BP and W. The appearance
of the net loan network is different from that of the nonloan network, since the
same nodes participate in only a few loan flows. Therefore, the position of a
node in the network image is strongly influenced by the number of connections
of that node. Some of the poorly connected nodes are placed at the periphery
despite the fact that they possess large flows. The four largest nodes (D, T,
BP, AV) are always positioned at the center of the network.
Network variability
The net nonloan flow network is extremely volatile in terms of flow value and
direction. For example, a A$109 flow from D to BP on Monday transforms
into a A$3.2×109 flow in the same direction on Tuesday, only to be replaced by
a A$6.3 × 108 flow in the opposite direction on Wednesday, which diminishes
further to A$2.5×109 on Thursday. Nodes T and BP display a similar pattern
of reversing flows between Tuesday and Wednesday. On the other hand, the
net flow between T and AV maintains the same direction, but the flow value
is strongly fluctuating. In particular, a moderate A$4.8×108 flow on Monday
53
2. Loan and nonloan flows in the Australian interbank network
(a) 19-02-2007 (b) 20-02-2007
(c) 21-02-2007 (d) 22-02-2007
Figure 2.13: Networks of daily net nonloan flows for D, AV, BP, T, W, BA,AH, AF, U, AP, P, A, BG. All the other nodes and the flows to and from themare combined in a single new node called “others”. The size of the nodes andthe thickness of the edges are proportional to the logarithm of value of theimbalances and the net flows respectively. The value of the flows and theimbalances can be gauged by referencing a network shown in the middle,where the values of the flows are indicated in units of A$1 billion.
54
2. Loan and nonloan flows in the Australian interbank network
(a) 19-02-2007 (b) 20-02-2007
(c) 21-02-2007 (d) 22-02-2007
Figure 2.14: Networks of daily net loan flows. The same nodes as in Fig-ures 2.13a–2.13d are used. The scale of the loan flows, the imbalances, andthe positions of the nodes are the same as those used for the nonloan flows inFigures 2.13a–2.13d to simplify visual comparison.
55
2. Loan and nonloan flows in the Australian interbank network
rises to A$1.9×109 on Tuesday, then falls sharply to A$2×108 on Wednesday
and again rises to A$2.2× 109 on Thursday.
Considering any three nodes, we observe that circular and transitive flows
are present on most days, the latter being more common. The most obvious
example is a circular flow between D, T, and BP on Thursday and a transitive
flow involving BG, T, and AV on the same day. The circular flows are unstable
in the sense that they do not persist over two days or more.
The net loan flow network exhibits similar characteristics. Few net loan
flows persist over the four days. For example, the flow from AV to T has the
same direction and is similar in value on all four days. Circular loan flows are
also present, as the flow between AV, T, and BP on Thursday demonstrates.
2.8 Conclusions
In this paper, we study the properties of the transactional flows between Aus-
tralian banks participating in RITS. The value distribution of transactions
is approximated well by a mixture of two log-normal components, possibly
reflecting the different nature of transactions originating from SWIFT and
Austraclear. For the largest banks, the value distributions of incoming and
outgoing transactions are similar. On the other hand, the central bank dis-
plays a high asymmetry between the incoming and outgoing transactions, with
the former clearly dominating the latter for transactions below A$106.
Using a matching algorithm for reversing transactions, we successfully sep-
arate transactions into loans and nonloans. For overnight loans, we estimate
the identification rate at 98%. The mean derived interest rate is within 0.01%
of the central banks’ target rate of 6.25%, while the standard deviation is
about 0.07%. We find a strong anti-correlation between loan and nonloan im-
balances (Pearson coefficient is about 0.9 on most days). A likely explanation
is that nonloan flows create surpluses in some banks. The banks lend the
surplus to banks in deficit, creating loan flows that counteract the imbalances
due to the nonloan flows. Hence, loan and nonloan imbalances of individual
banks are roughly equal in value and opposite in sign on any given day.
The flow networks are structurally variable, with 20% of nonloan flows
and 50% of loan flows replaced every day. Values of persistent flows, which
maintain the same source and destination over at least two consecutive days,
vary significantly from day to day. Some flow values change by several orders
of magnitude. Persistent flows increase in value several-fold between Monday
56
2. Loan and nonloan flows in the Australian interbank network
and Tuesday. Individual flow values can change by several orders of magnitude
on the following day. Overall, there is a reasonable correlation between the
flow values on consecutive days (Pearson coefficient is 0.65 for nonloans and
0.76 for loans on average). We also find that larger banks tend to sustain
larger loan flows, in accord with the intuitive expectations. However, there is
no correlation between loan and nonloan flows.
We examine visually the topology of the net loan and nonloan flow net-
works. The centre of both networks is dominated by the big four banks.
Twelve banks form a complete nonloan sub-network, in which each bank is
connected to every other bank in the sub-network. The three largest banks
form a complete sub-network even if the net flows are restricted to values
above A$108. Our examination reveals that the network topology of net flows
is complicated, with even the largest flows varying greatly in value and direc-
tion on different days.
Our findings suggest a number of avenues for future research on interbank
networks. Firstly, the relationships we uncovered can be used to constrain
analytical models and numerical simulations of interbank flows in financial
networks. In particular, our explanation of the link between the loan and
nonloan imbalances needs to be tested in numerical simulations. Secondly, it
is necessary to analyse interbank markets in other countries to establish what
elements of our results are signatures of general dynamics and what aspects
are specific to the epoch and location of this study. Even when high qual-
ity data are available, most previous studies concentrate on analysing static
topological properties of the networks or their slow change over time. The
internal dynamics of monetary flows in interbank networks has been largely
ignored. Importantly, one must ask whether the strong anti-correlation be-
tween loan and nonloan imbalances is characteristic of RTGS systems whose
institutional setup resembles the Australian one or whether it is a general
feature. For instance, in Italy a reserve requirement of 2% must be observed
on the 23rd of each month, which may encourage strong deviations between
loan and nonloan imbalances on the other days.
57
Chapter 3
Laplace transform analysis of
a multiplicative asset transfer
model
We analyze a simple asset transfer model in which the transfer amount is
a fixed fraction f of the giver’s wealth. The model is analyzed in a new
way by Laplace transforming the master equation, solving it analytically and
numerically for the steady-state distribution, and exploring the solutions for
various values of f ∈ (0, 1). The Laplace transform analysis is superior to
agent-based simulations as it does not depend on the number of agents, en-
abling us to study entropy and inequality in regimes that are costly to address
with simulations. We demonstrate that Boltzmann entropy is not a suitable
(e.g. non-monotonic) measure of disorder in a multiplicative asset transfer sys-
tem and suggest an asymmetric stochastic process that is equivalent to the
asset transfer model.
3.1 Introduction
A vibrant research theme in econophysics is the analysis of asset exchange
models. In these models, a large number of agents iteratively exchange assets,
typically representing monetary amounts. In the simplest model that has been
considered, the transfer amount is constant and independent of the agent’s
wealth, producing an exponential wealth distribution in the steady state (see
Yakovenko and Rosser Jr. (2009) for a review). More complicated fractional
59
3. Laplace transform analysis of a multiplicative asset transfer model
exchange models have also been considered by several authors Chatterjee and
Chakrabarti (2007); Hayes (2002), in which the size of each transfer is a linear
function of the wealths of the agents involved in the exchange.
It has been found both analytically and numerically that the steady-state
wealth probability distribution function ps(w) in fractional exchange models
depends strongly on the parameters that characterize the exchange Matthes
and Toscani (2008). Certain parameter values or exchange rules yield a
strongly peaked distribution with an exponential tail, while other values yield
a broad distribution with Pareto-like qualities. The dichotomy is exemplified
by two simple models. If the transfer amount is a fixed fraction f of the
giver’s wealth (the giver is the agent who surrenders the asset in the transfer),
then the resulting steady-state distribution is strongly peaked and decays ex-
ponentially in the tail. If, on the other hand, the transfer amount is a fixed
fraction f of the poorer agent’s wealth, then one finds a broad steady-state
distribution, which can be fitted well by a power law with exponent −1 across
a broad interval of wealths. In this paper, we refer to these two models as the
giver scheme and the poorer scheme respectively.1
In some of the asset exchange models considered in Chakraborti and
Chakrabarti (2000) and several other studies Chatterjee and Chakrabarti
(2007); Chatterjee et al. (2005), the fractional exchange amount is a random
linear combination of the wealths of the participating agents. The controlling
parameter is the saving propensity, λ, which determines the fraction of the
agents’ wealths that they do not offer to exchange. Comparing the output of
simulations for the giver scheme and the exchange schemes based on the sav-
ing propensity, one observes that the schemes are closely related, with f ≈ 0
corresponding to λ ≈ 1. If the saving propensity is the same for all the agents,
then the resulting steady-state distributions are similar to those obtained for
the giver scheme. On the other hand, if the saving propensity is uniformly
distributed, the steady-state distribution is a power law, ps(w) ∝ w−2. In
a recent study based on numerical simulations Saif and Gade (2007), it was
found that a combination of the poorer and giver schemes in one simulation
results in a power-law wealth distribution whose exponent depends on the
relative contributions of the two schemes. The more agents follow the giver
scheme, the greater the exponent.
Asset exchange models can be treated analytically via a kinetic or mas-
1They are also known as the theft-and-fraud and yard-sale models respectively (see Hayes(2002)).
60
3. Laplace transform analysis of a multiplicative asset transfer model
ter equation, which tracks the rate of change of the number of agents at any
given wealth. In particular, the master equation for the giver scheme has been
derived in Ispolatov et al. (1998). These authors found an expression for the
second moment in the steady state, which agrees with the expression found
in Angle (2006) for a similar model by assuming the gamma distribution of
wealth. The standard deviation converges to its steady-state value exponen-
tially on a time scale ∼ [f(1− f)]−1. The authors also found the asymptotic
behaviour of the wealth distribution at small values of wealth. The master
equation for the poorer scheme was derived recently Moukarzel et al. (2007),
but its solutions have not yet been studied. In Chatterjee et al. (2005), the au-
thors derived the kinetic equation for the case of uniformly distributed saving
propensity. They demonstrated that the solution follows a power law with the
same exponent as in the simulations. The kinetic equation approach was also
used in Slanina (2004) to analyze self-similar solutions of a non-conservative
asset exchange system. The author found a closed-form solution in the limit
of continuous trading by means of the Laplace transform and observed that
the distribution exhibits power-law behaviour at large wealths.
The dependence of the relaxation time on the exchange parameters has
been investigated numerically in Patriarca et al. (2007) for the models con-
sidered in Chatterjee and Chakrabarti (2007); Chatterjee et al. (2005). The
relaxation time-scale was found numerically to scale as ∼ (1 − λ)−1. This is
consistent with the values found analytically in the giver scheme for the stan-
dard deviation. The authors also considered how the relaxation time depends
on the number of agents but failed to find any significant trend.
Recently, much effort has been directed profitably at developing more so-
phisticated and realistic multi-agent models to be analyzed by means of numer-
ical simulations. In the present paper we take the opposite tack and return
instead to one of the simplest multiplicative models, the giver scheme. We
show that its master equation can be solved efficiently by a Laplace transform
technique. Armed with this new tool backed by multi-agent simulations, we
identify the following new properties of the system. (1) We get precise val-
ues of various quantities such as the steady-state entropy as a function of the
model parameter f , independently of the number of agents. (2) We explore
the thinly studied regime 1/2 < f < 1 and identify its unusual properties,
e.g. oscillations in ps(w). (3) Using multi-agent simulations, we investigate
how the Boltzmann entropy evolves with time as the system approaches equi-
librium and argue that the Boltzmann entropy is not a suitable entropy for
61
3. Laplace transform analysis of a multiplicative asset transfer model
the giver scheme, even though the system is closed and conservative. (4) We
propose a simple asymmetric stochastic process that is equivalent to the giver
scheme. (5) We investigate how the degree of inequality, characterized by
the Gini coefficient, depends on f . (6) Finally, we apply phase-space tech-
niques from statistical mechanics to the giver scheme in order to illuminate
the difficulties and opportunities that this asset transfer model presents.
3.2 Giver scheme
We consider a simple asset transfer model, in which the transfer amount is
equal to a fixed fraction of the giver’s wealth.2 If wg is the giver’s wealth and
wr is the receiver’s wealth prior to the transfer, then their wealths after the
transfer are given by wg−∆w and wr+∆w respectively, with ∆w = fwg and
f ∈ (0, 1). The model comprises a large number of agents, who are assigned
wealths initially according to some distribution. The transfers are assumed to
take place over a fixed time interval ∆t. At each discrete time ti, the agents
are divided randomly into pairs and the transfer formula is applied to each
pair. The transfers are complete by the time ti+1 = ti + ∆t and the process
repeats at the time ti+1. The probability of drawing any pair is the same.
In each pair, the giver is assigned randomly regardless of the wealths of the
agents.
The master equation for this system was derived in Ispolatov et al. (1998)
and is given by
∂p(w, t)
∂t= −p(w, t)+ 1
2(1 − f)p
(
w
1− f, t
)
+1
2f
∫ w
0dw′ p
(
w − w′
f, t
)
p(w′, t).
(3.1)
It is easy to verify that the mean of the distribution, µ1 =∫∞0 dw wp(w, t),
does not depend on time. Upon integrating by parts, one arrives at the evo-
lution equationdµ2(t)
dt= −f(1− f)µ2 + fµ21 (3.2)
for the second moment, µ2(t) =∫∞0 dw w2p(w, t), first reported in Ispolatov
et al. (1998). This equation can be solved for the variance
σ2(t) = µ2(t)− µ21 =
(
µ2(0) −µ21
1− f
)
e−f(1−f)t +fµ211− f
. (3.3)
2The giver is also called the payer or the loser in the literature.
62
3. Laplace transform analysis of a multiplicative asset transfer model
In the steady state, one has σs = σ(t→ ∞) = µ1[f/(1−f)]1/2. For simplicity,
we assume henceforth that the mean of p(w, t) equals unity.3 In the following
sections, we are mostly concerned with the steady-state distribution ps(w) =
p(w, t → ∞).
