On the Emergence of Shortest Paths by Reinforced Random Walks

TL;DR

This paper introduces a model where biased random walks on networks evolve edge weights to favor shortest paths, showing that over time, navigation converges to optimal routes with a power-law decay of non-shortest path usage.

Contribution

The study provides a theoretical analysis demonstrating that biased random walks naturally evolve to traverse only shortest paths, revealing insights into network navigation and structure-function co-evolution.

Findings

Biased random walks eventually only traverse shortest paths.

Probability of non-shortest path traversal decays as a power-law.

The model highlights exploration-convergence trade-offs.

Abstract

The co-evolution between network structure and functional performance is a fundamental and challenging problem whose complexity emerges from the intrinsic interdependent nature of structure and function. Within this context, we investigate the interplay between the efficiency of network navigation (i.e., path lengths) and network structure (i.e., edge weights). We propose a simple and tractable model based on iterative biased random walks where edge weights increase over time as function of the traversed path length. Under mild assumptions, we prove that biased random walks will eventually only traverse shortest paths in their journey towards the destination. We further characterize the transient regime proving that the probability to traverse non-shortest paths decays according to a power-law. We also highlight various properties in this dynamic, such as the trade-off between exploration and convergence, and preservation of initial network plasticity. We believe the proposed model and results can be of interest to various domains where biased random walks and decentralized navigation have been applied.