Performance bound analysis of linear consensus algorithm on strongly connected graphs using effective resistance and reversiblization

TL;DR

The paper addresses the problem of bounding the LQ performance cost $J(P)$ for linear consensus on directed, strongly connected graphs, extending reversible-case analyses to nonreversible settings through reversiblization with $P^*P$. It introduces the back-and-forth path and pivot node to relate directed dynamics to effective resistance and proves upper and lower bounds for $J(P)$ and $J_{\mathrm{w}}(P)$ in terms of the average effective resistance of $C_{P^*P}$, with refinements based on the graph structure. The results apply to important graph families, including Cayley graphs (where $P^*P=PP^*$) and geometric graphs, and yield asymptotic scaling insights (e.g., on $\mathbb{Z}_n^d$) that illuminate how topology shapes convergence performance. The work provides a general framework for analyzing nonreversible consensus beyond traditional reversibility assumptions, with practical implications for directed networks and distributed coordination tasks.

Abstract

We study the performance of the linear consensus algorithm on strongly connected directed graphs using the linear quadratic (LQ) cost as a performance measure. In particular, we derive bounds on the LQ cost by leveraging effective resistance and reversiblization. Our results extend previous analyses-which were limited to reversible cases-to the nonreversible setting. To facilitate this generalization, we introduce novel concepts, termed the back-and-forth path and the pivot node, which serve as effective alternatives to traditional techniques that require reversibility. Moreover, we apply our approach to Cayley graphs and random geometric graphs to estimate the LQ cost without the reversibility assumption. The proposed approach provides a framework that can be adapted to other contexts where reversibility is typically assumed.

Figures (10)

• Figure 1: Inclusion properties among classes of consensus matrices.
• Figure 1: The values of $J(P_{\varepsilon})$ for $\varepsilon\in[0.001,0.5]$. The left panel shows the lower and upper bounds in \ref{['mainres1']}. The right panel shows the lower and upper bounds in \ref{['mainres2']}.
• Figure 2: Notions of back-and-forth path, pivot, and new edge.
• Figure 2: Consensus algorithm on a circle.
• Figure 3: Graphs used to evaluate $\bar{\mathcal{R}}(\mathcal{G}(P^*P))$.

Theorems & Definitions (34)

• Lemma 2.1
• Definition 2.2: mainp
• Lemma 2.3
• Remark 2.4
• Lemma 3.1
• Proof 1
• Proposition 3.2
• Proof 2
• Theorem 3.3
• Lemma 3.4