arXiv subjects

Kevin McGoff

Publications and source records attributed to Kevin McGoff.

At least 19 recordsLinked to original sources

arXiv2026

Ergodic Optimization with Linear Constraints

Let $T : X \to X$ be a continuous map of a compact metrizable space, and let $\phi : X \to \mathbb{R}$ be a continuous function. The ergodic optimization problem is to maximize the integral $\int \phi \, d\mu$ as $\mu$ ranges over all $T$-invariant Borel probability measures on $X$. In this paper we consider a constrained version of the ergodic optimization problem. Given a `constraint set' $\mathcal{C}\subset C(X)$, let $M_\mathcal{C}(X,T)$ be the set of $T$-invariant Borel probability measures $\mu$ on $X$ such that $\int g \, d\mu = 0$ for all $g \in \mathcal{C}$. We investigate the problem of maximizing the integral $\int \phi \, d\mu$ over the constrained set $M_\mathcal{C}(X,T)$. We address basic properties of this optimization problem, beginning with nonemptiness of $M_\mathcal{C}(X,T)$ and existence of optimal solutions. Additionally, we establish the generic and prevalent uniqueness of optimal measures, we provide a realization result, and we give a characterization of the dual problem. This framework provides a common generalization of several previously considered optimization problems in dynamical systems and optimal transport.

math.DS↗

arXiv2026

Graph Disjointness with Applications to Reversible Markov Chains

The correspondence between weighted undirected graphs and reversible Markov chains via vertex random walks is simple and well known. Leveraging this correspondence and ideas from the theory of dynamical systems, we study the structural discordance of graphs and Markov chains by means of graph joinings. Informally, a joining of graphs $G$ and $H$ is a graph on the product of their vertex sets giving rise to a coupling of their random walks. Graphs $G$ and $H$ are strongly disjoint if their only joining is the tensor product, and they are weakly disjoint if the degree function of every joining is equal to the degree function of the tensor product. We establish close connections between graph joinings, disjointness, and graph factors. Our first principal result characterizes weak disjointness of graphs in terms of the spectral overlap of their Markov transition matrices. The second establishes that two graphs without self loops are strongly disjoint if and only if they are weakly disjoint and exactly one of the graphs is a tree. The third shows that the strong or weak disjointness of graphs is essentially determined by their vertex and edge sets, without regard to edge weights. Translating these results into the language of Markov chains yields new insights into the rigidity and structure of reversible couplings of reversible Markov chains.

math.ST↗

arXiv2025

Toward Practical Forecasts of Public Sentiments via Convexification for Mean Field Games: Evidence from Real World COVID-19 Discussion Data

We apply a convexification-based numerical method to forecast public sentiment dynamics using Mean Field Games (MFGs). The theoretical foundation for the convexification approach, established in our prior work, guarantees global convergence to the unique solution to the MFG system. The present work demonstrates the practical potential of this framework using real-world sentiment data extracted from social media public discussion during the COVID-19 pandemic. The results show that the MFG model with appropriate parameters and convexification yields sentiment density predictions that align closely with observed data and satisfy the governing equations. While current parameter selection relies on manual calibration, our findings establish the first proof-of-concept evidence that MFG models can capture complex temporal patterns in public sentiment, laying the groundwork for future work on systematic parameter identification methods, i.e. solutions of coefficient inverse problems for the MFG system.

math.NA↗

arXiv2025

Optimal graph joining with applications to isomorphism detection and identification

We introduce an optimal transport based approach for comparing undirected graphs with non-negative edge weights and general vertex labels, and we study connections between the resulting linear program and the graph isomorphism problem. Our approach is based on the notion of a joining of two graphs $G$ and $H$, which is a product graph that preserves their marginal structure. Given $G$ and $H$ and a vertex-based cost function $c$, the optimal graph joining (OGJ) problem finds a joining of $G$ and $H$ minimizing degree weighted cost. The OGJ problem can be written as a linear program with a convex polyhedral solution set. We establish several basic properties of the OGJ problem, and present theoretical results connecting the OGJ problem to the graph isomorphism problem. In particular, we examine a variety of conditions on graph families that are sufficient to ensure that for every pair of graphs $G$ and $H$ in the family (i) $G$ and $H$ are isomorphic if and only if their optimal joining cost is zero, and (ii) if $G$ and $H$ are isomorphic, the the extreme points of the solution set of the OGJ problem are deterministic joinings corresponding to the isomorphisms from $G$ to $H$.

math.CO↗

arXiv2025

Forecasting Public Sentiments via Mean Field Games

Motivated by the goal of forecasting public sentiments, we consider a forecasting problem in the context of the Mean Field Games theory. We develop a numerical method, which is a version of the so-called convexification method. We provide theoretical convergence analysis that establishes global convergence of the method with a convergence rate. We also conduct numerical experiments that demonstrate the accurate performance of the convexification technique and highlight some promising features of this approach.

math.NA↗

arXiv2024

Homomorphisms from aperiodic subshifts to subshifts with the finite extension property

