Paper Final

How do Complex Macroeconomic phenomena emerge from simple Microeconomic behaviour?

Emanuel Zuckerberger1,2 and Sorin Solomon3

1Institute for Economic Forecasting, Bucharest, Romania2Israel Electric Corporation, Haifa, Israel

3Racah Institute of Physics, Hebrew University, Jerusalem, Israel

Abstract

For a long while it was customary in many branches of science to formalize a collection of many similar objects in terms of "mean field" continuous functions representing the average of their individual properties. Often, this "representative agent" / "mean field" / continuum / linear way of thinking, is what kept the classical sciences as separated sub-cultures.

Indeed, the great conceptual barriers separating the scientific disciplines and the accompanying paradoxes connected to the emergence of life, intelligence, trade, firms, credit market and economies arise exactly from the failure of the "mean field" / "representative agent" assumptions.

When "More Is Different" (the title of the article published 30 years ago by Nobel Laureate Phil Anderson) the singular, extreme, very rare events and interactions become crucial and so life emerges from chemistry, conscience from life, social institutions /firms /markets from conscious interacting individuals, economies from firms etc.

Our basic approach is that of modeling the economy as a system of heterogeneous interacting agents with simple rules of behavior and with simple rules governing their interaction.

In a series of models representing such heterogeneous interacting agents (say firms with their rules of behaviour at a microeconomic level), Solomon and co-workers have discovered, using analytical, simulation and empirical methods, the emergence of adaptive exceptionally resilient collective objects (complex macroeconomic phenomena) out of the interactions of the agents.

Among the many analytical, simulation and empirical data processing methods used to study the emergence in economic systems (statistical mechanics, stochastic processes, phase transitions, percolation, belief propagation, opinion dynamics), we present here a group of ideas which is the most promising and likely to advance a long range vision by enhancing the predictability, intervention and planning abilities in social-economic systems. This group of ideas has developed around the discovery that random autocatalytic elements may self-organize spontaneously in highly resilient localized collective objects and change dramatically the naively expected behaviour of the entire system.

1. Introduction

For a long while, most models in economics and in finance assumed homogeneous and rational agents. Furthermore they assumed that the agents act in isolation and are linked only through the market mechanism.

These assumptions are highly problematic since: (i) people are obviously heterogeneous in their beliefs and preferences, (ii) it has been extensively documented that people systematically deviate from rational choice and (iii) individuals directly interact with each other in many ways.

Yet these assumptions have become standard in economic modelling, not because they have been inferred from observations of behaviour, but rather because they provide a convenient analytically tractable framework. Even when it is asserted that people, in reality, behave “irrationally” it is only meant that their behaviour does not satisfy the restrictive axioms which have become standard in economic theory.

This led Nobel Laureate Robert Solow to remark: "Maybe there is in human nature a deep-seated perverse pleasure in adopting and defending a wholly counterintuitive doctrine that leaves the uninitiated peasant wondering what planet he or she is on".

1

We consider that one could and (respectively) should employ microscopic (agent-based) methods to analyze economic systems, without the necessity of making these homogeneity and rationality assumptions. By so doing one could allow for the direct interaction between agents and give ultimately a meaningful micro-foundation to macro-economics instead of the pseudo-foundation provided by the representative agent.

The basic approach envisaged by us is that of modelling the economy as a system of heterogeneous interacting agents with simple rules of behaviour and with simple rules governing their interaction. In some notable cases there were proved analytically very salient and surprising features such as the spontaneous emergence of adaptive collective entities that increase dramatically the resilience of the entire system (market) (Yaari et al., 2008; Dover et al., 2009) the equality of the market fluctuations' scaling exponent to the Pareto wealth distribution exponent (Solomon and Levi, 2003) and the finite size corrections to the power law of firms (Malcai et al., 1999; Blank and Solomon, 2000).

However, in order to predict and control particular applications (individual credit status, stability of particular markets) and to extend the results to more complicated systems it is often necessary to use simulations.

