Russell Merris

Bio: Russell Merris is an academic researcher from California State University. The author has contributed to research in topics: Laplacian matrix & Matrix function. The author has an hindex of 25, co-authored 100 publications receiving 4826 citations. Previous affiliations of Russell Merris include University of California, Santa Barbara & National University of Singapore.

The Laplacian spectrum of a graph

Abstract: Let G be a graph. The Laplacian matrix $L(G) = D(G) - A(G)$ is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects of the spectrum of $L(G)$ are investigated. Particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications of G.

The Laplacian Spectrum of a Graph II

Abstract: Let G be a graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then L(G) = D(G) - A(G) is the Laplacian matrix of G. The first section of this paper is devoted to properties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers. The second section relates the degree sequence and the Laplacian spectrum through majorization. The third section introduces the notion of a d-cluster, using it to bound the multiplicity of d in the spectrum of L(G).

Split graphs

Abstract: The main topics of this article are split graphs, their degree sequences, and the place of these "split partitions" at the top of the partially ordered set of graphic partitions. One application is that threshold covered partitions are unigraphic.

Consensus problems in networks of agents with switching topology and time-delays

Abstract: In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

Consensus and Cooperation in Networked Multi-Agent Systems

Abstract: This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations

Information flow and cooperative control of vehicle formations

Abstract: We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability.

Information consensus in multivehicle cooperative control

Abstract: The purpose of this article is to provide a tutorial overview of information consensus in multivehicle cooperative control. Theoretical results regarding consensus-seeking under both time invariant and dynamically changing communication topologies are summarized. Several specific applications of consensus algorithms to multivehicle coordination are described