Decay towards the overall-healthy state in SIS epidemics on networks

The decay rate of SIS epidemics on the complete graph $K_{N}$ 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 $N$ nodes is not larger (but tight for $K_{N}$) 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 $N$ and for an effective infection rate $\tau=\frac{\beta}{\delta}$ above the epidemic threshold $\tau_{c}$. Our order estimate of $E\left[ T\right]$ sharpens the order estimate $E\left[ T\right] =O\left( e^{bN^{a}}\right)$ 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 $\tau>\tau_{c}$ is $E\left[ T\right] =O\left( e^{c_{G}N}\right)$, where $c_{G}>0$ depends on the topological structure of the graph $G$ and $\tau$

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 $N$-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