3.3 Laplace transform of the master equation
The Laplace transform of the master equation in the steady state is given by
g(z) =1
2g(z − fz) +
1
2g(z)g(fz), (3.4)
with g(z) =∫∞0 dw e−zwps(w). Note that the functional equation (3.4) applies
to any integral transform whose kernel depends only on the product of the
arguments of the function and its transform. For f = 1/2, the functional
equation has a closed form solution
g(z) =1
1 + Cz, (3.5)
where C is a complex-valued constant. Using the definition of the transform,
we have g(0) = 1 from the normalization of ps(w) and g′(0) = −1 from the
assumption that ps(w) has unit mean, which gives C = 1. Applying the
inverse Laplace transform to this solution gives the exponential distribution,
ps(w) = e−w, which was obtained in Ispolatov et al. (1998) by substituting
simple “test” functions into the master equation. No closed-form solutions
have been found for other values of f ∈ (0, 1).
The Taylor expansion of g(z) at z = 0 can be derived by substituting the
expansion in (3.4) and using g(0) = 1 and g′(0) = −1. For the first four terms
of the expansion, this procedure gives
g(z) = 1− z +1
2(1− f)z2 − 1 + f
6(1− f)2z3 +O(z4). (3.6)
In general, for g(z) =∑∞
n=0 an(−z)n/n!, a0 = 1, and a1 = 1, we obtain
an =
n−1∑
k=1
(
n
k
)
fkakan−k
1− fn − (1− f)nfor n > 1. (3.7)
Since g(−z) is the moment-generating function for the distribution ps(w),
the n-th moment of the distribution µn equal an, i.e. all moments of the
3If the mean µ1 6= 1, one can consider the function q(x) = µ1p(µ1x), which is normalizedand has unit mean.
63
3. Laplace transform analysis of a multiplicative asset transfer model
steady-state wealth distribution can be computed for any f using the recursive
formula (3.7).
Using the Taylor expansion, one has an → 1 as f → 0 and hence g(z) →e−z. Note that the functional equation (3.4) becomes an identity for f = 0
and g(0) = 1. Taking the inverse Laplace transform of g(z) = e−z, formally
one gets ps(w) = δ(w− 1). However, this wealth distribution is never reached
because the relaxation time scale tr = [f(1 − f)]−1 determined from (3.3)
tends to infinity in this limit. Indeed, ps(w) is equal to the initial distribution
if f = 0. On the other hand, the Taylor expansion does not have a limit as
f → 1. The functional equation (3.4) has the solution g(z) = 1 when f = 1,
but it does not satisfy the condition g′(0) = −1. It appears that ps(w) does
not have a proper limit as f → 1. Note also that the relaxation time tends to
infinity as f approaches unity as well.
The asymptotic behaviour of g(z) at infinity can also be deduced readily
from the functional equation. In the directions in the complex plane for which
one has g(z) → 0 as |z| → ∞, the equation
g(z − fz) = 2g(z) (3.8)
must be approximately true for large enough |z|. Assuming a power-law shape
|g(z)| ∝ |z|−α as |z| → ∞, equation (3.8) gives
α =−1
log2(1− f). (3.9)
By Watson’s lemma Davies (2002), this is consistent with the asymptotic
behaviour p(w) ∝ wα−1 as w → 0 that was found in Ispolatov et al. (1998) by
the method of dominant balance.
The functional equation (3.4) can be solved iteratively when it is cast in
the form
gi+1(z) =gi(z − fz)
2− gi(fz), (3.10)
where gi(z) is the i-th iteration. Experimentation shows that the choice
g0(z) = 1/(1 + z) works well for all f . A detailed description of the com-
putational procedure is given in 3.7; there are some subtleties involved in the
choice of grid and interpolation method. An example of the numerical solution
for f = 0.1 is presented in Figures 3.1a and 3.1b. The power-law behaviour
at large |z|, with the exponent given by (3.9), is confirmed numerically for
f = 0.1 (right panel of Figure 3.1a) and a range of other values. The itera-
tions converge in the negative half-plane, Re(z) < 0, despite the complicated
64
3. Laplace transform analysis of a multiplicative asset transfer model
θ=
90◦
θ=
60◦
θ=
0◦
log10|g(z)|
log10
r
Im[g(z)]
log10
r
Re[g(z)]
log10
r
−2
−1
01
23
−2−1
01
23
−2−1
01
23
−16
−14
−12
−10
−8
−6
−4
−20
−1
−0.50
0.51
−1
−0.50
0.51
(a)
Im
(z)
Re(z)
Im
[g(z)]
Im
(z)
Re(z)
Re[g
(z)]
−5
05
10
−5
05
10
−10
−5
05
10
−10
−5
05
10
−1
−0.50
0.51
−1
−0.50
0.51
(b)
Figure
3.1:
TheLap
lace
tran
sformg(z)forf=
0.1ob
tained
bysolving(3.4)iteratively.
(a)Re[g(z)]
(top
left
pan
el),
Im[g(z)]
(bottom
left
pan
el),
and|g(z)|
(rightpan
el)versusralon
gthereal
axis
(dashed
curve),theim
aginaryax
is(solid
curve),an
dthelineinclined
atθ=
60◦to
thereal
axis
(dottedcu
rve).Thevariab
lesran
dθaredefi
ned
byz=re
iθ.(b)A
view
ofthe
real
(top
)an
dim
aginary(bottom)parts
ofg(z)(values
above1an
dbelow
−1havebeencu
toff
).
65
3. Laplace transform analysis of a multiplicative asset transfer model
structure of g(z), illustrated in Figure 3.1b, as it gradually approaches e−z
for decreasing values of f . The convergence does not depend on the initial
function g0(z); e.g. g0(z) = e−z works just as well for small values of f .
3.4 Steady-state wealth distribution by Laplace
inversion
The steady-state probability distribution function ps(w) can be obtained by
inverting its Laplace transform g(z) numerically. A number of inversion al-
gorithms were reviewed recently in Hassanzadeh and Pooladi-Darvish (2007)
and Abate and Whitt (2006). The reviewers advised that at least two different
algorithms should be used as a cross-check, because different algorithms work
well for specific classes of functions and none of the algorithms is universally
accurate. Fortunately, the algorithms are easy to implement. We test four (re-
ferred to as the Euler, Talbot, Stehfest, and Zakian algorithms in Hassanzadeh
and Pooladi-Darvish (2007); Abate and Whitt (2006)) and find that the first
two give accurate results over a wider range of w. Euler has an additional ad-
vantage over Talbot: it samples g(z) in the positive half-plane only, where the
function g(z) has a simpler structure, as one sees in Figure 3.1b. The results
of the inversion are presented in Figures 3.2a and 3.2b for 0.025 ≤ f ≤ 0.5
and Figures 3.3a and 3.3b for 0.5 ≤ f ≤ 0.9. The exponential analytic solu-
tion is recovered numerically for f = 0.5. From the output of the inversion
algorithms, we compute the moments of the distribution (µ0, µ1, and µ2)
and find agreement with the analytical results to 8 significant digits. In Fig-
ures 3.4a and 3.4b for f = 0.95 and f = 0.05 respectively, we compare the
wealth distributions obtained from the Laplace transform (curves) and from
the agent-based simulations (crosses). We find excellent agreement between
these two methods for all values of f that we consider.
The algorithms that perform the inverse Laplace transform suffer from
truncation errors. Maximum precision is achieved near the peak of ps(w). In
the tail, the precision decreases until the results are completely dominated
by the truncation errors below a threshold value of ps(w). For example, we
perform all computations with 16 significant digits and achieve ∼ 8 significant
digits of precision at the peak of ps(w), but the algorithms break down at
ps(w) . 10−8.
The wealth distributions that we find for f < 1/2 are characterized by
66
3. Laplace transform analysis of a multiplicative asset transfer model
f=
0.025
f=
0.05
f=
0.1
f=
0.25
f=
0.5
ps(w)
w
00.5
11.5
22.5
30
0.51
1.52
2.53
(a)linearscale
f=
0.025
f=
0.05
f=
0.1
f=
0.25
f=
0.5
log10ps(w)
w
01
23
45
67
8−
7
−6
−5
−4
−3
−2
−101
(b)log-linearscale
Figure
3.2:
Thesteady-state
wealthprobab
ilitydistribution
functionps(w
)ob
tained
byinvertingtheLap
lace
tran
sformg(z)
forthefollow
ingvalues
ofthetran
sfer
fraction
:f
=0.5(boldsolidcu
rve),0.25
(dash-dot
curve),0.1(dottedcu
rve),0.05
(dashed
curve),an
d0.025(thin
solidcu
rve).
67
3. Laplace transform analysis of a multiplicative asset transfer model
f=
0.9
f=
0.8
f=
0.7
f=
0.6
f=
0.5
log10 ps(w)
log10w
−3
−2.5
−2
−1.5
−1
−0.5
00.5
11.5
2−
5
−4
−3
−2
−1 0 1 2
(a)log-lo
gsca
le
f=
0.9
f=
0.8
f=
0.7
f=
0.6
f=
0.5
log10 ps(w)
w
05
10
15
20
25
30
35
40
45
50
−5
−4
−3
−2
−1 0 1
(b)log-lin
earsca
le
Figu
re3.3:
Thestead
y-state
wealth
prob
ability
distrib
ution
function
ps (w
)ob
tained
byinvertin
gtheLap
lacetran
sformg(z)
forthefollow
ingvalu
esof
thetran
sferfraction
:f=
0.5(bold
solidcu
rve),0.6
(dash
-dot
curve),
0.7(dotted
curve),
0.8(dash
edcu
rve),an
d0.9
(thin
solidcu
rve).
68
3. Laplace transform analysis of a multiplicative asset transfer model
log10n(w)
log10w
(m.u.)
−1
−0.5
00.5
11.5
22.5
33.5
4−
10123456
(a)f=
0.95
log10n(w)w
(m.u.)
0100
200
300
400
500
600
700
−1
−0.50
0.51
1.52
2.53
3.54
(b)f=
0.05
Figure
3.4:
Thepop
ulation
distribution
n(w
),show
nwithcrosses,
asafunctionof
wealthw,measuredin
ficticiousmon
etary
units(m
.u.)
usedin
theagent-based
simulation
s.Thedistribution
iscomputedas
thenumber
ofagents
ineveryunit
wealth
interval
after100step
sin
thesimulation
ofthegiverschem
ewithtotalnumber
ofagentsN
=4×
105an
dtran
sfer
param
eter
(a)f=
0.95
and(b)f=
0.05.Theinitialdistribution
isuniform
inthewealthinterval
(a)[0,100
m.u.]an
d(b)[0,500
m.u.].
Thecorrespon
dingsolution
ofthesteady-state
masterequationforthesamefis
show
nwithasolidcu
rve,
withps(w
)scaled
toconform
withthedefi
nitionofn(w
)accordingto
Nps(w/〈w〉)/〈w〉where〈w
〉is
themeanwealth.Boththeagent-based
simulation
san
dthemasterequationpredictoscillationsin
thewealthdistribution
in(a)butnot
in(b).
69
3. Laplace transform analysis of a multiplicative asset transfer model
power-law behaviour at w ≪ 1, in accord with the analytical results, and
approximately exponential tails at large w. A careful examination of the tails
confirms that the asymptotic behaviour at large wealths is not exactly ex-
ponential. However, we have not been able to find a closed-form expression
for it. The distribution becomes tightly concentrated around its peak as f
decreases; the peak of the distribution gradually shifts towards w = 1. On
the other hand, the peak shifts towards w = 0 as f increases; the distribu-
tions eventually turns into an exponential function for f = 1/2. This overall
behaviour is similar to that observed in the asset exchange models based on
the saving propensity with 0 < λ < 1 Chatterjee and Chakrabarti (2007).
The structure of the steady-state solutions for f > 1/2 is very different.
The asymptotic approximation ps(w) ∝ wα−1, with α = −1/ log2(1 − f), is
valid for f > 1/2 as well and indicates that the distribution diverges at w = 0
(as α < 1). For values of f sufficiently close to 1, the wealth distribution
acquires a shape that is akin to a power law, ps(w) ∝ w−1, with overlaid
oscillations that become more prominent as f increases. This power-law be-
haviour cuts off exponentially at some critical wealth that increases slowly as
f approaches 1. At the same time the exponential drop-off at large wealths
becomes shallower as evident from Figure 3.3b. The oscillations of ps(w) ap-
pear to be periodic on a logarithmic scale, with the period depending on f .
For example, the periods for f = 0.9 and f = 0.99 are roughly one and two
decades respectively. This is directly related to the fact that all givers retain
1% of their wealth for f = 0.99 and 10% for f = 0.9.
3.5 Discussion
We now use the Laplace transform tools developed in Section 3.4 to address
two questions that are costly to explore with agent-based simulations: the
nature of disorder (entropy) and its evolution in the giver scheme, and the
degree of inequality in the steady state.
Entropy and the approach to equilibrium
According to Boltzmann, states with higher entropy are more probable be-
cause they correspond to a larger number of microscopic configurations of the
system. A closed system evolves to a state of maximum entropy, i.e. maxi-
mum disorder, which is characterized by the Boltzmann-Gibbs distribution.
70
3. Laplace transform analysis of a multiplicative asset transfer model
The exponential distribution observed in models where the transfer amount
is fixed and constant (see section II.C in Yakovenko and Rosser Jr. (2009))
is thus consistent with entropy maximization ideas. On the other hand, it is
argued in Yakovenko and Rosser Jr. (2009) that multiplicative asset exchanges
may lead to non-exponential distributions because of the broken time-reversal
symmetry, whereas, in models with fixed additive exchanges, the time-reversal
symmetry is preserved. Despite this, the entropy maximization technique has
been applied in Chakraborti and Patriarca (2008) to an asset exchange model
described by a Hamiltonian quadratic in wealth variables. It predicts a gamma
distribution of wealth, but ps(w) in multiplicative asset exchange models is
not a gamma distribution in general.
