Metastability of the contact process on fast evolving scale-free networks

Abstract

We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks we identify and analyse four possible strategies how the infection can survive for a long time. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density.MSc Classification: Primary 05C82; Secondary 82C22.

The contact process on random hyperbolic graphs: metastability and critical exponents

“…It is worth noting that such result has been shown in only a very limited number of other examples. Indeed, to our knowledge so far it was only established for the configuration model [12,15,35], and the so-called Pólya point graph [10] (which is the local limit of preferential attachment graphs [5]), as well as for certain classes of dynamical networks [26]. We shall comment further on the similarities and differences between all these results a bit later; in particular the exponent in the power of λ seems to be a universal constant only depending on the degree distribution, while the power of the logarithmic correction seems on the contrary to be model dependent.…”

The contact process on dynamical scale-free networks

“…Metastable densities have been calculated for static networks for a case of factor connection probabilities by Mountford et al [15] and for preferential attachment probabilities by Van Hao Can [2]. In [10] we started our project of studying metastability for evolving networks by looking at networks evolving by fast updating of all edges adjacent to a vertex simultaneously. This leads to a completely different phase diagram compared to Figure 1 and, in particular, the local survival strategy is not present in the strong form described above.…”

“…This leads to a completely different phase diagram compared to Figure 1 and, in particular, the local survival strategy is not present in the strong form described above. The fast and simultaneous updating of edges in the setup of [10] enables the use of methods relying on the fast mixing of large parts of the network. These methods are unavailable for significant parts of the proof of Theorem 1.…”

“…In the present paper we therefore focus on these parts and develop new techniques that succeed without such fast mixing assumptions. We omit or only give brief hints for the parts of the proof of Theorem 1 that follow by straightforward extension of the methods developed in [10].…”

Spread of Infection over P.A. random graphs with edge insertion

“…An accurate model for spread of infections must accommodate somehow the randomness of the environment, since concrete situations such as human societies produce networks which evolve in time according to random rules. In this direction, random graph models whose rule of evolution combines the so-called preferential attachment rule became a natural environment to the investigation of infectious processes [7,9,12]. The reason is that this mechanism of attachment which is driven by popularity, that is, individuals tends to get connected with the more popular ones, proved that it is capable of capturing network properties shared by many networks in real-file [6,13].…”