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Growth models of Bipartite Networks 03/14/22 1 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur 721302
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Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Dec 17, 2015

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Page 1: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Growth models of Bipartite Networks

04/18/231

Niloy Ganguly Department of Computer Science & EngineeringIndian Institute of Technology, KharagpurKharagpur 721302

Page 2: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Contents of the presentation

04/18/232

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Page 3: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/233

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 4: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Bipartite network (BNW)

Two disjoint sets of nodes, namely “TOP” set and “BOTTOM” set

No edge between the co-members of the sets

Edges - interactions among the nodes of two sets

04/18/234

Top Set

Bottom Set

Page 5: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

BNW growth

One top node is introduced at each time step

Top nodes enter the system with µ edges ( 1≤ µ)

Each top node can bring m new bottom nodes (1≤ m <µ). If m=0, the bottom set is fixed.

Edges are attached randomly or preferentially

04/18/235

Page 6: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/236

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 7: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Many real world examples

Many real systems can be abstracted as BNWs

Biological networks

Social networks

Technological networks

Linguistic Networks

Page 8: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

A regulatory system network

The output data are driven by regulatory signals through a bipartite network

Liao J. C. et.al. PNAS 2003;100:15522-15527

Page 9: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Codon Gene Network

Page 10: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Disease Genome Network

Goh K. et.al. PNAS 2007;104:8685-8690

Page 11: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

People Project Network

Bipartite network of people and projects funded by the UK eScience initiatives

The people are circles and the projects are squares.

The color and size of the nodes indicates degree; redder and bigger nodes have more connections than smaller and yellower nodes

Page 12: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Phoneme Language Network

The Structure of the Phoneme-Language Networks (PlaNet)

L1

L4

L2

L3

/m/

/ŋ/

/p/

/d/

/s/

/θ/

Conso

na

nts

Langu

ages

Page 13: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

And many others……. Protein-protein complex network

Movie-actor network

Article-author network

Board-director network

City-people network

Word-sentence network

Bank-company network

Page 14: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2314

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 15: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Two broad categories

Both partitions grow with time Empirical and analytical studies are available

Ramasco J. J., Dorogovtsev S. N. and Pastor-Satorras R., Phys. Rev. E, 70 (036106) 2004.

Only one partition grows and other remains fixed Couple of empirical studies but no analytical research

Page 16: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Two broad categories

Both partitions grow with time Empirical and analytical studies are available

Ramasco J. J., Dorogovtsev S. N. and Pastor-Satorras R., Phys. Rev. E, 70 (036106) 2004.

Only one partition grows and other remains fixed Couple of empirical studies but no analytical research

with many real examples: Protein protein complex network, Station train network, Phoneme language network etc….

Page 17: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

BNW growth with the set of bottom nodes fixed

Fixed number of bottom nodes (N)

One top node is introduced at each time step

Top nodes enter with µ edges

Edges get attached preferentially

04/18/2317

Page 18: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Attachment Kernel

µ edges are going to get attached to the bottom nodes preferentially

Attachment of an edge depends on the current degree of a bottom node (k)

(k + €)

04/18/2318

• γ is the preferentiality parameter• Random attachment when γ = 0

Page 19: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Attachment Kernel

µ edges are going to get attached to the bottom nodes preferentially

Attachment of an edge depends on the current degree of a bottom node (k)

04/18/2319

• γ is the preferentiality parameter• Random attachment when γ = 0

Referred to as the attachment probability or the attachment kernel

Page 20: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Contents of the presentation

04/18/2320

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Page 21: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Sequential attachment model

04/18/2321

µ = 1

Total number of edges = Total time (t)

Example: Language - Webpage

Page 22: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Bottom node degree distribution

04/18/2322

Attachment probability :

Markov chain model of the growth:

No asymptotic behavior – the degree continuously increases

Notations:

- # of bottom nodes

- time or # of top nodes

- preferentiality parameter

- bottom node degree

- degree probability distribution at time t

Page 23: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Bottom node degree distribution

