Scale-Free Structures Emerging from Co-evolution of a Network and the Distribution of a Diffusive Resource on it

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Scale-Free Structures Emerging from Co-evolution of a Network and the Distribution of a Diffusive Resource on it

Takaaki Aoki and Toshio Aoyagi

Phys. Rev. Lett. 109, 208702 – Published 12 November 2012

See Synopsis: Networks Evolving on Two Fronts

Abstract

Co-evolution exhibited by a network system, involving the intricate interplay between the dynamics of the network itself and the subsystems connected by it, is a key concept for understanding the self-organized, flexible nature of real-world network systems. We propose a simple model of such coevolving network dynamics, in which the diffusion of a resource over a weighted network and the resource-driven evolution of the link weights occur simultaneously. We demonstrate that, under feasible conditions, the network robustly acquires scale-free characteristics in the asymptotic state. Interestingly, in the case that the system includes dissipation, it asymptotically realizes a dynamical phase characterized by an organized scale-free network, in which the ranking of each node with respect to the quantity of the resource possessed thereby changes ceaselessly. Our model offers a unified framework for understanding some real-world diffusion-driven network systems of diverse types.

Received 4 August 2011

DOI:https://doi.org/10.1103/PhysRevLett.109.208702

Synopsis

Networks Evolving on Two Fronts

Published 12 November 2012

A new network dynamics model captures the simultaneous evolution of both the nodes of a network and the connections between them.

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Authors & Affiliations

Takaaki Aoki^1,* and Toshio Aoyagi^2,3

¹Faculty of Education, Kagawa University, Takamatsu 760-8521, Japan
²Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
³CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan

^*takaaki.aoki.work@gmail.com

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Issue

Vol. 109, Iss. 20 — 16 November 2012

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Images

Figure 1
(a) A schematic illustration of coevolving dynamics of a network and the distribution of a diffusive resource on it. (b), (c), (d) Typical features of the network that emerge through the coevolving dynamics. The cumulative distributions of the quantity of the resource, $x_{i}$ , the weights, $w_{i j}$ , and the strengths, $s_{i}$ , converge to power-law forms. The topology of the network, represented by the adjacency matrix $a_{i j}$ , is chosen as an Erdös-Rényi random graph with $N = 16384$ and $⟨ k ⟩ = 10$ . The initial distribution of the resource and the initial weights were generated according to a normal distribution with mean $μ = 1$ and standard deviation $σ = 0.1$ . Other parameter values are as follows: $κ = 0.05$ , $D = 0.34$ , $ϵ = 0.01$ .Reuse & Permissions
Figure 2
In the extreme case $κ = 0$ , in which the resource diffuses over the network without dissipation, we find two regimes, corresponding to different types of dependence of $w_{i j} (t)$ on $x_{i} (t)$ and $x_{j} (t)$ , characterized by the value of a parameter $α$ [see Eq. (3)]. In these two regimes, qualitatively different network structures are realized in the asymptotic state. (a) The coevolving network dynamics yields a scale-free structure with a power-law resource distribution for $α = 1.25$ and a non-scale-free structure for $α = 0.5$ . The underlying topology, $a_{i j}$ , is given by a regular random graph ( $N = 512$ , $k = 5$ ). (b) The dependence of the asymptotic strength distribution on the imposed topology of the network and the type of the weight dynamics. The top three graphs correspond to three types of weight dependence of $Δ w_{i j}$ on $x_{i}$ and $x_{j}$ : $Δ w_{i j} \sim (x_{i} x_{j})^{α}$ with $α = 0.75$ , 1 and 1.25. The other parameter values are as follows: $N = 16384$ , $⟨ k ⟩ = 10$ , $D = 0.02$ , $ϵ = 0.01$ .Reuse & Permissions
Figure 3
(a) Dependence of the exponent of the cumulative distribution of the resource on the decay constant, $κ$ , and the diffusion constant, $D$ . The other parameter values are the same as in Fig. 1. The points labeled “(b)” and “(c)” indicate the parameter values used in the corresponding cases. (b) Time evolution of the quantities of the resource at the 20 most resource-rich nodes in the asymptotic state for a system with dissipation ( $κ = 0.05$ ). In this case with nonzero $κ$ , the distribution of the resource over the network realizes a steady state asymptotically, but, as seen here, the quantity of the resource at each node and the rankings of the nodes continue to change indefinitely. (c) Same as in (b), but with no dissipation ( $κ = 0$ ). In this case, the quantities of the resource at the top 20 nodes remain fixed.Reuse & Permissions

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