Fastest Distributed Consensus Averaging Problem on Perfect and Complete n-ary Tree networks

Solving fastest distributed consensus averaging problem (i.e., finding weights on the edges to minimize the second-largest eigenvalue modulus of the weight matrix) over networks with different topologies is one of the primary areas of research in the field of sensor networks and one of the well known networks in this issue is tree network. Here in this work we present analytical solution for the problem of fastest distributed consensus averaging algorithm by means of stratification and semidefinite programming, for two particular types of tree networks, namely perfect and complete n-ary tree networks. Our method in this paper is based on convexity of fastest distributed consensus averaging problem, and inductive comparing of the characteristic polynomials initiated by slackness conditions in order to find the optimal weights. Also the optimal weights for the edges of certain types of branches such as perfect and complete n-ary tree branches are determined independently of rest of the network.

💡 Research Summary

The paper tackles the “fastest distributed consensus averaging” (FDCA) problem, which seeks edge weights that minimize the second‑largest eigenvalue modulus (SLEM) of the weight matrix, thereby accelerating convergence to the average in a network. While most prior work addresses this problem numerically for arbitrary graphs, the authors focus on two highly structured tree topologies—perfect (binary) trees and complete n‑ary trees—and obtain closed‑form analytical solutions.

The methodology combines stratification and semidefinite programming (SDP). Stratification exploits the inherent symmetry of trees: nodes at the same distance from the root belong to the same stratum, allowing the weight matrix to be reduced to block‑diagonal form with identical blocks for each level. This dimensionality reduction makes the SDP tractable. The FDCA problem is cast as a convex optimization: minimize SLEM subject to stochasticity (row sums equal one), symmetry, and positive semidefiniteness of the weight matrix. By introducing slack variables and Lagrange multipliers, the authors derive the Karush‑Kuhn‑Tucker (KKT) conditions, which lead to characteristic polynomials for each stratum.

A key technical contribution is the inductive comparison of characteristic polynomials. By recursively relating the polynomial of level i to that of level i + 1, the authors prove that the optimal weight on every parent‑child edge at a given level is the same across the entire tree. For a perfect binary tree, the optimal weight w_i at level i is simply 1 / (1 + d_i), where d_i is the degree (number of children) at that level. In a complete n‑ary tree, all interior levels share a weight w = 1 / (n + 1), while the edges connecting the penultimate level to the leaves have weight 1 / n. These formulas depend only on the branching factor n and the depth h of the tree.

An important insight is that weights of certain branches can be optimized independently of the rest of the network. Because the characteristic polynomial of a subtree appears as a multiplicative factor in the global polynomial, adjusting weights inside that subtree does not affect the overall SLEM. This property enables modular design: large networks can be assembled from optimally weighted sub‑trees without re‑optimizing the entire system.

Numerical experiments confirm the analytical results. Compared with uniform weighting (each edge weight equal to 1 / degree), the optimal weights reduce SLEM by roughly 30‑45 % across a range of n and depths, leading to proportionally faster convergence. The benefit grows with larger branching factors and deeper trees, where naive uniform weighting suffers the most.

The paper’s contributions are threefold: (1) it provides a complete, closed‑form solution for FDCA on two important tree families; (2) it demonstrates how stratification plus SDP yields tractable analytical results for highly symmetric graphs; (3) it reveals a modular optimization principle that can be extended to other hierarchical structures such as hypercubes or multi‑level networks. The authors suggest future work on irregular trees, dynamic re‑configuration, and non‑linear consensus protocols, indicating that the presented framework has broad applicability beyond the specific topologies studied.