In this section, we explore the applicability of the Boltzmann entropy to
the giver scheme. Using the Laplace transformed solutions of the master
equation (3.1), we compute the steady-state Boltzmann entropy4
Ss = −∫ ∞
0dw ps(w) log[ps(w)] (3.11)
for several values of f in the range 0.01 ≤ f ≤ 0.9 and plot the results
in Figure 3.5a. Entropy maximization arguments Kapur (1989) imply that
the entropy defined by (3.11) leads uniquely to the exponential distribution,
p(w) = e−w with Ss = 1, if the only condition is that the mean of the dis-
tribution is fixed to µ1 = 1. In our model, however, the transfer fraction f
places additional constraints on how the distribution of wealth evolves with
time. Therefore, it is not surprising that we observe a range of steady-state
entropies Ss 6= 1 corresponding to different values of f . The exponential distri-
bution for f = 1/2 appears to be the most disordered state of the system with
the highest entropy Ss = 1. For all other values of f , the entropy is smaller
and it becomes negative as f approaches 0 or 1. Reading off the graph, we
find that Ss is negative for 0 < f < 0.058 and 0.836 < f < 1. The entropy
tends to negative infinity as f → 0 or f → 1, which is in accord with the
behaviour of ps(w) in these limits.
4We stress that this definition applies to a normalized distribution with unit mean. Un-like the case of discrete probability distributions, continuous entropy can be negative andit is not invariant with respect to the change of variable. It may be more appropriate toconsider the Kullback-Leibner divergence Ds =
∫∞
0dw ps(w) log[ps(w)/m(w)], which is a
measure of the divergence between ps(w) and the reference distribution m(w). It is conve-nient to take m(w) = e−w, in which case the divergence essentially reverts to Boltzmannentropy because Ds = 1− Ss.
71
3. Laplace transform analysis of a multiplicative asset transfer model
Ss
σ2s
10−
210−
1100
101
−1
−0.8
−0.6
−0.4
−0.2 0
0.2
0.4
0.6
0.8 1
(a)stea
dy-sta
teen
tropy
S(t)
t(step
s)
010
20
30
40
50
60
70
80
90
100
0
0.0
5
0.1
0.1
5
0.2
0.2
5
(b)en
tropyevolutio
n
Figu
re3.5:
(a)Boltzm
ann
entrop
ySsof
thestead
y-state
distrib
ution
asafunction
ofthevarian
ceσ2s=f/(1
−f).
The
criticalvalu
esσ2s=
0.062,1,
and5.098,
correspon
dingtof=
0.058,0.5,
and0.836
respectively,
areindicated
with
thedotted
lines.
(b)Entrop
yas
afunction
oftim
efor
theinitial
distrib
ution
givenby(3.12)
with
f=
0.058,com
puted
fromthemulti-
agentsim
ulation
ofthegiver
schem
e.For
thesim
ulation
,thedistrib
ution
(3.12)was
scaledupto
giveN
=337123
agents
in0≤w
≤1421.
Tocom
pute
theentrop
y,thepop
ulation
distrib
ution
produced
bythesim
ulation
was
norm
alizedto
aprob
ability
distrib
ution
with
unitmean
.
72
3. Laplace transform analysis of a multiplicative asset transfer model
Negative values of Ss are already a warning that the Boltzmann entropy
may not be a faithful measure of disorder in a multiplicative asset transfer
system like the giver scheme. However, the situation worsens when we look
at how S(t) = −∫∞0 dw p(w, t) log[p(w, t)] evolves with time by conducting
multi-agent simulations. In many cases, it decreases instead of increasing. For
example, if we choose p(w, 0) = e−w initially, S(t) decreases with time for all
f 6= 1/2 and remains constant for f = 1/2. Moreover, we can easily find
realistic situations where S(t) does not change monotonically with time, as
the experiment described below shows. Consider an initial distribution of the
form
p(w, 0) =
p1, 0 ≤ w ≤ 1,
p2, 1 < w ≤ w2,
0, otherwise,
(3.12)
with the parameters p1, p2, and w2 chosen to give S(0) = 0 (p1 ≈ 0.296, p2 ≈1.669, and w2 ≈ 1.421). The evolution of entropy for this initial distribution
with f = 0.058 is plotted in Figure 3.5b. The entropy grows initially but
after about ten steps in the simulation it turns over and begins to decrease,
eventually reaching Ss = 0 as expected for f = 0.058. This is in marked
contrast to the behaviour of S(t) in an ideal gas, where one has dS(t)/dt ≥ 0
according to Boltzmann’s H-theorem.
The population distribution is determined by dividing the wealth axis into
small bins and computing the number of agents that fall in each bin. One can
define the multiplicityW as the number of permutations of the agents between
different wealth bins such that the occupation numbers of the bins do not
change. The definition of entropy, Ss = logW , leads to the expression (3.11) in
the continuous limit. Entropy maximization under the assumption that total
wealth is conserved gives an exponential distribution. However, this ignores
the global constraints on the probability distribution ps(w) imposed by the
transfer fraction f . The maximization procedure must take these constraints
into account to derive the steady-state wealth distribution appropriate to the
giver scheme. Unfortunately, at the time of writing, we have been unable to
derive these additional constraints, and they do not appear in the literature.
73
3. Laplace transform analysis of a multiplicative asset transfer model
log10 ξ(w)
w
00.5
11.5
22.5
3−
5
−4
−3
−2
−1 0 1
(a)
wi
i
0100
200
300
400
500
600
700
800
900
1000
0
0.2
0.4
0.6
0.8 1
1.2
1.4
1.6
1.8 2
(b)
Figu
re3.6:
(a)Thelim
itingprob
ability
distrib
ution
ξ(w),show
nwith
crosses,ob
tained
fromon
erealization
oftheasy
mmetric
random
process
(3.13)for
f=
0.05after
106iteration
s.Themean
is0.9998
andthevarian
ceis
0.0519(cf.
0.0526theoretically
fromthemaster
equation
).Thestead
y-state
distrib
ution
ps (w
)for
thesam
efob
tained
byLap
laceinversion
isshow
nas
asolid
curve
forcom
parison
.(b)Thefirst
1000valu
esof{w
i }.
74
3. Laplace transform analysis of a multiplicative asset transfer model
Random process
The evolution of wealth in the giver scheme can be analyzed in terms of a
random process defined by
wi+1 = wi +∆wi, (3.13)
where w1 = 1 and ∆wi = +f or ∆wi = −fwi with equal probability. This
process is asymmetric, i.e. multiplicative in the negative direction and addi-
tive in the positive. To illuminate the relationship between the giver scheme
and the random process we note that (1) an agent’s loss of wealth is always
proportional to his wealth, i.e. it is multiplicative, and (2) an agent’s gain
of wealth can originate from any other agent in the population and therefore
equals f〈w〉 on average, or simply f if we set 〈w〉 = 1. We compute the lim-
iting distribution ξ(w) of this process by applying (3.13) a sufficiently large
number of times and then constructing a histogram of all {wi}. By the ergodic
assumption, this is equivalent to computing a large number of realizations of
this random process and using the final values in each realization to find the
limiting distribution.
Figures 3.6a and 3.6b display one particular realization of the random
process (3.13) and the corresponding limiting distribution. Given the close
link between the transfer model and the random process, it is not surprising
that the limiting distribution ξ(w) of the random process (3.13) appears to be
identical to the steady-state distribution ps(w) of the giver scheme. We obtain
similar results for other values of f ≪ 1. Note that the slight discrepancy
between ξ(w) and ps(w) at w > 2 is due to insufficient sampling of large
wealths by the random process. The agreement improves as the number of
iterations increases.
Inequality of wealth
A traditional measure of inequality in economic systems is the Gini coefficient,
defined asG = 1−2∫ 10 L(X)dX, where L(X) is the Lorenz curve. For a contin-
uous distribution, we have L(w) =∫ w0 dw′ w′ps(w
′), X(w) =∫ w0 dw′ ps(w
′),5
and hence
Gs = 1− 2
∫ ∞
0dw ps(w)
∫ w
0dw′ w′ps(w
′), (3.14)
5 Note that we have L(X) ≤ X for all X because L(0) = 0, L(1) = 1, and dL/dX is amonotonically increasing function.
75
3. Laplace transform analysis of a multiplicative asset transfer model
Gs
σ2s
10−2 10−1 100 1010
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.7: Gini coefficient of the steady-state distribution ps(w) as a functionof the variance σ2s = f/(1− f).
such that Gs = 0 corresponds to perfect equality, and Gs = 1 to perfect
inequality.
In Figure 3.7 we plot Gs versus the steady-state variance σ2s = f/(1 − f)
for 0.01 ≤ f ≤ 0.9. As expected, Gs increases monotonically with σ2s , since
both quantities are measures of dispersion. Interestingly, however, there is
an inflection point in the Gs(σ2s) curve at σ2s = 1, Gs = 1/2, corresponding
to the exponential distribution (i.e. f = 1/2). For f → 0, we have ps(w) →δ(w− 1), which corresponds to perfect equality since all agents have the same
wealth. On the other hand, for f → 1, the distribution ps(w) becomes sharply
peaked near w = 0, while the standard deviation approaches infinity. This
corresponds to the situation where most agents have zero wealth, except for
one who has everything, i.e. perfect inequality.
We can understand the evolution towards inequality in terms of the state
vector of the system. Consider N agents whose wealths are characterized
by random variables wi, i = 1, 2, . . . , N . The state of the system can be
described by the phase-space vector w = (w1, w2, . . . , wN ). The constraints
that define the phase-space are (1) 0 ≤ wi ≤ 1 for all i, and (2)∑N
i=1 wi = 1
(we assume for convenience that the total wealth is unity). These constraints
76
3. Laplace transform analysis of a multiplicative asset transfer model
define a segment of the (N − 1)-dimensional hyperplane embedded in N -
dimensional space. Without any additional constraints, entropy maximization
Kapur (1989) gives gi(wi) = (N−1)(1−wi)N−2 for the probability distribution
of the wealth wi of the i-th agent, with mean 〈wi〉 = 1N and variance σ2wi
=N−1
N2(N+1). However, in our system, the asset transfer process and the value of
the parameter f place additional restrictions on the evolution of w. For the
increment of the phase-space vector w from time tk to time tk+1, i.e. after one
generation of asset transfers, we have
|∆w|2 =N∑
i=1
[wi(tk+1)− wi(tk)]2. (3.15)
The terms in the sum on the right hand side can be split into two groups,
associated with the givers and the receivers. Since the transfer amount is
proportional to the giver’s wealth, we get
|∆w|2 = 2f2∑
i∈givers
[wi(tk)]2. (3.16)
Therefore the following inequality must always be satisfied:
|∆w| ≤ 21/2f |w|. (3.17)
In addition, we have
N−1/2 ≤ |w| ≤ 1 (3.18)
due to the restrictions of the phase-space itself. Note that the state wi = 1/N
for all i is the nearest point to the origin.
When f is small, the norm of the increment |∆w| is also small compared
with the maximum linear extent of the phase space (which equals 21/2); the
evolution of w is gradual. Furthermore, |∆w| is also constrained by |w|, whichcan be very small if N is large and all agents are clustered as close to the origin
as possible. So, if the dispersion in wealth is modest, w moves slower through
the phase space than if there is great inequality. On the other hand, w changes
more rapidly on average if |w| is close to unity, which corresponds to large
inequality. The states of equality are therefore more probable, which explains
why the steady-state distribution tends towards a delta function for f → 0.
Even if the initial wealth distribution is very unequal, w drifts quickly towards
the states of near equality.
When f is close to 1, |∆w| can be comparable to the size of the phase
space. Since f is large, the gains in wealth of the individual agents can be
77
3. Laplace transform analysis of a multiplicative asset transfer model
large as well. This leads to the situation where a few agents own most of the
wealth. These agents retain their large wealth for a short time only (typically a
few time steps) before they become givers and pass their large wealth to other
agents. In the extreme case f = 1, one agent possesses all the wealth at any
instant, while all the other agents have zero wealth. This maximum wealth is
passed from agent to agent frequently. This corresponds to w jumping from
one corner of the phase space to another. For f ≈ 1, w evolves similarly, with
ps(w) peaking strongly at zero wealth.
3.6 Conclusions
We develop a new technique for computing the steady-state probability dis-
tribution of a multiplicative asset transfer model, which we call the giver
scheme, by Laplace transforming the associated master equation to give a
functional equation for the characteristic function of the distribution. In the
giver scheme, the transfer amount fwg is proportional to the giving agent’s
wealth wg, so the model depends on a single parameter f ∈ (0, 1). We develop
an efficient iterative method to solve the functional equation for any f , and
we employ several Laplace inversion algorithms to recover the steady-state
distribution ps(w).
We comprehensively explore the dependence of the wealth distribution on
the value of f , especially the thinly studied regime 1/2 ≤ f < 1. We find
a stark qualitative difference between the distributions for f ≈ 0 (sharply
peaked distribution centred around the mean wealth) and f ≈ 1 (broad dis-
tribution of approximately power-law shape with overlaid oscillations). These
two extremes correspond to near-perfect equality and inequality respectively,
as characterized by the Gini coefficient. Both extremes are also characterized
by negative Boltzmann entropy. While the regime f ≈ 0 is generally thought
to represent to some extent the exchange processes occuring in the real econ-
omy, the regime f ≈ 1 is probably less applicable to realistic economic systems,
except perhaps in situations involving extreme leverage. The regime f ≈ 1
may also be relevant to the analysis of gambling, where transitory fortunes
are made and lost frequently.
We show that the Boltzmann entropy is unlikely to be a faithful measure of
disorder in a multiplicative asset transfer system, since it does not vary mono-
tonically as a function of time, assuming the second law of thermodynamics.
This is an important and counterintuitive result, because the system in the
78
3. Laplace transform analysis of a multiplicative asset transfer model
giver scheme is closed and the microscopic transfer rules conserve wealth, in a
manner reminiscent of the microcanonical ensemble in statistical mechanics.
In a multiplicative transfer system, the correlations between various subsys-
tems (e.g. subclasses corresponding to a particular historical sequence of giving
and receiving) and the time-reversal asymmetry of the microscopic rules are
crucial to the system’s dynamics and, therefore, cannot be ignored.
