Expanded Borel Cayley Graphs (Ex-BCGs): A novel communication topology for multi-agent systems

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Abstract

In our early work, we reported Borel Cayley Graphs (BCGs) have the best topological and graph-spectral properties compared to toroidal and diagonal mesh networks and small-world networks. However, BCGs' dependence on generating parameters in size selection and network characteristics has been a challenge in network applications. In this paper, we propose Expanded Borel Cayley Graphs (Ex-BCGs) as a communication topology for multi-agent systems. Ex-BCGs are group-theoretically expanded graphs from BCGs with fixed generating parameters. Ex-BCGs not only conserve constant nodal degree and node labeling scheme but also inherit the communication efficiency of the original BCGs. With experimental results, we show the proposed Ex-BCGs no longer require new generating parameters for new sizes. Moreover, we show resolution of this constraint to produce efficient topological and graph-spectral properties and fast convergence speed when compared to BCGs of similar sizes. We also present the Cut-Through Rewiring (CTR) algorithm as a fault-tolerant methodology against node-failures by regarding pruning nodes as node-failures and utilizing the CTR as a way of recovering connections for neighbors of the pruned nodes in multi-agent systems. The numerical results show that resized networks from Ex-BCGs by the CTR algorithm are more fault-tolerant with robust connectivity, efficient topological properties and faster convergence speed than BCGs.

Introduction

A multi-agent networked system consists of multiple autonomous agents that interact with each other to achieve common objectives and their own objective (Wooldridge and Jennings, 1995, Wooldridge, 2002). The multi-agent systems are used in several applications such as vehicle/spacecraft formation control (Kang and Yeh, 2002), robot trajectory tracking control (Wu et al., 2007) and disaster response (Schurr et al., 2005). As a fundamental aspect of the multi-agent systems, the agents are required to share information by continuously communicating with each other. Thus, the importance of designing efficient communication topology is the primary issue in multi-agent systems (Zhang et al., 2008, Axtell, 2000, Rafiee and Bayen, 2012, Olfati-Saber, 2005). In this paper, we focus on constructing an efficient underlying network topology to enable fast and reliable communication between agents.

Among the methodologies for evaluating communication efficiency of multi-agent systems, consensus problems are studied in many literatures (Saber and Murray, 2003, Olfati-Saber et al., 2007, Olfati-Saber, 2005, Ren et al., 2005). The consensus problems address a cooperative control by making an agreement between agents in multi-agent systems which are involved with frequent system environmental changes. When the agents of the systems asymptotically reach a common value (make an agreement), it is called consensus. Consensus in multi-agent systems is an important research field, since agents need to share information with other agents for coordination such as decision making, formation and position synchronization (Ren et al., 2005). Accordingly, the convergence speed or how fast the agents reach a consensus is an important quantitative metric for the efficiency of a communication topology in multi-agent systems.

Over the last few decades, much effort to formulate an efficient underlying network topology for multi-agent systems has been made. Rafiee and Bayen (2012) present a design methodology to construct the optimal communication graph satisfying (1) the lowest communication cost and (2) the fastest convergence to achieve a consensus. A dynamic topology formulation scheme is proposed in Ye et al. (2012) relying on a composite self-organization mechanism for agents to dynamically detect a change in the underlying network and seek an efficient task allocation.

In particular, graph theory based communication topology formulation has been applied to multi-agent systems for data communication (Zhang et al., 2007, Hatano and Mesbahi, 2005, Yu et al., 2009a). As an effort to apply graph theory in forming a communication topology, Olfati-Saber demonstrates ultrafast convergence speed of small-world networks in multi-agent systems (Olfati-Saber, 2005). An extended work to construct an effective communication topology based on small-world networks by optimizing the number of long distant (large number of hops) edges and the agents improves the communication efficiency with enhanced diameter and average path length (Wu and Wang, 2009).

In previous work, we reported that Borel Cayley Graphs (BCGs), a family of group theoretic and pseudo-random graphs, are a promising underlying topology for network applications. BCGs are symmetric and dense networks, in particular, known as the densest logical topology with a limited constant nodal degree (Chudnovsky et al., 1988). From topological and graph-spectral properties analysis (Yu et al., 2009a, Yu et al., 2009b), BCGs were shown to have the best performance when compared to toroidal mesh and diagonal mesh networks and small-world networks over a variety of sizes. Specifically, the experimental result of diameter, average path length and algebraic connectivity demonstrates the effectiveness of BCGs as a communication topology in network applications. Moreover, the ultrafast convergence speed of the average consensus protocol highlights remarkable capabilities of BCGs when used as the communication topology of multi-agent networked systems (Yu et al., 2010a).

