# In-homogeneous Virus Spread in Networks

## TL;DR

This paper extends the NIMFA virus spread model to heterogeneous networks, characterizing steady states and thresholds using generalized Laplacians and eigenvalues, and analyzing the effects of curing rates on infection probabilities.

## Contribution

It introduces a full heterogeneous extension of NIMFA, defines a generalized Laplacian for metastable states, and characterizes the epidemic threshold via eigenvalues of a modified adjacency matrix.

## Key findings

- Steady-state infection probabilities are convex in own curing rates.
- Critical threshold is determined by the largest eigenvalue of a modified adjacency matrix.
- The model applies to any network with N nodes, generalizing previous homogeneous models.

## Abstract

Our $N$-intertwined model (now called NIMFA) for virus spread in any network with $N$ nodes is extended to a full heterogeneous setting. The metastable steady-state nodal infection probabilities are specified in terms of a generalized Laplacian, that possesses analogous properties as the classical Laplacian in graph theory. The critical threshold that separates global network infection from global network health is characterized via an $N$ dimensional vector that makes the largest eigenvalue of a modified adjacency matrix equal to unity. Finally, the steady-state infection probability of node $i$ is convex in the own curing rate $\delta_{i}$, but concave in the curing rates $\delta_{j}$ of the other nodes $1\leq j\neq i\leq N$ in the network.