3.7 Iterative procedure
We assume that the computations are carried out with 16 significant digits.
For a given complex argument z, define a uniform grid u = {uk}K1 that covers
the interval [−4, log10(|z|)]. The approximation (3.6) gives sufficient precision
for |z| < 10−4 for computations with 16 significant digits. Choose the num-
ber of points K such that there are a large number of points in every unit
interval, say, 103 logarithmic grid points per decade in [10−4, |z|]. Define two
auxiliary grids, u(f) = log10(f) + u and u(1−f) = log10(1 − f) + u, and ini-
tialize the iterations with g0(10uz/|z|) = 1/(1 + 10uz/|z|). For a given set of
values gi(10uz/|z|), defined on the grid u, find the corresponding values on
the auxiliary grids by performing a spline interpolation or using the approxi-
mation (3.6) where appropriate. Then use these values in equation (3.10) to
find gi+1(10uz/|z|). Continue iterating until the convergence criterion is met
(we find that the convergence spreads gradually from zero to |z|). Typically
the convergence requires a few dozens of iterations for |z| ∼ 100 in the posi-
tive half-plane. Once the convergence is reached, apply a spline interpolation
to find g(z′) for any z′ along the same direction in the complex plane as z,
provided that |z′| < |z|.The obvious disadvantage of the procedure outlined above is that it relies
on interpolation. Its precision is therefore limited by the number of points in
the grid u, i.e. the discretization of the interval [0, |z|]. It is possible, however,to avoid interpolation altogether by defining a special non-uniform grid that
is invariant with respect to division by f and (1 − f). This gives rise to an
alternative procedure for computing the iterations.
Define a grid rk,m = fk(1 − f)m with 0 ≤ k ≤ K and 0 ≤ m ≤ M ,
where K = ⌈log(10−4/|z|)/ log(f)⌉ and M = ⌈log(10−4/|z|)/ log(1 − f)⌉ are
defined such that |z|rK,0 < 10−4 and |z|r0,M < 10−4. The function g(z) on
the grid zk,m = rk,mz, defined according to gk,m = g(rk,mz), has the following
3. Laplace transform analysis of a multiplicative asset transfer model
gk,m+1. Therefore the iteration rule becomes
gk,m =gk,m+1
2− gk+1,m, (3.19)
for 0 ≤ k ≤ K − 1 and 0 ≤ m ≤ M − 1. For gK,M one can use the ap-
proximation (3.6). In fact, the Taylor expansion can be used for any point
|zk,m| < 10−4. Thus, no interpolation is required and the iterations can be
computed more efficiently. However, unlike the approach based on interpola-
tion, this procedure must be repeated for different arguments even if they lie
in the same direction in the complex plane.
80
Chapter 4
A note on the theory of fast
money flow dynamics
The gauge theory of arbitrage was introduced by Ilinski in Ilinski (1997) and
applied to fast money flows in Ilinskaia and Ilinski (1999); Ilinski (2001). The
theory of fast money flow dynamics attempts to model the evolution of cur-
rency exchange rates and stock prices on short, e.g. intra-day, time scales. It
has been used to explain some of the heuristic trading rules, known as tech-
nical analysis, that are used by professional traders in the equity and foreign
exchange markets. A critique of some of the underlying assumptions of the
gauge theory of arbitrage was presented by Sornette in Sornette (1998). In
this paper, we present a critique of the theory of fast money flow dynamics,
which was not examined by Sornette. We demonstrate that the choice of the
input parameters used in Ilinski (2001) results in sinusoidal oscillations of the
exchange rate, in conflict with the results presented in Ilinski (2001). We
also find that the dynamics predicted by the theory are generally unstable in
most realistic situations, with the exchange rate tending to zero or infinity
exponentially.
4.1 Introduction
Fast money flows are analyzed in Ilinskaia and Ilinski (1999); Ilinski (2001) in
terms of the lattice gauge theory of arbitrage developed in Ilinski (1997). The
main idea of the theory is that the dynamics should only depend on gauge
invariant quantities rather than the exchange rates themselves. Changing the
units in which stocks of currency are denominated obviously changes the nom-
81
4. A note on the theory of fast money flow dynamics
inal exchange rate. However, it is obvious that such changes of scale, i.e. gauge
transformations, should have no effect on its dynamics. Some assumptions of
the theory have been criticized in Sornette (1998); for example, the lack of
justification for the exponential form of the weight of a given market config-
uration. However, the results of the theory reported in Ilinskaia and Ilinski
(1999); Ilinski (2001) seem impressive, reproducing in particular some of the
phenomenological rules of technical trading employed by professional traders.
Hence the theory appears to be a promising tool for analyzing the markets.
In this note, we present our analysis of the theory of fast money flow dy-
namics and re-examine the results presented in Ilinskaia and Ilinski (1999);
Ilinski (2001). In Sect. 4.2, we present the derivation of the dynamical equa-
tions of the theory. In Sect. 4.3, we examine the dynamics predicted by the
theory for various initial conditions. We highlight certain inconsistencies in
the theory, the unstable dynamics for most realistic values of the parameters
and initial conditions, and the resulting problems in applying the theory to
technical trading. In Sect. 4.4, we revisit the action and demonstrate that the
expression used in Ilinskaia and Ilinski (1999); Ilinski (2001) is inconsistent
with the evolution operator resulting from the lattice formulation.
4.2 Lattice gauge theory and fast money flow
dynamics
In analogy with quantum electrodynamics, Ilinski identified the exchange rate
S between two currencies with the field and the trading agents with matter.
In general, the exchange rate dynamics depends on the interest rates of the
underlying currencies. However, since we are interested in intra-day dynamics
only, we consider the special case of zero interest rates. Ilinski tacitly assumed
that the interest rates of the two currencies are identical, i.e r1 = r2. In this
paper we set r1 = r2 = 0 and assume that transaction costs are zero.
The part of the action s1 that describes the dynamics of the field on its
own is formulated by identifying arbitrage on the lattice with the curvature,
which gives
s1 = − 1
2σ2
∫ T
0dt
(
dy
dt
)2
. (4.1)
In Eq. (4.1), T is the investment horizon and σ2 is the volatility (presumed
to be constant in the interval 0 ≤ t ≤ T ). This expression is equivalent to a
Gaussian random walk in y = lnS.
82
4. A note on the theory of fast money flow dynamics
The effect of the field y on “matter”, i.e. the trading agents, is described
by the Hamiltonian
H(ψ1, ψ+1 , ψ2, ψ
+2 ) = H21ψ
+1 ψ2 +H12ψ
+2 ψ1, (4.2)
where ψ+k and ψk are creation and annihilation operators for agents in currency
k (k = 1, 2), and the coefficients H21 and H12 depend on y. According to
Ilinski, H21 = heβy and H12 = he−βy, where h and β are constants (we
discuss the motivation behind these formulas in Sect. 4.4). Following the
standard treatment of a quantum harmonic oscillator (see, e.g. Slavnov and
Faddeev (1980)), Ilinski Ilinski (2001) derived a path-integral expression for
the evolution operator in terms of the coherent states ψ1 and ψ2, which are
the eigenstates of the annihilation operators ψ1 and ψ2 respectively. From the
evolution operator one can obtain the expression for the part of the action s2
that represents the field’s effect on matter:
s2 =
∫ T
0dt
[
ψ1dψ1
dt+ ψ2
dψ2
dt+H(ψ1, ψ1, ψ2, ψ2)
]
, (4.3)
where the overbar denotes complex conjugation.
Finally, departing from the electrodynamics analogy, Ilinski introduced
Farmer’s term F to describe the effect of matter on the field. As a result, the
action s1 is replaced by
s1F = − 1
2σ2
∫ T
0dt
[
d(y + F )
dt
]2
, (4.4)
where
F =f
M(ψ1ψ1 − ψ2ψ2), (4.5)
M is the total number of agents, and f is a constant (Ilinski (2001) uses α in
place of f).
The total action is given by
s = s1F + s2. (4.6)
Following Ilinski, we introduce new variables η = βy and τ = ht, and replace
complex-valued ψk with φk and ρk, defined by ψk = (Mρk)1/2e−iφk (k = 1, 2)
and ρ1 + ρ2 = 1. Ilinski identifies Mρk with the number of agents in currency
k; the total number of agents is conserved. The action can be written as
s =M
∫ hT
0dτ L, (4.7)
83
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.1: Re-creation of Ilinski’s solution of Eqs. (4.9–4.11) given on page 169of Ilinski (2001) for α1 = 1.5, α2 = 10, C0 = 0, and the initial conditions:η(0) = 0.2, υ(0) = 0, ρ(0) = 0.5. The factor α1 in Eq. (4.10) is replaced withunity to match Ilinski’s Euler-Lagrange equations. The displayed quantitiesare as follows: ρ− 1/2 (solid), υ + η (dashed), η (dot-dashed), υ (dotted).
where the Lagrangian L is given by
L = −(2α2)−1(
η′ + α1ρ′)2
+ ρυ′ + φ′2+
+ 2[ρ(1 − ρ)]1/2 cosh(υ + η), (4.8)
with α1 = 2βf , α2 =Mβ2σ2/h, ρ = ρ1, υ = φ1−φ2. A prime denotes a deriva-
tive with respect to τ . Due to the unique structure of the Lagrangian (4.8),
the resulting Euler-Lagrange equations can be simplified to the following first
84
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.2: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.1. The factor α1 in Eq. (4.10) is restored.
However, some of the second-order nature of the Euler-Lagrange equations is
retained in the constant C0 = η′(0) + α1ρ′(0) + α2[ρ(0) − 1/2], whose value
depends explicitly on the derivatives ρ′(0) and η′(0). The equation for φ2 is
85
4. A note on the theory of fast money flow dynamics
trivial and we omit it. To solve Eqs. (4.9–4.11), one needs to specify the initial
conditions η(0), υ(0), ρ(0), and η′(0), which uniquely determine C0 (note that
ρ′(0) is given by Eq. (4.11)). Alternatively, one can set η(0), υ(0), ρ(0), and
C0, which uniquely determine η′(0).
4.3 Analysis of the Euler-Lagrange equations
Missing coefficient
By introducing new variables, ρ = ρ − 1/2 and η = υ + η, and linearizing
(|ρ| ≪ 1, |η| ≪ 1), we obtain η = ρ′ and
ρ′′ + (α2 − 4)ρ = C0. (4.12)
For α2 > 4, the general solution is
ρ = A sin(2πνt+ θ) + C0(α2 − 4)−1, (4.13)
η = 2πνA cos(2πνt+ θ), (4.14)
with ν = (α2−4)1/2/2π (A and θ are found from the initial conditions). This is
inconsistent with the solutions presented in Ilinskaia and Ilinski (1999); Ilinski
(2001), which exhibit oscillations decaying slowly with time. The origin of this
inconsistency can be traced to a simple algebraic mistake in the derivation of
the equations of motion given in Ilinskaia and Ilinski (1999); Ilinski (2001).
On page 168 of Ilinski (2001), the second term on the right-hand side of the
equation for υ′ is missing a factor α1. The same coefficient is also missing in the
equations given in Ilinskaia and Ilinski (1999). This is essentially equivalent
to replacing α1 in our Eq. (4.10) with unity, while keeping α1 in our Eq. (4.9)
intact.
We verify the above by numerically solving Eqs. (4.9–4.11) in their incor-
rect form (with α1 missing from one of the equations as in Ilinskaia and Ilinski
(1999); Ilinski (2001)) and in their correct form derived in this paper. We are
able to perfectly reproduce1 the plots presented on page 169 of Ilinski (2001)
by solving the incorrect equations (see Fig. 4.1). Note that we have α1 = 1.5
and α2 = 10 for the parameters used in Ilinski (2001). Ilinski claimed to set
η′(0) = 0 (dy(0)/dt = 0 in his notation), but this is obviously incorrect; the
solutions he presented are obtained for C0 = 0, which gives η′(0) ≈ −0.3020.
1In the caption of figure 7.2 in Ilinski (2001), it is claimed that one of the quantitiesdisplayed is η (y in Ilinski’s notation), but actually η + υ is plotted.
86
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.3: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.2, except α1 = 0.
As anticipated by the linearized analysis, the correct nonlinear equations of
motion do not show any decay in the oscillation amplitude (see Fig. 4.2).
Furthermore, we do not observe any enhancement of oscillations for smaller
values of α1, as Farmer’s term becomes less important. In fact, the solutions
for α1 = 0 plotted in Fig. 4.3 are only slightly different from those for α1 = 1.5
(cf. the plots given on page 171 of Ilinski (2001)). After some exploration, we
conclude that Farmer’s term does not have any critical effect on the dynamics
of the system; it only affects the amplitude of oscillations of η and υ, and their
phase shift from ρ.
87
4. A note on the theory of fast money flow dynamics
Unstable solutions
In Sect. 4.3, we explored the dynamics of η = η + υ in the case C0 = 0.
However, there is no a priori reason why the initial conditions should conspire
to give C0 = 0. In this section, we briefly examine the dynamics of η = β lnS
in the more general case C0 6= 0.
Linearizing Eqs. (4.9) and (4.10) gives
η′ = −α2ρ− α1η + C0, (4.15)
υ′ = 4ρ+ α1η. (4.16)
We find that the solutions for η and υ are also harmonic oscillations plus an
extra term linear in time. The average value of η changes linearly with time
at a rate −4C0(α2−4)−1, while the average of υ changes at the same rate but
with the opposite sign. This behaviour is illustrated in Figs. 4.4 and 4.5 (note
that ρ and η remain small, so the linearization assumption is not broken).
Thus, for C0 > 0, the exchange rate S decays exponentially to zero, whereas
for C0 < 0, it grows exponentially. In both cases the exponential time-scale is
given by τc = 0.25(α2 − 4)|C0|−1.
