The Deformed Consensus Protocol

International audienceThis paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the communication graph with the so-called deformed Laplacian. The deformed Laplacian is a second-degree matrix polynomial in the real variable s which reduces to the standard Laplacian for s equal to unity. The stability properties of the ensuing deformed consensus protocol are studied in terms of parameter s for some special families of undirected and directed graphs, and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems. Examples and simulation results are provided to illustrate our theoretical findings

The Second-order Parametric Consensus Protocol

International audienceIn this paper we extend the parametric consensus protocol recently introduced by the author, to more realistic agents modeled as double integrators and interacting over an undirected communication network. The stability properties of the new protocol in terms of the real parameter "s" are studied for some relevant graph topologies, and the connection with the notion of bipartite consensus is highlighted. The theory is illustrated with the help of two worked examples, dealing with the coordination of a team of quadrotor UAVs and with cooperative temperature measurement in an indoor environment

The Second-order Parametric Consensus Protocol

International audienceIn this paper we extend the parametric consensus protocol recently introduced by the author, to more realistic agents modeled as double integrators and interacting over an undirected communication network. The stability properties of the new protocol in terms of the real parameter "s" are studied for some relevant graph topologies, and the connection with the notion of bipartite consensus is highlighted. The theory is illustrated with the help of two worked examples, dealing with the coordination of a team of quadrotor UAVs and with cooperative temperature measurement in an indoor environment

The deformed graph Laplacian and its applications to network centrality analysis

We introduce and study a new network centrality measure based on the concept of nonbacktracking walks; that is, walks not containing subsequences of the form uvu where u and v are any distinct connected vertices of the underlying graph . We argue that this feature can yield more meaningful rankings than traditional walk-based cent rality measures. We show that the resulting Katz-style centrality measure may be computed via the so -called deformed graph Laplacian?a quadratic matrix polynomial that can be associated with any graph. By proving a range of new results about this matrix polynomial, we gain insights into the behavior of the algorithm with respect to its Katz-like parameter. The results also inform implementation issues. In particular we show that, in an appropriate limit, the new measure coincide s with the nonbacktracking version of eigenvector centrality introduced by Martin, Zhang and New man in 2014. Rigorous analysis on star and star-like networks illustrates the benefits of the new approach, and further results are given on real networks

Generating functions of non-backtracking walks on weighted digraphs: radius of convergence and Ihara's theorem

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In [P. Grindrod, D. J. Higham, V. Noferini, The deformed graph Laplacian and its application to network centrality analysis, SIAM J. Matrix Anal. Appl. 39(1), 310--341, 2018], the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed unweighted graphs, showing that it depends on the number of cycles in the undirectization of the graph. For weighted graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound. Finally, we consider also backtracking-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case

Forest Generating Functions of Directed Graphs

A spanning forest polynomial is a multivariate generating function whose variables are indexed over both the vertex and edge sets of a given directed graph. In this thesis, we establish a general framework to study spanning forest polynomials, associating them with a generalized Laplacian matrix and studying its properties. We introduce a novel proof of the famous matrix-tree theorem and show how this extends to a parametric generalization of the all-minors matrix-forest theorem. As an application, we derive explicit formulas for the recently introduced class of directed threshold graphs. We prove that multivariate forest polynomials are, in general, irreducible and we define a number of specializations that may be compactly expressed in terms of various factors. A specialization in this context is an identification of some of the variables of the polynomial, for example evaluating f(x,y,z) as f(x,x,z). This allows us to derive results that generalize and extend many known properties of the traditional Laplacian matrix in algebraic graph theory. We analyze the connection between the matrix algebra generated by the traditional Laplacian matrix and certain matrices of forest polynomials. Using this analysis, we derive explicit formulas for these matrices in the cases of Cartesian products of complete graphs and de Bruijn graphs. More generally, we derive an explicit formula relating spanning forest polynomials of a graph to the numbers of D-lazy walks in the graph. These are walks that may choose to remain at a given vertex if that vertex is not of maximum degree D. This leads us to the study of externally equitable partitions (EEPs), which are objects of recent interest in the control theory literature. We prove that for graphs with EEPs satisfying an additional criteria, the specialized forest polynomials may be factored into a product of forest polynomials of related quotient graphs. We apply this theorem to complete multipartite graphs, hypercube graphs, directed line graphs, and others

On the properties of the deformed consensus protocol

Abstract—This paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the undirected communication graph with the so-called deformed Laplacian. The deformed Laplacian is a second-degree matrix polynomial in the real variable s which reduces to the standard Laplacian for s equal to unity. The sta-bility properties of the ensuing deformed consensus protocol are studied in terms of parameter s for some special families of undirected graphs, and for graphs of arbitrary topology by leveraging the spectral theory of quadratic eigenvalue problems. Examples and simulation results are provided to illustrate our theoretical findings. I