Emergence of Multiscaling in Heterogeneous Complex Networks

Abstract

In this paper we provide numerical evidence of the richer behavior of the connectivity degrees in heterogeneous preferential attachment networks in comparison to their homogeneous counterparts. We analyze the degree distribution in the threshold model, a preferential attachment model where the affinity between node states biases the attachment probabilities of links. We show that the degree densities exhibit a power-law multiscaling which points to a signature of heterogeneity in preferential attachment networks. This translates into a power-law scaling in the degree distribution, whose exponent depends on the specific form of heterogeneity in the attachment mechanism.

Robustness of heterogeneous complex networks

“…In order to analyze the network tolerance we need to specify particular models within this general class. The so-called threshold model [15] is a simple case of heterogenous PA based on the assumption that the affinity between nodes is inversely related to the distance between their states as defined by a certain space metric. The rationale for this choice is that the inverse relationship between affinity and state distance may be a reasonable proxy for many real networks where PA can be considered as the most relevant linking mechanism, such as protein interactions, web page hyperlinks, scholar citations or social relationships.…”

Local affinity in heterogeneous growing networks

“…Given the averaging role of the width on the topology of the threshold networks [35], the study will focus on the threshold µ, assuming that the individual affinities are step functions ( i = 0 ∀i). This leaves two choices to fix the attachment probabilities {Π i }.…”

Connectivity degrees in the threshold preferential attachment model

“…The addition of nodes is iterated until a network with a size N is achieved. The Barabási-Albert model can be easily extended to heterogeneous networks [34] by imposing a metric structure on the node states, while preserving the original mechanisms of growth and preferential attachment. In the heterogeneous models, network elements can be described by state variables that allow us to quantify the affinities between them.…”

“…Heterogeneous preferential attachment models [33,34] have been proposed to extend the Barabási-Albert model to heterogeneous networks. In these models the affinity between node states biases the degree of the network nodes to yield their visibility measure.…”