Technical trading
Ilinski justified certain rules employed in technical trading (see Ilinskaia and
Ilinski (1999) and pages 170–173 of Ilinski (2001)), e.g., the use of positive
and negative volume indices (PVI and NVI respectively), by appealing to the
solutions of the equations of motion. The relevant figures are presented in
Ilinski (2001) on pages 170 (figure 7.3) and 172 (figure 7.7). We identify the
trading volume V with |ρ′| and the return R with η′/β = S′/S. In Ilinski
(2001), the derivative of η = υ+ η is used incorrectly instead of η to compute
the return (see also footnote 1). For comparison, we plot the volume and the
return curves in Fig. 4.6, computed using the correct equations of motion and
C0 = 0. The quantities plotted in figure 7.7 of Ilinski (2001) are not specified,
nor are the parameters and initial conditions, so we do not comment on that
figure’s validity.
Ilinski used the trading volume and the return curves to construct con-
tinuous2 versions of PVI and NVI. The details of the construction are left
unspecified. However, the PVI and NVI are usually computed from daily re-
turns, not from continuous intra-day variables. In any event, the resulting
2In technical trading, these quantities are discrete and defined by recursive formulas.
88
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−1
−0.5
0
0.5
1
Figure 4.4: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.2, except with C0 = 0.1 instead of C0 = 0. The curvesare coded as in Figs. 4.1 and 4.2.
construction must depend strongly on the time-scale that is chosen, since the
indices are defined recursively. Examining figure 7.7 in Ilinski (2001), one
observes that, for instance, the continuous PVI is constant if the trading vol-
ume V is decreasing with time and changes linearly if V is increasing, with a
slope of +1 where the return curve R is positive and −1 where R is negative.
However, this simple trend is inconsistent with the recursive definitions of the
PVI and NVI employed in technical trading.
Moreover, the constant-amplitude solutions we employed in this section
only exist for C0 = 0. In all other cases, the exchange rate converges to
89
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−1
−0.5
0
0.5
1
Figure 4.5: As for Fig. 4.4, with C0 = −0.1.
zero or diverges to infinity exponentially on a short time scale. The condition
C0 = 0 requires precise alignment between the initial values ρ(0), v(0), η(0),
and η′(0). There is no reason to expect that such a precise alignment will
be observed at any time in the real market. Therefore, the lattice gauge
model predicts unrealistic behaviour (e.g., exponential divergence if C0 < 0)
of the exchange rate under most circumstances. Given the issues raised in
this section, it is premature to conclude that the technical trading schemes
employed by market participants can be justified by the lattice gauge model.
90
4. A note on the theory of fast money flow dynamics
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.6: The trading volume V (bold solid curve) and the return R (bolddot-dashed curve) for the same parameters as in Fig. 4.2. For comparison, wealso display ρ− 1/2 (thin solid curve) and η (thin dot-dashed curve).
4.4 Revisiting the action
We conclude by re-examining the derivation of the action s given by Eq. 4.6.
Consider two currencies, referred to as currency 1 and currency 2, linked by
an exchange rate S(t) that depends on time t, such that the amount C2 of
currency 2 at time t corresponds to the amount C1 = S(t)C2 of currency 1. We
assume that the currencies can only be exchanged at the discrete times tn =
n∆t (n = 0, . . . , N) and define Sn = S(tn). At any given time tn, an agent can
decide to either exchange his stock of currency for the counterpart currency
91
4. A note on the theory of fast money flow dynamics
or keep his position, in which case his stock of currency remains unchanged
(recall that we neglect interest rates completely since we are interested in the
intra-day dynamics). We display these possibilities in Fig. 4.7, showing part
of the lattice from time tn to time tn+1.
✲Sn
✛S−1n
✲Sn+1
✛S−1
n+1
✻
❄
✻
❄
Currency 2 Currency 1
tn
tn+1
Figure 4.7: Lattice diagram for the intra-day foreign exchange trading in twocurrencies. Interest rates are ignored.
The returns on arbitrage along the closed loops of the elementary plaquette
shown in Fig. 4.7 are given by S−1n Sn+1−1 for the clockwise loop and SnS
−1n+1−
1 for the counter-clockwise loop. The total return SnS−1n+1 + S−1
n Sn+1 − 2 is
identified in Ilinski (1997) with the curvature on the lattice and, therefore, the
corresponding discrete action is given by
A1 =N∑
n=0
an(SnS−1n+1 + S−1
n Sn+1 − 2). (4.17)
Assuming that for any n we have an∆t → 1/2σ2 in the limit ∆t → 0, we
obtain the continuous action s1 given by (4.1). No justification is given in
Ilinski (1997, 2001) for why the limit of an∆t must be finite. The expression
for Farmer’s term was derived in Ilinski (2001), but we omit it because its
inclusion has no critical effect on the dynamics (see Sect. 4.3).
In order to derive the Hamiltonian given by Eq. (4.2) and the expressions
for the coefficients H12 and H21, Ilinski considered the case of a single trader
first and then generalized to multiple traders by using creation and annihi-
lation operators. In the case of a single trader, Ilinski postulated that the
probability of a given path Q through the lattice from t0 to tN is exponen-
tially weighted with respect to s(Q) = ln(U1U2 . . . UJ), where {Uj} are the
parallel transport coefficients on the lattice (note that J > N for most paths).
Thus, for a given path Q, the probability is given by
P (Q) ∼ eβs(Q). (4.18)
92
4. A note on the theory of fast money flow dynamics
Depending on the path, a given Uj can be Sn, S−1n , or unity (note that Ilinski
introduces a new gauge, under which the exchange rates remain unchanged,
except at t0 and tN where they equal unity; see pages 131–132 of Ilinski (2001)
for more details). The state of the trader is characterized by the probabilities
p1 and p2 of being in currency 1 and currency 2 respectively. The evolution
of the state vector ( p1p2 ) can be described by the transition matrix
P (tn; tn−1) =
(
1 Sβn
S−βn 1
)
, (4.19)
which Ilinski essentially identifies3 with the discrete version of the continuous
evolution operator U(t, t′) that satisfies
∂U
∂t= HU, (4.20)
where H is the Hamiltonian and U(0, 0) is the identity matrix. Ilinski claim
that the expression for the transition matrix (4.19) and the formula (4.20)
result in
H =1
∆t
(
0 Sβ
S−β 0
)
. (4.21)
Finally, identifying the parameter h with 1/∆t, we obtain the expressions for
H12 and H21, the Hamiltonian H given by (4.2), and the action s2.
In deriving the action Ilinski considered a more general case of non-zero
interest rates, but this does not nullify the two issues pointed out below.
Firstly, we note that the Hamiltonian given by (4.21) becomes infinite in
the limit ∆t → 0. It is stated in Ilinski (2001) that ∆t in the continuous-
time calculations “stands for the smallest time-scale of the theory, the time
cut-off” (see page 133). However, if ∆t is retained in the finite form in the
Hamiltonian and, therefore, the action s2, it must also appear in the finite
form in the expression for the action s1 for consistency. Secondly, we observe
that the transition matrix P (tn; tn−1) is degenerate; its determinant is zero.
Therefore, it cannot possibly be identified with the evolution operator. We
conclude that the justification provided for the Hamiltonian (4.2) in Ilinski
(2001) is insufficient.
3 In the case of non-zero interest rates, P (tn; tn−1) is related to U(tn; tn−1) by a simplematrix transform (see page 132 of Ilinski (2001)); however, P (tn; tn−1) = U(tn; tn−1) ifr1 = r2 = 0 and the transaction costs are zero.
93
4. A note on the theory of fast money flow dynamics
4.5 Conclusions
We have examined the theory of fast money flow dynamics developed in Ilin-
skaia and Ilinski (1999); Ilinski (2001) and uncovered errors in 1) the derivation
and the analysis of the equations of motion based on the theory, and 2) the
justification of the action based on the lattice gauge formalism.
The equations of motion presented in Ilinskaia and Ilinski (1999); Ilin-
ski (2001) are missing the coefficient α1 in one term, crucially modifying the
dynamics of the system. We also find that most of the solutions of the equa-
tions of motion, in their correct form derived in this paper, are unstable with
respect to the initial conditions, resulting in unrealistic behaviour of the ex-
change rate. We show that the justification of the technical trading given in
Ilinski (2001) is based on an erroneous interpretation of the variables related
to the exchange rate and on the stability predicted by the incorrect equations
of motion.
The theory of fast money flows relies on a particular form of the Hamilto-
nian that describes the effect of the exchange rate on the actions of the agents.
We demonstrate that this form is not consistent with the lattice gauge formu-
lation and diverges in the continuum limit.
94
Chapter 5
Memory on multiple
time-scales in an Abelian
sandpile
We report results of a numerical analysis of the memory effects in two-dimen-
sional Abelian sandpiles. It is found that a sandpile forgets its instantaneous
configuration in two distinct stages: a fast stage and a slow stage, whose du-
rations roughly scale as N and N2 respectively, where N is the linear size
of the sandpile. We confirm the presence of the longer time-scale by an in-
dependent diagnostic based on analysing emission probabilities of a hidden
Markov model applied to a time-averaged sequence of avalanche sizes. The
application of hidden Markov modeling to the output of sandpiles is novel. It
discriminates effectively between a sandpile time series and a shuffled control
time series with the same time-averaged event statistics and hence deserves
further development as a pattern-recognition tool for Abelian sandpiles.
5.1 Introduction
The Abelian sandpile is a simple open dynamical system governed by deter-
ministic rules, which demonstrates interesting emergent behaviour. Its charac-
teristic feature is the presence of avalanches, whose scale-free size distribution
is bounded above by the size of the sandpile. Sandpiles have been studied ex-
tensively in the past two decades both computationally and analytically, but
many of their properties remain unexplained Dhar (2006); Pruessner (2012).
95
5. Memory on multiple time-scales in an Abelian sandpile
Most research has concentrated on the microstructure and properties of the
avalanches or on long-term average properties of the recurrent configurations.
The intermediate time-scale, that covers many avalanches but not so many as
to reach the long-term averages, has not been investigated as thoroughly due
to its complexity. In this paper we present the results of numerical experiments
on Abelian sandpiles on the intermediate time-scale.
Starting from an arbitrary configuration in the recurrent regime, subse-
quent grain drops and the avalanches they cause gradually modify the dis-
tribution of charges that defines the starting configuration. This results in a
gradual loss of the memory of the starting configuration. Some features are
lost quickly while others persist for a long time until eventually all memory of
the starting configuration is lost. Even though the basic features of memory
loss are known Dhar (2006), the details of this process are not well understood.
For instance, we do not know how the rate of memory loss changes with time,
nor do we know which features of the starting configuration are responsible
for maintaining memory on different time scales. These issues are important
for developing a deeper understanding of the dynamical properties of Abelian
sandpiles. We explore these issues by conducting numerical simulations of a
two-dimensional sandpile and analysing the output in two independent ways:
(i) the distribution of occupation numbers in absolute difference maps, and
(ii) emission probabilities in a hidden Markov model (HMM). Hidden Markov
models Rabiner (1989) owe their success to ease of implementation and ef-
fectiveness in capturing temporal patterns. They have become one of the
standard tools in such diverse areas as speech recognition Pieraccini (2012)
and bioinformatics Durbin (1998). Since hidden Markov models incorporate
Markov chains in their setup, they allow one to examine statistically the suc-
cession of events, or states, an important capability missing in standard tech-
niques of statistical analysis. To our knowledge, this technique has not been
applied to sandpiles. We show here by way of a control experiment that it
captures hidden patterns in the succession of time-averaged avalanche sizes.
We discuss the relevant properties of Abelian sandpiles and introduce site
occupancy fractions in Section 5.2; their statistical properties are briefly ex-
plored in Section 5.3. Section 5.4 defines absolute difference maps and presents
results on the dual-time-scale evolution of sandpile memory. Hidden Markov
models are introduced in Section 5.5, where we emphasize their pattern recog-
nition and classification capacities. Their application to time-averaged se-
quences of avalanche sizes as a memory diagnostic is described in Section 5.6.
96
5. Memory on multiple time-scales in an Abelian sandpile
5.2 Abelian sandpiles
A sandpile is a cellular automaton introduced in Bak et al. (1987) as an exam-
ple of a slowly driven system with dissipative boundaries that spontaneously
evolves to a critical state, which is characterised by scale-free distributions. It
does not require any parameter adjustment to achieve criticality, unlike (say)
the Ising model. Hence this system and its numerous derivatives demonstrate
what has become known as self-organised criticality Jensen (1998). The orig-
inal model described in Bak et al. (1987) is closely based on its prototype,
an actual pile of sand onto which grains of sand are dropped. The slope of
the pile increases, as grains are added, until an avalanche occurs, whereupon
the slope decreases. The time of the next avalanche is unpredictable and the
distribution of avalanche sizes is approximated by a power law. A sandpile
is an open dynamical system with random driving and deterministic rules for
toppling.
In Dhar (1990), a generalisation of the original toppling rules was proposed,
which has the property that the outcome of an avalanche does not depend on
the order in which the critical sites topple. Most research in the last two
decades has concentrated on this model known as the Abelian sandpile rather
than the original model where an avalanche does depend on the order of
the toppling. The properties of Abelian sandpiles have been investigated by
means of computer simulations and analytical approaches including branching
processes and spanning trees, Abelian operators, loop-removed random walks,
mean-field theory, renormalisation techniques and the logarithmic conformal
field theory (see Pruessner (2012) for a comprehensive review).
Toppling rules
In a typical simulation of a two-dimensional sandpile one considers a square
grid of sites described by a matrix zij , i, j = 1, . . . , N , where N is frequently
chosen to be a multiple of two. The values of zij are variously known as
height, slope, or charge. Initially the sandpile is assumed to be empty, with
zij = 0. At each step of the evolution, the charge of a randomly selected site
is increased by one, representing a grain drop at this site. That is, the charge
of a site can be thought as the number of grains located at the site. A site
is considered stable if zij < 4 and unstable or critical if zij ≥ 4. If a grain
drop creates an unstable site, an avalanche begins and further drops are halted
97
5. Memory on multiple time-scales in an Abelian sandpile
until the avalanche finishes. A critical site contains at least four grains and
it topples by distributing four of its grains into the four neighbouring sites,
i.e. if (i, j) is a critical site, then it topples according to the following rule:
zij → zij − 4, zi±1,j → zi±1,j + 1, and zi,j±1 → zi,j±1 + 1. If a critical site
is at the edge of the sandpile, it lacks one or two neighbours and the grains,
which would have gone to those neighbouring sites, are removed (i.e. they fall
off the edges of the pile). The avalanche continues until there are no critical
sites left, after which regular evolution of the sandpile consisting of random
grain drops resumes. The number of topples in an avalanche is referred to as
the size (or power) of the avalanche.
