Abstract
Using a heterogeneous mean-field network formulation of the Bass innovation diffusion model and recent exact results on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We compare our results with those obtained for the SIS epidemic model with the spectral method applied to adjacency matrices. It turns out that diffusion times on finite Barabasi-Albert networks are at a minimum. This may be due to a little-known property of these networks: whereas the value of the assortativity coefficient is close to zero, they look disassortative if one considers only a bounded range of degrees, including the smallest ones, and slightly assortative on the range of the higher degrees. We also find that if the trickle-down character of the diffusion process is enhanced by a larger initial stimulus on the hubs (via a inhomogeneous linear term in the Bass model), the relative difference between the diffusion times for BA networks and uncorrelated networks is even larger, reaching, for instance, the 34% in a typical case on a network with 104 nodes.
Highlights
The study of epidemic diffusion is one of the most important applications of network theory [1,2,3], the absence of an epidemic threshold on scale-free networks being perhaps the best known result [4]
In [8] the dependence of the epidemic threshold and diffusion time on the network assortativity was investigated, using a degreepreserving rewiring procedure which starts from a BarabasiAlbert network and analysing the spectral properties of the resulting adjacency matrices
In this paper we mainly focus on Barabasi-Albert (BA) networks [9], using the exact results by Fotouhi and Rabbat on the degree correlations [10]
Summary
The study of epidemic diffusion is one of the most important applications of network theory [1,2,3], the absence of an epidemic threshold on scale-free networks being perhaps the best known result [4]. We compare these times with those for uncorrelated scale-free networks with exponent γ = 3 and with those for assortative networks whose correlation matrices are mathematically constructed and studied in another work. Our research objectives in this work have been the following: (a) Analyse the assortativity properties of BA networks, using the correlation functions recently published These properties are deduced from the Newman coefficient r and from the average nearest neighbor degree function knn(k). (c) Briefly discuss the validity of the mean-field approximation, compared to a first-moment closure of the Bass model on BA networks described through the adjacency matrix.
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