Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Abstract

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution.We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1 − p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (αn) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold.The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.Other models for different real world networks request the use of different growth paradigms and do not present the scale free property. For example, some networks exhibit the small-world phenomenon, in which sub-networks have connections between almost any two nodes within them. Furthermore, most pairs of nodes are connected by at least one short path. On the other hand, one of the most studied models, the Erdös-Rényi random graph, does not exhibit either the power-law behavior for the degree distribution of its nodes, nor the small-world phenomenon [4,12,13,14,15]. Mathematically, the growth of networks can be modeled through random graph processes, i.e. a family (G t ) t∈N of random graphs (defined on a common probability space), where t is interpreted as time. Different features of the model are then described as properties of the corresponding random graph process. In particular, the interest often focuses on the degree, deg(v, t), of a vertex v at time t, that is on the total number of incoming and (or) outgoing edges to and (or) from v, respectively. In this framework, new nodes of the Barabási-Albert model link with higher probabilities with nodes of higher degree. An important feature of preferential attachment models is an asymptotic power-law degree distribution: the fraction P (k) of vertices in the network with degree k, goes as P (k) ∼ k −γ , with γ > 0, for large values of k. Real world modeling instances motivated the proposal of generalizations of the Barabási-Albert model, see e.g. [3,6,7,10,16,17,18,20,22]. A common characteristic of many of these models is the presence of the...

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