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Distributed Graph Augmentation Protocols to Achieve Strong Connectivity in Multi-Agent Networks

Guilherme Ramos, Diogo Poças, Sérgio Pequito

Abstract

In multi-agent systems, strong connectivity of the communication network is often crucial for establishing consensus protocols, which underpin numerous applications in decision-making and distributed optimization. However, this connectivity requirement may not be inherently satisfied in geographically distributed settings. Consequently, we need to find the minimum number of communication links to add to make the communication network strongly connected. To date, such problems have been solvable only through centralized methods. This paper introduces a fully distributed algorithm that efficiently identifies an optimal set of edge additions to achieve strong connectivity, using only local information. The majority of the communication between agents is local (according to the digraph structure), with only a few steps requiring communication among non-neighboring agents to establish the necessary additional communication links. A comprehensive empirical analysis of the algorithm's performance on various random communication networks reveals its efficiency and scalability.

Distributed Graph Augmentation Protocols to Achieve Strong Connectivity in Multi-Agent Networks

Abstract

In multi-agent systems, strong connectivity of the communication network is often crucial for establishing consensus protocols, which underpin numerous applications in decision-making and distributed optimization. However, this connectivity requirement may not be inherently satisfied in geographically distributed settings. Consequently, we need to find the minimum number of communication links to add to make the communication network strongly connected. To date, such problems have been solvable only through centralized methods. This paper introduces a fully distributed algorithm that efficiently identifies an optimal set of edge additions to achieve strong connectivity, using only local information. The majority of the communication between agents is local (according to the digraph structure), with only a few steps requiring communication among non-neighboring agents to establish the necessary additional communication links. A comprehensive empirical analysis of the algorithm's performance on various random communication networks reveals its efficiency and scalability.

Paper Structure

This paper contains 7 sections, 10 theorems, 1 equation, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Distributed Graph Augmentation Protocols to Achieve Strong Connectivity in Multi-Agent Networks
  4. Illustrative Examples
  5. Pedagogical Examples
  6. Random Graphs
  7. Conclusions

Key Result

Lemma 1

Given a weakly connected digraph $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ with $\alpha$ s-SCCs and $\beta$ t-SCCs, any solution to MCDAP has $\gamma(\mathcal{G}):=\max\{\alpha, \beta\}$ edges.

Figures (4)

  • Figure 1: The digraph with black edges is the one to be augmented (MCDAP), which decomposition in SCCs is represented by boxes of nodes. The red edges represent the augmentation computed using the scheme in Algorithm \ref{['fig:schematics']}. The edges' labels are the round in which the edge is created.
  • Figure 2: The digraph with black edges in (a) is the digraph to be augmented (MCDAP), which decomposition in SCCs is represented by boxes of nodes. $\mathcal{E}^\ast=\{(4,1)(2,3),(1,9)\}$.
  • Figure 3: The digraph with black edges is the one to be augmented (MCDAP), which decomposition in SCCs is represented by boxes of nodes. $\mathcal{E}^\ast=\{(9,1)(12,5),(5,1)\}$.
  • Figure 4: Average number of rounds of Algorithm \ref{['fig:schematics']} on random digraphs, using different random generation models.

Theorems & Definitions (18)

  • Lemma 1: eswaran1976augmentation
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 8 more