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Constructing Stochastic Matrices for Weighted Averaging in Gossip Networks

Erkan Bayram, Mohamed-Ali Belabbas

TL;DR

We address the problem of constructing gossip matrices that yield multiple consensus clusters with a prescribed averaging weight on a given topology. The authors propose to use holonomy concepts and derived graphs to test admissible partitions and to design local stochastic matrices whose infinite product converges to a finite limit set determined by the weight vector $w$ and partition $\pi$. An explicit algorithm constructs rate matrices $\mathcal{B}^{ij}_{kl}(w)$ and can insert permutation blocks to realize left eigenvector $w$, ensuring $w$-holonomy on a $2$-edge-connected simple graph $G$. This work solves the open problem of realizing gossip matrices for a given network and partition, enabling efficient distributed consensus with predefined clustering and weights, with potential applications in decentralized optimization and federated learning.

Abstract

The convergence of the gossip process has been extensively studied; however, algorithms that generate a set of stochastic matrices, the infinite product of which converges to a rank-one matrix determined by a given weight vector, have been less explored. In this work, we propose an algorithm for constructing (local) stochastic matrices based on a given gossip network topology and a set of weights for averaging across different consensus clusters, ensuring that the gossip process converges to a finite limit set.

Constructing Stochastic Matrices for Weighted Averaging in Gossip Networks

TL;DR

We address the problem of constructing gossip matrices that yield multiple consensus clusters with a prescribed averaging weight on a given topology. The authors propose to use holonomy concepts and derived graphs to test admissible partitions and to design local stochastic matrices whose infinite product converges to a finite limit set determined by the weight vector and partition . An explicit algorithm constructs rate matrices and can insert permutation blocks to realize left eigenvector , ensuring -holonomy on a -edge-connected simple graph . This work solves the open problem of realizing gossip matrices for a given network and partition, enabling efficient distributed consensus with predefined clustering and weights, with potential applications in decentralized optimization and federated learning.

Abstract

The convergence of the gossip process has been extensively studied; however, algorithms that generate a set of stochastic matrices, the infinite product of which converges to a rank-one matrix determined by a given weight vector, have been less explored. In this work, we propose an algorithm for constructing (local) stochastic matrices based on a given gossip network topology and a set of weights for averaging across different consensus clusters, ensuring that the gossip process converges to a finite limit set.

Paper Structure

This paper contains 5 sections, 3 theorems, 23 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Proof of Main Results
  5. Conclusion

Key Result

Lemma 1

If $P$ is a permutation matrix with order ${|\pi_0|}$ such that $w_0 P \neq w_0$ and $M$ is a stochastic matrix such that $\tilde{w} M=\tilde{w}$, that is, it holds that (up to labelling): then the $w$-order of the cycle $C$ is strictly greater than $1$.

Figures (2)

  • Figure 1: The graph $\vec{G}$
  • Figure 2: Derived graphs $\mathcal{D}^G(\pi_0)$, $\mathcal{D}^G(\pi_1)$, and $\mathcal{D}^G(\pi_2)$

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 3.1: Derived Graph $\mathcal{D}^G(\cdot)$ Function
  • Example 3.1
  • Lemma 1
  • Remark 1
  • Definition 3.2: Rate Matrix Set on $w$
  • Remark 2
  • Theorem 1: Correctness of Algorithm \ref{['alg:state_transition']}
  • Example 3.2: Cont.
  • proof : Proof of Lemma \ref{['lem:P_c_block']}
  • ...and 3 more