How scale-free networks and large-scale collective cooperation emerge in complex homogeneous social systems

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How scale-free networks and large-scale collective cooperation emerge in complex homogeneous social systems

Wei Li, Xiaoming Zhang, and Gang Hu

Phys. Rev. E 76, 045102(R) – Published 22 October 2007

Abstract

We study how heterogeneous degree distributions and large-scale collective cooperation in social networks emerge in complex homogeneous systems by a simple local rule: learning from the best in both strategy selections and linking choices. The prisoner’s dilemma game is used as the local dynamics. We show that the social structure may evolve into single-scale, broad-scale, and scale-free (SF) degree distributions for different control parameters. In particular, in a relatively strong-selfish parameter region the SF property can be self-organized in social networks by dynamic evolutions and these SF structures help the whole node community to reach a high level of cooperation under the poor condition of a high selfish intention of individuals.

Received 30 July 2007

DOI:https://doi.org/10.1103/PhysRevE.76.045102

Authors & Affiliations

Wei Li¹, Xiaoming Zhang², and Gang Hu^1,*

¹Department of Physics, Beijin Normal University, Beijing 100875, China
²College of Physics Science, Shenzhen University, Shenzhen 518060, China

^*ganghu@bnu.edu.cn

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Vol. 76, Iss. 4 — October 2007

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Images

Figure 1
Schematic structure of a LLN system. Each node connects with $n$ local neighbor nodes in 2D network with fixed links and has one long-range active link to a node randomly chosen (arrow from a given node to a target node). For example, link $A B$ is an active link of node $A$ and a passive link of node $B$ . And in the figure node $B$ has one long-range active link, three passive links, and four fixed local links.Reuse & Permissions
Figure 2
Behaviors and evolutions of cooperations for different network structures. $β = 0.1$ for this and all the following figures. (a) Average percentages of cooperators in the stationary states of various network systems as the functions of the payoff parameter $b$ (averages over $10^{3}$ random initial strategies with $p_{C} = 0.5$ and random long-range links). (b) $p_{C}$ ’s evolve with $t$ for various systems at $b = 2.7$ . System size of $100 \times 100$ is taken for (a) and (b).Reuse & Permissions
Figure 3
Cumulative stationary degree distributions of passive links $P (k)$ [(a),(b),(c)] and asymptotic stationary patterns of ALLN systems for different parameter $b$ 's[(d),(e),(f)]. Arbitrary initial strategy distributions with $p_{C} = 0.5$ are used. (a), (d) $b = 1.1$ ; (b), (e) $b = 2.2$ ; (c), (f) $b = 2.7$ . In (d), (e), and (f), open circles (엯) represent cooperators, solid circles (●) denote defectors, and solid gray lines show LRLs. The sizes of the open and solid circles are proportional to the logarithms of the numbers of links of the given nodes in each figure, and different scales are used for different figures. The maximum hubs have links 17 in (a), 273 in (b), and 609 in (c). System size $N = 40 \times 40$ is used in Figs. 3, 4.Reuse & Permissions
Figure 4
Self-organization of scale-free network in a dynamic coevolution of individual strategies and network structure with PD and LFB rules. $b = 2.7$ . (a)–(e) The same as Figs. 3d, 3e, 3f with pattern snapshots at different times plotted. (a) $t = 0$ . Initial random strategy and LRL distributions. (b) $t = 2$ , $p_{C} = 0.052$ . The number of defectors (cooperators) increases (decreases). (c) $t = t_{m} = 4$ , $p_{C} = 0.0125$ . The percentage of defectors arrives at the maximum. Small cooperator clusters remain. (d) $t = 7$ , $p_{C} = 0.15$ . Some defectors adopt cooperative strategy and cooperator clusters expand by LFB. (e) $t = 30$ . Stationary pattern with scale-free scaling and $p_{C} = 0.66$ is approached. The maximum cooperator hubs have links (a) 6, (b) 18, (c) 40, (d) 241, and (e) 442. (f) Cumulative degree distributions of passive long-range links at time $t = 0, 2, 4, 7, 30$ .Reuse & Permissions

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