Site occupancy
If the sandpile starts from an empty configuration, initially there are no or few
critical sites and the charge increases steadily at most sites. The sandpile is in
the transient regime. After about 2.125N2 steps the mean charge reaches the
expected value of 2.125, after which the influx of grains is balanced on average
by the efflux over the edges due to avalanches. At this stage the sandpile is
in the recurrent regime. Once the sandpile is in a recurrent configuration, it
can only move to another recurrent configuration; given enough time any re-
current configuration is recreated eventually. The so-called burning algorithm
Dhar (1990) can be used to establish whether a configuration is recurrent or
transient. It recursively removes every site, whose charge is equal to or greater
than the number of its neighbours, starting from the edges of the sandpile.
The number of recurrent configurations is estimated to be ∼ 3.21N2
Dhar
(2006), which for a large sandpile is a vanishingly small fraction of the total
number of configurations 4N2
.
The problem of determining the probability pk that a randomly chosen
site has charge k in a recurrent configuration has been solved analytically in
the limit N → ∞ using various techniques such as uniform spanning trees
Priezzhev (1994); Caracciolo and Sportiello (2012), loop-erased random walks
Kenyon and Wilson (2011), and the dimer model Poghosyan et al. (2011). The
The number of sites nk with charge k = 0, 1, 2, 3 can be easily computed for
a given configuration zij. Then, fk = nk/N2 gives the fraction of sites with
charge k = 0, 1, 2, 3, and one has∑3
k=0 fk = 1. Considering recurrent configu-
rations only, the ensemble-averaged quantities 〈fk〉 approach the probabilities
98
5. Memory on multiple time-scales in an Abelian sandpile
pk as N → ∞. For finite sandpiles, the ensemble-averaged fractions 〈fk〉 devi-ate from pk, and the discrepancy increases as N decreases. The mean sandpile
charge can be estimated numerically by∑3
k=1 k〈fk〉 for finite N and is given
by∑3
k=1 kpk = 2.125 in the limit N → ∞.
Toppling waves
Avalanches in an Abelian sandpile can be viewed as waves of toppling Ivashke-
vich et al. (1994), where each wave is defined by allowing all critical sites, bar
the original critical site, to topple until there are no critical sites left, at which
point the original critical site is toppled again thereby launching the next
wave. The region encompassed by a wave consists of a set of contiguous sites
(without holes) each of which has toppled once. After a wave stops, the sites
whose charge differs from the original charge before the wave passes through
are located at the boundary of this region, while all internal sites regain their
original charge. The boundary itself consists of two thin layers; the wave
causes the charge of the sites at the outer layer to increase and the charge of
the sites at the inner layer to decrease. Avalanches that consist of multiple
waves can be thought of as a series of consecutive waves; subsequent waves
can be larger or smaller than preceding waves, and there is no restriction on
the number of waves in a given avalanche (note that the number of waves
is equal to the number of times the trigger site topples Fey et al. (2010)).
The structure of a multi-wave avalanche is therefore more complex than that
of a single-wave avalanche. Furthermore, the interaction of the waves in a
multi-wave avalanche is likely to be a factor in the observed deviation in two
dimensions of the avalanche size distribution from the power-law predicted by
the mean-field approximation Abdolvand and Montakhab (2010).
Temporal correlations
In this paper, we concentrate on the problem of sandpile memory, as mani-
fested in the temporal sequence of avalanche sizes. Detailed, time-averaged
statistics of output sequences have not received much attention in the litera-
ture. Most of the emphasis has been on the power spectral density of sandpile
“noise”. Indeed, the original motivation for studying sandpiles was to model
the long-range temporal correlations with a 1/f -type power spectrum exhib-
ited by some dynamical systems. Further studies revealed that the power
spectrum of the sandpile’s temporal activity has 1/f2 behaviour (see Jensen
99
5. Memory on multiple time-scales in an Abelian sandpile
0.0 0.1 0.2 0.3 0.4 0.5fk, k = 0, 1, 2, 3
100
101
102
p(f
k)
Figure 5.1: Probability density function p(fk) of the fraction fk of sites withcharge k = 0, 1, 2, 3 (left to right) for two sandpile simulations on a squaregrid with N = 32 (blue) and N = 64 (red), based on samples consisting of103N2 steps in the recurrent regime. The curves show the normal distributionsfor the corresponding values of the mean and standard deviation; they agreewell with the data. The first 5N2 steps of the simulations are discarded toeliminate transient configurations.
(1998) for details). Studies of temporal and spatial correlations in sandpiles
typically use some modification of the standard rules of the Abelian sandpile,
e.g. Davidsen and Paczuski (2002); Barrat et al. (1999), or confine the in-
vestigation to sandpiles consisting of narrow strips (a quasi-one-dimensional
geometry), e.g. Yadav et al. (2012); Maslov et al. (1999). As loss of memory in
sandpiles is related to relaxation due to grain movement caused by avalanches,
we also note the study of residence time of grains Dhar and Pradhan (2004)
where the scaling function of probability distribution of residence time is de-
rived explicitely for a simplified one dimensional model and a constraint is
obtained in the general case. In this paper, in contrast, we use the standard
Abelian sandpile model to investigate memory effects concerned with specific,
time-ordered changes in the global structure of a given sandpile configuration,
as the sandpile evolves away from it.
5.3 Site occupancy fraction distributions
The simulations we perform follow the standard recipe for a two-dimensional
Abelian sandpile on a square grid withN2 sites. We start each simulation from
the zero configuration zij = 0 (the empty sandpile). A single grain is added
at each step at a random position; no grains are added during an avalanche.
100
5. Memory on multiple time-scales in an Abelian sandpile
Table 5.1: Mean values and standard deviations for the samples shown inFigure 5.1. The bottom row gives the analytical estimates of probabilities pk,obtained in the limit of an infinitely large sandpile.
Dissipation happens at the edges of the sandpile, where grains fall out of the
system. The first 5N2 steps are excluded from the statistical analysis to avoid
transient configurations. Following each grain drop the sandpile is allowed to
relax, if the drop causes an avalanche. The sandpile configuration at time-step
t is given by the matrix zij after t grain drops and relaxations.
The first set of simulations explores the baseline behaviour of the fractions
fk introduced earlier. The probability density function p(fk) for each k is
presented in Figure 5.1 for a simulation consisting of 103N2 time-steps in
the recurrent regime. The numerical output can be fitted reasonably well
(see Figure 5.1) by Gaussian distributions with the parameters recorded in
Table 5.1. The standard deviations decrease by approximately a factor of
two as N doubles. As N increases the mean values µk = 〈fk〉 approach the
analytical estimates pk obtained in the limit N → ∞.
5.4 Short- and long-term memory in site
occupancy
One way to study memory in an Abelian sandpile is to measure the difference
between two configurations separated by a certain time interval. The details
of this approach are as follows. For a given t0 > 5N2 we compute the absolute
difference matrix Dij(t, t0) = |zij(t)−zij(t0)| between the initial configuration
zij(t0) at time-step t0 and a subsequent configuration zij(t) at t > t0; this
approach is somewhat reminiscent of how damage is measured in (Stapleton
et al., 2004) even though in our case no damage is introduced but two con-
figuration at different times are compared. Since for each position (i, j) we
have zij(t) ∈ {0, 1, 2, 3} at any time-step t, entries in the matrix Dij(t, t0) are
restricted to the same set of values, and we have Dij(t0, t0) = 0. Dij(t, t0)
can be used to describe the rate and dynamics of how the sandpile forgets
the configuration at t0 as t increases, but it contains too much information to
101
5. Memory on multiple time-scales in an Abelian sandpile
be practical. It is more manageable to consider the fractions δk of sites with
value k in the absolute difference matrix Dij . This is the same calculation
applied previously to the matrix zij to find fk but now applied to Dij to find
δk. The fractions δk are functions of the time delay t − t0 and can change
significantly after an avalanche. As we do not wish to be biased towards a
particular avalanche history, we consider the ensemble-averaged fractions 〈δk〉based on a large sample of trials.
As a control experiment, one can compute the fractions δk for the absolute
difference matrix of two sandpile configurations taken randomly from the set
of recurrent configurations. Given a large enough sample of such configura-
tions, one obtains distributions for δk and mean values 〈δk〉, which can also
For instance, a site (i, j) with Dij = 3 arises only if the corresponding location
in the two configurations used to compute the absolute difference has 0 in
one configuration and 3 in the other, or vice versa. Since the probabilities
to observe charge 0 and 3 at a given location are given by p0 and p3, the
probability to find 0 at a given location in one configuration and 3 at the same
location in the other configurations is given by p0p3, which implies d3 = 2p0p3.
The ensemble-averaged fractions 〈δk〉 converge to dk once enough time elapses,
such that the configuration at t loses all memory of the initial configuration
at t0 as t→ ∞. Note that for a relatively small sandpile such as 32× 32, it is
more relevant to use the mean values µk in place of pk to compute dk, since
the probabilities pk are determined in the limit N → ∞.
To test for this loss of memory, we conduct a second set of simulations,
where we compute δk as a function of t− t0 for a large number of trials. We
use several sizes ranging from N = 16 to N = 128 and run each trial for at
least N2 time-steps, after which the value of t0 is reset to the current time-
step t for the next trial. The number of trials varies from 105 for N = 16
to 100 for N = 128. The results for N = 32 with 104 trials are shown in
Figures 5.2a and 5.2b. We observe that the fractions 〈δk〉 converge to dk on
the time-scale ∼ N2, which is close to the time-scale over which a sandpile
starting with zij = 0 approaches the recurrent regime. Note that the ensemble-
averaged curves shown in Figure 5.2 eliminate most of the jitter characteristic
102
5. Memory on multiple time-scales in an Abelian sandpile
100
101
102
103
t−
t 0
0.0
0.2
0.4
0.6
0.8
1.0
〈δk〉
k=0
k=1
k=2
k=3
(a)
0200
400
600
800
1000
t−t 0
10−4
10−3
10−2
10−1
100
log10|〈δk〉−dk|
k=0
k=1
k=2
k=3
(b)
Figure
5.2:
Siteoccupan
cyan
alysisof
anAbeliansandpile.
(a)Ensemble-averagedsite
occupan
cyfraction
s〈δ
k〉v
ersusthetime
delay
t−t 0,whereδ k
isthefraction
ofsiteswithchargek=
0,1,2,3in
theab
solute
differen
cematrixD
ij=
|z ij(t)−z ij(t
0)|.
Theen
semble-averagedcu
rves
arebased
on10
4trials
(onerepresentative
trialis
show
nin
black).
Thestan
darderrorof
the
ensemble-averagedvalues,whichis
∼0.001,
isnot
show
n.Thedashed
lines
representtheexpectedvaluesdkfort→
∞an
dN
=32.(b)Approachof
〈δk〉t
otheexpectedvaluesdkshow
non
alogarithmic
scale.
103
5. Memory on multiple time-scales in an Abelian sandpile
of individual trials, the level of which (estimated visually to be ∼ 0.1) gives
∼ 0.1/√104 = 0.001 for the standard error of the ensemble-averaged curves.
Interestingly, Figure 5.2a reveals that the memory of a given configuration is
lost in two stages: a fast stage on the time-scale ∼ N , characterised by rapid
changes in 〈δk〉, and a slow stage on the long time-scale, where the fractions
〈δk〉 vary slowly. The transition from the fast stage to the slow stage is followed
by a small bump or depression in 〈δk〉 near t − t0 ≈ 100 superimposed over
approximately exponential curves. Moreover, 〈δ2〉 approaches d2 noticeably
quicker than the other fractions.
The loss of memory during the fast and slow stages is roughly exponential,
and fitting the sum of two exponentials (Koehler, 2012) to each memory curve
shown in Figure 5.2b yields the following decay times: τ0 = 21, τ1 = 20,
τ2 = 22, τ3 = 24 for the fast stage and τ0 = 190, τ1 = 200, τ2 = 100,
τ3 = 230 for the slow stage, where the subscript refers to the charge k. The
large uncertainty in the exponential fits does not allow us to infer accurate
scaling dependence of the decay times, but we find it broadly consistent with
the approximate scaling dependence of the fast and slow stages, i.e. ∼ N and
∼ N2.
At the beginning of the fast stage, each avalanche contributes fully (or
nearly fully) to the departure from the original configuration and therefore to
rapid memory loss. Even though some of the sites affected by an avalanche
may have already been changed by previous avalanches, they constitute a
small fraction of the affected sites. However, the fraction of such sites tends to
increase with each avalanche, until eventually the majority of sites affected by
a new avalanche have already been changed by previous avalanches. At this
stage, each avalanche contributes marginally to memory loss and therefore
the memory loss slows down. We hypothesise that this corresponds to the
transition from the fast stage to the slow stage.
5.5 Hidden Markov model
The short- and long-term memory loss displayed in Figure 5.2 pertains to
global quantities, such as the site occupancy fractions 〈δk〉. Another global
characteristic, which can be used to address the problem of memory loss, is the
sequence of avalanches and their associated sizes. By averaging avalanche size
over a fixed time interval we obtain a quantity describing the average activity
of the sandpile during that period. The patterns in the sequence of average
104
5. Memory on multiple time-scales in an Abelian sandpile
sizes contain information about sandpile memory. To access this information,
we now conduct an experiment using hidden Markov methods. The method
is described in Section 5.5 and the results of the experiments in Section 5.6.
