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We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.

In this paper we address two questions about the synchronization of coupled oscillators in the Kuramoto model with all-to-all coupling. In the first part we use some classical results in convex geometry to prove bounds on the size of the frequency set supporting the existence of stable, phase locked solutions and show that the set of such frequencies can be expressed by a seminorm which we call the Kuramoto norm. In the second part we use some ideas from extreme order statistics to compute upper and lower bounds on the probability of synchronization for very general frequency distributions. We do so by computing exactly the limiting extreme value distribution of a quantity that is equivalent to the Kuramoto norm., Keywords: Kuramoto model, convex analysis, permutahedron, extreme-value statistics

Given an admissible map γ f for a homogeneous network N , it is known that the Jacobian D γ f ( x ) around a fully synchronous point x = ( x 0 , … , x 0 ) is again an admissible map for N . Motivated by this, we study the spectra of linear admissible maps for homogeneous networks. In particular, we define so-called network multipliers. These are (relatively small) matrices that depend linearly on the coefficients of the response function, and whose eigenvalues together make up the spectrum of the corresponding admissible map. More precisely, given a network N , we define a finite set of network multipliers ( Λ l ) l = 1 k and a class of networks C containing N . This class is furthermore closed under taking quotient networks, subnetworks and disjoint unions. We then show that the eigenvalues of an admissible map for any network in C are given by those of (a subset of) the network multipliers, with fixed multiplicities ( m l ) l = 1 k and independently of the given (finite dimensional) phase space of a node. The coefficients of all the network multipliers of C are furthermore linearly independent, which implies that one may find the multiplicities ( m l ) l = 1 k by simply expressing the trace of an admissible map as a linear combination of the traces of the multipliers. In particular, we will give examples of networks where the network multipliers need not be constructed, but can be determined by looking at small networks in C . We also show that network multipliers are multiplicative with respect to composition of linear maps.

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.

We present a variety of results analyzing the behavior of a class of stochastic processes---referred to as piecewise deterministic Markov processes (PDMPs)---for the infinite-time interval and determine general conditions on when the moments of such processes will, or will not, be well-behaved. We also characterize the types of finite-time blowups that are possible for these processes, and obtain bounds on their probabilities.

The Kuramoto–Sakaguchi model is a generalization of the well-known Kuramoto model that adds a phase-lag paramater or “frustration” to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks the gradient structure, significantly complicating the analysis of the model. We present several results determining the stability of phase-locked configurations: the first of these gives a sufficient condition for stability, and the second a sufficient condition for instability. In fact, the instability criterion gives a count, modulo 2, of the dimension of the unstable manifold to a fixed point and having an odd count is a sufficient condition for instability of the fixed point. We also present numerical results for both small ( N≤10) and large ( N=50) collections of Kuramoto–Sakaguchi oscillators.

We present and analyze a nonabelian version of the Kuramoto system, which we call the Quantum Kuramoto system. We study the stability of several classes of special solutions to this system, and show that for certain connection topologies the system supports multiple attractors. We also present estimates on the maximal possible heterogeneity in this system that can support an attractor, and study the effect of modifications analogous to phase-lag.

The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applications of our results to competition models in ecology and rapidly mixing Markov Chains.

We present and analyze a model of opinion formation on an arbitrary network whose dynamics comes from a global energy function. We study the global and local minimizers of this energy, which we call stable opinion configurations, and describe the global minimizers under certain assumptions on the friendship graph. We show a surprising result that the number of stable configurations is not necessarily monotone in the strength of connection in the social network, i.e. the model sometimes supports more stable configurations when the interpersonal connections are made stronger.

The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If $A$ is a symmetric matrix, by writing $A = B + x{\bf 1}$, where ${\bf 1}$ is the matrix with unit entries, we consider the problem of choosing $x$ to give the optimal Gershgorin bound on the eigenvalues of $B$, which then leads to one-sided bounds on the eigenvalues of $A$. We show that this $x$ can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the "top two" or "bottom two" degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices., Comment: 18 pages, 7 figures

We consider the Moran process with two populations competing under an iterated Prisoner’s Dilemma in the presence of mutation, and concentrate on the case where there are multiple evolutionarily stable strategies. We perform a complete bifurcation analysis of the deterministic system which arises in the infinite population size. We also study the Master equation and obtain asymptotics for the invariant distribution and metastable switching times for the stochastic process in the case of large but finite population. We also show that the stochastic system has asymmetries in the form of a skew for parameter values where the deterministic limit is symmetric.

There are a number of models of coupled oscillator networks where the question of the stability of fixed points reduces to calculating the index of a graph Laplacian. Some examples of such models include the Kuramoto and Kuramoto--Sakaguchi equations as well as the swing equations, which govern the behavior of generators coupled in an electrical network. We show that the index calculation can be related to a dual calculation which is done on the first homology group of the graph, rather than the vertex space. We also show that this representation is computationally attractive for relatively sparse graphs, where the dimension of the first homology group is low, as is true in many applications. We also give explicit formulae for the dimension of the unstable manifold to a phase-locked solution for graphs containing one or two loops. As an application, we present some novel results for the Kuramoto model defined on a ring and compute the longest possible edge length for a stable solution., 18 pages, 2 figures, 1 table

We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called graph fibrations give rise to maps of dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.

We develop a theoretical framework for analyzing ecological models with a multidimensional niche space. Our approach relies on the fact that ecological niches are described by sequences of symbols, which allows us to include multiple phenotypic traits. Ecological drivers, such as competitive exclusion, are modeled by introducing the Hamming distance between two sequences. We show that a suitable transform diagonalizes the community interaction matrix of these models, making it possible to predict the conditions for niche differentiation and, close to the instability onset, the asymptotically long time population distributions of niches. We exemplify our method using the Lotka-Volterra equations with an exponential competition kernel.

We study the problem that arises in a class of stochastic processes referred to as Stochastic Hybrid Systems (SHS) when computing the moments of the states using the generator of the process and Dynkin's formula. We focus on the case when the SHS is at equilibrium or approaching equilibrium. We present a family of such processes for which infinite-dimensional linear-system analysis tools are ineffective, and discuss a few differing perspectives on how to tackle such problems by assuming that the SHS state distribution is such that its entropy is maximum. We also provide a numerical algorithm that allows us to efficiently compute maximum entropy solutions, and compare results with Monte Carlo simulations for some illustrative SHS.