Emergence of scaling in random networks

TL;DR

A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

Abstract

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scale-free distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

Figures (2)

• Figure 1: The distribution function of connectivities for various large networks. ( A) Actor collaboration graph with $N=212,250$ vertices and average connectivity $\langle k\rangle =28.78$; ( B) World wide web, $N=325,729$, $\langle k\rangle=5.46$ ( 6); ( C) Powergrid data, $N=4,941$, $\langle k \rangle=2.67$. The dashed lines have slopes ( A) $\gamma_{actor}=2.3$, ( B) $\gamma_{www}=2.1$ and ( C) $\gamma_{power}=4$.
• Figure 2: ( A) The power-law connectivity distribution at $t=150,000$ (o) and $t=200,000$ ($\Box$) as obtained from the model (see text), using $m_0=m=5$. The slope of the dashed line is $\gamma=2.9$. ( B) The exponential connectivity distribution for model A, in the case of $m_0=m=1$ (o), $m_0=m=3$ ($\Box$), $m_0=m=5$ ($\Diamond$) and $m_0=m=7$ ($\triangle$). ( C) Time evolution of the connectivity for two vertices added to the system at $t_1=5$ and $t_2=95$. The dashed line has slope $0.5$.