Metastable densities for the contact process on power law random graphs

Mountford, Thomas; Valesin, Daniel; Yao, Qiang

doi:10.1214/ejp.v18-2512

Cited by 52 publications

(91 citation statements)

References 13 publications

(28 reference statements)

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“…in agreement with the exact mathematical results of Mountford et al [24]. As discussed above, this prediction is, however, impossible to verify numerically, because the onset of the asymptotic regime could be seen only for exceedingly large networks.…”

Section: Sis Prevalence As a Function Of λsupporting

confidence: 86%

“…Concerning finite size effects, for γ > 3 as only the first scaling regime is observed, the condition setting the effective threshold is Eq. (24). A direct numerical verification of it for CMP is very hard, as practically all non-isolated nodes are part of the CMPGC and finite clusters (upon which methods to determine the position of the threshold are based) are extremely rare.…”

Section: Numerical Testmentioning

confidence: 99%

See 1 more Smart Citation

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

Castellano

Pastor-Satorras

2020

Phys. Rev. X

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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]

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Section: Sis Prevalence As a Function Of λsupporting

confidence: 86%

Section: Numerical Testmentioning

confidence: 99%

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

Castellano

Pastor-Satorras

2020

Phys. Rev. X

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show abstract

“…For the construction of a 0 on the complement of this set, we note that all of the functions a have the form λ e +o(1) , and since λ < 1, for each λ small the dominant survival strategy (that is, the one which gives the largest lower bound for the density) corresponds to the expression with the smallest exponent. If η < 1/2 this gives for the values of the parameter where a 0 is defined gives the lower bound in (11).…”

Section: Application To the Factor Kernelmentioning

confidence: 99%

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

Jacob¹,

Linker²,

Mörters³

2019

Ann. Appl. Probab.

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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.

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“…(in fact, (6.3) and (6.4) are slightly different from the mentioned results in [22], but are readily seen to follow from their proof. We spare the reader the details.)…”

Section: Extinction Time For the Configuration Modelmentioning

confidence: 66%

Exponential extinction time of the contact process on finite graphs

Mountford

Mourrat

Valesin

et al. 2016

Stochastic Processes and their Applications

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International audienceWe study the extinction time τ of the contact process started with full occupancy on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on Z, then, uniformly over all trees of degree bounded by a given number, the expectation of τ grows exponentially with the number of vertices. Additionally, for any increasing sequence of trees of bounded degree, τ divided by its expectation converges in distribution to the unitary exponential distribution. These results also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree, and provide the basis for powerful coarse-graining arguments. To demonstrate this, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett [CD09], we show that, for any non-zero infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices

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Metastable densities for the contact process on power law random graphs

Cited by 52 publications

References 13 publications

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

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

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

Exponential extinction time of the contact process on finite graphs

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