Restrictions and stability of time-delayed dynamical networks

Linear Algebra and its Applications

et al. 2020

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices. *

Section: Introductionmentioning

confidence: 99%

Characterizing cospectral vertices via isospectral reduction

Kempton

Sinkovic

Linear Algebra and its Applications

et al. 2020

“…Consider the graph G inFigure 1whose adjacency matrix A = A(G) is also shown, which has the separable automorphism φ = (2, 5, 8)(3,6,9,4,7,10).…”

mentioning

confidence: 99%

“…We again consider the graph G from Example 3.5 with φ = (2, 5, 8)(3,4,6,7,9,10). Here the graph's adjacency matrix A = A(G) is both irreducible and nonnegative.…”

mentioning

confidence: 99%

Extensions and applications of equitable decompositions for graphs with symmetries

Francis

Linear Algebra and its Applications

Sorensen

et al. 2017

We extend the theory of equitable decompositions introduced in [2], where it was shown that if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism φ, it is possible to use φ to decompose a matrix M appropriately associated with the graph. The result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. We show here that a large class of automorphisms, which we refer to as separable, can be realized as a sequence of basic automorphisms, allowing us to equitably decompose M over any such automorphism. We also show that not only can a matrix M be decomposed but that the eigenvectors of M can also be equitably decomposed. Additionally, we prove under mild conditions that if a matrix M is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix M. Last, we describe how an equitable decomposition effects the Gershgorin region Γ(M) of a matrix M, which can be used to localize the eigenvalues of M. We show that the Gershgorin region of an equitable decomposition of M is contained in the Gershgorin region Γ(M) of the original matrix. We demonstrate on a real-world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix' Gershgorin region.

“…Additionally, we show that the eigenvector centrality of the base vertices of a network remain unchanged as the network is specialized. In terms of dynamic properties we prove that if a dynamical network is intrinsically stable, which is a stronger form of a standard notion of stability (see [28] for more details), then any specialized version of this network will also be intrinsically stable. Hence, network growth given by our models of specialization will not destabilize the network's dynamics if the network has this stronger version of stability.…”

Section: Introductionmentioning

confidence: 91%

Specialization Models of Network Growth

Bunimovich

Journal of Complex Networks

Webb

2018

Self Cite

One of the most important features observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain subnetworks begin to specialize the function(s) they perform. Here, we introduce a class of models of network growth based on this notion of specialization and show that as a network is specialized using this method its topology becomes increasingly sparse, modular, and hierarchical, each of which are important properties observed in real networks. This procedure is also highly flexible in that a network can be specialized over any subset of its elements. This flexibility allows those studying specific networks the ability to search for mechanisms that describe the growth of these particular networks. As an example, we find that by randomly selecting these elements a network's topology acquires some of the most well-known properties of real networks including the small-world property, disassortativity, power-law like degree distributions, and power-law like clustering coefficients. As far as the authors know, this is the first such class of models that creates an increasingly modular and hierarchical network topology with these properties.