# Network Formation under Random Attack and Probabilistic Spread

## TL;DR

This paper analyzes how networks form under the threat of random attacks and cascading infections, revealing that equilibrium networks are sparse and that social welfare remains high in less dense networks despite the risks.

## Contribution

It characterizes the edge density of equilibrium networks under attack and demonstrates that social welfare is preserved in moderately dense networks.

## Key findings

- Equilibrium networks have at most O(n log n) edges.
- There exist equilibrium networks with Ω(n) edges, showing the bound is tight.
- Social welfare remains high (Θ(n^2)) in networks with O(n) edges despite attacks.

## Abstract

We study a network formation game where agents receive benefits by forming connections to other agents but also incur both direct and indirect costs from the formed connections. Specifically, once the agents have purchased their connections, an attack starts at a randomly chosen vertex in the network and spreads according to the independent cascade model with a fixed probability, destroying any infected agents. The utility or welfare of an agent in our game is defined to be the expected size of the agent's connected component post-attack minus her expenditure in forming connections. Our goal is to understand the properties of the equilibrium networks formed in this game. Our first result concerns the edge density of equilibrium networks. A network connection increases both the likelihood of remaining connected to other agents after an attack as well the likelihood of getting infected by a cascading spread of infection. We show that the latter concern primarily prevails and any equilibrium network in our game contains only $O(n\log n)$ edges where $n$ denotes the number of agents. On the other hand, there are equilibrium networks that contain $\Omega(n)$ edges showing that our edge density bound is tight up to a logarithmic factor. Our second result shows that the presence of attack and its spread through a cascade does not significantly lower social welfare as long as the network is not too dense. We show that any non-trivial equilibrium network with $O(n)$ edges has $\Theta(n^2)$ social welfare, asymptotically similar to the social welfare guarantee in the game without any attacks.