951 research outputs found
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
Global Dynamics of Pulse-Coupled Oscillators
Networks of pulse-coupled oscillators can be used to model systems from firing neurons to blinking fireflies. Many past studies have focused on numerical simulations and locating the synchronous state of such systems. In this project, we construct a Poincare map for a system of three pulse-coupled oscillators and use rigorous computational techniques and topological tools to study both synchronous and asynchronous dynamics. We present sample results, including the computed basin of attraction for the synchronous state as well as a depiction of gradient-like dynamics in the remainder of the phase space. In the future, we hope to automate this process so that it can be applied to a wide range of network topologies and parameter values
Graph Gradient Diffusion
We analyze graph dynamical systems with odd analytic coupling. The set of
equilibria is union of manifolds, which can intersect and have different
dimension. The geometry of this set depends on the coupling function, as well
as several properties of the underlying graph, such as homology, coverings,
connectivity and symmetry. Equilibrium stability can change among the manifolds
of equilibria and within a single manifold of equilibria. The gradient
structure and topological bifurcation theory can be leveraged to establish
Lyapunov, asymptotic, and linear stability.Comment: 22 pages, 6 figure
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