Heterogeneous pair-approximation for the contact process on complex networks

Abstract

Recent works have shown that the contact process running on the top of highly heterogeneous random networks is described by the heterogeneous mean-field theory. However, some important aspects such as the transition point and strong corrections to the finite-size scaling observed in simulations are not quantitatively reproduced in this theory. We develop a heterogeneous pair-approximation, the simplest mean-field approach that takes into account dynamical correlations, for the contact process. The transition points obtained in this theory are in very good agreement with simulations. The proximity with a simple homogeneous pair-approximation is elicited showing that the transition point in successive homogeneous cluster approximations moves away from the simulation results. We show that the critical exponents of the heterogeneous pairapproximation in the infinite-size limit are the same as those of the one-vertex theory. However, excellent matches with simulations, for a wide range of network sizes, are obtained when the sub-leading finite-size corrections given by the new theory are explicitly taken into account. The present approach can be suited to dynamical processes on networks in general providing a profitable strategy to analytically assess and fine-tune theoretical corrections.

Collective versus hub activation of epidemic phases on networks

“…This collective transition is consistent with the finite threshold numerically observed in the CP on SF networks [20,21], in agreement with HMF predictions [19,21]. Interestingly in the case of the CP, the QMF prediction [30] coincides with the HMF theory [19], λ c = 1, indicating a threshold completely independent of the network structure.…”

“…Interestingly in the case of the CP, the QMF prediction [30] coincides with the HMF theory [19], λ c = 1, indicating a threshold completely independent of the network structure. This theoretical prediction it is not fully observed in numerical simulations, which show a constant threshold but that is modulated by network heterogeneity [19][20][21] C. KJI model…”

“…The fact that the lifetime τ rec k is diverging with degree k faster than τ inf k,k for any value of λ is the ultimate cause of the null epidemic threshold in the SIS model [10,14,15]. However, other epidemic and dynamical models on SF networks, in particular the contact process (CP) [19], defined by an infection rate inversely proportional to the degree of the infected node, λ/k, possess a finite threshold which can be better captured in terms of a degree-based HMF theory [20,21]. This observation claims for an understanding of the mechanisms ruling epidemic transitions, regarding in particular the conditions under which the threshold is either constant or vanishing.…”

Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks

“…Percolation [2], epidemic spreading [3], and spin systems [4] are only a few examples of breakthrough in the investigation of critical phenomena in complex networks. Absorbing state phase transitions [5] have become a paradigmatic issue in the interplay between nonequilibrium systems and complex networks [6][7][8][9][10], being the epidemic spreading a prominent example where high complexity emerges from very simple dynamical rules on heterogeneous substrates [3,[11][12][13][14][15].…”

Optimal allocation of resources for suppressing epidemic spreading on networks

“…An interesting generalization is how to solve the present constraint optimization problem based on other existing theoretical methods, such as pair mean-field method that takes into account the role of dynamical correlations between neighboring nodes [31][32][33][34][35][36][37][38][39]. Moreover, the method presented here could be applied to a number of other optimization problems, for example, controlling opinion dynamics in social networks [40].…”