# A spatial small-world graph arising from activity-based reinforcement

## TL;DR

This paper introduces a spatial hierarchical random graph model with activity-based reinforcement, demonstrating its convergence and small-world properties, inspired by synaptic plasticity, differing from classical static models.

## Contribution

It presents a novel spatial hierarchical random graph model with reinforcement dynamics, establishing convergence and small-world characteristics, and motivated by neural synaptic plasticity.

## Key findings

- The reinforcement mechanism converges.
- The resulting graph exhibits small-world properties.
- The model is motivated by synaptic plasticity.

## Abstract

In the classical preferential attachment model, links form instantly to newly arriving nodes and do not change over time. We propose a hierarchical random graph model in a spatial setting, where such a time-variability arises from an activity-based reinforcement mechanism. We show that the reinforcement mechanism converges, and prove rigorously that the resulting random graph exhibits the small-world property. A further motivation for this random graph stems from modeling synaptic plasticity.