# The Spread of Voting Attitudes in Social Networks

## TL;DR

This paper explores how voting attitudes spread in social networks using an extended power index and a dynamic model, revealing conditions for influence cascades and connections to cellular automata like Rule 90.

## Contribution

It introduces a novel extension of the Shapley-Shubik index to graph-based networks and models voter influence spread with cellular automata connections.

## Key findings

- Small voter groups can influence entire networks under certain conditions
- The process can be modeled with cellular automata, specifically Rule 90
- The model exhibits arbitrarily long periodic behaviors

## Abstract

The Shapley-Shubik power index is a measure of each voters power in the passage or failure of a vote. We extend this measure to graphs and consider a discrete-time process in which voters may change their vote based on the outcome of the previous vote. We use this model to study how voter influence can spread through a network. We find conditions under which a vanishingly small portion of consenting voters can change the votes of the entirety of the network. For a particular family of graphs, this process can be modelled using cellular automata. In particular, we find a connection between this process and the well-studied cellular automata, Rule 90. We use this connection to show that such processes can exhibit arbitrarily-long periods.