Given a sequence of observed states characterising some system, a hidden
Markov model seeks to represent the observed sequence in terms of 1) an un-
derlying finite-state Markov chain that cannot be observed and 2) an emission
model, which gives the probability of each observed state when the Markov
chain is in a given hidden state (see Rabiner (1989)). In this paper we only
consider HMMs with discrete observed and hidden states. At each time-step
the system is described by one of the observed states and is assumed to be
in one of the hidden states. The meanings (i.e. the physical significance) of
the hidden states are not known a priori. The Markov chain is described by a
set of transition probabilities, which form an n × n matrix A, where n is the
number of hidden states. The emission model is described by a set of emission
probabilities, which form an n ×m matrix B, where m > n is the number of
observed states. To complete the HMM, one also needs to define a vector π of
initial probabilities of the hidden states. The set {A,B, π} defines the HMM.
The Baum-Welch forward-backward algorithm is a likelihood-maximisation
technique that recursively adjust the probabilities of the HMM in such a way
that the probability of the observed sequence increases Rabiner (1989). The
recursive adjustment, also referred to as training (or learning), continues until
the probability of the observed sequence ceases to increase, and the HMM pa-
rameters stabilise. To start training, one specifies some initial HMM, i.e. some
values for transition probabilities, emission probabilities, and initial probabili-
ties of the hidden states. The rate of convergence and the limit of convergence
depend strongly on the initial specification. There is no guarantee of conver-
gence to an optimal model, so one performs training for a range of initial
HMMs. The training algorithm is easy to implement and its description can
be found in e.g. Rabiner (1989); Li et al. (2000).
To illustrate the classification potential of an HMM consider the following
example Cave and Neuwirth (1980); Stamp (2004). A text in English can be
treated as a sequence of 27 states, consisting of 26 characters of the alphabet
and the symbol for space, after all punctuation marks and other symbols are
removed. An HMM with two hidden states, whose meanings are not known
a priori, can be trained on a given text. Remarkably, the resulting emission
probabilities reveal that one hidden state, say, H0, corresponds to vowels and
the other, say, H1, to consonants. Specifically, if the system is in the hidden
105
5. Memory on multiple time-scales in an Abelian sandpile
H0 H1
H0 0.32 0.68H1 0.65 0.35
O0 O1 O2
H0 0.00 0.32 0.68H1 0.65 0.35 0.00
Table 5.2: Transition probabilities Aij (left) between the hidden states i =0, 1 (H0,H1) and emission probabilities Bij (right) from the hidden states toobserved states j = 0, 1, 2 (O0, O1, O2) for the averaging time-scale Ta = 32.
state H0 at a certain position in the text, the probability of the observed
state in that position is high for vowels and low for consonants. The reverse
is true for the hidden state H1, i.e. the probability is high for consonants
and low for vowels. Since an HMM strives to capture the foremost statistical
properties of the succession of observed states, one concludes that the most
prominent statistical feature of an arbitrary English text in terms of the two-
state classification is the dichotomy between vowels and consonants.
5.6 HMM analysis of long-term memory
We harness the classifying power of HMMs to investigate memory loss in a
sandpile. At each step a sandpile can be characterised by the output size of
the released avalanche; if no avalanche is initiated, the size is zero. The power
released by an avalanche after each drop depends in an unpredictable way on
where exactly the dropped grain lands. Since the grains are dropped randomly,
this translates into shot noise in the observed avalanche size, which has little
to do with the intrinsic internal structure of the sandpile. To suppress the
shot noise, we average the avalanche size over an averaging interval Ta and
bin the average size into three bins (low, medium, high), such that the number
of samples in each bin is the same.
We apply an HMM to the average size time-series generated by the 32 ×32 sandpile (N = 32), specifically to a sequence of 103N3 time-steps in the
recurrent regime (the first 5N2 steps are discarded). We consider several
averaging time-scales spaced logarithmically in the range N ≤ Ta ≤ N3.
For each averaging time-scale we feed a sequence of 103 samples of average
avalanche size into the HMM; that is, the observed sequence contains 103
terms for each averaging time-scale Ta. The observed sequence consists of
three states. We assume there are two hidden states. For each Ta we use 20
random initial HMMs and train each model for several thousand steps, which
is sufficient for the best models to stabilise.
106
5. Memory on multiple time-scales in an Abelian sandpile
3226
2728
291024
211
212
213
214
215
Ta
0
0.250.5
0.751
Bij
(a)sandpilesequen
ce
3226
2728
291024
211
212
213
214
215
Ta
0
0.250.5
0.751
Bij
(b)shuffled
sandpilesequen
ce
Figure
5.3:
Hidden
Markovan
alysisof
anAbeliansandpile.
(a)Emission
probab
ilitiesB
ijfrom
hidden
statei=
0(red
)an
dhidden
statei=
1(blue)
toob
served
statesj=
0,1,2versustheaveragingtime-scaleTaforthe32×32
sandpile.
Adiscrete
HMM
withtw
ohidden
states
andthreeob
served
states
isused.Observed
states
aredefi
ned
bybinningtheavalan
chesize
into
high,med
ium,an
dlow
bins,
such
that
thereareequal
number
ofsamplesin
each
bin
foreach
averaginginterval.(b)Results
ofacontrol
experim
entwherethesameinputdatais
shuffled
random
lybeforebeingfedinto
theHMM.
107
5. Memory on multiple time-scales in an Abelian sandpile
The transition and emission probabilities of the final HMM with the high-
est likelihood are shown in Table 5.2 for Ta = 32. We observe that the hidden
state i = 0 is biased towards the lower values of the averaged size and i = 1
is biased towards the higher values. In other words, when the sandpile is in
the hidden state i = 0 for the duration of the averaging period Ta = 32, the
avalanches have higher power on average (loud state) as compared to when
the sandpile is in the hidden state i = 1, when avalanches have lower power
on average (quiet state). We also observe that a transition from one to the
other hidden state is roughly twice as likely as the lack of a transition.
The emission probabilities Bij of the best HMM at the end of training
for other values of Ta are displayed in Figure 5.3a. The values Bij are fairly
stable in the range N ≤ Ta ≤ 2N2, where N = 32, and the same applies to the
transition probabilities. This indicates that there is no fundamental difference
in the statistics of the average size sequence on these time-scales. However,
for Ta > 2N2 the final HMMs are qualitatively different. Loud and quiet
states are no longer distinguishable, as can be seen in Figure 5.3a, and the
likelihood of a transition between the hidden states increases markedly. This
result is consistent with the memory horizon observed in the memory plots
discussed previously (Figure 5.2). It is unlikely that any memory will persist
over a time interval longer than ∼ N2 time steps. Indeed, a square sandpile
with N2 sites takes about 2.125N2 steps to reach the recurrent regime from
an empty configuration, where 2.125N2 is the expected mean number of sand
grains in the sandpile. In the recurrent regime, one might expect the sandpile
to migrate to a completely different configuration on the same time-scale.
The results of a control experiment are shown in Figure 5.3b, where we
apply the same procedure as above to randomly shuffled sequences of sizes, i.e.
the temporal pattern is shuffled without changing the probability distribution
of sizes. We find that the final HMM resembles that obtained in Figure 5.3a
for Ta > 2N2. That is, shuffling the sequence of sizes completely destroys the
classification pattern found by the HMM on the original unshuffled sequences
of average sizes.
5.7 Conclusion
Our simulations indicate that an Abelian sandpile forgets a given configuration
in two stages: a fast stage on the time-scale ∼ N and a slow stage on the
time-scale ∼ N2. The details of memory loss are embedded in the particular
108
5. Memory on multiple time-scales in an Abelian sandpile
sequence of grain drops and avalanches, which is reflected in the behaviour of
site occupancy fractions as functions of time. By taking the ensemble averages
of the site occupancy curves over many configurations, we eliminate the details
of a particular sequence of random drops and derive the smooth memory curves
characteristic of a given Abelian sandpile. We find that memory loss is roughly
exponential during the fast and slow stages, with the site occupancy fraction
for charge 2 decaying much faster than the other fractions. We also observe a
hint of an oscillation in the memory curves following the transition from the
fast to the slow stage.
An independent analysis based on hidden Markov modeling confirms that
memory extends up to the time scale ∼ N2. The analysis identifies the hidden
states with quiet and loud periods in the sandpile’s evolution, during which
the avalanche power is respectively low or high on average. We note a re-
markable consistency in the output of the hidden Markov models obtained
from avalanche size sequences averaged on different time-scales ranging from
N to N2. This indicates that the statistical properties of the succession of the
observed states, and by implication the hidden states too, are similar across a
broad range of time-scales. As the waiting time between individual avalanches
is exponential, the temporal patterns detected by the hidden Markov model
must be driven by longer term structural changes in the sandpile. The tem-
poral patterns disappear if the observed sequence is shuffled, even though the
distribution of avalanche sizes remains the same.
Our HMM analysis focuses on the avalanche size, since this is the simplest
and most prominent characteristic of the sandpile evolution. The patterns
detected by the HMM are interesting but difficult to interpret without further
insight into the underlying structure of the sandpile. Of course, the sandpile
possesses many other characteristics that can be fed, after averaging, into an
HMM. Future studies could address and exploit sandpile properties like the
waiting time between avalanches, number of waves per avalanche, the mean
charge of the sandpile, or the occupancy numbers, just to name a few. More-
over, one can combine several characteristics into a single observed quantity to
take advantage of correlations between variables. We do not know at present
which combination of parameters will be the most successful in detecting the
underlying patterns of the sandpile, but future numerical experiments should
be able to address these questions.
Other promising avenues for future studies include 1) improving statisti-
cal significance of the reported results by increasing the number of samples
109
5. Memory on multiple time-scales in an Abelian sandpile
used for computing the memory curves and in the hidden Markov analysis, 2)
extending the simulations to larger two-dimensional and higher-dimensional
sandpiles, as well as other sandpile models such as Manna and Oslo models
(Manna, 1991; Christensen et al., 1996), 3) developing a dynamical model of
memory loss, and possibly based on the mean-field approximation or other
analytical techniques, which explains the evolution of the ensemble-averaged
site occupancy fractions.
110
Chapter 6
Conclusion
The main objective of this work is to contribute to our understanding of
the economy and financial system by conducting a number of projects in the
emerging field of econophysics, which uses techniques developed in statistical
physics to model and analyse economic and financial systems. The thesis ad-
dresses the following issues: 1) empirical investigation of transactional flows
between commercial banks and their network properties, 2) analytical and
agent-based investigation of wealth distributions and income inequality, 3) an
analysis of lattice-gauge theories of fast money flows on the foreign exchange
market, 4) numerical study of memory effects in stochastic dynamical systems
and the application of hidden Markov models to discover temporal patterns
in such systems. An important component of this work was to create a foun-
dation for future research in the field of econophysics at the University of
Melbourne. This was achieved by exploring a broad range of diverse issues of
interest and methods employed in econophysics research such as multi-agent
simulations and network science. The over-arching theme that ties the issues
explored in the thesis is the nature and properties of monetary instruments
and their role in the modern economy.
6.1 The interbank network
Empirical investigation of the Australian interbank transactional flows and
their network properties described in chapter 2 contributes to the growing
body of literature on the interbank network properties in other countries,
their network topologies, and properties of the flows. It is based on the data
provided by the Reserve Bank of Australia through their real-time gross set-
111
6. Conclusion
tlement system that captures all high-value transactions between banks in
Australia. The study offers a unique view of the structure and variability of
daily monetary flows in the Australian banking system and is the first study
to report on the Australian interbank network and its variability. Another
unique feature of this study is simultaneous investigation of the transactional
flows due to overnight loans and the flows due to other (nonloan) payments.
The sample of interbank transactions provided by the RBA consists of all
payments including overnight loans; the loans have been found by following
Furfine (2003). The procedure involves comparing transactions on two con-
secutive days and detecting those transactions that reverse on the next day
with the same amount plus interest, which closely matches the central bank’s
target rate. Just under 900 overnight loans have been identified over the
course of four days (a week in February 2007) out of over 95000 transactions
over the same period. The overnight loans account for less than 1% of all
transactions and about 6.5% of value of all payments. The interest rate of the
overnight loans is found to lie within 0.1% per annum of the target rate set
by the RBA, which was 6.5% per annum during this period. The distribution
of the nonloan transactions is well represented by a mixture of two lognormal
components, which are likely to correspond to the transactions arising from
the SWIFT and Austraclear feeds of the gross settlement system.
A major finding of the study is strong anti-correlation (whose value is
about −0.9 on most days) between the daily imbalances of overnight loans
and nonloans. The daily imbalance is computed for each bank and represents
the cumulative change in the reserve account of the bank, i.e. it is equal to
the difference in value of all incoming and outgoing transactions on a given
day. The payments recorded by the RBA correspond to financial and other
transactions between the customers of commercial banks and to a smaller
extent between the banks themselves. The daily stream of such payments is
largely random and results in an unpredictable change in the reserve accounts
of the commercial banks, with some banks’ accounts increasing in value while
the others decreasing. The money held in the reserve accounts attracts smaller
interest than the target rate set by the RBA, which encourages the banks with
the positive imbalances to lend the excess in the short term money market. At
the same time, banks whose reserves are depleted seek to eliminate the deficit,
and the implied liquidity risk, by acquiring overnight loans. This creates
an interesting dynamics in the interbank network, whereby flows of nonloan
transactions create imbalances of the banks’ reserves, which in turn engender
112
6. Conclusion
flows of overnight loans to remove the imbalances. The study reported in
Chapter 2 confirms this dynamics and provides empirical constraints on the
extent of the connection between loan and nonloan flows.
Comparing the interbank networks on different days during the week re-
veals that about 80% of nonloan flows persist for two days or more, although
the amount of persistent flows can change significantly. For the overnight loan
flows, only 50% of the flows persist. Furthermore, persistent loan flows carry
about 65% of the total value of the loan flows, whereas persistent nonloan
flows account for as much as 96% of the total value of nonloan flows. There
is no significant correlation between individual loan and nonloan flows, i.e.
overnight loans and other payments are linked via imbalances only.