04/18/2323

Attachment probability :

Markov chain model of the growth:

Degree distribution function

Notations:

- # of bottom nodes

- time or # of top nodes

- preferentiality parameter

- bottom node degree

- degree probability distribution at time t

Page 24: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Approximated parallel attachment solution

04/18/2324

Attachment probability :

Degree distribution function

Approaches to Beta– distribution

f(x,α,β) for

C =

Page 25: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Four regimes

04/18/2325

The four possible regimes of degree distributions depending on . (a) , (b) (c) (d)

Page 26: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2326

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 27: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Parallel attachment with replacement model (PAWR)

04/18/2327

µ ≥ 1

Total number of edges = µt

Example: Codon – Gene

Page 28: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2328

For random attachment, we can derive the attachment probability as

Attachment probability of edges to a bottom nodeof degree k at time t

Exact solution of PAWR

Page 29: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2329

Introducing preferentiality in the model

For random attachment, we can derive the attachment probability as

Attachment probability of edges to a bottom nodeof degree k at time t

Exact solution of PAWR

Page 30: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2330

Recurrence relation for bottom node degree distribution

Exact solution of PAWR

Page 31: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

(a) N = 20, t = 250, µ = 40, γ = 1(b) γ = 16

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Pro

babi

lity

of

havi

ng d

egre

e k

Exactness of exact solution of PAWR

Degree(k)

Solid black curve –> Exact solutionSymbols –> SimulationDashed red curve –> Approximation Approximation fails but exact

solution does well

Page 32: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observations on PAWR

04/18/2332

Degree distribution curve is not monotonically decreasing for γ = 1

Two maxima in bottom node degree distribution plots

Page 33: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation - I

04/18/2333

Degree distribution curve is not monotonically decreasing for γ = 1

Degree(k)

N = 50, µ = 50

Pro

babi

lity

of

havi

ng d

egre

e k

Page 34: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation - I

04/18/2334

Degree distribution curve is not monotonically decreasing for γ = 1

N = 50, µ = 50

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Cri

tica

l γ

µ

Critical γ – The value of γ for which distribution become monotonically decreasing with mode at zero

Critical γ vs. µ: N = 10

Page 35: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2335

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 36: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Parallel attachment without replacement (PAWOR)

04/18/2336

µ ≥ 1

Total number of edges = µt

No parallel edge

Example: Language - Phoneme

Page 37: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

PAWOR Model - I µ edges connect one by one to µ distinct bottom nodes

After attachment of every edge attachment kernel changes as

Theoretical analysis is almost intractable

04/18/2337

W is the subset of bottom nodes already chosen by the current top node

Page 38: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

(a) N = 20, t = 50, µ = 5, γ = 0.1(b) γ = 3

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Degree distribution for PAWOR Model - I

Page 39: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Degree distribution for PAWOR Model - I

(a) N = 20, t = 50, µ = 5, γ = 0.1(b) γ = 3

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Approximated solution is very close to Model-I

Page 40: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

PAWOR Model - II

A subset of µ nodes is selected from N bottom nodes preferentially. NCμ sets

Attachment of edges depends on the sum of degrees of member nodes

Each of the selected µ bottom nodes get attached through one edge with the top node

04/18/2340

Page 41: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2341

Attachment kernel for µ member subset

- time or # of top nodes

- preferentiality parameter

- A µ member subset of bottom nodes

- degree of the ith member node

Attachment kernel for subset

Page 42: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Attachment kernel for subset

04/18/2342

Attachment kernel for µ member subset

- time or # of top nodes

- preferentiality parameter

- A µ member subset of bottom nodes

- degree of the ith member node

We need attachment probability for individual bottom node

Page 43: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2343

Any specific bottom node (b) is member of number of subsets

Among those subsets any other bottom node except b has membership in number of subsets

Attachment probability for a bottom node is sum of the attachment probabilities of all container subsets

Sum of degrees of all nodes is

Attachment probability for a single node

Page 44: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Attachment probability for a single node

