# Asymptotical properties of social network dynamics on time scales

## TL;DR

This paper investigates the stability of social network dynamics modeled by Hopfield neural networks on various time scales, linking network structure and node properties to asymptotic behavior.

## Contribution

It extends stability analysis of social networks to dynamic systems on arbitrary time scales, incorporating network parameters and node characteristics.

## Key findings

- Derived stability conditions connecting network size and node degree with time scale properties
- Established correspondence between original and Hopfield models' asymptotic behavior
- Generalized stability results for social networks on diverse time scales

## Abstract

In this paper we develop conditions for various types of stability in social networks governed by Imitation of Success principle. Considering so-called Prisoner's Dilemma as the base of node-to-node game in the network we obtain well-known Hopfield neural network model. Asymptotic behavior of the original model and dynamic Hopfield model has a certain correspondence. To obtain more general results, we consider Hopfield model dynamic system on time scales. Developed stability conditions combine main parameters of network structure such as network size and maximum relative nodes' degree with the main characteristics of time scale, nodes' inertia and resistance, rate of input-output response.