Complex Cooperative Networks from Evolutionary Preferential Attachment Complex Cooperative Networks from Evolutionary Preferential Attachment Jesús Gómez.

Complex Cooperative Complex Cooperative Networks from Networks from

Evolutionary Preferential Evolutionary Preferential AttachmentAttachment

Jesús Gómez GardeñesJesús Gómez GardeñesUniversitat Rovira i VirgiliUniversitat Rovira i Virgili & & Scuola Superiore di Scuola Superiore di

CataniaCatania & & BIFIBIFI

Net-Works 08, Pamplona, June 9-11 2008 Net-Works 08, Pamplona, June 9-11 2008

Collaborators:- Luis Mario Floría (UZ, BIFI)- Luis Mario Floría (UZ, BIFI)- Yamir Moreno (BIFI)- Yamir Moreno (BIFI)- Anxo S- Anxo Sánchez (UC3M, BIFI)ánchez (UC3M, BIFI)- Julia Poncela (BIFI)Julia Poncela (BIFI)- Manuel CampilloManuel Campillo

10 years of net-working10 years of net-working1998 Structural studies of real complex systems1998 Structural studies of real complex systems

R. Albert, A.L. Barabàsi, Rev. Mod. Phys. (2002)

Statistical Characterization: Small-World, Scale-free, universality??

Models: clustering coefficient, degree-degree correlations,…

Community detection algorithms - Networks coarse graining

Function? Dynamics on networksFunction? Dynamics on networks

S. Boccaletti et al. Phys.Rep. (2006)

Diffusion (Technological/Social): Random Walks, Routing of data, Epidemic Spreading,…

Dynamical Systems (Biological): Synchronization transition & Linear Stability (MSF), Multistability (Dyn. reliability),..

Evolutionary Dynamics (Biology/Social): Survival of cooperation

10 years of net-working10 years of net-working

DYNAMICSDYNAMICSInfected, congested, synchronized,

evolutionary fitness

STRUCTURESTRUCTURESF, ER, clustering, correlations,

communities

OutlineOutline

FEEDBACK STRUCTURE- DYNAMICS:

Complex Networks from Evolutionary Preferential AttachmentComplex Networks from Evolutionary Preferential Attachment- Cooperative behavior

- Topological properties

STRUCTURE AFFECTS DYNAMICS:

Cooperative behavior in Complex NetworksCooperative behavior in Complex Networks- Regular versus homogeneous networks - Microscopic organization of Cooperation

Evolutionary Dynamics on graphsEvolutionary Dynamics on graphs

2-Strategies game:2-Strategies game: Fernado and Lewis have to chose one strategy: A or BFernado and Lewis have to chose one strategy: A or B They chose simultaneously and obtain a payoff given They chose simultaneously and obtain a payoff given

by the matrix:by the matrix:

⎟⎟⎠

⎞⎜⎜⎝

⎛db

ca

B

A

BA Fernando’s strategy

Lew

is’

stra

tegy

Lewis’payoff

Social DilemmasSocial Dilemmas

Population of N agents playing the game all-2-Population of N agents playing the game all-2-all:all:

Fraction Fraction x with strategy A ( with strategy A ( (1-x) with B) with B) Payoffs:Payoffs:

PA =ax+ c(1−x) PB =bx+ d(1−x)Natural selection:Natural selection: Evolution of strategies:Evolution of strategies:x

•=x PA − xPA + (1−x)PB( )⎡⎣ ⎤⎦

Social DilemmasSocial Dilemmas

Social Dilemmas:Social Dilemmas: A means A means cooperationcooperation

B means B means defectiondefection

⎟⎟⎠

⎞⎜⎜⎝

⎛

>>>

010

18

D

C

DC

dcab

Players prefer unilateral

defection to mutual cooperation

Prisoner’s DilemmaPrisoner’s DilemmaPrisoner’s DilemmaPrisoner’s DilemmaHawk-DoveHawk-DoveSnowdriftSnowdrift

Hawk-DoveHawk-DoveSnowdriftSnowdriftStag HuntStag HuntStag HuntStag Hunt

⎟⎟⎠

⎞⎜⎜⎝

⎛

>>>

18

010

D

C

DC

cdba

Players prefer mutual defection to unilateral

cooperation

⎟⎟⎠

⎞⎜⎜⎝

⎛

>>>

110

08

D

C

DC

cdab

Both tensions are incorporated

Social Dilemmas appear as a collection of Social Dilemmas appear as a collection of paradigmatic models accounting for diverse paradigmatic models accounting for diverse situations:situations:

Companies competing for a market Companies competing for a market (Economy)(Economy) Individuals cooperating for a common goal Individuals cooperating for a common goal

(Sociology)(Sociology) Animals hunting preys Animals hunting preys (Biology)(Biology)

Social DilemmasSocial Dilemmas

- In the well-mixed population hypothesis cooperation does not survive when prisoners dilemma is considered.

