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Complex Cooperative Networks from Evolutionary Preferential Attachment Complex Cooperative Networks from Evolutionary Preferential Attachment Jesús Gómez Gardeñes Universitat Rovira i Virgili & Scuola Superiore di Catania & BIFI Net-Works 08, Pamplona, June 9-11 2008 Collaborators: - Luis Mario Floría (UZ, BIFI) - Yamir Moreno (BIFI) - Anxo S á nchez (UC3M, BIFI) - Julia Poncela (BIFI) - Manuel Campillo

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10 years of net-working 1998 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 networks Function? 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

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10 years of net-working DYNAMICS Infected, congested, synchronized, evolutionary fitness STRUCTURE SF, ER, clustering, correlations, communities

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Outline FEEDBACK STRUCTURE- DYNAMICS: Complex Networks from Evolutionary Preferential Attachment - Cooperative behavior - Topological properties STRUCTURE AFFECTS DYNAMICS: Cooperative behavior in Complex Networks - Regular versus homogeneous networks - Microscopic organization of Cooperation Evolutionary Dynamics on graphs

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2-Strategies game: Fernado and Lewis have to chose one strategy: A or B Fernado and Lewis have to chose one strategy: A or B They chose simultaneously and obtain a payoff given by the matrix: They chose simultaneously and obtain a payoff given by the matrix: Fernandos strategy Lewis strategy Lewis payoff Social Dilemmas Population of N agents playing the game all-2-all: Fraction with strategy A ( with B) Fraction x with strategy A ( (1-x) with B) Payoffs: Payoffs: Natural selection: Evolution of strategies: Evolution of strategies:

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Social Dilemmas Social Dilemmas: A means cooperation B means defection B means defection Players prefer unilateral defection to mutual cooperation Prisoners Dilemma Hawk-DoveSnowdriftHawk-DoveSnowdrift Stag Hunt Players prefer mutual defection to unilateral cooperation Both tensions are incorporated

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Social Dilemmas appear as a collection of paradigmatic models accounting for diverse situations: Social Dilemmas appear as a collection of paradigmatic models accounting for diverse situations: Companies competing for a market (Economy) Companies competing for a market (Economy) Individuals cooperating for a common goal (Sociology) Individuals cooperating for a common goal (Sociology) Animals hunting preys (Biology) Animals hunting preys (Biology) Social Dilemmas

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- 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??? Social Dilemmas

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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, P i. Each agent, i, compares its payoff with a single agent, i, picked up at random from its neighborhood. Strategy update rule: - If P i > P j, i keeps its strategy. - If P i < P j, i takes the strategy of j with probability Social Dilemmas on Complex Networks

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Homogeneous networks Heterogeneous networks Social Dilemmas on Complex Networks … we let the system evolve and compute the average fraction of cooperators: b bbb c F.C. Santos et al., PNAS 103, 3490 (2006). F.C. Santos and J. Pacheco PRL 95, 098104 (2005).

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Social 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

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Dynamical States Star-like graph: Linear Chain: 2 N I.C. 1 2 N -1 c c c ccc DDD D DD 2 N I.C. Central node + P peripheral nodes P Int[b]+1 Otherwise Prisoners Dilemma on CN

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F C D Cs - 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 k F - The size of B is at least Int[k F (b-1)+b+1] nFnF 2 I.C. Dynamical States Prisoners Dilemma on CN Pure Cooperators Fluctuating Pure Defectors Three asymptotic states: No trajectory inside F evolves to an equilibrium configuration

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b Scale-FreeErdös-Rènyi b Fraction of Pure Cooperators Fraction of Fluctuating Fraction of Pure Defectors Look at the contribution of each dynamical class to the asymptotic state of the population Cooperation Evolution Prisoners Dilemma on CN

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Emergent Clusters - Clusters are defined by the nodes that share a common strategy (PC, F, PD) and the links among them. Prisoners Dilemma on CN - 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 ρ d <1 J. Gomez-Gardeñes et al., Phys. Rev. Lett. 98,108103 (2007).

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Different internal organization of C and D cores Two different paths from cooperation to defection Emergent Clusters Prisoners Dilemma on CN

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How are PC cluster exposed to Fluctuating players? Fluctuating players? Clusters Topology Prisoners Dilemma on CN 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

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log(k) b FluctuatingsPure Cooperators b P CP / 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 Prisoners Dilemma on CN

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Dynamical characterization of Fluctuations C D time Distribution of cooperation intervals times, Distribution of total cooperation times of a fluctuating node, T C J. Gomez-Gardeñes et al., J. Theor. Biol. (2008). Prisoners Dilemma on CN

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SF (b=3.0): SF (b=2.1): Clusters Topology Prisoners Dilemma on CN PCPD PCs F

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Conclusions Known Facts: SF topology promotes cooperation Quantitative differences are observed (b) Three classes of agents: Fluctuating players Pure Cooperators Pure Defectors Fluctuating 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) - N CC Indicators: & - N DC - Hubs role Confirmed by: & - Surface of PC clusters

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SF enhance the survival of cooperation The differences with homogeneous structure rely on the structural organization of strategies. 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? 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!!

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Evolutionary Preferential Attachment The network grows by adding new nodes every T time steps: Two channels of evolution: Network Growth and Evolutionary Dynamics We explore two cases: T = D and D =10 T 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 D time steps

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Evolutionary Preferential Attachment Coupling Dynamics And Growth: The new node attaches to nodes following a preferential attachment to nodes with high evolutionary fitness, f i (t): Two channels of evolution: Network Growth and Evolutionary Dynamics 0: Weak selection limit. 1: Strong selection limit. Parameters: b,, T / D Outputs: + Topology

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Degree Distribution b =1.5

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Degree Distribution

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

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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??

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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

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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 enhanced!!!

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Summary Structure is shaped by Dynamics Structure is shaped by Dynamics: It is possible to build up complex networks using a dynamical feedback mechanism that shapes the systems structure. The model provides an evolutionary explanation of the features of real networks: Scale-free and clustering Dynamics is affected Dynamics 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!

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http://neptuno.unizar.es/jgg/ Related Publications: J. G ó mez-Gardeñes et al., Phys. Rev. Lett. 98, 108103 (2007) J. Poncela et al., New J. Phys. 9, 184 (2007). J. G ó mez-Garde ñ es et al., J. Theor. Biol., in press (2008). J. Poncela et al., PLoS ONE, in press (2008). G. Szab ó and G. F á th, Physics Reports 446, 97 (2007).

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