$G$ Method and Finite-Time Consensus

TL;DR

This paper extends the $G$ method to analyze finite-time consensus in the DeGroot model and distributed systems, providing new results on reaching consensus and distributed averaging in finite steps across various graph structures.

Contribution

The paper introduces an extended $G$ method for finite-time consensus analysis in the DeGroot model and distributed systems, including new results and specific graph-based averaging procedures.

Findings

Finite-time consensus results for homogeneous and nonhomogeneous DeGroot models.

Discovery of a subset/subgroup property in a special case of the DeGroot model.

Distributed averaging in $m$ steps on graphs with $2^m$ vertices containing an $m$-cube subgraph.

Abstract

We give an extension of the $G$ method, with results, the extension and results being partly suggested by the finite Markov chains and specially by the finite-time consensus problem for the DeGroot model and that for the DeGroot model on distributed systems. For the (homogeneous and nonhomogeneous) DeGroot model, using the $G$ method, a result for reaching a partial or total consensus in a finite time is given. Further, we consider a special submodel/case of the DeGroot model, with examples and comments -- a subset/subgroup property is discovered. For the DeGroot model on distributed systems, using the $G$ method too, we have a result for reaching a partial or total (distributed) consensus in a finite time similar to that for the DeGroot model for reaching a partial or total consensus in a finite time. Then we show that for any connected graph having $2^{m}$ vertices, $m\geq 1,$ and a spanning subgraph isomorphic to the $m$-cube graph, distributed averaging is performed in $m$ steps -- this result can be extended -- research work -- for any graph with $n_{1}n_{2}...n_{t}$ vertices under certain conditions, where $t,$ $n_{1},n_{2},...,n_{t}\geq 2,$ and, in this case, distributed averaging is performed in $t$ steps.