The idea of microscopic simulation as developed in Levi, Levi and Solomon (1994, 2000) is simple: one can investigate a complex economic system with many heterogeneous and possibly bounded-rational interacting agents by representing each agent individually in a computer program, and by monitoring and recording the system dynamics resulting from the individuals’ actions and interactions. This approach enables one to extend the modelling capabilities, and to investigate the effects of heterogeneity, bounded rationality, learning and the direct interaction of agents. In principle one could represent an economy (as well as any other system) arbitrarily faithfully within a computer simulation. However this per se would not necessarily lead to a significant improvement of the understanding of the system: instead of one natural system that one does not understand their will be two systems, one in nature and one in the computer, which one does not understand. Thus the definition of the models representing a system should only capture well defined features that are believed to be crucial for reproduction of the stylized facts of the system under study. One can then analyse the model analytically, numerically or by computer experiments and decide, by comparison with the empirical facts, if indeed the hypotheses implicit in the model definition explain, reproduce and predict the actual system behaviour (Solomon and Levi, 2003) and its reactions to experimental manipulations.

Among the many analytical, simulation and empirical data processing methods which have been used by Solomon and co-workers to study the emergence in economic systems (statistical mechanics, stochastic processes, phase transitions, percolation, belief propagation, opinion dynamics) this paper describes the most promising, likely to advance our long range vision by enhancing the predictability, intervention and planning abilities in social-economic systems: the group of ideas which has developed around the discovery that random autocatalytic elements may self-organize spontaneously in highly resilient localized collective objects and change dramatically the naively expected behaviour of the entire system. We describe the theoretical results, the application to the post-liberalization Polish economy, the extension to generic after-shock dynamics and the role of the generalized Leontief I/O matrix in the divergence-convergence-alignment of coupled economic systems at the national and international level.

2. Logistic systems: from Malthus to Eigen and Schuster

Logistic systems, containing the essential elements of proliferation, death and competition were long recognized as a very compelling representation of a wide range of systems.

Elliott W. Montroll, one of the fathers of statistical mechanics, wrote a book dedicated to this subject and stated: "almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth" (Montroll, 1978).

Lord Robert May of Oxford, former President of the Royal Society wrote in Nature: “I would urge that people be introduced to the logistic equation early in their education… Not only in research but also in the everyday world of politics and economics…” (May, 1976).

In 1798, T.R. Malthus wrote the first equation describing the dynamics of a population of auto-catalytically proliferating individuals (Malthus, 1798):

dN(t)/dt = r·N(t) (1)

with its obvious exponential solution N(t) = N0·ert, where r defines the growth rate. This equation may represent a very wide range of phenomena in various fields: proliferation in biology, behavior adoption in sociology, capital returns in economics, or proselytizing in politics.

2

The impact of the prediction of an exponential increase in the population was so great, that everybody breathed with relief when in 1838 P.F. Verhulst offered a way out of it – the logistic equation (Verhulst, 1838):

dN(t)/dt = r·N(t)( 1- N(t)/k) (2)

where k defines the carrying capacity. The nonlinear interaction term -r·N2(t)/k may represent confrontation over limited resources in biology, finite population in sociology, competition in economics, or limited constituency in politics. By including this term the solution saturates at a constant asymptotic value N → k (rather than increasing indefinitely).

The belief in the logistic equation was strong enough already at the beginning of the 20th century as to extend its premises for more practical (and vital) problems: Sir Ronald Ross wrote a system of differential coupled equations to describe the course of malaria in humans and mosquito (Ross, 1911).

This model was taken up by Lotka in (Lotka, 1923) where the system of equations generalizing the logistic equation was introduced:

dn1(t)/dt = a1·n1(t) - a11·n12(t) - a12·n1(t)·n2(t) (3)

dn2(t)/dt = a2·n2(t) - a22·n22(t) - a21·n2(t)·n1(t)  

Vito Voltera (Voltera, 1931) advocated independently the use of equations in biology and social sciences [40] and re-deduced the logistic curve by reducing the Verhulst equation (2) to a variational principle that maximized a function that he named “quality of life”.