- The structure of interaction between players is set by a Graph.

- This is a realistic assumption, e.g. Social networks.

How does the structure of the graph affect the survival of cooperation???How does the structure of the graph affect the survival of cooperation???

Social DilemmasSocial Dilemmas

Numerical recipe for Replicator Dynamics on graphs:Numerical recipe for Replicator Dynamics on graphs:

At each time step (generation) each agent, i, plays once with all the agents in its neighborhood, .

The agents accumulate their obtained payoffs, Pi .

Each agent, i, compares its payoff with a single agent, i, picked up at random from its neighborhood.

Strategy update rule: - If Pi> Pj , i keeps its strategy.

- If Pi< Pj , i takes the strategy of j with probability

ij Γ∈

Social Dilemmas on Complex Social Dilemmas on Complex NetworksNetworks

bkk

PPPP

ji

ijijji },max{)(

−=−=Π → β

Homogeneous networks Heterogeneous networks

Social Dilemmas on Complex Social Dilemmas on Complex NetworksNetworks

… we let the system evolve and compute the average fraction of cooperators:

b b b b

c

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

1

0;1

b

c

D

C

DC

dba

F.C. Santos et al., PNAS 103, 3490 (2006).F.C. Santos and J. Pacheco PRL 95, 098104 (2005).

Social Dilemmas on Complex NetworksSocial Dilemmas on Complex Networks

Again, we find surprinsing results when analyzing the impact of SF topology on wide variety of dynamics:

- Epidemic Spreading

- Synchronization

→ Absence of epidemic threshold

→ Enhancement of Synchronizability

Why???

Social dilemmas in synthetic networks show an extremely high promotion of cooperative behavior on Scale-Free networks

compared to that found in homogeneous (all-2-all or ER) graphs

Dynamical States

Star-like graph:

Linear Chain:

2N I.C.1

2N-1

c c c c c c

D D D DD D

2N I.C.

Central node+

P peripheral nodes

P ≥ Int[b]+1

Otherwise →

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

F C DCs

- Nodes 1 & 2 are linked to ALL elements in F- Node 2 is also connected to ALL the nodes in B- The elements of F are arbitrarily linked between them

- The maximal degree of a node of F with other elements of F is kF

- The size of B is at least Int[kF(b-1)+b+1]

nF2 I.C.

Dynamical States

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

Pure Cooperators Fluctuating Pure Defectors

Three asymptotic states:

No trajectory inside F evolves to an equilibrium configuration

b

Scale-Free Erdös-Rènyi

b

Fraction of Pure Cooperators

Fraction of Fluctuating <c>Fraction of Pure Defectors

Look at the contribution of each dynamical class to the asymptotic state of the population

Cooperation Evolution

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

Emergent Clusters

- Clusters are defined by the nodes that share a common strategy (PC, F, PD) and the links among them.

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

- There may coexist several disjoint clusters simultaneously

Number of simultaneous clusters of pure players?

Number of Pure Cooperators

Clusters

SF networks show a unique PC Cluster

Number of Pure Defectors

Clusters

PD Clusters collapse into a single

one in ER graphs at ρd<1

J. Gomez-Gardeñes et al., Phys. Rev. Lett. 98,108103 (2007).

Different internal organization of C and D cores

Two different paths from cooperation to defection

Emergent Clusters

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

How are PC cluster exposed to How are PC cluster exposed to Fluctuating players?Fluctuating players?

Clusters’ Topology

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

J. Gomez-Gardeñes et al., Phys. Rev. Lett. 98,108103 (2007).

Measure of the effective “surface” of PC clusters

Pure Cooperators are more frequently connected (exposed) to Fluctuating nodes

in ER graphs

log(

k)

b

FluctuatingsPure Cooperators

b

PCP / F

J. Gomez-Gardeñes et al., J. Theor. Biol. (2008).

All the nodes of an ER graph are topologically equivalent but…

where are PC, F, PD located in a SF heterogeneous network?

