scholarly journals Explicit bounds for critical infection rates and expected extinction times of the contact process on finite random graphs

Bernoulli ◽  
2021 ◽  
Vol 27 (3) ◽  
Author(s):  
E. Cator ◽  
H. Don
Keyword(s):  
Random Graphs ◽  
Contact Process ◽  
Infection Rates ◽  
Explicit Bounds

scholarly journals Contact process with exogenous infection and the scaled SIS process

2017 ◽  
Vol 5 (5) ◽  
pp. 712-733 ◽  
Author(s):  
June Zhang ◽  
José M.F. Moura ◽  
June Zhang
Keyword(s):  
Equilibrium Distribution ◽  
Equilibrium Configuration ◽  
Contact Process ◽  
Finite Size ◽  
Infection Rates ◽  
Contact Processes ◽  
Arbitrary Topology ◽  
Extended Contact ◽  
Complete Networks

Abstract Propagation of contagion in networks depends on the graph topology. This article is concerned with studying the time-asymptotic behaviour of the extended contact processes on static, undirected, finite-size networks. This is a contact process with nonzero exogenous infection rate (also known as the $\epsilon$-susceptible-infected-susceptible model). The only known analytical characterization of the equilibrium distribution of this process is for complete networks. For large networks with arbitrary topology, it is infeasible to numerically solve for the equilibrium distribution since it requires solving the eigenvalue-eigenvector problem of a matrix that is exponential in $N$, the size of the network. We derive a condition on the infection rates under which, depending on the degree distribution of the network, the equilibrium distribution of extended contact processes on arbitrary, finite-size networks is well approximated by a closed-form formulation. We confirm the goodness of the approximation with small networks answering inference questions like the distribution of the percentage of infected individuals and the most-probable equilibrium configuration. We then use the approximation to analyse the equilibrium distribution of the extended contact process on the 4941-node US Western power grid.

scholarly journals Metastable densities for the contact process on power law random graphs

2013 ◽  
Vol 18 (0) ◽  
Author(s):  
Thomas Mountford ◽  
Daniel Valesin ◽  
Qiang Yao
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Contact Process

scholarly journals The contact process on scale-free networks evolving by vertex updating

2017 ◽  
Vol 4 (5) ◽  
pp. 170081 ◽  
Author(s):  
Emmanuel Jacob ◽  
Peter Mörters
Keyword(s):  
Phase Transition ◽  
Power Law ◽  
Mathematical Proof ◽  
Contact Process ◽  
Network Size ◽  
Mean Field Approximation ◽  
Infection Rates ◽  
Scale Free ◽  
Power Law Exponent ◽  
Scale Free Networks

We study the contact process on a class of evolving scale-free networks, where each node updates its connections at independent random times. We give a rigorous mathematical proof that there is a transition between a phase where for all infection rates the infection survives for a long time, at least exponential in the network size, and a phase where for sufficiently small infection rates extinction occurs quickly, at most polynomially in the network size. The phase transition occurs when the power-law exponent crosses the value four. This behaviour is in contrast with that of the contact process on the corresponding static model, where there is no phase transition, as well as that of a classical mean-field approximation, which has a phase transition at power-law exponent three. The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which the infection spreads in the network, and decrease the time at which the infection spends in metastable states.

scholarly journals Coupling on weighted branching trees

2016 ◽  
Vol 48 (2) ◽  
pp. 499-524 ◽  
Author(s):  
Ningyuan Chen ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Fixed Points ◽  
Random Graphs ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Linear Functions ◽  
Dependence Structure ◽  
Branching Trees ◽  
Explicit Bounds ◽  
Weighted Branching Process

Abstract In this paper we consider linear functions constructed on two different weighted branching processes and provide explicit bounds for their Kantorovich–Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich–Rubinstein distance of linear functions. We focus on the case where the limits are endogenous fixed points of suitable smoothing transformations.

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