As for the net flows, the number of transactions per net flow is approx-
imated well by a power law with the exponent α = −1 for nonloans and
α = −1.4 for overnight loans. Out of about 470 net nonloan flows per day
there are 110 flows that consist of a single transaction (usually these occur be-
tween small banks); more than 1000 transactions can be present in a net flow
between two large banks. Similarly, out of about 60 loan flows per day there
are as many as 40 flows that consist of a single transaction (between small
banks); loan flows between large banks can consist of 30 individual loans or
more. Given the small number of commercial banks in Australia compared
to other countries, where the interbank networks have been studied, it is not
surprising that the shape of the degree distribution of the Australian inter-
bank network is difficult to infer. It is inconsistent with a power law, which
has been observed in many countries, and is close to an exponential distribu-
tion, although this could not be rigorously confirmed. The loan and nonloan
networks are disassortative with an assortativity coefficient of about −0.4 for
nonloans and −0.1 for loans on average. The topology of the net flows is
found to be highly variable, with many circular and transitive flow structures
present.
The study of the interbank network based on five days provides a valuable
insight into the dynamics of the interbank network. However, to understand
statistical properties of the network and its variability requires follow-up stud-
ies based on longer sequences of data. A longer sequence of data will also allow
to constrain the contribution from interbank loans with two-day maturity or
longer. In addition, the intraday timing of transactions will allow to inves-
tigate the dynamics at more finely grained time scales, which is significant
for understanding the interbank network’s reaction to external events such
113
6. Conclusion
as changes in target interest rates or other relevant economic news. These
future studies are essential for uncovering the dynamical laws that govern the
dynamics of the interbank flows composed of high-value transactions, particu-
larly given that the internal dynamics of monetary flows in interbank networks
has been neglected so far. The relationship between loan and nonloan flows via
the daily imbalances of the banks that was reported in Chapter 2 is a natural
one but it has not been addressed quantitatively in the studies of the banking
networks in other countries. Therefore, there is a need for similar studies in
order to confirm this relationship in other countries where the properties and
the institutional design of the banking system may be different.
Another area of future studies that could stem from research reported in
Chapter 2 concerns multi-agent numerical simulations of transactional flows
in banking systems. These simulations are of particular interest as they al-
low one to probe various mechanism that determine bank’s behaviour in the
overnight loan market in response to changes in their reserve accounts. An
appropriate model for nonloan transactions also needs to be investigated. A
naive approach where nonloan payments are assumed to be randomly taken
from a suitable distribution may be incorrect. Indeed, a significant reduction
in the reserve account of a bank indicates the preponderance of outgoing pay-
ments (by value) over the incoming payments. If this occurs, it may reduce
the likelihood of further outgoing payments, since the funds of the bank’s
customers have been reduced. Similarly, a significant increase in the reserve
account of a bank may increase the likelihood of outgoing payments. Since
the overnight loans have to be paid back (with interest) on the next day (or
the day after next in case of two-day loans), the largely random dynamics
of nonloan payments has an effect on the following days and therefore the
dynamics on a given day cannot be considered independent of previous days.
This may have serious implications for stability of the interbank network and
the continuous operation of the real time gross settlement system.
6.2 Wealth distributions
The studies reported in Chapter 3 seek to refine our understanding of asset
exchange systems, and the giver scheme in particular, by proposing an efficient
technique for numerically computing the shape of the wealth distribution in
the steady state. The technique also provides an interesting example of em-
ploying a numerical inverse Laplace transform to solve the master equation
114
6. Conclusion
that describes the detailed balance of the giver scheme by matching influx and
outflow of agents at every value of wealth. In the steady state, the Laplace
transform of the master equation yields a functional equation that is amenable
to analysis, e.g. by Taylor series, which gives a recursive expression for the
moments of the wealth distribution. The functional equation is solved numer-
ically by iterations in the complex plane; numerical experiments reveal that
the convergence does not depend on the shape of the initial approximation or
details of the grid. The solution of the functional equation thus obtained is
then fed into the numerical inverse Laplace transform, which yields the wealth
distribution for the specified value of the transfer parameter.
The procedure for computing the wealth distribution in the giver scheme
compares favourably with the direct approach of estimating the distribution by
running a multi-agent simulation of the asset exchanges. It is computationally
faster and provides much better precision than estimating the distribution
by computing the histogram of the agents’ wealth. In particular, it gives a
handle on the asymptotic behaviour of the distribution function at extreme
values of wealth where the number of agents is small. It is confirmed that the
asymptotic behaviour is not exponential, even though a closed-form expression
for the tail has not been found. The dependence of the wealth distribution on
the value of the transfer parameter and the corresponding changes in wealth
inequality are investigated in Chapter 3. The Gini coefficient, a common
measure of income inequality, is computed for a range of different values of
the transfer parameter.
The wealth distributions obtained for small values of the transfer parame-
ter are found to be qualitatively different from those for values close to unity,
which corresponds to the case when the givers concede most of their wealth in
a single exchange (the gambling scenario). Namely, the former distributions
(small values of the transfer parameter) are peaked around the mean of the
distribution and are characterised by low level of inequality (Gini coefficient
close to zero), whereas the latter (values close to unity) are approximately
power-law in shape with overlaid oscillations and show high level of inequality
(Gini coefficient close to unity). This unexpected oscillatory pattern in the
wealth distribution is found to be approximately periodic on logarithmic scale
with the period inversely proportional to the fraction of wealth retained by
the givers.
The giver scheme is a closed system with constant amount of total wealth
and no friction. It can be expected to exhibit the usual traits of Boltzmann
115
6. Conclusion
entropy, which measures the level of disorder in the system, i.e. the entropy
is expected to rise steadily as the system evolves towards the steady state.
However, multi-agent simulations reported in Chapter 3 demonstrate that the
entropy of the system evolves in a non-monotonic fashion, in stark conflict
with the expectations based on its behaviour in physical systems such as an
ideal gas. The Boltzmann H-theorem, according to which entropy cannot
decrease, is not applicable to the giver scheme since its microscopic rules of
exchange are not symmetric with respect to time reversal. Indeed, the exact
rules that reconstruct past behaviour of system in the reverse time order can
be worked out easily but they are different from the rules describing exchange
when time order is normal. Therefore, Boltzmann entropy is not a faithful
measure of disorder in a multiplicative asset transfer system.
The study reported in Chapter 3 raises a number of questions, which could
inform future research efforts in this area. Firstly, the technique of using the
master equation and the Laplace transform is applicable to asset transfer sys-
tems with rules different from those employed in the giver scheme. Secondly,
since Boltzmann entropy fails to behave properly in the giver scheme, other
measures on entropy, e.g. Tsallis or Renyi entropy, need to be investigated
in the context of asset exchange systems that lack time symmetry. Thirdly,
an intriguing area of future research concerns exchange systems where the
total amount of money in the systems is not conserved and, moreover, can be
produced endogenously by the system’s participants.
The giver scheme and most other exchange models investigated by econo-
physicists view money as a commodity that cannot be created by the agents,
which is consistent with the perspective commonly accorded to consumers
and companies in mainstream economics. According to the mainstream view,
base money (currency) is created by the central bank and is multiplied by
commercial banks through fractional reserves, such that the banks can loan
the deposited money as long as they retain a certain fraction of all deposits
(the reserve). However, this picture conflicts with how money is actually cre-
ated by commercial banks, which create new money when they make loans
(McLeay and Radia, 2014). The actual process of money creation is endoge-
nous and is the opposite of what is implied by the money multiplier theory.
In practise, loans created by the bank engender new deposits, which in turn
raise the amount of base money as the central banks attempts to accommo-
date the demand for cash in the economy. Modeling the banking system and
its impact on the economy in this light presents a timely opportunity to make
116
6. Conclusion
a contribution which is both theoretically interesting and rich with significant
social implications in terms of the institutional design of the banking system.
6.3 Critique of fast money flow theory
The objective of Chapter 4 is to analyse the lattice gauge model of fast money
flow dynamics. The assumptions of the theory are reviewed; a careful deriva-
tion of the Euler-Lagrange equations that determine the model dynamics are
given. It is found that the dynamics of the model reported by Ilinski (2001)
is inconsistent with the Lagrangian derived there. For instance, Ilinski (2001)
observes that the oscillations in exchange rate are slowly decaying with time
for a certain combination of initial values of the model variables. However, it
is shown in Chapter 4 that the oscillations persist indefinitely with no decay
in this case. The inconsistency is traced to an algebraic error in the deriva-
tion of the Euler-Lagrange equations given in Ilinski (2001). Furthermore, it
is shown that the constraint on the initial values of model variables used by
Ilinski (2001) is unrealistic. If the constraint is relaxed, the dynamics become
unstable with the exchange rate either growing exponentially or decaying ex-
ponentially to zero. In light of these results, the implications for technical
analysis are re-evaluated and it is found that the model provides no support
for technical trading.
Furthermore, it is observed in Chapter 4 that the continuous form of the
action has not been sufficiently motivated. The part of the action that de-
scribes the dynamics of the exchange rate, which is identified with a field, is
obtained by taking the limit ∆t → 0, whereas the part that represents the
effect of the exchange rate on the number of agents in each currency uses a
finite value of ∆t as one of the input parameters of the model. The transition
from discrete evolution of the state vector, which represents the number of
agents in each currency at each time step, to continuous evolution described
by a Hamiltonian is unjustified, since the transition matrix can be degenerate
and therefore its action cannot be identified with the evolution operator.
6.4 On memory in sandpiles
In chapter 5, the Abelian sandpile is introduced as a toy model that captures
some of the features of financial markets in order to explore the connection
between observed changes in the market, e.g. price changes in the foreign ex-
117
6. Conclusion
change (FX) market, and underlying structural features of the market, which
are not observed. The emphasis of the study is on analysing Abelian sandpiles
rather than establishing a plausible model of the FX market. To that end,
chapter 5 investigates memory effects in the sandpile. It examines the hidden
structural changes resulting from grain drops and avalanches and relates those
changes to the observed avalanches by analysing the sandpile evolution with
a hidden Markov model to capture patterns in the average intensity of the
avalanches.
The investigation reported in chapter 5 concerns two-dimensional Abelian
sandpiles on a square grid. The quantitative measure of memory loss employed
in the study relies on computing site occupancy fractions, which are equal to
the number of sites occupied by a specific charge (ranging from 0 to 3 in
a two-dimensional sandpile) normalised by the total number of sites. Site
occupancy fractions are closely related to the probabilities that a given site
has a specific charge; the two concepts coincide in the limit of the infinite
sandpile in a recurrent configuration. Each of the four occupancy fractions
varies as the sandpile evolves. The distribution of each occupancy fraction
computed from a large number of configurations is found to be approximated
well by a Gaussian.
The memory loss is determined by comparing sandpile configurations sep-
arated by a certain time delay measured in grain drops. Occupancy fractions
are computed for the absolute difference of these configurations and the frac-
tion’s variation with time delay is investigated. As time delay increases, the
fractions approach constant values characteristic of the absolute difference
maps of unrelated configurations. The analysis described in chapter 5 shows
that the most common value in the absolute difference map is 1, followed by
0 and then 2, with 3 being the least common. It is interesting to note that
as far as the sandpile configuration is concerned the most common charge is
3, followed by 2, 1, and 0, in order of frequency. The dynamics of memory
loss is revealed most conveniently by comparing the fractions for the absolute
difference maps of time-delayed configurations with their expected values for
the absolute difference maps of unrelated configurations, for which time delay
is effectively infinite.
The simulations show that the memory of a given configuration is lost
in two stages, each characterised by approximately exponential decay as mea-
sured by the rate of approach of the fractions to the expected values. The first
stage is characterised by a faster decay; its time-scale is proportional to the
118
6. Conclusion
linear size of the sandpile. The second stage scales with the number of sites
in the sandpile (linear size squared) and consequently lasts much longer than
the first stage. It is not clear at present what is responsible for the two-stage
pattern of memory loss. It is left to future research to uncover the detailed
mechanisms responsible for this behaviour and develop an analytical model of
the fractions as a function of the time delay from the initial configuration.
Chapter 5 also attempts to answer the following question: can memory
loss be detected in the sequence of observed quantities that characterise the
sandpile’s activity, e.g. avalanche size (which in the financial context may cor-
respond to large exchange rate movements in the foreign exchange market, for
example). To eliminate “shot noise” due to random grain drops, the average
avalanche size over a specific time window is fed as a sequence into a hidden
Markov model. The hidden Markov model looks for patterns in the time series
of average avalanche activity; the analysis is repeated for a range of averaging
periods. The hidden Markov output demonstrates that the sandpile retains
memory on time scales that are less than the long time scale. On longer time
scales, the output depends randomly on the time scale, which is qualitatively
similar to the output obtained from the same time series shuffled randomly.
In other words, no pattern is detected by the hidden Markov model on time
scales exceeding the long time scale. The hidden Markov analysis demon-
strates that all memory is lost on long time scales, in accord with the previous
study based on the occupation fractions.
The work presented in chapter 5 contributes to the line of research devoted
to analysing temporal correlations in the Abelian sandpiles by introducing the
innovation of hidden Markov analysis. It can be extended easily to other sand-
pile models like the Manna model or the Oslo model, and conceivably to other
dynamical systems that exhibit self-organised criticality. Even in the context
of the standard Abelian sandpile model, it is desirable to extend the study’s
results by considering larger sandpiles, longer sequences of data, and higher-
dimensional sandpiles. An important direction for future research is to identify
the internal elements of the sandpile that are responsible for the dynamics of
memory loss exhibited by ensemble-averaged fractions. Due to averaging over
many realisations this dynamics is independent of random driving by grain
drops and therefore genuinely reflects the internal structure of the sandpile.
The new idea of analysing sandpile dynamics with hidden Markov models mer-
its future development. Such models can be used in a number of ways, one of
which is to feed to the model other sandpile characteristics besides avalanche
119
6. Conclusion
size, or even combinations of two or more quantities.
Unlike the study based on the occupation fractions, the hidden Markov
analysis is agnostic with respect to the internal structure of the dynamical
system it is applied to, yet it is sensitive to the memory effects, which depend
on the internal structure of the system, as the above results demonstrate.
Therefore, hidden Markov models can in principle be used as a diagnostic tool
of the internal structure of financial systems, where the internal structure is
hidden as in the case of the FX markets.
120
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