Minimal Construction of Graphs with Maximum Robustness

TL;DR

This paper establishes tight necessary conditions on the number of edges that an undirected graph with an arbitrary number of nodes must have to achieve maximum $r$- and $(r,s)$-robustness and constructs two classes of undirected graphs, referred as to $\gamma$- and $\gamma,\gamma)$-Minimal Edge Robust Graphs (MERGs), that provably achieve maximum robustness with minimal numbers of edges.

Abstract

The notions of $r$-robustness and $(r,s)$-robustness of a network have been earlier introduced in the literature to achieve resilient consensus in the presence of misbehaving agents. However, while higher robustness levels enable networks to tolerate a higher number of misbehaving agents, they also require dense communication structures, which are not always desirable for systems with limited communication ranges, energy, and resources. Therefore, this paper studies the fundamental structures behind $r$-robustness and $(r,s)$- robustness properties in two ways. (a) We first establish tight necessary conditions on the number of edges that an undirected graph with an arbitrary number of nodes must have to achieve maximum $r$- and $(r,s)$-robustness. (b) We then use these conditions to construct two classes of undirected graphs, referred as to $γ$- and $(γ,γ)$-Minimal Edge Robust Graphs (MERGs), that provably achieve maximum robustness with minimal numbers of edges. We demonstrate the effectiveness of our method via comparison against existing robust graph structures and a set of simulations.

Figures (7)

• Figure 1: These figures illustrate the first two iterations in the proof of \ref{['lem:complete']}. Figure (a) shows the initial configuration, where node $v_1\in \mathcal{V}$ has $\gamma$ neighbors (colored in yellow). Figure (b) shows the first iteration where node $v_2 \in \mathcal{S}_1$ is connected to $\gamma-1$ purple nodes and to $v_1$ in $\mathcal{S}_2$. The green nodes $v_1$ and $v_2$ form a $2$-clique. Note we maintain $|\mathcal{S}_1|=\gamma-1$ and $|\mathcal{S}_2|=\gamma$ from figures (b) to (e). Figure (c) shows $\mathcal{S}_1$ and $\mathcal{S}_2$ after the completion of the first step, where $\gamma-1$ nodes in $\mathcal{S}_1$ (including $v_2$) have been swapped with $\gamma-1$ purple nodes in $\mathcal{S}_2\setminus\{v_1\}$. Then, the start of the second step is shown in Figure (d), where another node $v_3 \in \mathcal{S}_1$ has edges with $\gamma-2$ red nodes and with $v_1,v_2$ in $\mathcal{S}_2$. Here, the green nodes $v_1,v_2,$ and $v_3$ form a $3$-clique. Figure (e) shows $\mathcal{S}_1$ and $\mathcal{S}_2$ after $\gamma-2$ nodes in $\mathcal{S}_1$ (including $v_3$) have been swapped with $\gamma-2$ red nodes in $\mathcal{S}_2\setminus\{v_1,v_2\}$. This process continues until $(\gamma+1)$-clique is formed.
• Figure 2: Visualizations of $5$-MERGs with $9$ (left) and $10$ (right) nodes.
• Figure 3: This figure visualizes the contradiction shown in the proof for \ref{['lem:min_addition4']}. The yellow solid lines represent edges, whereas the colored dotted lines represent the absences of edges. There are two disjoint non-empty subsets $\mathcal{S}_1,\mathcal{S}_2 \subset \mathcal{V}$, in which all nodes only have $\gamma-1$ neighbors outside of the subsets, necessitating at least $\lceil \frac{\gamma}{2} \rceil$ of the colored dotted lines to become actual edges to resolve the contradiction.
• Figure 4: Visualizations of $(5,5)$-MERGs with $9$ (left) and $10$ (right) nodes.
• Figure 5: Simulation results for the first set of simulations. We run consensus on $25$-MERGs with (a) $49$ and (b) $50$ nodes, and $(25,25)$-MERGs with (c) $49$ and (d) $50$ nodes, demonstrating resilient consensus through the W-MSR algorithm under $F=12$ and $F=24$ malicious agents, respectively. The normal agents' states are solid colored lines, while malicious agents' states are shown as red dotted lines.

Theorems & Definitions (38)

• Definition 1: Malicious agent
• Definition 2: Byzantine agent
• Definition 3: $\mathbf F$-total
• Definition 4: $\mathbf F$-local
• Definition 5: $\mathbf r$-reachable
• Definition 6: $\mathbf r$-robust
• Definition 7: $\mathbf {(r,s)}$-reachable
• Definition 8: $\mathbf {(r,s)}$-robust
• Definition 9: $\mathbf r$-MERG
• Definition 10: $\mathbf {(r,s)}$-MERG