Despite these promising properties, BCGs have limitations when used in real network applications. Theoretically, the selection of BCG's size is determined by two constrained generating parameters (i.e., a prime number and its factor). Thus, BCG of an arbitrary network size is not always feasible. Although BCG resizing algorithm is introduced (Yu et al., 2010a), it is impossible to expand the size of BCGs beyond the predetermined size without reselecting different parameters. Moreover, changing the generating parameters for resizing may result in unpredictable topological and spectral properties and consensus performance.1 This inconsistent performance is due to the intensive dependence of BCG network characteristics on the generating parameters (Yu et al., 2009b). Although the BCG random expansion algorithm increases the size of BCGs without performance degradation (Yu et al., 2010b), nodes and edges are randomly added and this makes network control more difficult.

In this paper, we propose Expanded BCGs (Ex-BCGs) that are deterministically ed graphs from BCGs without changing the generating parameters. Our Ex-BCGs consist of a node ID space expansion (Expanded Borel subgroup) and well defined connection rules between the subgroup elements. Ex-BCGs preserve the constant nodal degree and node labeling scheme independent of size expansion. As a result, we eliminate the penalty occurred from changing the generating parameters and preserve the communication efficiency of the original BCGs. The resulting Ex-BCGs display more predictable and efficient topological/spectral properties and fast convergence speed of the average consensus protocol.

To improve size flexibility of Ex-BCGs, we apply the Cut-Through Rewiring (CTR) algorithm, which was originally introduced to resize BCGs (Yu et al., 2010a). The CTR algorithm of Ex-BCGs is capable of resizing Ex-BCGs to an arbitrary size while preserving a constant nodal degree and connection rule. Moreover, we propose using the CTR algorithm as a topological fault-tolerant methodology. We regard pruned nodes as failed nodes and rewiring as a connection recovering scheme for multi-agent systems against node-failures. We compare the networks resized from Ex-BCGs with similarly resized BCGs. The numerical results proved that the target Ex-BCGs have smaller diameter, shorter average path length and faster convergence speed of the average consensus protocol while remaining fully connected even after resizing to its 10%40% size. We measure the size flexibility and fault-tolerance of Ex-BCGs and BCGs of similar sizes by reducing the size until the network is disconnected. The simulation result demonstrates that Ex-BCGs have more robust connectivity, efficient topological properties and faster convergence speed of the average consensus protocol.

This paper is organized as follows. In Section 2, we review complex network topologies for multi-agent systems: regular, random and small-world network topologies. And we present the average consensus protocol used to evaluate the communication efficiency of our Ex-BCGs and benchmark topologies. Then, Borel Cayley graphs and its resizing algorithm are provided. Section 3 introduces the metrics used to evaluate our graphs and the benchmark network topologies for comparison study. Section 4 presents Ex-BCGs and investigates its topological and spectral properties. In Section 4, Ex-BCGs are also compared with BCGs of similar sizes. In Section 5, we explore the CTR algorithm in Ex-BCGs to enhance size flexibility and fault-tolerance of the resized Ex-BCGs. Section 6 provides our conclusions.

Section snippets

Background

In this section, we review the complex network topologies for multi-agent systems, the average consensus protocol, Borel Cayley graphs and its resizing algorithm. Table 1 summarizes the notations used throughout the paper.

Benchmark network topologies

Yu et al. (2010a) report that BCGs have the smallest diameter, shortest average path length, largest algebraic connectivity and fastest convergence speed of the average consensus protocol in comparison with toroidal and diagonal mesh networks and small-world networks in a variety of sizes. By comparing with BCGs of similar sizes, we can make a relative comparison to toroidal and diagonal mesh networks and small-world networks of similar sizes. Thus, we select BCGs of close size to that of Ex

Expanded Borel Cayley graphs

In this section, we show how to deterministically expand BCGs and highlight its properties beneficial to multi-agent systems network applications.

Resizing & fault-tolerant methodology for Ex-BCGs

In the previous work (Yu et al., 2010a, Yu et al., 2010c, Kim et al., 2011), the Cut-Through Rewiring (CTR) algorithm was used to solve the size inflexibility of BCGs and establishing wireless connections in WSNs applications. We identified the feasibility of CTR algorithm in Ex-BCGs in Section 4.2. In other words, Ex-BCGs can be resized to an arbitrary size while preserving a constant degree.

Conclusions and discussion

In this paper, we proposed Ex-BCGs to formulate an efficient and reliable communication topology for multi-agent systems. Ex-BCGs are based on the node ID space expansion and redefined connection rule without the need to change the generating parameters. Ex-BCGs preserve a constant nodal degree and a node labeling scheme; and retain the communication efficiency of original BCGs. As a result, Ex-BCGs efficiently resolve the dependence of BCGs on the generating parameters. Through the extensive

Acknowledgment

The authors are partially supported by the National Science Foundation under Grant no. CNS 0829656 and IIP 0917956. Any opinions, findings and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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