Generation of scale-free networks using a simple preferential-rewiring dynamics

Abstract

We propose a simple dynamical model that generates networks with power-law degree distributions with the exponent 2 through rewiring only. At each time step, two nodes, i and j, are randomly selected, and one incoming link to i is redirected to j with the rewiring probability R, determined only by degrees of two nodes, k i and k j , while giving preference to high-degree nodes.To take the structure of networks into account, we also consider what types of networks are of interest, whether links are directed or not, and how we choose a rewiring link out of all incoming links to i, as a result, specifying 24 different cases of the model. We then observe numerically that networks will evolve to steady states with power-law degree distributions when parameters of the model satisfy certain conditions.

Network rewiring dynamics with convergence towards a star network

“…Given the smallest degree node p connected to i the probability that the edge e ip will be rewired to node j is given by R. In addition, to ensure the non-degenerate case, the following conditions must also be true: p = j and the edge e pj cannot already exist. If either of these conditions are true, then the rewiring procedure is aborted [16]. The steady-state scaling of this model is related to parameter β in equation (2.1) and the mean degreek of N. Ree shows [15] that this relationship results in either a steady-state exponential or power-law graph.…”

“…The curved distribution suggests an exponential distribution rather than scale free, although for the purposes of this approach it is not likely that a true scale-free network is required to examine the properties of a process. In addition, the non-degenerate assumption for the network means that scale-free distributions are difficult to create, as implied by the strict conditions stated by Ree [16]. However, the inset plot of figure 4 when r = 0.37 suggests that the distribution is not exactly exponential and appears to have some power-law properties.…”

“…The algorithm presented here is based on the work of Ree [15,16], and so a brief review of the Ree algorithm is appropriate. Given a graph G with N nodes (vertices) and E edges, the Ree algorithm commences by selecting two random nodes i and j with degrees k i and k j .…”

“…Variations of this model have been used to study the effect of network structure on processes such as genetic drift [12] and military communication [13], although this model often produced disconnected networks, and the convergence properties were not well defined. Other models for fixed network architectures that have stationary scale-free distributions under a range of conditions include Park & Lai [14], Ree [15,16] and Xie et al [17]. The field of statistical mechanics has also been used to examine network construction [18,19], using the metaphor of temperature and total energy to determine the phase of the network.…”

Convergence speed of consensus problems over undirected scale-free networks

“…A rewiring model which does not change the total size of the graph is introduced in Ref. [11]. It can only provide γ = 2 approximately.…”