Abstract
Scale-free networks, in which the distribution of the degrees obeys a power-law, are ubiquitous in the study of complex systems. One basic network property that relates to the structure of the links found is the degree assortativity, which is a measure of the correlation between the degrees of the nodes at the end of the links. Degree correlations are known to affect both the structure of a network and the dynamics of the processes supported thereon, including the resilience to damage, the spread of information and epidemics, and the efficiency of defence mechanisms. Nonetheless, while many studies focus on undirected scale-free networks, the interactions in real-world systems often have a directionality. Here, we investigate the dependence of the degree correlations on the power-law exponents in directed scale-free networks. To perform our study, we consider the problem of building directed networks with a prescribed degree distribution, providing a method for proper generation of power-law-distributed directed degree sequences. Applying this new method, we perform extensive numerical simulations, generating ensembles of directed scale-free networks with exponents between 2 and 3, and measuring ensemble averages of the Pearson correlation coefficients. Our results show that scale-free networks are on average uncorrelated across directed links for three of the four possible degree-degree correlations, namely in-degree to in-degree, in-degree to out-degree, and out-degree to out-degree. However, they exhibit anticorrelation between the number of outgoing connections and the number of incoming ones. The findings are consistent with an entropic origin for the observed disassortativity in biological and technological networks.
Highlights
The use of networks is fundamental to model the structure and the dynamics of a vast number of systems found throughout the natural and engineered worlds. Their main appeal lies in allowing the reduction of a complex system to a discrete set of elements, the nodes, that interact across links
Scale-free networks have been observed in citation distributions [7,8,9], Internet and WWW topology [10,11], biological systems [12,13], technological, economic and social systems [14,15], and transport processes [16,17], and they have been the subject of a considerable body of research
The connectivity of a node is no longer represented by a single scalar, as each node has a number of incoming connections and a number of outgoing connections
Summary
The use of networks is fundamental to model the structure and the dynamics of a vast number of systems found throughout the natural and engineered worlds. The simplest global structural attribute of a network is its degree distribution P(k), which expresses the probability of having a node with k links. A important case is that of scale-free networks, in which the degree distribution obeys a power-law PðkÞ*k{c [3,4,5,6].
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