Spectral Theory for Networks with Attractive and Repulsive Interactions

There is a wealth of applied problems that can be posed as a dynamical system defined on a network with both attractive and repulsive interactions. Some examples include: understanding synchronization properties of nonlinear oscillator;, the behavior of groups, or cliques, in social networks; the study of optimal convergence for consensus algorithm; and many other examples. Frequently the problems involve computing 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 one of the most 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. In certain situations, when the homology group is trivial, the index of the operator is rigid and is determined only by the topology of the network and is independent of the strengths of the interactions. In general these constraints give upper and lower 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 present some numerical studies of this problem for large random matrices.Comment: 27 pages; 9 Figure

A New Perspective on the Average Mixing Matrix

We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.Comment: 14 page

The Relativized Second Eigenvalue Conjecture of Alon

We prove a relativization of the Alon Second Eigenvalue Conjecture for all $d$-regular base graphs, $B$, with $d\ge 3$: for any $\epsilon>0$, we show that a random covering map of degree $n$ to $B$ has a new eigenvalue greater than $2\sqrt{d-1}+\epsilon$ in absolute value with probability $O(1/n)$. Furthermore, if $B$ is a Ramanujan graph, we show that this probability is proportional to $n^{-{\eta_{\rm \,fund}}(B)}$, where ${\eta_{\rm \,fund}}(B)$ is an integer depending on $B$, which can be computed by a finite algorithm for any fixed $B$. For any $d$-regular graph, $B$, ${\eta_{\rm \,fund}}(B)$ is greater than $\sqrt{d-1}$. Our proof introduces a number of ideas that simplify and strengthen the methods of Friedman's proof of the original conjecture of Alon. The most significant new idea is that of a ``certified trace,'' which is not only greatly simplifies our trace methods, but is the reason we can obtain the $n^{-{\eta_{\rm \,fund}}(B)}$ estimate above. This estimate represents an improvement over Friedman's results of the original Alon conjecture for random $d$-regular graphs, for certain values of $d$

Virus Propagation in Multiple Profile Networks

Suppose we have a virus or one competing idea/product that propagates over a multiple profile (e.g., social) network. Can we predict what proportion of the network will actually get "infected" (e.g., spread the idea or buy the competing product), when the nodes of the network appear to have different sensitivity based on their profile? For example, if there are two profiles $\mathcal{A}$ and $\mathcal{B}$ in a network and the nodes of profile $\mathcal{A}$ and profile $\mathcal{B}$ are susceptible to a highly spreading virus with probabilities $\beta_{\mathcal{A}}$ and $\beta_{\mathcal{B}}$ respectively, what percentage of both profiles will actually get infected from the virus at the end? To reverse the question, what are the necessary conditions so that a predefined percentage of the network is infected? We assume that nodes of different profiles can infect one another and we prove that under realistic conditions, apart from the weak profile (great sensitivity), the stronger profile (low sensitivity) will get infected as well. First, we focus on cliques with the goal to provide exact theoretical results as well as to get some intuition as to how a virus affects such a multiple profile network. Then, we move to the theoretical analysis of arbitrary networks. We provide bounds on certain properties of the network based on the probabilities of infection of each node in it when it reaches the steady state. Finally, we provide extensive experimental results that verify our theoretical results and at the same time provide more insight on the problem