Construction of the Sparsest Maximally r-Robust Graphs

TL;DR

This work addresses resilient consensus under misbehaving agents by focusing on $r$-robustness and seeks the sparsest graph topologies that still achieve maximum robustness. It derives tight edge-count lower bounds for $r$-robust graphs at the minimal node counts $2r-1$ and $2r$, and then constructs two classes of graphs—$(2r-1,r)$-robust and $(2r,r)$-robust—that attain these bounds. Theoretical proofs rely on clique-based structures and $r$-reachability arguments, while simulations validate both resilience under the W-MSR algorithm and the sparsity of the proposed constructions. The results provide concrete design guidelines for robust, communication-efficient networked robotics systems and suggest directions for extending the framework to larger graphs and control synthesis to maintain robustness with reduced communication overhead.

Abstract

In recent years, the notion of r-robustness for the communication graph of the network has been introduced to address the challenge of achieving consensus in the presence of misbehaving agents. Higher r-robustness typically implies higher tolerance to malicious information towards achieving resilient consensus, but it also implies more edges for the communication graph. This in turn conflicts with the need to minimize communication due to limited resources in real-world applications (e.g., multi-robot networks). In this paper, our contributions are twofold. (a) We provide the necessary subgraph structures and tight lower bounds on the number of edges required for graphs with a given number of nodes to achieve maximum robustness. (b) We then use the results of (a) to introduce two classes of graphs that maintain maximum robustness with the least number of edges. Our work is validated through a series of simulations.

Figures (7)

