Brief paperStability analysis of diagonally equipotent matrices☆
Introduction
At the boundary of the set of diagonally dominant matrices lies a special class of matrices which we call diagonally equipotent, meaning that for them the diagonal dominance is never strict in any coordinate. This class includes as special cases the Laplacian matrices of directed graphs used in studying the consensus problem (Mesbahi and Egerstedt, 2010, Ren et al., 2007) but also those that can be obtained when the consensus problem is relaxed to include competing interactions (modeled as negative weights of the adjacency matrix), see Altafini (2013). It also occurs for example in chemical reaction network theory, where biochemical reactions are represented as mass-action ODEs (Jayawardhana, Rao, & van der Schaft, 2012) (although in this case only the positive orthant is usually of interest). More generally, it occurs whenever in a linear system the off-diagonal terms (of any sign) are exactly compensated (in absolute value) by the diagonal entry of each row.
The scope of this paper is to show that the class of diagonally equipotent matrices admits a complete classification for what concerns their stability properties. In particular, irreducible diagonally equipotent matrices with negative diagonal entries are -matrices (Berman and Plemmons, 1994, Horn and Johnson, 1991), and as such they are at least diagonally semistable (Hershkowitz, 1992, Hershkowitz and Schneider, 1985). For them, nonsingularity corresponds to asymptotic stability, while singularity corresponds to critical stability. In order to discriminate the two cases, it is not possible to use any of the standard criteria for diagonally dominant matrices (Hershkowitz, 1992, Huang, 1998, Kaszkurewicz and Bhaya, 2000), nor arguments inspired by Geršgorin theorem (Horn & Johnson, 1985). In fact, diagonally equipotent matrices have always singular comparison matrices, because by construction the latter have always zero row sums. However, this does not imply that diagonally equipotent matrices are singular. In Kolotilina (2003) a necessary and sufficient condition for nonsingularity of such matrices is provided. Combining this condition with diagonal stability, for the case of negative diagonal entries (most common in the applications mentioned above) we show in this paper that diagonally equipotent matrices are asymptotically stable if and only if all the cycles of formed by their graph have positive sign (i.e., have an even number of negative edges).
Such a condition on the sign of the cycles appears under different names in different domains. For example it is used in linear algebra (Engel and Schneider, 1973, Fiedler and Ptak, 1969), in the theory of signed graphs (Zaslavsky, 1982), in systems and control (Sezer and Siljak, 1994, Willems, 1976) and in the theory of monotone dynamical systems (Smith, 1988). It is also used in different contexts in other disciplines, spanning from social network theory (where it is known under the name of “structural balance”, see Cartwright & Harary, 1956) to statistical physics (where it has to do with the presence or less of “frustration” in spin glasses Binder & Young, 1986).
The use of this cycle sign property in the context of stability analysis is however new. Most remarkably, for diagonally equipotent matrices, asymptotic stability does not depend on the numerical value of the entries of a matrix but only on their sign. As such it can be considered a “qualitative” condition (Brualdi and Shader, 1995, Maybee and Quirk, 1969), i.e., it determines the (singularity and stability) properties of the entire class of matrices having the same sign pattern, under the additional constraint of diagonal equipotence. When compared to the classical results for sign-pattern matrices (Brualdi and Shader, 1995, Hall and Li, 2006, Maybee and Quirk, 1969), the resulting qualitative conditions are remarkably different. For example, if for unconstrained matrices sign nonsingularity (i.e., nonsingularity of the class of matrices carrying a given sign pattern) corresponds to having all negative cycles on the graph of the matrix, for diagonally equipotent matrices this reduces to at least one negative cycle of . The same condition is necessary and sufficient for qualitative stability (i.e., asymptotic stability of the entire class of matrices carrying a given sign pattern), under the constraint of diagonal equipotence.
Section snippets
Graphs associated to a matrix
Given a matrix , consider the directed graph of where is the set of nodes, is the set of directed edges (we use the convention that is the head of the arrow and its tail) and is its weighted adjacency matrix. A directed path is a sequence of edges in , its length is the number of nodes it touches (i.e., ), and its sign is the sign of . The paths are always
Hurwitz stability of diagonally equipotent matrices
Formally, the problem investigated in the paper is the following. Problem 1 Determine the stability character of irreducible and with negative diagonal entries.
It follows from Proposition 1 that a matrix irreducible with negative diagonal entries is at least critically stable. However, as the following example shows, may or may not be singular. Example Consider the two matrices Even if are equimodular, their spectra are
Sign pattern properties of diagonally equipotent matrices
Recall (see e.g. Brualdi & Shader, 1995) that a matrix determines a qualitative class of all matrices in having the same sign pattern as , denoted . A square matrix is sign nonsingular if every matrix in is nonsingular, and sign singular if every matrix in is singular. It is said to be qualitatively stable if every matrix in is Hurwitz stable. A classical result concerning sign nonsingularity is the following. Theorem 6 Considerwith. Thenis signBrualdi & Shader, 1995, Theorem 3.2.1
Conclusion
Diagonally equipotent matrices admit a complete classification for what concerns the stability property. While the classical arguments of diagonal dominance and of location of the Geršgorin disks are inadequate in determining the asymptotically stable cases, a purely graphical criterion, the sign of the cycles of the graph associated with the given matrix, provides instead necessary and sufficient conditions. These conditions are “qualitative” i.e., they hold for the entire class of matrices
Acknowledgments
The author would like to thank the reviewers for constructive criticisms, and in particular for pointing out the Ref. Kolotilina (2003).
Claudio Altafini received a Master degree (“Laurea”) in Electrical Engineering for the University of Padova, Italy, in 1996 and a Ph.D. in Optimization and Systems Theory from the Royal Institute of Technology, Stockholm, Sweden in 2001.
He then moved to the International School for Advanced Studies (SISSA) in Trieste, Italy, where he is now an Assistant Professor. Dr. Altafini serves as an Associate Editor for the IEEE Transactions on Automatic Control.
His research interests are in the areas of
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Claudio Altafini received a Master degree (“Laurea”) in Electrical Engineering for the University of Padova, Italy, in 1996 and a Ph.D. in Optimization and Systems Theory from the Royal Institute of Technology, Stockholm, Sweden in 2001.
He then moved to the International School for Advanced Studies (SISSA) in Trieste, Italy, where he is now an Assistant Professor. Dr. Altafini serves as an Associate Editor for the IEEE Transactions on Automatic Control.
His research interests are in the areas of nonlinear systems analysis and control, with applications to quantum mechanics, systems biology and complex networks.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Constantino M. Lagoa under the direction of Editor Roberto Tempo.
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