# Analysis of the susceptible-infected-susceptible epidemic dynamics in networks via the non-backtracking matrix

## TL;DR

This paper introduces a new, computationally efficient lower bound on the decay rate of infection probabilities in SIS epidemic models on networks, improving upon previous bounds and relating to the network's non-backtracking matrix.

## Contribution

The paper presents a novel lower bound based on the non-backtracking matrix that outperforms existing bounds and is more computationally feasible for sparse networks.

## Key findings

- The new bound is tighter than previous bounds.
- It is computationally less expensive for sparse networks.
- The epidemic threshold is expressed in terms of the line graph and non-backtracking matrices.

## Abstract

We study the stochastic susceptible-infected-susceptible model of epidemic processes on finite directed and weighted networks with arbitrary structure. We present a new lower bound on the exponential rate at which the probabilities of nodes being infected decay over time. This bound is directly related to the leading eigenvalue of a matrix that depends on the non-backtracking and incidence matrices of the network. The dimension of this matrix is N+M, where N and M are the number of nodes and edges, respectively. We show that this new lower bound improves on an existing bound corresponding to the so-called quenched mean-field theory. Although the bound obtained from a recently developed second-order moment-closure technique requires the computation of the leading eigenvalue of an N^2 x N^2 matrix, we illustrate in our numerical simulations that the new bound is tighter, while being computationally less expensive for sparse networks. We also present the expression for the corresponding epidemic threshold in terms of the adjacency matrix of the line graph and the non-backtracking matrix of the given network.