A simplified form of (3), when a11 = a22 = 0, is well known as Lotka-Voltera predator-prey equations:

dn1(t)/dt = n1(t)·(a1 - a12·n2(t)) (4) dn2(t)/dt = n2(t)·(a2 – a21·n1(t))

Later, R. A. Fisher (Fischer, 1937) extended equation (2) to spatial distributed systems and expressed it in terms of partial differential equations:

∂N(x,t)/ ∂t = a·N(x,t) – b·N2(x,t) + D·ΔN(x,t) (5)

He applied this equation to the spread of a mutant superior gene within a population and showed that as opposed to usual diffusion, the propagation consists of a sharp frontier („Fisher wave”) which advances with constant speed.

A crucial step was taken by Eigen and Schuster (Eigen, 1971; Eigen and Schuster, 1979) who generalized the Lotka system (3) of two equations for two populations to an arbitrary number of equations and populations. They used the new system in the study of the Darwinian selection and evolution in pre-biotic environments. They considered “quasi-species” of auto-catalytic molecules which can undergo mutations. Each sequence i self-replicates at a rate ai

and undergoes mutations to other sequences j at rates aij. The resulting system of equation is:

dni(t)/dt = ai·ni(t) + ΣNj=1 aij·nj(t) - ΣN

j=1 aij·ni – ΣNj=1 aiij·ni(t)·nj(t) (6)

3. Discrete extended logistic systems

3.1 The generalized Lotka-Voltera (GLV) system

Considering a uniform interaction in (6), the (discrete) generalized Lotka-Voltera system was introduced (Solomon and Levi, 1996; Solomon, 2000; Levi, Levi and Solomon, 2000; Solomon and Richmond, 2001a; Solomon and Richmond, 2001b):

wi(t+1) = ai·wi(t) + α·w(t) − b·w(t)·wi(t) (7)

where w(t) is the average value of the wi(t)’s. It was shown that (i) the system has a steady state for the normalized quantity xi(t) = wi(t)/w(t); (ii) the steady state distribution of the x i can be calculated analytically and (iii) the fluctuations of the average w(t) have a wide distribution with a power-law tail that is closely connected with the value of the steady state distribution.

Obviously, as there is no explicit space in this system, one cannot see localization effects. However, the fluctuations of the average value are enormous but changing around a fixed value.

The possible interpretations of such a model are very diverse: w i(t) can represent (i) the annual income of each

3

individual in the society if income distribution is considered; or (ii) the value of a specific stock 9at the closing time of the market, for example) if the stock market is considered. Then, the α·w(t) is connected to (i) the social benefits one gets from being part of the society, such as social security, charity and minimum wage; or to (ii) correlations among different stocks in the market. The a i’s stand for (i) the relative change between the current year and the previous one; or for (ii) the relative change between the value of the current day and the previous one. Finally, b(w(t),t)·wi(t) represents (i) the overall trend of the society; or (ii) the overall trend of the market.

3.2 The discrete spatially extended logistic system: the “AB Model”

Since the late 90's the theory of discrete spatially extended logistic systems (Shnerb et al., 2000; Shnerb et al., 2001; Louzoun, Solomon, Goldenberg and Mazursky, 2003; Louzoun, Shnerb and Solomon, 2007) was introduced and studied, In the simplest model, the “AB Model”, the catalysts dynamics is assumed to be independent of the proliferating agent dynamics. This model turned out to be analytically solvable by statistical field theory techniques and later by branching random walks techniques. The “AB model” is the simplest model that explains the emergence and survival of adaptive spatio-temporal collective objects in terms of the rare events connected with the fortuitous variations of the spatial distribution of growth factors.

The “AB Model” is actually a reaction-diffusion system which has two types of agents: catalyst agents A and proliferating agents B (Yaari, Stauffer and Solomon, 2009). It is a discrete system, both in space and in the fields it describes (A and B in any spatial point are natural numbers, never negative) and as such needs to be described with a set of rate equations. Then the agents may go through the following two possible processes with the corresponding rates:

(i) Diffusion: at each time step, with probabilities Da/2d and Db/2d, respectively, an A or B moves to one of the nearest neighbour site on a d-dimensional lattice;

(ii) Reaction: at each step, with probabilities μ and λ· NA, a single B dies or gives birth to a new B, respectively, where NA is the number of A’s in the same location. A never "die" or "get born".

Naively, this system can be mapped into two partial differential equations:

∂B(x,t)/ ∂t = Db·ΔB(x,t) + (λ·A(x,t) − μ) · B(x,t) (8)

∂A(x,t)/ ∂t = Da·ΔA(x,t) (9)

It is tempting to say that we can solve equation (9) to get:

A(x,t) → nA (10)

in long times and then to plug it into equation (8) to say that depending on the parameter m = (nA·λ − μ) the total number of B’s will either increase exponentially (if m > 0) ore decrease exponentially (if m < 0). It turns out that this “mean-field” treatment is totally wrong and as was shown in (Shnerb et al., 2000) in low enough dimensions the B’s will asymptotically increase exponentially no matter what the rest of the parameters are! The intuitive explanation for this surprising result is that the B’s somehow adapt themselves to be localized around regions with good conditions (large number NA’s of A)

The B population survives because its spatial distribution turns out to adapt spontaneously to the fluctuations of the catalysts density. This is an emergent (not “put by hand”) property of the B population in spite of the fact that the individual B composing it do not posses such a property.

Due to this very surprising property, the model has been explored in great detail using analytical field theoretical techniques, numerical simulations and strong coupling analysis. It turns out that for d ≤ 2, the system exhibits an active phase at any growth rate at the continuum limit. For d > 2 there is a kinetic phase transition at some positive m. In fact the condition for survival is λ / DA > 1-Pd where Pd is the Polya constant in d-dimensions. For d=1 and d=2, Pd= 1 and there is no condition.

In the last 10 years, many theoretical, computer experimental and empirical bits of understanding on this issue were gathered. This knowledge turned out to constitute a conceptual framework that applies to many biological, cognitive and socio-economic systems. In the present paper we limit ourselves to economic applications. In this framework, one inputs some of the most fundamental socio-economic “microscopic” dynamical facts and obtains as an output several of the well known economic macroscopic patterns. In the next section we present a few examples of such patterns that the model predicts under a unified conceptual paradigm.

4

4. Applying the “AB Model” to economic systems

4.1 Localized high activity sectors/ regions that drive the growth of the entire system. As was shown in (Shnerb et al., 2000) the system (self)-organizes itself in islands-like structures. These islands have strong adaptive properties and are much more efficient in exploiting the available resources, as a result the system can survive and eventually reaches a steady growth (see islands growth in sequence of snapshots at the upper panel of figure 1) even when the average /continuity approximations would predict total extinction. This result has been rigorously proved both by field theory techniques (see renormalization group flow diagram in figure 1) (Shnerb et al., 2000) and by branching random walk proofs by mathematicians (Sidoravicius and Kesten, 2003).

4.2 Globalization leads to instability. In (Louzoun et al. 2003) it was shown that when the competition between different agents is of long range, the system exhibits wide inequalities between spatial regions, sectors, individuals. It was also predicted that globalization would lead to large fractal temporal fluctuations and to systemic instability and crisis. Needless to mention the relevance of this study to the current crisis. In fact the level of aggregation that is optimal for a given system has been computed: a too short interaction (competition) radius leads to a stable but low intensity economy that does not exploit efficiently the various opportunities and does not transfer efficiently activity form the places that became disadvantageous (the gray line at the bottom of figure 2). On the other hand, in the systems which are completely globalized, the activity is completely localized in one contiguous island (snapshots in figure 2) and its collapse results in the collapse of the system as a whole. As a consequence, the system has an optimal number of islands that ensures it's growth while minimizes its fluctuations.

5

Figure 1: The “AB Model”- discrete stochastic spatially extended logistic system: the A's are the catalysts that make the conditions for the reactants B's to grow. Both types of agents (A and B) come in discrete quantities (0,1,2,3,..). One can see in the upper panel that the B's self arranged in an islands like spatial structure. The lower panel is the flow chart of the renormalization group calculation: the main result is that in low enough dimensions (two or less), regardless of the other parameters of the problem (diffusion rates, birth and death rates), the B's population will grow asymptotically provided that the lattice is big enough. Or as was written in the title of the paper: “Life always win on the surface!”

