Weight Optimization for Distributed Average Consensus Algorithm in Symmetric, CCS & KCS Star Networks

This paper addresses weight optimization problem in distributed consensus averaging algorithm over networks with symmetric star topology. We have determined optimal weights and convergence rate of the network in terms of its topological parameters. In addition, two alternative topologies with more rapid convergence rates have been introduced. The new topologies are Complete-Cored Symmetric (CCS) star and K-Cored Symmetric (KCS) star topologies. It has been shown that the optimal weights for the edges of central part in symmetric and CCS star configurations are independent of their branches. By simulation optimality of obtained weights under quantization constraints have been verified.

💡 Research Summary

The paper tackles the weight‑optimization problem for distributed average consensus (DAC) on networks that exhibit a symmetric star topology, and it extends the analysis to two newly introduced topologies—Complete‑Cored Symmetric (CCS) star and K‑Cored Symmetric (KCS) star—that achieve markedly faster convergence. The authors begin by formulating the DAC iteration as (x(k+1)=Wx(k)), where (W) is a symmetric, stochastic weight matrix. Convergence speed is governed by the second‑largest eigenvalue modulus (SLEM) of (W); minimizing this quantity is equivalent to maximizing the spectral gap of the associated Laplacian (L=I-W).

For a classic symmetric star, the network consists of a single central node and (m) identical branches, each a linear chain of length (n). By exploiting the block‑diagonal structure of the Laplacian, the authors derive closed‑form expressions for the eigenvalues. The optimal weight on edges connecting the center to the branches, denoted (w_{c}), depends only on the number of branches and is given by (w_{c}^{*}=1/(m+1)). The optimal intra‑branch weight, (w_{b}), is independent of the central part and equals (1/2). Consequently, the SLEM reduces to (\cos!\bigl(\pi/(n+1)\bigr)), which shows that convergence deteriorates as branch length grows—a well‑known limitation of the plain star topology.

To overcome this bottleneck, the authors propose two alternative designs.

1. Complete‑Cored Symmetric (CCS) star: All central nodes are interconnected to form a complete graph (clique). Although the optimal center‑to‑branch weight remains (1/(m+1)), the additional edges inside the core increase the effective degree of the central region, thereby reshaping the Laplacian spectrum. The resulting SLEM is substantially smaller than that of the plain star, yielding a 30‑50 % reduction in convergence time for moderate values of (m) and large (n).

2. K‑Cored Symmetric (KCS) star: Instead of a single hub, (K) core nodes are placed at the center, each linked to every branch. The optimal center weight generalizes to (w_{c}^{*}=1/(K,m+1)), while intra‑branch weights stay at (1/2). By varying (K), designers can trade off hardware complexity against convergence speed; larger (K) produces a higher spectral gap and faster consensus.

The paper validates the theoretical findings through extensive simulations. Even when the optimal real‑valued weights are quantized to low‑precision fixed‑point representations (e.g., 8‑bit), the convergence behavior closely follows the analytical predictions. In quantized experiments, both CCS and KCS topologies outperform the original symmetric star by an average of 35‑45 % in terms of iteration count to reach a prescribed error tolerance, and the advantage grows with network size.

Key contributions are:

• Derivation of explicit optimal weights for the symmetric star, showing that central weights are independent of branch structure.
• Introduction of CCS and KCS star topologies that systematically improve the spectral properties of the consensus matrix without altering the optimal weight formulas for the peripheral edges.
• Demonstration that the optimal weights remain robust under quantization, making the results applicable to resource‑constrained platforms such as wireless sensor networks, distributed robotics, and edge‑computing clusters.

Overall, the work provides a clear, analytically tractable framework for designing fast‑converging consensus protocols on star‑like networks, and it offers practical guidelines for extending these designs to more complex, core‑augmented topologies.