[Paper Review] Optimal Weights of Certain Branches of an Arbitrary Connected Network for Fastest Distributed Consensus Averaging Problem

Saber Jafarizadeh|arXiv (Cornell University)|Apr 29, 2010

Distributed Control Multi-Agent Systems3 citations

TL;DR

This paper presents an analytical method to determine optimal weights for specific network branches—path, lollipop, semi-complete, and ladder—regardless of the rest of the network, using graph stratification and semidefinite programming (SDP) with slackness condition analysis. The key contribution is the derivation of optimal weights that maximize convergence speed in distributed consensus averaging, validated through numerical simulations.

ABSTRACT

Solving fastest distributed consensus averaging problem over networks with different topologies has been an active area of research for a number of years. The main purpose of distributed consensus averaging is to compute the average of the initial values, via a distributed algorithm, in which the nodes only communicate with their neighbors. In the previous works full knowledge about the network's topology was required for finding optimal weights and convergence rate of network, but here in this work for the first time the optimal weights are determined analytically for the edges of certain types of branches, namely path branch, lollipop branch, semi-complete Branch and Ladder branch independent of the rest of network. The solution procedure consists of stratification of associated connectivity graph of branch and Semidefinite Programming (SDP), particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Several Examples and numerical simulations are provided to confirm the validity of the obtained results.

Motivation & Objective

• To address the challenge of achieving the fastest convergence in distributed consensus averaging over arbitrary connected networks.
• To determine optimal edge weights for specific branch types without requiring full knowledge of the entire network topology.
• To develop an analytical framework that enables weight optimization for path, lollipop, semi-complete, and ladder branches.
• To validate the proposed method through numerical simulations demonstrating convergence speed improvements.

Proposed method

• Applying graph stratification to decompose the connectivity graph of each branch type into analyzable structural components.
• Formulating the weight optimization problem as a semidefinite program (SDP) to ensure convergence speed maximization.
• Solving the SDP by analyzing slackness conditions to derive closed-form expressions for optimal weights.
• Using inductive comparison of characteristic polynomials derived from slackness conditions to determine weight distributions.
• Validating the analytical results through numerical simulations on representative network topologies.

Experimental results

Research questions

• RQ1 Can optimal weights for specific network branch types be derived independently of the rest of the network?
• RQ2 What analytical method enables the determination of optimal weights for path, lollipop, semi-complete, and ladder branches?
• RQ3 How do slackness conditions in semidefinite programming contribute to deriving optimal weights?
• RQ4 To what extent do the derived weights improve convergence speed in distributed consensus averaging?

Key findings

• Optimal weights for path, lollipop, semi-complete, and ladder branches are derived analytically without requiring knowledge of the global network structure.
• The method relies on SDP formulation and slackness condition analysis to systematically determine optimal weight distributions.
• Characteristic polynomial induction enables inductive comparison to identify optimal weight configurations.
• Numerical simulations confirm the validity and effectiveness of the derived optimal weights in accelerating consensus convergence.