Fastest Distributed Consensus on Path Network

Providing an analytical solution for the problem of finding Fastest Distributed Consensus (FDC) is one of the challenging problems in the field of sensor networks. Most of the methods proposed so far deal with the FDC averaging algorithm problem by numerical convex optimization methods and in general no closed-form solution for finding FDC has been offered up to now except in [3] where the conjectured answer for path has been proved. Here in this work we present an analytical solution for the problem of Fastest Distributed Consensus for the Path network using semidefinite programming particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions.

💡 Research Summary

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The paper addresses the problem of fastest distributed consensus (FDC) on a path‑shaped sensor network, seeking the set of edge weights that yields the quickest convergence of the average‑consensus algorithm. The authors begin by formulating the consensus iteration as x(t + 1) = W x(t), where W is a symmetric tridiagonal weight matrix reflecting the network topology. The convergence rate is governed by the second‑largest eigenvalue in magnitude (the SLEM) of W; minimizing this eigenvalue is equivalent to solving a convex optimization problem.

Previous work solved the FDC problem for a path only through numerical semidefinite programming (SDP) and could not provide a closed‑form solution. In contrast, this study derives an exact analytical expression for the optimal weights by exploiting the SDP duality framework and the complementary slackness conditions. The authors first rewrite the SDP in standard primal form, introduce dual variables (a symmetric matrix Z and a scalar y), and derive the complementary slackness condition Z·(W − (1/n)11ᵀ) = 0 together with Z ⪰ 0 and Tr(Z)=1.

To satisfy these conditions analytically, the weight matrix W is expanded in a basis of elementary tridiagonal matrices {B_k}, each representing a single edge in the path. The coefficients α_k of this expansion become the unknown weights. By substituting the expansion into the slackness equations, the authors obtain a set of linear relations among the α_k that can be interpreted as equalities between the coefficients of characteristic polynomials of certain matrix blocks.

A key insight is that these relations can be solved recursively: the end‑edge weights are shown to be proportional to cos θ, while interior weights follow a sinusoidal pattern. Defining θ = π/(n + 1) (where n is the number of nodes), the optimal weights are expressed as

 w_i =