Mandelbrot Law of Evolving Networks

Abstract

Degree distributions of many real networks are known to follow the Mandelbrot law, which can be considered as an extension of the power law and is determined by not only the power-law exponent, but also the shifting coefficient. Although the shifting coefficient highly affects the shape of distribution, it receives less attention in the literature and in fact, mainstream analytical method based on backward or forward difference will lead to considerable deviations to its value. In this Letter, we show that the degree distribution of a growing network with linear preferential attachment approximately follows the Mandelbrot law. We propose an analytical method based on a recursive formula that can obtain a more accurate expression of the shifting coefficient. Simulations demonstrate the advantages of our method. This work provides a possible mechanism leading to the Mandelbrot law of evolving networks, and refines the mainstream analytical methods for the shifting coefficient.PACS numbers: 89.75.Hc, 89.75.Fb, 02.50.-r Many systems can be described as networks [1][2][3][4], in which, the nodes correspond to the elements and the links to the relations between elements. Uncovering the mechanisms underlying the structural features of real networks is one of the most significant challenges in network science. Two pioneering models, respectively for small-world [5] and scalefree networks [6], give explanations for many real phenomena, such as, the logarithmic growth of average distance, the power-law degree distribution, and the high clustering coefficient. With the idea of 'rich get richer', the Barabási-Albert (BA) network [6] embodies two mechanisms: growth and preferential attachment. That is, at each time step, a new node is added and connected to a few old nodes with probability proportional to their degree as:

An evolving model of online bipartite networks

“…bookmarks, music, movies), as well as the underlying mechanisms. Despite many previous studies demonstrated that both exponential and power-law degree distribution could be obtained by corresponding models, empirical analysis of some online bipartite networks shows that the user degree distribution actually follows shifted power-law, so-called Mandelbrot law [51,52], instead of purely exponential or power-law decay, while the object degree distribution always obeys power-law [19,50], and it can not be fully explained by previous models. Therefore, We propose an evolutionary model to consider the proactive selection activity of users and the passive pattern of objects.…”

“…From Eq. 12, we know that the user degree distribution is a shifted power-law distribution [53,54], which is also familiar as Mandelbort law [51,52].…”

Punctuated equilibrium dynamics in human communications

“…In fact, strict power-law distribution is rarely observed in empirical studies albeit bursts and heavy tails are widespread. In addition to the widespread bimodal distribution, human activity patterns may be power-law distribution followed by distinct cutoff [10,[27][28][29][30], multimodal distribution of power-law with different scaling exponents [30][31][32][33], in some instances it is more consistent with Mandelbrot distribution [34], and so on. Human activity patterns exhibit such a wealth of statistical properties.…”

The Asymptotic Mandelbrot Law of Some Evolution Networks

“…To understand the origins of such Mandelbrot law, Ren, Yang and Wang [11] proposed a interesting growing network that is generated with linear preferential attachment. In such networks, there exits a recursive dependence relationship between every two consecutive degrees…”

“…To derive the asymptotic of the degree distribution, Ren, Yang and Wang [11] studied the following three kinds of approximations: I) forward-difference approximation, assuming…”