# Complex Contagions with Timers

## TL;DR

This paper introduces timers into threshold models of social influence to account for response delays, showing how heterogeneous timers can alter the speed and order of contagion spread in networks.

## Contribution

It presents a novel timer-based extension to threshold models, analyzing the effects of homogeneous and heterogeneous delays on contagion dynamics.

## Key findings

- Heterogeneous timers can accelerate or decelerate contagion spread.
- Analytical approximation aligns well with numerical simulations.
- Empirical network data confirms model applicability.

## Abstract

A great deal of effort has gone into trying to model social influence --- including the spread of behavior, norms, and ideas --- on networks. Most models of social influence tend to assume that individuals react to changes in the states of their neighbors without any time delay, but this is often not true in social contexts, where (for various reasons) different agents can have different response times. To examine such situations, we introduce the idea of a timer into threshold models of social influence. The presence of timers on nodes delays the adoption --- i.e., change of state --- of each agent, which in turn delays the adoptions of its neighbors. With a homogeneous-distributed timer, in which all nodes exhibit the same amount of delay, adoption delays are also homogeneous, so the adoption order of nodes remains the same. However, heterogeneously-distributed timers can change the adoption order of nodes and hence the "adoption paths" through which state changes spread in a network. Using a threshold model of social contagions, we illustrate that heterogeneous timers can either accelerate or decelerate the spread of adoptions compared to an analogous situation with homogeneous timers, and we investigate the relationship of such acceleration or deceleration with respect to timer distribution and network structure. We derive an analytical approximation for the temporal evolution of the fraction of adopters by modifying a pair approximation of the Watts threshold model, and we find good agreement with numerical computations. We also examine our new timer model on networks constructed from empirical data.