135 research outputs found
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
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
Predicting Dynamics on Networks Hardly Depends on the Topology
Processes on networks consist of two interdependent parts: the network
topology, consisting of the links between nodes, and the dynamics, specified by
some governing equations. This work considers the prediction of the future
dynamics on an unknown network, based on past observations of the dynamics. For
a general class of governing equations, we propose a prediction algorithm which
infers the network as an intermediate step. Inferring the network is impossible
in practice, due to a dramatically ill-conditioned linear system. Surprisingly,
a highly accurate prediction of the dynamics is possible nonetheless: Even
though the inferred network has no topological similarity with the true
network, both networks result in practically the same future dynamics
Exact Coupling Threshold for Structural Transition in Interconnected Networks
Interconnected networks are mathematical representation of systems where two
or more simple networks are coupled to each other. Depending on the coupling
weight between the two components, the interconnected network can function in
two regimes: one where the two networks are structurally distinguishable, and
one where they are not. The coupling threshold--denoting this structural
transition--is one of the most crucial concepts in interconnected networks.
Yet, current information about the coupling threshold is limited. This letter
presents an analytical expression for the exact value of the coupling threshold
and outlines network interrelation implications
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