Given a countable group $G$ and two subshifts $X$ and $Y$ over $G$, a continuous, shift-commuting map $ϕ: X \to Y$ is called a homomorphism. Our main result states that if every finitely generated subgroup of $G$ has polynomial growth, $X$ is aperiodic, and $Y$ has the finite extension property (FEP), then there exists a homomorphism $ϕ: X \to Y$. By combining this theorem with a previous result of Bland, we obtain that if the same conditions hold, and if additionally the topological entropy of $X$ is less than the topological entropy of $Y$ and $Y$ has no global period, then $X$ embeds into $Y$. We also establish some facts about subshifts with the FEP that may be of independent interest.

math.DS↗

arXiv2024

Generalized Measures of Population Synchrony

Synchronized behavior among individuals is a ubiquitous feature of populations. Understanding mechanisms of (de)synchronization demands meaningful, interpretable, computable quantifications of synchrony, relevant to measurements that can be made of dynamic populations. Despite the importance to analyzing and modeling populations, existing notions of synchrony often lack rigorous definitions, may be specialized to a particular experimental system and/or measurement, or may have undesirable properties that limit their utility. We introduce a notion of synchrony for populations of individuals occupying a compact metric space that depends on the Fréchet variance of the distribution of individuals. We establish several fundamental and desirable mathematical properties of this synchrony measure, including continuity and invariance to metric scaling. We establish a general approximation result that controls the disparity between synchrony in the true space and the synchrony observed through a discretization of state space, as may occur when observable states are limited by measurement constraints. We develop efficient algorithms to compute synchrony in a variety of state spaces, including all finite state spaces and empirical distributions on the circle, and provide accessible implementations in an open-source Python module. To demonstrate the usefulness of the synchrony measure in biological applications, we investigate several biologically relevant models of mechanisms that can alter the dynamics of synchrony over time, and reanalyze published data concerning the dynamics of the intraerythrocytic developmental cycles of $\textit{Plasmodium}$ parasites. We anticipate that the rigorous definition of population synchrony and the mathematical and biological results presented here will be broadly useful in analyzing and modeling populations in a variety of contexts.

q-bio.PE↗

arXiv2024

Alignment and Comparison of Directed Networks via Transition Couplings of Random Walks

We describe and study a transport based procedure called NetOTC (network optimal transition coupling) for the comparison and alignment of two networks. The networks of interest may be directed or undirected, weighted or unweighted, and may have distinct vertex sets of different sizes. Given two networks and a cost function relating their vertices, NetOTC finds a transition coupling of their associated random walks having minimum expected cost. The minimizing cost quantifies the difference between the networks, while the optimal transport plan itself provides alignments of both the vertices and the edges of the two networks. Coupling of the full random walks, rather than their marginal distributions, ensures that NetOTC captures local and global information about the networks, and preserves edges. NetOTC has no free parameters, and does not rely on randomization. We investigate a number of theoretical properties of NetOTC and present experiments establishing its empirical performance.

cs.LG↗

arXiv2022

Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups

In this work we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group $G$, if $X$ is a $G$-SFT with positive topological entropy $h(X) > 0$, then the entropies of the SFT subsystems of $X$ are dense in the interval $[0, h(X)]$. In fact, we prove a "relative" version of the same result: if $X$ is a $G$-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs $Z$ for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic $G$-shifts.

math.DS↗

arXiv2021

Estimation of Stationary Optimal Transport Plans

We study optimal transport for stationary stochastic processes taking values in finite spaces. In order to reflect the stationarity of the underlying processes, we restrict attention to stationary couplings, also known as joinings. The resulting optimal joining problem captures differences in the long run average behavior of the processes of interest. We introduce estimators of both optimal joinings and the optimal joining cost, and we establish consistency of the estimators under mild conditions. Furthermore, under stronger mixing assumptions we establish finite-sample error rates for the estimated optimal joining cost that extend the best known results in the iid case. Finally, we extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem.

math.ST↗

arXiv2021

Optimal Transport for Stationary Markov Chains via Policy Iteration

We study the optimal transport problem for pairs of stationary finite-state Markov chains, with an emphasis on the computation of optimal transition couplings. Transition couplings are a constrained family of transport plans that capture the dynamics of Markov chains. Solutions of the optimal transition coupling (OTC) problem correspond to alignments of the two chains that minimize long-term average cost. We establish a connection between the OTC problem and Markov decision processes, and show that solutions of the OTC problem can be obtained via an adaptation of policy iteration. For settings with large state spaces, we develop a fast approximate algorithm based on an entropy-regularized version of the OTC problem, and provide bounds on its per-iteration complexity. We establish a stability result for both the regularized and unregularized algorithms, from which a statistical consistency result follows as a corollary. We validate our theoretical results empirically through a simulation study, demonstrating that the approximate algorithm exhibits faster overall runtime with low error. Finally, we extend the setting and application of our methods to hidden Markov models, and illustrate the potential use of the proposed algorithms in practice with an application to computer-generated music.

