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Graph Theoretic Investigations on Inefficiencies in Network Models

Pietro Cenciarelli, Daniele Gorla, Ivano Salvo

TL;DR

The paper investigates three notions of inefficiency across network models: weakness in depletable channels with node charges, edge-weakness in edge-capacitated flow networks, and vulnerability in Wardrop latency models. It provides precise graph-theoretic characterizations—weakness via revisiting a minimal vertex separator, edge-weakness via revisiting a connected cut’s edge-set, and vulnerability via subgraphs homeomorphic to the Wheatstone network—and establishes polynomial-time methods to detect these properties. In particular, weakness is decidable in polynomial time with a chain-based algorithm, and vulnerability has a polynomial bound of $O(|V|^5)$, while edge-weakness is efficiently handled in the acyclic case. The results reveal Braess-like paradoxes across models, linking energy depletion, standard flows, and selfish routing, and show how removing edges can rectify inefficiencies, with practical implications for network design and analysis.

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.

Graph Theoretic Investigations on Inefficiencies in Network Models

TL;DR

The paper investigates three notions of inefficiency across network models: weakness in depletable channels with node charges, edge-weakness in edge-capacitated flow networks, and vulnerability in Wardrop latency models. It provides precise graph-theoretic characterizations—weakness via revisiting a minimal vertex separator, edge-weakness via revisiting a connected cut’s edge-set, and vulnerability via subgraphs homeomorphic to the Wheatstone network—and establishes polynomial-time methods to detect these properties. In particular, weakness is decidable in polynomial time with a chain-based algorithm, and vulnerability has a polynomial bound of , while edge-weakness is efficiently handled in the acyclic case. The results reveal Braess-like paradoxes across models, linking energy depletion, standard flows, and selfish routing, and show how removing edges can rectify inefficiencies, with practical implications for network design and analysis.

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.

Paper Structure

This paper contains 13 sections, 15 theorems, 1 equation, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Inefficiency in Depletable Channels: Weakness
  3. Checking Weakness
  4. Inefficiency in Flow Networks: Edge-weakness
  5. Inefficiency in Traffic Networks: Vulnerability
  6. Comparing Weakness, Edge-Weakness, and Vulnerability
  7. Conclusion
  8. Proofs
  9. Proofs of Section \ref{['sec:weak']}
  10. Proofs of Section \ref{['sec:complexity']}
  11. Proofs of Section \ref{['sec:edgeweak']}
  12. Proofs of Section \ref{['sec:vuln']}
  13. Proofs of Section \ref{['sec:compare']}

Key Result

theorem 1

A graph is weak if and only if there exists an mvs $T$ and a directed walk $a\leadsto b$ with $a,b\in T$.

Figures (6)

  • Figure 1: (a) Four communication devices with their communication radius: (b) the associated directed graph.
  • Figure 2: Two channels
  • Figure 3:
  • Figure 5:
  • Figure 7: The inclusion diagram for cyclic (top-left) and acyclic (top-right) directed graphs ($W$ is the Wheatstone graph -- see Fig. \ref{['fig:example']}(b))
  • ...and 1 more figures

Theorems & Definitions (22)

  • definition 1
  • theorem 1
  • lemma 1
  • theorem 2
  • theorem 3
  • theorem 4
  • definition 2
  • theorem 5
  • definition 3
  • theorem 6
  • ...and 12 more