Distributed Mechanism for Detecting Average Consensus with Maximum-Degree Weights in Bipartite Regular Graphs

Jozef Kenyeres,Martin Kenyeres

doi:10.3390/math9233020

Jozef Kenyeres, Martin Kenyeres

Open Access

https://doi.org/10.3390/math9233020

Copy DOI

Abstract

In recent decades, distributed consensus-based algorithms for data aggregation have been gaining in importance in wireless sensor networks since their implementation as a complementary mechanism can ensure sensor-measured values with high reliability and optimized energy consumption in spite of imprecise sensor readings. In the presented article, we address the average consensus algorithm over bipartite regular graphs, where the application of the maximum-degree weights causes the divergence of the algorithm. We provide a spectral analysis of the algorithm, propose a distributed mechanism to detect whether a graph is bipartite regular, and identify how to reconfigure the algorithm so that the convergence of the average consensus algorithm is guaranteed over bipartite regular graphs. More specifically, we identify in the article that only the largest and the smallest eigenvalues of the weight matrix are located on the unit circle; the sum of all the inner states is preserved at each iteration despite the algorithm divergence; and the inner states oscillate between two values close to the arithmetic means determined by the initial inner states from each disjoint subset. The proposed mechanism utilizes the first-order forward and backward finite-difference of the inner states (more specifically, five conditions are proposed) to detect whether a graph is bipartite regular or not. Subsequently, the mixing parameter of the algorithm can be reconfigured the way it is identified in this study whereby the convergence of the algorithm is ensured in bipartite regular graphs. In the experimental part, we tested our mechanism over randomly generated bipartite regular graphs, random graphs, and random geometric graphs with various parameters, thereby identifying its very high detection rate and proving that the algorithm can estimate the arithmetic mean with high precision (like in error-free scenarios) after the suggested reconfiguration.

Highlights

IntroductionWireless sensor networks (WSNs), a technology applicable to the low-power measurement and controlling, pose a substantial contribution in completing this magnificent technological shift [1,2]
We identify in this subsection that only the largest eigenvalue (λ1 (W(G)) and the smallest eigenvalue (λn (W(G)) of the weight matrix W(G) are on the unit circle, causing that the inner states oscillate between two values close to the arithmetic means determined by the initial inner states from each disjoint set, and the sum of all the inner states is preserved at each iteration despite the divergence of the algorithm
In Theorem 3, we identify that the sum of all the inner states is preserved at each iteration even though the algorithm diverges when AC with MD weights is applied in bipartite regular connected graphs

Summary

Introduction

Wireless sensor networks (WSNs), a technology applicable to the low-power measurement and controlling, pose a substantial contribution in completing this magnificent technological shift [1,2]. This technology is formed by tiny affordable autonomous sensor devices (often referred to as sensor nodes/sensors) able to concurrently sense physical quantities of interest, process the sensed information, and wirelessly exchange messages with each other [3,4]. Figure 1) consists of the following five basic components, namely, [3,5]: Communication device: used to transfer/to receive messages to/from other sensor nodes in the network.

Methods

Findings

Discussion

Conclusion