TY - JOUR
T1 - Spectral theory for dynamics on graphs containing attractive and repulsive interactions
AU - Bronski, Jared C.
AU - Deville, Lee
PY - 2014
Y1 - 2014
N2 - Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number of eigenvalue crossings. The homology group also gives a natural decomposition of the dynamics into "fixed" degrees of freedom, whose index does not depend on the edge weights, and an orthogonal set of "free" degrees of freedom, whose index changes as the edge weights change. We also explore the spectrum of the Laplacians of signed random matrices.
AB - Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number of eigenvalue crossings. The homology group also gives a natural decomposition of the dynamics into "fixed" degrees of freedom, whose index does not depend on the edge weights, and an orthogonal set of "free" degrees of freedom, whose index changes as the edge weights change. We also explore the spectrum of the Laplacians of signed random matrices.
KW - Dynamics on networks
KW - Graph Laplacian
KW - Social networks
KW - Spectral graph theory
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U2 - 10.1137/130913973
DO - 10.1137/130913973
M3 - Article
AN - SCOPUS:84931569479
SN - 0036-1399
VL - 74
SP - 83
EP - 105
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 1
ER -