193 research outputs found
Die-out Probability in SIS Epidemic Processes on Networks
An accurate approximate formula of the die-out probability in a SIS epidemic
process on a network is proposed. The formula contains only three essential
parameters: the largest eigenvalue of the adjacency matrix of the network, the
effective infection rate of the virus, and the initial number of infected nodes
in the network. The die-out probability formula is compared with the exact
die-out probability in complete graphs, Erd\H{o}s-R\'enyi graphs, and a
power-law graph. Furthermore, as an example, the formula is applied to the
-Intertwined Mean-Field Approximation, to explicitly incorporate the
die-out.Comment: Version2: 10 figures, 11 pagers. Corrected typos; simulation results
of ER graphs and a power-law graph are added. Accepted by the 5th
International Workshop on Complex Networks and their Applications, November
30 - December 02, 2016, Milan, Ital
Decentralized Protection Strategies against SIS Epidemics in Networks
Defining an optimal protection strategy against viruses, spam propagation or
any other kind of contamination process is an important feature for designing
new networks and architectures. In this work, we consider decentralized optimal
protection strategies when a virus is propagating over a network through a SIS
epidemic process. We assume that each node in the network can fully protect
itself from infection at a constant cost, or the node can use recovery
software, once it is infected.
We model our system using a game theoretic framework and find pure, mixed
equilibria, and the Price of Anarchy (PoA) in several network topologies.
Further, we propose both a decentralized algorithm and an iterative procedure
to compute a pure equilibrium in the general case of a multiple communities
network. Finally, we evaluate the algorithms and give numerical illustrations
of all our results.Comment: accepted for publication in IEEE Transactions on Control of Network
System
Containing epidemic outbreaks by message-passing techniques
The problem of targeted network immunization can be defined as the one of
finding a subset of nodes in a network to immunize or vaccinate in order to
minimize a tradeoff between the cost of vaccination and the final (stationary)
expected infection under a given epidemic model. Although computing the
expected infection is a hard computational problem, simple and efficient
mean-field approximations have been put forward in the literature in recent
years. The optimization problem can be recast into a constrained one in which
the constraints enforce local mean-field equations describing the average
stationary state of the epidemic process. For a wide class of epidemic models,
including the susceptible-infected-removed and the
susceptible-infected-susceptible models, we define a message-passing approach
to network immunization that allows us to study the statistical properties of
epidemic outbreaks in the presence of immunized nodes as well as to find
(nearly) optimal immunization sets for a given choice of parameters and costs.
The algorithm scales linearly with the size of the graph and it can be made
efficient even on large networks. We compare its performance with topologically
based heuristics, greedy methods, and simulated annealing
Decay towards the overall-healthy state in SIS epidemics on networks
The decay rate of SIS epidemics on the complete graph is computed
analytically, based on a new, algebraic method to compute the second largest
eigenvalue of a stochastic three-diagonal matrix up to arbitrary precision. The
latter problem has been addressed around 1950, mainly via the theory of
orthogonal polynomials and probability theory. The accurate determination of
the second largest eigenvalue, also called the \emph{decay parameter}, has been
an outstanding problem appearing in general birth-death processes and random
walks. Application of our general framework to SIS epidemics shows that the
maximum average lifetime of an SIS epidemics in any network with nodes is
not larger (but tight for ) than E\left[ T\right]
\sim\frac{1}{\delta}\frac{\frac{\tau}{\tau_{c}}\sqrt{2\pi}% }{\left(
\frac{\tau}{\tau_{c}}-1\right) ^{2}}\frac{\exp\left( N\left\{
\log\frac{\tau}{\tau_{c}}+\frac{\tau_{c}}{\tau}-1\right\} \right) }{\sqrt
{N}}=O\left( e^{N\ln\frac{\tau}{\tau_{c}}}\right) for large and for an
effective infection rate above the epidemic
threshold . Our order estimate of sharpens the
order estimate of Draief and
Massouli\'{e} \cite{Draief_Massoulie}. Combining the lower bound results of
Mountford \emph{et al.} \cite{Mountford2013} and our upper bound, we conclude
that for almost all graphs, the average time to absorption for
is , where depends on
the topological structure of the graph and
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