Network Discovery by Generalized Random Walks

TL;DR

This paper analyzes network exploration using random walks, deriving formulas for edge discovery, and introduces an adaptive walk model that improves discovery efficiency on dense networks.

Contribution

It provides an exact formula for edge discovery in stationary walks and introduces the Edge Explorer Model for better exploration of dense graphs.

Findings

Edge exploration lags behind site exploration in graphs with many loops.

Stationary transition probability walks follow a specific scaling law.

The Edge Explorer Model enhances network discovery on dense networks.

Abstract

We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as function of time. We show that for STP walks site and edge exploration obey the same scaling $\sim n^{\lambda}$ as function of time $n$. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the Edge Explorer Model, which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks.