[Paper Review] Fastest Distributed Consensus Problem on Branches of an Arbitrary Connected Sensor Network

Saber Jafarizadeh, Abbas Jamalipour|arXiv (Cornell University)|Apr 28, 2010

Distributed Control Multi-Agent Systems27 references3 citations

TL;DR

This paper presents an analytical solution for determining optimal weights in distributed consensus algorithms on specific branch structures within arbitrary connected sensor networks, independent of the global topology. By leveraging graph stratification and Semidefinite Programming (SDP) with slackness condition analysis, it derives optimal convergence rates for these branches, validated through numerical examples and characteristic polynomial comparisons.

ABSTRACT

This paper studies the fastest distributed consensus averaging problem on branches of an arbitrary connected sensor network. In the previous works full knowledge about the sensor network's connectivity topology was required for determining the optimal weights and convergence rate of distributed consensus averaging algorithm over the network. Here in this work for the first time, the optimal weights are determined analytically for the edges of certain types of branches, independent of the rest of network. The solution procedure consists of stratification of associated connectivity graph of the branches 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 results are provided to confirm the optimality of the obtained weights.

Motivation & Objective

• To address the challenge of achieving the fastest convergence in distributed consensus algorithms on arbitrary sensor network topologies.
• To determine optimal edge weights for specific branch structures without requiring full knowledge of the global network topology.
• To develop a method that enables analytical derivation of optimal weights for branches, independent of the rest of the network.
• To validate the optimality of derived weights through numerical examples and characteristic polynomial analysis.

Proposed method

• Applying graph stratification to decompose the connectivity structure of the branch into hierarchical levels.
• Formulating the consensus problem as a Semidefinite Programming (SDP) optimization problem to minimize convergence time.
• Solving the SDP using slackness conditions to derive analytical expressions for optimal weights.
• Using inductive comparison of characteristic polynomials derived from slackness conditions to determine weight values.
• Validating the solution through numerical simulations and graphical analysis of convergence behavior.
• Ensuring the solution is topology-independent by isolating branch structures from the broader network.

Experimental results

Research questions

• RQ1 Can optimal consensus weights be derived analytically for specific branch structures in sensor networks without full network knowledge?
• RQ2 What mathematical framework enables the determination of fastest convergence rates for distributed consensus on arbitrary branches?
• RQ3 How do slackness conditions in SDP contribute to deriving optimal weights in consensus algorithms?
• RQ4 To what extent can branch-level optimization improve overall consensus convergence speed in large-scale sensor networks?
• RQ5 Can characteristic polynomial comparisons reliably verify the optimality of derived weights in consensus systems?

Key findings

• The proposed method successfully derives optimal weights for specific branch types using only local network information, independent of the global topology.
• The solution achieves the fastest possible convergence rate for the given branch structures by solving the SDP with slackness conditions.
• Numerical results confirm the optimality of the derived weights through convergence rate comparisons and stability analysis.
• The use of characteristic polynomial inductive comparison provides a rigorous analytical path to weight determination.
• The approach is scalable to various branch configurations and maintains optimality across tested examples.
• The method reduces reliance on global network knowledge, enabling faster and more efficient distributed consensus in large sensor networks.