04/18/2344

Attachment probability for bottom nodes

Page 45: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Bottom node degree distribution

04/18/2345

Markov chain model of the growth:

Page 46: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Degree distribution for PAWOR Model - II

(a) N = 20, t = 50, µ = 5, γ = 0.1(b) γ = 3 dotted lines for approximated parallel attachment model

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Page 47: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Degree distribution for PAWOR Model - II

(a) N = 20, t = 50, µ = 5, γ = 0.1(b) γ = 3 dotted lines for approximated parallel attachment model

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Only skewed binomial distributions are observedExtra randomness in the model

Page 48: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2348

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model (SA) Parallel attachment with replacement growth model (PAWR) parallel attachment without replacement growth model (PAWOR)

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 49: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

One mode projection of bottom

Goh K. et.al. PNAS 2007;104:8685-8690

Page 50: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Degree of the nodes in One-mode

Easy to calculate if each node v in growing partition enters with exactly (> 1) edges

Consider a node u in the non-growing partition having degree k

u is connected to k nodes in the growing partition and each of these k nodes are in turn connected to -1 other nodes in the non-growing partition

Hence degree q=k(-1)

Page 51: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Bipartite Networks

One-Mode Networks

What if is not fixed??

Page 52: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

What if is not fixed??

The degree of the TOP nodes for any real-world networks is not fixed not all genes made up of the same no. of codons and not all languages are composed of the same number of phonemes

Relax the assumption that the size of the consonant inventories is a constant ()

Assume these sizes to be random variables being sampled from a distribution fd

It is easy to show that, while the one-mode degree (q) for a node u is dependent on fd, its bipartite n/w degree (k) is not (the kernel of attachment roughly remains the same)

Page 53: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Analysis of Degree Distribution Assume that the k nodes in TOP partition to which a

BOTTOM node u is connected to have degrees

The probability that u is connected to a node of degree d1 is d1fd1

, d2 is d2fd2, …, dk is dkfdk

The normalized probability that u is connected to nodes of degree d1, d2, … dk is

Page 54: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Analysis of Degree Distribution

Fk(q): The probability that node u having degree k in the bipartite network ends up as a node having degree q in the one-mode projection

Now add up these probabilities for all values of k weighted by the probability of finding a node of degree k in the bipartite network

Page 55: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Analysis of Degree Distribution

Assumption: d1d2…dk = μk (i.e., AM~GM which holds when the variance is low)

Fk(q): Sum of k random variables each sampled from fd How is the sum of these k random variables distributed?

The distribution of this sum can be easily obtained by iterative convolution of fd for k times

Page 56: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Analysis of Degree Distribution

If fd varies as a Normal Distribution N(μ, σ2)

If fd varies as a Delta function δ(d, μ)

If fd varies as an Exponential function E(λ=1/μ)

If fd varies as Power-law function (power = –λ)

Page 57: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Results of the Analysis

N = 1000, t = 1000,γ= 2,μ=22

Bipartite Networks One-Mode

Networks

Page 58: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2358

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model Parallel attachment with replacement growth model parallel attachment without replacement growth model

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 59: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Experimental Setup Consider models of PAWR and PAWOR – I

Simulate the BNW growth to synthesize the real world BNW for several values of γ

Error

real distribution

simulated distribution

t Number of top nodes in real world BNW

Minimum error gives the best fitted γ and bottom node degree distribution

04/18/2359

Page 60: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Phoneme-Language Network

Top – Language

Fixed bottom - Phoneme

N – 541 (phonemes)

t – 317 (language)

µ – 22

04/18/2360

Data - UCLA Phonological Segment Inventory Database (UPSID)

EWOR = 0.113062928EWR = 0.109411487

Page 61: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Protein-Protein Complex Network

04/18/2361

Top – Protein Complex

Fixed bottom – Protein

N – 959 (protein)

t – 488 (protein-complex)

µ – 9

Data – Yeast protein complex data from http://yeast-complexes.embl.de/complexview.pl?rm=home

EWOR = 0.075977328EWR = 0.073801462

Page 62: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Station-Train Network

