Flows on Graphs with Random Capacities

TL;DR

This paper studies the behavior of maximum flows in graphs with randomly assigned capacities, deriving probability distributions and thresholds for different graph structures, including trees and complete graphs.

Contribution

It introduces a method to analyze flow distributions on graphs with random capacities, extending from trees to more complex networks.

Findings

Maximum flow probability distribution for binary trees derived.

Threshold behavior identified for infinite trees.

Method generalized to graphs with loops and complete graphs.

Abstract

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold that depends on the distribution of capacities. We then examine the maximal total flux from the root to the leaves. Our methods generalize to simple graphs with loops, e.g., to hierarchical lattices and to complete graphs.