Figure 2: The same type of model as in Figure 1 (the “AB Model”). In this instance, the effects of different ranges of competition (the saturation term of Verhulst) ha been studied. The grey line corresponds to local competition, the black one has a radius of competition of 10 (in 100x100 lattice) and the blue line stands for “infinite” radius of competition (globalization). One sees the dramatic effect of this parameter on the stability of the system: the fluctuations get bigger and bigger. The three insets are snapshots of the case with infinite radius of competition: one sees how eventually all the activity remains in one island. As a consequence each change in the leading island leads to macroscopic catastrophes

4.3 Growth alignment effect (GAE). In Dover et al. (2009) and Yaari et al.(2008), in various discrete spatially extended logistic systems (at various aggregation levels), clusters of agents who share the same growth rate were found. In figure 3 (Dover et al., 2009), the prediction of the equalization of growth rate is shown and tested versus world data from various countries. One can observe clusters of countries that share the same growth rates regardless of the absolute value of income (or GDP) per capita similar to the theoretical prediction coming from the model. Unlike the usual neoclassical theories of growth convergence (σ-convergence) GAE implies only the equalization ("alignment") of the growth rates, not of the absolute values of the economic quantities. This saves the ages-long debate whether convergence is a real effect.

4.4

4.4 J-curve effect. Yaari et al., 2008 have used a compact minimal version of the renormalization group results (figure 1) of the generic “AB Model” to reproduce a very fundamental economic macroscopic pattern that follows generically major shocks / crises. The result of the shock (even of beneficial reforms) is always an initial decline that is followed by an upturn. Unlike the usual macroeconomics explanations (adjustment, etc), in this model the effect is due to the in-homogeneity of the microscopic elements: The main idea is that following dramatic events, large sectors previously dominating the economy start fading away while previously undeveloped sectors take over. Grouping all the growing sectors on the one hand and all the fading ones on the other hand, the recession-recovery process is then completely determined by the value of the returns for the two aggregate sectors and the transfer rate of economic activity between them (figure 4 exemplifies it in the case of Portugal). This understanding was exploited in order to find an optimal dynamic schedule for the effective transfer rate between the fading and the taking-over sectors. In figure 6A one sees the effects of different intervention policies.

Following the theoretical predictions of the minimalist model which says that after an external shock the overall activity of the system will follow a J-shape, the GDP data of all countries in the last 50 years was studied empirically. It turned out that using only 3 parameters (!) it was possible to fit all the cases identified to exhibit an external shock. The 3 parameters were shown to be related to the respective growth rates of the former and future leading sectors and to the interaction between them (Fig. 5), (Dover et al., 2009). Furthermore, this model allows one to both detect shocks (which can stand for laws/regime changes or wars) by looking at the countries GDP and to make quantitative predictions as for what will be the depth of the crisis, when will the economy start to grow again, when will it equalize the initial value and how one can optimize the process.

6

Figure 3: Real GDPs (in units of 1990 US dollars) for the years 1965 − 2003, of the 12 European countries and the Real GDPs of Spain, Greece and Portugal. One sees that although these three countries strengthen their economic connections with the rest of the European countries since the 1970's, their GDP didn't approach the value of the rest of Europe but rather their growth rate became aligned with the average rate of the rest of Europe.

Figure 4: Example of the J-shape of Growth Rate of Economic sectors within Portugal during the 1970’s and 1980’s. The two solid lines represent Value Added (equivalent to sectorial GDP) of the aggregations of the sectors into two groups, group A contains Construction, Retail, Transport, Communications and Finance and Group B contains Hotels, Restaurants, Services, Health, Education and Public administration. The crossing (switch between dominating groups of sectors) happens at around 1975 as is also evident in Portugal’s real GDP per capita (blue line) where the J-curve can be seen.