• Figure 1: This shows the snapshots of the initial constructions of $S_1$ and $S_2$ from start of the first to end of the second step in the proof of Lemma \ref{['lem:complete']}. Figure (a) shows that any node $v_1$ (colored in green) will have $r$ neighbors (colored in yellow). Note that Figure (b) indicates $|S_1|=r-1$ and $|S_2|=r$, which are fixed for the figures (c), (d), and (e). Figure (b) shows the start of the first step where a node $v_2 \in S_1$ is connected to all the $r-1$ purple nodes and $v_1$ in $S_2$. The green nodes $v_1$ and $v_2$ together form a $2$-clique. Figure (c) shows $S_1$ and $S_2$ of the end of the first step after we swap $r-1$ nodes including $v_2$ in $S_1$ with $r-1$ purple nodes in $S_2\setminus\{v_1\}$. Then, the start of the second step is shown in Figure (d), where another node $v_3 \in S_1$ has edges with $r-2$ red nodes and $v_1,v_2$ in $S_2$. Figure (e) shows $S_1$ and $S_2$ after we swap $r-2$ nodes in $S_1$ including $v_3$ with $r-2$ red nodes in $S_2\setminus\{v_1,v_2\}$. At the end of second step, we have a $3$-clique (colored in green in Figure (e)). This process continues until $(r+1)$-clique is formed.
• Figure 2: This presents the snapshots of initial developments of $S_1$ and $S_2$ from the start of the first step to the end of the second step in the proof for Theorem \ref{['thm:min_addition']}. The first step starts with $|S_1|=r-2$ and $|S_2|=r+1$, where $S_2$ contains a $(r+1)$-clique from Lemma \ref{['lem:complete']} whose nodes are colored in green. Then, since $S_1$ is $r$-reachable, a node $v_1 \in S_1$ must have edges (colored in yellow) with any $r$ nodes in $S_2$. This requires $r$ additional edges. However, once $v_1 \in S_1$ is moved to $S_2$ (shown Figure (b)), at second step, it has a node $v_2$ that has edges (colored in yellow) with any $r$ nodes in $S_2$. This is because $S_1$ is $r$-reachable even without $v_1$. This requires $r$ additional edges again, as shown on Figure (c). Lastly, the second step ends after $v_2 \in S_1$ is moved to $S_2$, which is shown on Figure (d). After two steps, we needed $2r$ additional edges. This process continues until $S_1$ becomes empty.
• Figure 3: This presents the snapshots of two stages of inductive developments of $S_1$ and $S_2$ in the proof for Lemma \ref{['lem:complete2']}. Figure (a) indicates $|S_1|=r-1$ and $|S_2|=r+1$, which are fixed, and thus omitted in the other figures. In Figure (a), $S_2$ contains $r+1$ nodes including $k$-clique $C_{k,1}=\{v_1,\cdots,v_k\}$ which are colored in green, while $S_1$ contains the remaining $r-1$ nodes. In the first stage, since $S_1$ is $r$-reachable, a node $u_1 \in S_1$ has edges with at least $r$ nodes in $S_2$. Figure (a) shows the third case scenario where the yellow node $u_1$ does not have an edge with $v_1$, which is highlighted in yellow. Then, $u_1$ forms a $k$-clique $C_{k,2}$ with $C_{k,1}\setminus\{v_1\}$. That implies that if any node $u_i \in S_1$, $i\neq 1$, has edges with at least $r$ nodes in $S_2$ but not with $v_1$ in any future steps, $u_i$ will form a $(k+1)$-clique with $C_{k,2}$. We then swap $u_1$ with a node in $S_2\setminus C_{k,1}$, as shown in Figure (b). Then in the second stage, since $S_1$ is $r$-reachable, a node $u_2 \in S_2$ (colored in red in the figure) has edges with at least $r$ nodes in $S_2$. Figure (c) presents the third case scenario where $u_2$ only has $r$ edges and does not have an edge with $v_2$, which is highlighted in red. Similar to before, if any $u_i \in S_2$, $i\neq 2$, in future steps has edges with $r$ nodes except for $v_2$ again, that forms a $(k+1)$-clique. Then, we swap $u_2$ with a node in $S_2\setminus U'$ where $U'=C_{k,1}\cup C_{k,2}$, as shown on Figure (d). Still, $S_1$ is $r$-reachable, and thus the process continues until a $(\lfloor\frac{r+3}{2}\rfloor)$-clique is formed.
• Figure 4: This presents the snapshots of two stages of inductive developments of $S_1$ and $S_2$ in the proof for Theorem \ref{['thm:min_addition2']}. Figure (a) indicates $|S_1|=r-1$ and $|S_2|=r+1$, which are fixed, and thus omitted in the other figures. In Figure (a), $S_2$ contains $r+1$ nodes including $(\lfloor{\frac{r+4}{2}}\rfloor)$-clique $C$ that are colored in green, while $S_1$ contains the remaining $r-1$ nodes. A gray box represents a vertex set $\mathcal{V}_S$ of $S$. Any node in the gray box is in $\mathcal{V}_S$. As $S_1$ is $r$-reachable, a node $v_1 \in S_1$ has edges with any $r$ nodes in $S_2$. Figure (a) shows the case where the yellow node $v_1$ does not have an edge with one node in $\mathcal{V}_S$, which is highlighted in yellow. We swap it with any node in $S_2\setminus \mathcal{V}_S$. Then, as $v_1$ becomes a vertex of subgraph $S$, as shown on Figure (b), $|\mathcal{E}_S|$ increases at least by $\lfloor{\frac{r+2}{2}}\rfloor$. Since $S_1$ is $r$-reachable, a node $v_2 \in S_1$ (colored in red in the figure) has edges with at least $r$ nodes in $S_2$. Figure (c) presents the case where $v_2$ only has $r$ edges and does not have one with a node in $\mathcal{V}_S$, which is highlighted in red. Similar to before, we swap $v_2$ with a node in $S_2\setminus \mathcal{V}_S$. Then, as we add $v_2$ to the subgraph $S$, as shown on Figure (d), $|\mathcal{E}_S|$ increases at least by $\lfloor{\frac{r+4}{2}}\rfloor$. This process continues until $S_2=\mathcal{V}_S$.
• Figure 5: This figure visualizes a $(10,5)$-robust graph. Each of the $r=5$ nodes in $K = \{v_1,v_2,v_3,v_4,v_5\}$ (colored in yellow) connects to all $9$ nodes, and $\delta=4$ of them (in square) lose one edge which is represented in a dotted line. Since $\delta$ is always even ($4$ in this case), we can always form $\frac{\delta}{2}$ disjoint pairs of two nodes from $\delta$ nodes ($2$ pairs i.e. $\{v_1,v_2\}$ and $\{v_3,v_4\}$ in this case). Then, we remove the edges between the pairs.

Theorems & Definitions (26)

• Definition 1: misbehaving agent
• Definition 2: $\mathbf F$-total
• Definition 3: $\mathbf F$-local
• Definition 4: $\mathbf r$-reachable
• Definition 5: $\mathbf r$-robust
• Definition 6: $\textbf{clique}$ clique
• Lemma 1
• proof
• Theorem 1
• proof