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

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.

💡 Research Summary

This paper investigates the performance of the linear consensus algorithm on strongly connected directed graphs using the linear‑quadratic (LQ) cost as a unified performance metric. While prior work on consensus performance has largely relied on the reversibility (detailed balance) assumption—allowing a direct analogy between Markov chains, electrical networks, and effective resistance—real‑world communication networks are often non‑reversible. To bridge this gap, the authors develop a framework that does not require reversibility.

The core technical contribution consists of two parts. First, they introduce a “reversibilization” technique: given a row‑stochastic, irreducible, aperiodic consensus matrix (P) with stationary distribution (\pi), they construct a symmetric matrix (P^{*}= \Pi^{1/2} P \Pi^{-1/2}) and define a conductance matrix (\Phi(P)= n \Pi P) (with (n) the number of agents). This mapping embeds the dynamics of a non‑reversible chain into a reversible electrical network, preserving the spectrum needed for performance analysis.

Second, because the standard “2‑fuzz” concept for undirected graphs does not extend to directed graphs, the authors propose two novel graph‑theoretic constructs:

• Back‑and‑forth path – a pair of forward and reverse walks between two vertices, capturing both directions of information flow in a directed graph.
• Pivot node – a distinguished vertex through which all back‑and‑forth paths can be routed, playing the role of a “central hub” that restores a form of symmetry.

Using these constructs, Lemma 3.10 establishes a relationship between the effective resistance of the reversibilized conductance matrix (C_{\text{rev}}) and the original directed graph. Consequently, they derive explicit upper and lower bounds on the LQ cost:

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