4.5 Post-liberalization Polish economy. The fall of the iron curtain was a unique event, which has provided a great opportunity to study a socio-economic system across rapid transition between socialist regimes and the free market regime (“shock therapy”). Using a data set which includes yearly data of the ~3000 counties of Poland for more than 15 years, Yaari et al. (2008) were able to demonstrate many of the theoretical predictions of the “AB model”. They have shown that growth is led by few singular "growth centers" (figure 6), that initially developed at a tremendous rate, followed by a diffusion process to the rest of the country and leading to a positive growth rate uniform across the counties. The very wide differences in the economic activity per capita of the counties persisted in the form of a fat-tailed distribution of economic activity in spite of the alignment of their growth rates (figure 7B). An interesting finding was the role of education in economic development: The analysis shows that the counties’ economic activity after the liberalization depend strongly on the education distribution (generated by the socialist regime) before liberalization (but NOT on the economic activity then).

5. Conclusions and future vision

The introduction of agent based models in social sciences was a bold move forced upon us by the reality that

7

Figure 7: The figure above shows the time evolution of the number of enterprises per capita in various counties (aggregated by their education level). (A, B) are empirical data while (C, D) are model predictions. (A) The time evolution of the counties growth rates. The data are aggregated according to the average education level in intervals of 0.5 education years per capita. The first point, corresponding to the growth between 1989 and 1990 is largely representative for the communist regime since the Balcerowicz reform was introduced in 1990. Then, immediately after the liberalization the growth rates of the different counties diverged strongly: in the growth centers, the economic activity more then tripled while in most other counties it halved. Later on, the growth rates of all the counties became similar. Nevertheless, the inequality between the growth centers and the rest of the counties continued to increase exponentially (the scale in this panel is Lin-log). (C) and (D) represent the same effects as generated by the theoretical model. One can see the agreement between the different parts (theory and reality).

Figure 6: The figure represents the autocatalytic growth of the Polish economy after liberalization (in the years 1989, 1994, 2004 respectively from left to right). One can see that rather than a uniform evolution, the dynamics is driven by a few singular locations.

Figure 5: The left panel shows the results of the minimalistic version of the model that consists of three parameters only. The toy model was used in order to demonstrate it's ability to serve as a tool for policy makers: the total economic output is plotted versus time for various values of the transfer rate (β) between the different parts of the economy. The right panel shows several empirical J-curves and their impressive fits with the assistance of three parameters only. In principal, one can predict in different stages of the process the resulting outcome: what will be the asymptotic growth rate? When will the economy recover its initial value?

the actual objects underlying social dynamics are individuals and their acts rather then continuous (space-) time functions governed by (partial-) differential equations. This quantum jump was not without perils: the dynamical stability of a bunch of discrete entities governed by discrete interactions is more difficult to establish than the Lyapunov exponent of time evolution of differential systems.

In fact it was found that in the presence of autocatalytic interactions, not only the numerical stability of such systems is in question but rather the actual dynamical stability of the real systems displays strong irregularities and intermittencies. Rather then an unpleasant nuisance, this singular behavior is the very root of the complex dynamics that makes possible stable human organizations, resilient economies and sustainable growth. The amplification of certain individual events to systemic self-sustaining changes is not the noise but rather the signal.

The studies reviewed in this article sum up to a baseline that provides the basis to the next quantum jump. More precisely one is lead to the conclusion that the generic criterion that separate phenomena doomed to remain local and buried in the noise from the ones destined to take over the system is auto-catalysis. In order to understand, predict and steer systemic changes one has to discover, identify and characterize the feed-back loops that sustain and amplify them. Our previous studies have identified a wide range of such mechanisms bridging over the many scales from the individual to the systemic level.

Examples of such autocatalytic mechanisms are contagion between interacting neighbors (or business partners), proliferation (of successful entities under appropriate conditions), generation by such entities of the very conditions that produce them (or makes them grow), interactions between various aggregation levels within the system (individual events contributing to the general mood that encourage their further emergence) etc.

The discovery, validation and practical use of those autocatalytic loops can be fully achieved only by a future intimate dialogue between the empirical studies and the theoretical ones (with significant assistance from the computational and numerical approaches).

8

References:

Blank A. and Solomon S. (2000) Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components), Physica A, 287, (1-2), 279-288

Dover Y., Moulet S., Solomon S. and Yaari G. (2009) Do all economies grow equally fast? The journal Risk and Decision Analysis, Vol. 1/3, special issue Black Swans, Rare and Persistent Events, Dependence and Default Models and the Modelling of Uncommon Risks.

