3,477 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
Spectral Perturbation and Reconstructability of Complex Networks
In recent years, many network perturbation techniques, such as topological
perturbations and service perturbations, were employed to study and improve the
robustness of complex networks. However, there is no general way to evaluate
the network robustness. In this paper, we propose a new global measure for a
network, the reconstructability coefficient {\theta}, defined as the maximum
number of eigenvalues that can be removed, subject to the condition that the
adjacency matrix can be reconstructed exactly. Our main finding is that a
linear scaling law, E[{\theta}]=aN, seems universal, in that it holds for all
networks that we have studied.Comment: 9 pages, 10 figure
Reverse Line Graph Construction: The Matrix Relabeling Algorithm MARINLINGA Versus Roussopoulos's Algorithm
We propose a new algorithm MARINLINGA for reverse line graph computation,
i.e., constructing the original graph from a given line graph. Based on the
completely new and simpler principle of link relabeling and endnode
recognition, MARINLINGA does not rely on Whitney's theorem while all previous
algorithms do. MARINLINGA has a worst case complexity of O(N^2), where N
denotes the number of nodes of the line graph. We demonstrate that MARINLINGA
is more time-efficient compared to Roussopoulos's algorithm, which is
well-known for its efficiency.Comment: 30 pages, 24 figure
- …