Cumulative Merging Percolation and the Epidemic Transition of the Susceptible-Infected-Susceptible Model in Networks

Abstract

We consider cumulative merging percolation (CMP), a long-range percolation process describing the iterative merging of clusters in networks, depending on their mass and mutual distance. For a specific class of CMP processes, which represents a generalization of degree-ordered percolation, we derive a scaling solution on uncorrelated complex networks, unveiling the existence of diverse mechanisms leading to the formation of a percolating cluster. The scaling solution accurately reproduces universal properties of the transition. This finding is used to infer the critical properties of the Susceptible-Infected-Susceptible (SIS) model for epidemics in infinite and finite power-law distributed networks, allowing us to rationalize and reconcile previously published results, thus ending a long-standing debate.1 In Ref. [25] it is demonstrated that the CMP threshold is a lower-bound for the epidemic threshold. Based on the physical picture, we expect the two quantities to coincide arXiv:1906.06300v1 [cond-mat.stat-mech]

Spectral properties and the accuracy of mean-field approaches for epidemics on correlated power-law networks

“…However, the theoretical threshold estimates depart from simulation results leading to decreasing ratios λ M F c /λ c in the large network limit. We expect this ratio to decrease asymptotically as 1/ ln(k max ) [45], in agreement with recent rigorous results [48]. Again, PQMF theory performs better than QMF.…”

“…Compared to the uncorrelated case, assortative networks (α > 0) have a smaller threshold, while the threshold is larger for α < 0, i.e., disassortative mixing, in agreement with the behavior of the LEV of the adjacency matrix [14,23]. In the case γ > 3, this phenomenology can be qualitatively explained by considering the mechanism of long-range mutual reinfection of hubs [25,45,47], which triggers the epidemic transition. According to this mechanism, the subgraph consisting of the hub plus its nearest-neighbors can sustain in isolation an active state for times long enough to permit the activation of other hubs, even if they are not directly connected.…”

“…Equation (9) factorizes the state of nearest neighbors and thus neglects dynamical correlations among them. These dynamical correlations actually transmit the infection from the localized PEV to the rest of the network, and thus may in principle transform the active but localized state just above λ QMF c into a full-fledged endemic state [25,45].…”

“…But actually the QMF approach neglects dynamical correlations, which have the effect of allowing mutual reinfection events among different hubs in the network. In this way an endemic global state can be established thanks to the long-range interactions among localized states [45] setting the actual threshold to an intermediate value:…”

Nonmassive immunization to contain spreading on complex networks

“…The mechanism for sustaining SIS activity on this kind of network can be summarized as follows [53]: A hub stays active in isolation for long times through a feedback mechanism where it infects its neighbors which in turn reinfect the hub. If this time is long enough and distances among hubs increase sufficiently slowly with size, rare fluctuations can promote the mutual activation of hubs in the thermodynamical limit even if they are not directly connected, triggering an endemic phase and leading to an asymptotically null epidemic threshold compatible with the QMF theory [51,53,55]. Otherwise, a finite epidemic threshold, compatible with HMF, would be observed [39,54].…”