Eigen M. (1971) Self-organization of matter and the evolution of biological macromolecules, Naturwissenschaften, 58 (10), 465-523.

Eigen M. and Schuster P. (1979) The Hypercycle, Springer, Berlin Heidelberg New York. Fischer R. A. (1937) The wave of advance of advantageous genes, Annals of Eugenics, 7, 355-369. Levy M., Levy H. and Solomon S. (1994) A microscopic model of the stock market; Cycles, booms and crashes,

Economics Letters, 45, 103-111.Levy M., Levy H. and Solomon S. (2000) Microscopic Simulation of Financial Markets: from investor behaviour

to market phenomena, Academic Press.Lotka A. J. (1923) Contribution to the analysis of malaria epidemiology, American Journal of Hygiene 3, 1-121. Louzoun Y., Shnerb M.N. and Solomon S. (2007) Microscopic noise, adaptation and survival in hostile

environments, Eur. Phys. J. B 56, 141-148. Louzoun Y., Solomon S., Goldenberg J. and Mazursky D. (2003) World-Size Global Markets Lead to Economic

Instability, Artificial Life, Vol. 9, 357–370.Malcai O., Biham O. and Solomon S. (1999) Power-law distributions and Levy-stable intermittent fluctuations in

stochastic systems of many autocatalytic elements, Physical Review E, 60, 1299-1305.Malthus T.R. (1798) An essay on the principle of population, first printed anonymously through J. Johnson, St

Paul's Churchyard, London 1798, reprinted by Macmillan et Co, 1894.May R. M. (1976) Simple mathematical models with very complicated dynamics, Nature 261, 459-467.Montroll, E. W. (1978) Social dynamics and the quantifying of social forces. Proc. Natl. Acad. Sci USA, Vol. 75

(10), 4633-4637.Ross R. (1911) The prevention of malaria, John Murray London.Shnerb M.N., Bettelheim E., Louzoun Y., Agam O. and Solomon S. (2001) Adaptation of Autocatalytic

Fluctuations to Diffusive Noise, Phys Rev E, Vol. 63(2).Shnerb M.N., Louzoun Y., Bettelheim E. and Solomon S. (2000) The importance of being discrete: Life always

wins on the surface, Proc. Natl. Acad. Sci USA, Vol. 97(19), 10322-10324.Sidovaricius V. and Kesten V. (2003) Branching random walk with catalysts, Electronic journal of probability,

Vol. 8, 5-51.Solomon, S. (2000) Generalized Lotka Voltera (GLV) models of stock markets, Applications of Simulation to So-

cial Sciences, pp. 301–322.Solomon S. and Levi M. (1996) Spontaneous Scaling Emergence in Generic Stochastic Systems, J. Mod. Phys. C

7 (5), 745-755.Solomon S. and Levy M. (2003) Pioneers on a new continent: on physics and economics, Quantitative Finance,

Vol. 3 (1), C12-C15.Solomon S. and Richmond P. (2001a) Power laws of wealth, market order volumes and market returns, Physica

A, 299, 1, 188-197.Solomon S. and Richmond P. (2001b) Stability of Pareto-Zipf law in non-stationary economies, Computing in

Economics and Finance, Society foe Computational Economics.Verhulst P.F. (1838) Notice sur la Loi que la Population Suit dans son Accroissement, Correspondence Mathema-

tique et Physique publie par A Quetelet, 10, (1838), pp 113-121, Mem. Acad Roy Bruxelles 28(1844); English trans -lation in D. Smith and N. Keyfiz, Mathematical Demography, Springer NY 1977, 333-337).

Voltera V. (1931) Variations and fluctuations of the number of individuals in animal species living together, Mc Graw Hill, NY.

Yaari G., Stauffer d. and Solomon S. (2009) Intermittency and Localization, in Encyclopedia of Complexity and Systems Science, edited by R Meyers, Springer, 4920-4930.

Yaari G., Solomon S., Rakocy K. and Nowak A. (2008) Microscopic Study Reveals the Singular Origins of Growth, European physics journal B, Vol. 62(4), 505-513.

9