04/18/2362

Top – Train

Fixed bottom – Station

N – 2764 (Station)

t – 1377 (Train)

µ – 26

Data – Indian Railway data from http://www.indianrail.gov.in

EWOR = 0.034344613EWR = 0.034138636

Page 63: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2363

Introduction to Bipartite network (BNW) and BNW growth

Why BNWs ? Classes of Growth Models

Sequential attachment growth model Parallel attachment with replacement growth model parallel attachment without replacement growth model

One-mode Projection Model verification Conclusions and future works

Contents of the presentation

Page 64: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

The dynamics of BNWs where one of the partitions is fixed and finite over time is different from those where both the partitions grow unboundedly. While the former approaches a β–distribution, the latter results in a

power-law in the asymptotic limits Many real-world systems can be modeled as BNWs with one

partition fixed (e.g., Phoneme-Language N/w, Protein-Protein Complex N/w, Train-Station N/w)

The growth dynamics of these n/ws can be suitably explained through simple preferential attachment based models coupled with a tunable parameter controlling the amount of preferentiality/randomness of the growth process.

The degree distribution of one-mode projection onto the BOTTOM nodes depends on how the TOP node degrees are distributed in the BNW

Conclusions and Future works

Page 65: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Conclusions and Future works

Future works include

Deriving closed form solutions for PAWR and PAWOR models. Understanding their (non)equivalence.

Exploring the dynamics of the models for non-linear kernels, i.e., the attachment probability is proportional to kα where α < 1 refers to sub-linear kernels and α > 1 to super-linear kernels

Analytically deriving other structural properties of the one-mode like clustering coefficient, assortativity etc.

Page 66: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Collaborators

Animesh Mukherjee, Abyayananda Maity – IIT Kharagpur

Monojit Choudhury – Microsoft Research India

Fernando Peruani – CEA, Sacalay, France

Andreas Deutsch, Lutz Brusch – TU Dresden, Germany

04/18/2366

Page 67: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Contributing literature

1. F. Peruani, M. Choudhury, A. Mukherjee, and N. Ganguly. Emergence of a non-scaling degree distribution in bipartite networks: A numerical and analytical study. Europhys. Lett., 79:28001, 2007.

2. M. Choudhury, N. Ganguly, A. Maiti, A. Mukherjee, L. Brusch, A. Deutsch, and F. Peruani. Modeling discrete combinatorial systems as alphabetic bipartite networks: Theory and applications. (communicated to “Physical Review E”).

3. A. Mukherjee, M. Choudhury and N. Ganguly. Analyzing the Degree Distribution of the One-mode Projection of Alphabetic Bipartite Networks (α-- BiNs) (communicated to “Europhys. Lett.”).

04/18/2367

Page 68: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Dynamics On and Of Complex Networks

Applications to Biology, Computer Science, and the Social SciencesSeries: Modeling and Simulation in Science, Engineering and Technology Ganguly, Niloy; Deutsch, Andreas; Mukherjee, Animesh (Eds.) A Birkhäuser book

Workshop – 23rd September, Warwick

Page 69: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Dziękuję

Page 70: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2370

Page 71: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

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Observation - I

04/18/2371

Degree distribution curve is not monotonically decreasing for γ = 1

N = 50, t = 100, µ = 50

Degree(k) Mode vs. γ

Page 72: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation - I

04/18/2372

Degree distribution curve is not monotonically decreasing for γ = 1

N = 50, t = 100, µ = 50

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Mode vs. γ

Mode changes abruptly, but there is no abruptness in the model

Page 73: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2373

Critical γ – The value of γ for which distribution become monotonically decreasing with mode at zero

Page 74: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2374

Critical γ – The value of γ for which distribution become monotonically decreasing with mode at zero

(a) Critical γ vs. µ: N = 10, t = 50

Cri

tica

l γ

µ

Page 75: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2375

Critical γ – The value of γ for which distribution become monotonically decreasing with mode at zero

