Dynamic scaling, data-collapse and self-similarity in Barabási–Albert networks

Abstract

In this article, we show that if each node of the Barabási-Albert (BA) network is characterized by the generalized degree q, i.e. the product of their degree k and the square root of their respective birth time, then the distribution function F (q, t) exhibits dynamic scaling F (q, t → ∞) ∼ t −1/2 φ(q/t 1/2 ) where φ(x) is the scaling function. We verified it by showing that a series of distinct F (q, t) vs q curves for different network sizes N collapse onto a single universal curve if we plot t 1/2 F (q, t) vs q/t 1/2 instead. Finally, we show that the BA network falls into two universality classes depending on whether new nodes arrive with single edge (m = 1) or with multiple edges (m > 1).PACS numbers: 61.43. Hv, 64.60.Ht, 68.03.Fg, 89.75.Da Many complex systems can be described as an interwoven web of a large network if the constituents are regarded as nodes or vertices and the interactions between constituents as links or edges. For example, cells of living systems are networks of molecules linked by chemical interaction [1][2][3], the Internet is a network of routers and computers linked by cables or wireless connections [4], the power-grid is a network of substations linked by transmission lines [5], and social networks are networks of individuals linked by friendships, professional ties, etc [6][7][8][9][10]. The first theoretical attempt to guide our understanding about complex network topology began with the seminal work of Paul Erdös and Alfréd Rényi in 1959 [11]. The main result of the Erdös-Rényi (ER) model is that the degree distribution P (k), defined as the probability that a randomly chosen node is connected to k other nodes by one edge, is Poissonian revealing that it is almost impossible to find nodes that have significantly higher or fewer links than the average degree. However, real networks are neither completely regular where all the nodes have the same degree k nor completely random where P (k) is Poissonian instead they are scale-free in character where P (k) obeys power-law.Just over a decade ago Barabási and Albert revolutionized the notion of the network theory by recognizing the fact that natural and man-made networks are not static rather they grow by continuous addition of new nodes. They further argued that the new nodes establish links to the well-connected existing ones preferentially rather than randomly known as the preferential attachment (PA) rule. It essentially embodies the intuitive idea of the rich get richer principle of the Matthew effect in sociology [12]. BA then presented a simple theoretical model incorporating both the ingredients and showed that the resulting network can reproduce the power-law degree distribution which most real life networks exhibit [13,14]. Recently, we have shown that random sequential partition of a square into contiguous and non-overlapping blocks can be described as a network with power-law degree distribution if blocks are regarded as nodes and common border between blocks as links [15].A power-law distribution function is considered ...

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