Highly connected nodes always plays as PC&

Fluctuating strategies spreads from low connectivity nodes

Clusters’ Topology

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

Dynamical characterization of FluctuationsC

Dtime

Distribution of cooperationintervals’ times,

Distribution of total cooperationtimes of a fluctuating node, TC

cτ

J. Gomez-Gardeñes et al., J. Theor. Biol. (2008).

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

SF (b=3.0):SF (b=2.1):

Clusters’ Topology

Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN

PC PDPC

s F

ConclusionsConclusionsKnown Facts: SF topology promotes cooperation

Quantitative differences are observed <c>(b)

Three classes of agents:

Pure Cooperators Pure Defectors Fluctuating playersFluctuating players

They act as borders

between PC & PD They may occupy a macroscopic

part of the network

Once defined and identified, one can unveil the internal organization of the three classes of agents

Qualitative differences are observed PC(b), F(b), PD(b)

- NCC

Indicators: & - NDC

- Hubs roleConfirmed by: &

- Surface of PC clusters

• SF enhance the survival of cooperation

• The differences with homogeneous structure rely on the structural organization of strategies.

However:However: If SF networks are best suited to support cooperation, where did they come from?

What are the mechanisms that shape the system structure?

What have we learned?What have we learned?

One cannot think of an optimized design…

Social networks are the result of a collection of many local decisions based on local

interactions.

Function affects structure and the other way around!!Function affects structure and the other way around!!

Evolutionary Preferential AttachmentEvolutionary Preferential Attachment

• The network grows by adding new nodes every τT time steps:

Two channels of evolution: Network Growth and Evolutionary Dynamics

We explore two cases: τT =τD and τD =10τT

The new node is added as C or D with equal probability.

• A Prisoners Dilemma round robin is played within the nodes of the every τD time steps

Evolutionary Preferential AttachmentEvolutionary Preferential Attachment

• Coupling Dynamics And Growth:

The new node attaches to nodes following a preferential attachment to nodes with high evolutionary fitness, fi(t):

Two channels of evolution: Network Growth and Evolutionary Dynamics

Πi (t) =1 − ε + ε fi (t)

1 − ε + ε f j (t)j =1

N (t )

∑0: Weak selection limit.1: Strong selection limit.

Parameters: b, , τT/ τD Outputs: <c> + Topology

Πi (t) =1 − ε + ε fi (t)

1 − ε + ε fi (t)j =1

N (t )

∑

Degree DistributionDegree Distribution

b =1.5τD = 10τ T

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

τD = τ T τD = 10τ T

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

Degree DistributionDegree Distribution

→ 1τD = τ T τD = 10τ T

⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

J. Poncela et al., PLoS ONE., (2008).

→ 1 τD = τ T τD = 10τ T⎟⎟⎠

⎞⎜⎜⎝

⎛

=>=

0

01

0;1

bD

C

DC

dba

• For a fixed b, strong selection (1) yields both the highest level of cooperation and scale-free behavior, all in one!

• Correlations are present, both in degree-degree and clustering-degree. A typical fingerprint in real networks.

Summary:

We have not imposed any maximization of cooperation level. The network is shaped by its dynamics.

Moreover, strong selection would seem to favor defective systems within the context of the PD game. It is just the opposite!!!

What about the organization of cooperation??

• Real hubs are defectors • Middle class are cooperators

J. Poncela et al., PLoS ONE., (2008).

Probability that a node of degree k plays as cooperator

This picture is radically different than that found for static SF networks

What would be the level of cooperation if the system stop growing?

J. Poncela et al., PLoS ONE., (2008).

Growing to Static Network

Cooperation is actually Cooperation is actually enhanced!!!enhanced!!!

Summary

• Structure is shaped by DynamicsStructure is shaped by Dynamics: It is possible to build up complex networks using a dynamical feedback mechanism that shapes the system’s structure.

The model provides an evolutionary explanation of the features of real networks: Scale-free and clustering

• Dynamics is affectedDynamics is affected: The model points out the many differences in the microscopic organization of strategist compared to the case in which the game evolves on static networks. Dynamics in a Growing is qualitatively different!

http://neptuno.unizar.es/jgg/

Related Publications:

• J. Gómez-Gardeñes et al., Phys. Rev. Lett. 98, 108103 (2007)• J. Poncela et al., New J. Phys. 9, 184 (2007).• J. Gómez-Gardeñes et al., J. Theor. Biol., in press (2008).• J. Poncela et al., PLoS ONE, in press (2008).

G. Szabó and G. Fáth, Physics Reports 446, 97 (2007).