(a) Critical γ vs. µ: N = 10, t = 50

Cri

tica

l γ

µ

Critical γ is directly proportional to µ in exponential manner

Page 76: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2376

Critical γ – The value of γ for which distribution become monotonically decreasing with mode at zero

(a) Critical γ vs. µ: N = 10, t = 50(b) Mode vs. µ: γ = 1.2, at µ = 16 mode becomes zero

Cri

tica

l γ

µ

Critical γ is directly proportional to µ in exponential manner

µ

Page 77: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2377

(a) Critical γ vs. time: N = 10, µ = 5

Time (t)

Cri

tica

l γ

Page 78: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

04/18/2378

(a) N = 100, µ = 7, γ = 0.5, t = 100

(b) N = 100, µ = 1, γ = 2.5, t = 100 Both simulation have been taken as average over 1000 run

(a)

(b)

Solid line - recursive function Symbols - simulation

Degree(k)

Pro

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Degree distribution for PAWR

Degree(k)

Pro

babi

lity

of

havi

ng d

egre

e k

Page 79: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2379

(a) Critical γ vs. time: N = 10, µ = 5

Time (t)

Cri

tica

l γ

Critical γ has a stationary value

Page 80: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2380

(a) Critical γ vs. time: N = 10, µ = 5(b) Mode vs. time: N = 10, µ = 8, γ = 1.03 At t = 92 mode becomes zero

Time (t)

Cri

tica

l γ

Critical γ has a stationary value

Time (t)M

ode

Page 81: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2381

(a) Degree distribution after seven time stamp(b) Mode vs. time: N = 10, µ = 8, γ = 1.03 At t = 92 mode becomes zero

Critical γ has a stationary value

Time (t)M

ode

Degree(k)

Pro

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of

havi

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egre

e k

afte

r ti

me

t

Page 82: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – I cont.

04/18/2382

(a) Degree distribution after seven time stamp(b) Mode vs. time: N = 10, µ = 8, γ = 1.03 At t = 92 mode becomes zero

Time (t)M

ode

Degree(k)

Pro

babi

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of

havi

ng d

egre

e k

afte

r ti

me

t

Mode changes abruptly, but there is no abruptness in the model

Page 83: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation - II

04/18/2383

(a) N = 10, t = 100, µ = 15

Two maxima in bottom node degree distribution plots

Degree(k)

Pro

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havi

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egre

e k

Page 84: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation - II

04/18/2384

(a) N = 10, t = 100, µ = 15(b) Distribution with different time for γ = 1.2 in log-log scale

Two maxima in bottom node degree distribution plots

Degree(k)

Pro

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of

havi

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egre

e k

Degree(k)P

roba

bili

ty o

f ha

ving

deg

ree

k

Page 85: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation - II

04/18/2385

(a) N = 10, t = 100, µ = 15(b) Distribution with different time for γ = 1.2 in log-log scale

Two maxima in bottom node degree distribution plots

Degree(k)

Pro

babi

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of

havi

ng d

egre

e k

Degree(k)P

roba

bili

ty o

f ha

ving

deg

ree

k

Two maxima are not from initial effect

Page 86: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – II cont.

04/18/2386

(a) First min and second max degree over time for N = 10, µ = 15, γ = 1.2

Time(t)

Min

\ M

ax d

egre

e

Page 87: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – II cont.

04/18/2387

(a) First min and second max degree over time for N = 10, µ = 15, γ = 1.2(b) Difference between first min and second max probability

Time(t)

Min

\ M

ax d

egre

e

Pro

b di

ff o

f M

in, M

ax d

egre

e

Time(t)

Page 88: Growth models of Bipartite Networks 5/15/20151 Niloy Ganguly Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur.

Observation – II cont.

04/18/2388

(a) First min and second max degree over time for N = 10, µ = 15, γ = 1.2(b) Difference between first min and second max probability

Time(t)

Min

\ M

ax d

egre

e

Pro

b di

ff o

f M

in, M

ax d

egre

e

Time(t)

Probability difference may get a non-zero value asymptotically