The Impact of Multi-group Multi-layer Network Structure on the Performance of Distributed Consensus Building Strategies

Abstract

We consider a structural approach to the consensus building problem in multi-group multi-layer (MGML) distributed sensor networks (DSNs) common in many natural and engineering applications. From among the possible network structures, we focus on bipartite graph structure as it represents a typical MGML structure and has a wide applicability in the real world. We establish exact conditions for consensus and derive a precise relationship between the consensus value and the degree distribution of nodes in a bipartite MGML DSN. We also demonstrate that for subclasses of connectivity patterns, convergence time and simple characteristics of network topology can be captured by explicit algebra. Direct inference of the convergence behavior of consensus strategies from MGML DSN structure is the main contribution of this paper. The insights gained from our analysis facilitate the design and development of large-scale DSNs that meet specific performance criteria. INTRODUCTIONConsensus problem widely appears in natural and engineering applications and has received extensive study in a multitude of fields including sensor networking, distributed computing, and decentralized control [1-5] among others. However, it has not been investigated extensively in networks with multi-group multi-layer (MGML) communication structures; that is, networks composed of multiple groups, in each of which some nodes are designated as leaders and given the responsibility of communicating with other groups through certain layered communication schemes. Such communication structure is commonly observed in nature and has tremendous potential for engineered systems. For instance, this structure is typically observed in bird flocking as studied in, for example, [6,7]. Similarly, in distributed sensor networks (DSNs), properly designed MGML communication structure is envisioned to be more secure and energy efficient [8,9].Network topology has been known to be crucial to the performance of consensus strategies, particularly to convergence time (see e.g., [10,11]), which impacts communication overhead and network lifetime in DSN applications. In designing MGML network structures for consensus building, one question is whether there exists a quantitative relationship between the network structure and its performance. This question is particularly important in large-scale networks, where scalability is of critical concern. We address this issue by studying the network design problem for some common MGML structures. Examples of such structures are illustrated in Figure 1. Bipartite graph structure (Figure 1(a)), which captures a wide arrange of 2-layer lead-follower communication topology, is widely used in coding theory [12,13] and has been recently adopted As is well known and can be found in, for example, [1,3], the convergence time (i.e., the number of iterations such that the difference between sensor value and final consensus value is within ı ‡ Ergodic implies both recurrent and aperiodic. T 1 3 1 3 1 3 /.I 3 3˝0.25 1 4 /xOEk. Explicit results charac...

Majorisations for the eigenvectors of graph-adjacency matrices

“…reach a consensus) and therefore to reduce energy consumption (e.g. Wan, Namuduri, Akula, & Varanasi, 2012;Xiao & Boyd, 2004). While there are some nascent efforts on designing a network's dynamics, much remains to be done -especially in the realistic circumstance that a network's topology can only be partially designed or modified.…”

Impact of mobility on convergence rate in a wireless sensor network

“…The system matrix shown above is a stochastic matrix, similar to the one presented in [5] and we shall refer to it as the "new" system matrix for the sake of differentiation. The new system matrix can capture the state of a disconnected network which was not the case in [5] and it is calculated by analyzing the graph directly instead of using routing matrix . If [0] represents the initial sensor values, the sensor data [ ] at iteration can be represented as follows:…”

“…This paper makes use of the concept of Virtual Fusion Centers (VFCs) as presented in [5]. VFC's are few special nodes in a graph which perform decision making calculations on information gathered from their immediate neighbors.…”

“…Olfati-Saber et al studied the consensus building in different topologies of directed and undirected networks with planar graph representation and established a direct connection between the algebraic connectivity (eigenvalue) and the performance of linear consensus protocol [4]. Wan et al established the conditions for convergence in a network represented by a two layered hierarchical graph [5].…”

“…A modified version of system matrix presented in [5] is used in this analysis. A strategy to choose a set of nodes, which can act as effective mobile agents and consequently increase the rate of convergence, is presented.…”