Random complex networks

Abstract

Exactly what is meant by a ‘complex’ network is not clear; however, what is clear is that it is something other than a random graph. Complex networks arise in a wide range of real social, technological and physical systems. In all cases, the most basic categorization of these graphs is their node degree distribution. Particular groups of complex networks may exhibit additional interesting features, including the so-called small-world effect or being scale-free. There are many algorithms with which one may generate networks with particular degree distributions (perhaps the most famous of which is preferential attachment). In this paper, we address what it means to randomly choose a network from the class of networks with a particular degree distribution, and in doing so we show that the networks one gets from the preferential attachment process are actually highly pathological. Certain properties (including robustness and fragility) which have been attributed to the (scale-free) degree distribution are actually more intimately related to the preferential attachment growth mechanism. We focus here on scale-free networks with power-law degree sequences—but our methods and results are perfectly generic.

Air Transport versus High-Speed Rail: An Overview and Research Agenda

“…National Public Transport Data Repository) and Innovata LLC (http://www.innovata-llc.com/). A weighted, directed, temporal, and multimodal transportation network could be constructed in order to analyze the characteristics of integrated public transport system, going beyond traditional views as simple networks or networks of networks[98][99][100].…”

A new method to reduce the number of time delays in a network

“…To understand the role of the topology we have centered our attention on two markers: the density ρ of the graph and the second moment of the degree distribution k 2 as a measure of degree heterogeneity. 13,14 The density of the graph is the ratio between the actual and the maximum number of edges possible for a given number of vertices. In average, it is equal to ρ = k /(n − 1) where k is the mean vertex degree.…”

“…The variance k 2 is modulated with a parameter α. The plots represent in (a) the ratio r z = n z /(n − 1), in (b) the ratio r s defined in Eq (13),. and in (c) the variance of the vertex degree k 2 .…”

An accelerated growth model to generate complex networks with connectivity distribution slope that varies with time

“…Many systems and their interactions can be described using Directed Complex Networks (DCN) which share similar properties [1][2][3]. In order to model a system as a complex network, a set of components in the system are defined as nodes, and the relationships between them as links.…”

“…However, research published in 1998 and 1999 [4][5][6] reported that some real networks have in-and out-degree distributions that follow a power-law function [7], P (k) ∼ k γ , which means that the properties observed in this type of networks may not be the result of simple random processes, hence they have been called complex networks (CN). Newer studies have uncovered that many other systems follow the CN mechanisms and properties [3]. For example: the power grid, airline networks, social-contact disease networks, neuronal networks, protein-protein interactions, scientific papers' citation networks [4], the WWW and the Internet at the autonomous system scale, to mention a few.…”