Graph Theoretic Investigations on Inefficiencies in Network Models

TL;DR

This paper investigates various network models to identify graph structures that ensure maximum efficiency under different flow and routing constraints, revealing that inefficiencies can often be remedied by edge removal, akin to Braess's paradox.

Contribution

It characterizes graph topologies guaranteeing maximum flow efficiency across multiple models and provides algorithms to identify such structures, linking different inefficiency classes.

Findings

Characterizes graphs ensuring maximum flow under energy constraints.

Identifies inefficiency classes in different network models.

Shows inefficiencies can be mitigated by edge removal, similar to Braess's paradox.

Abstract

We consider network models where information items flow %are sent from a source to a sink node. We start with a model where routing is constrained by energy available on nodes in finite supply (like in Smartdust) and efficiency is related to energy consumption. We characterize graph topologies ensuring that every saturating flow under every energy-to-node assignment is maximum and provide a polynomial-time algorithm for checking this property. We then consider the standard flow networks with capacity on edges, where again efficiency is related to maximality of saturating flows, and a traffic model for selfish routing, where efficiency is related to latency at a Wardrop equilibrium. Finally, we show that all these forms of inefficiency yield different classes of graphs (apart from the acyclic case, where the last two forms generate the same class). Interestingly, in all cases inefficient graphs can be made efficient by removing edges; this resembles a well-known phenomenon, called Braess's paradox.