math.OC↗

arXiv2021

Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we investigate whether they have various properties, including topological transitivity, topological mixing, dense periodic points, and specification. To each Markov multi-map, we associate a shift of finite type (SFT), and then our main results relate the properties of the SFT with those of the Markov multi-map. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

math.DS↗

arXiv2020

Shotgun identification on groups

We consider the problem of shotgun identification of patterns on groups, which extends previous work on shotgun identification of DNA sequences and labeled graphs. A shotgun identification problem on a group $G$ is specified by two finite subsets $C \subset G$ and $K \subset G$ and a finite alphabet $\mathcal{A}$. In such problems, there is a ``global" pattern $w \in \mathcal{A}^{CK}$, and one would like to be able to identify this pattern (up to translation) based only on observation of the ``local" $K$-shaped subpatterns of $w$, called reads, centered at the elements of $C$. We consider an asymptotic regime in which the size of $w$ tends to infinity and the symbols of $w$ are drawn in an i.i.d. fashion. Our first general result establishes sufficient conditions under which the random pattern $w$ is identifiable from its reads with probability tending to one, and our second general result establishes sufficient conditions under which the random pattern $w$ is non-identifiable with probability tending to one. Additionally, we illustrate our main results by applying them to several families of examples.

math.PR↗

arXiv2020

Ubiquity of entropies of intermediate factors

We consider topological dynamical systems $(X,T)$, where $X$ is a compact metrizable space and $T$ denotes an action of a countable amenable group $G$ on $X$ by homeomorphisms. For two such systems $(X,T)$ and $(Y,S)$ and a factor map $π: X \rightarrow Y$, an intermediate factor is a topological dynamical system $(Z,R)$ for which $π$ can be written as a composition of factor maps $ψ: X \rightarrow Z$ and $φ: Z \rightarrow Y$. In this paper we show that for any countable amenable group $G$, for any $G$-subshifts $(X,T)$ and $(Y,S)$, and for any factor map $ π:X \rightarrow Y$, the set of entropies of intermediate subshift factors is dense in the interval $[h(Y,S), h(X,T)]$. As a corollary, we also prove that if $(X,T)$ and $(Y,S)$ are zero-dimensional $G$-systems, then the set of entropies of intermediate zero-dimensional factors is equal to the interval $[h(Y,S), h(X,T)]$. Our proofs rely on a generalized Marker Lemma that may be of independent interest.

math.DS↗

arXiv2019

Entropy conjugacy for Markov multi-maps of the interval

We consider a class $\mathcal{F}$ of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of $[0,1]^{\mathbb{Z}_+}$. For a multi-map in $\mathcal{F}$, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in $\mathcal{F}$.

math.DS↗

arXiv2019

Gibbs posterior convergence and the thermodynamic formalism

In this paper we consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. This work establishes tight connections between Gibbs posterior inference and the thermodynamic formalism, which may inspire new proof techniques in the study of Bayesian posterior consistency for dependent processes.

math.ST↗

arXiv2018

Pressure and escape rates for random subshifts of finite type

In this work we consider several aspects of the thermodynamic formalism in a randomized setting. Let $X$ be a non-trivial mixing shift of finite type, and let $f : X \to \mathbb{R}$ be a Hölder continuous potential with associated Gibbs measure $μ$. Further, fix a parameter $α\in (0,1)$. For each $n \geq 1$, let $\mathcal{F}_n$ be a random subset of words of length $n$, where each word of length $n$ that appears in $X$ is included in $\mathcal{F}_n$ with probability $1-α$ (and excluded with probability $α$), independently of all other words. Then let $Y_n = Y(\mathcal{F}_n)$ be the random subshift of finite type obtained by forbidding the words in $\mathcal{F}_n$ from $X$. In our first main result, for $α$ sufficiently close to $1$ and $n$ tending to infinity, we show that the pressure of $f$ on $Y_n$ converges in probability to the value $P_X(f) + \log(α)$, where $P_X(f)$ is the pressure of $f$ on $X$. Additionally, let $H_n = H(\mathcal{F}_n)$ be the random hole in $X$ consisting of the union of the cylinder sets of the words in $\mathcal{F}_n$. For our second main result, for $α$ sufficiently close to one and $n$ tending to infinity, we show that the escape rate of $μ$-mass through $H_n$ converges in probability to the value $-\log(α)$ as $n$ tends to infinity.

math.DS↗

arXiv2018

Factoring onto $\mathbb{Z}^d$ subshifts with the finite extension property

We define the finite extension property for $d$-dimensional subshifts, which generalizes the topological strong spatial mixing condition defined by Briceño (2016), and we prove that this property is invariant under topological conjugacy. Moreover, we prove that for every $d$, every $d$-dimensional block gluing subshift factors onto every $d$-dimensional subshift which has strictly lower entropy, a fixed point, and the finite extension property. This result extends a theorem from Boyle, Pavlov, and Schraudner (2010), which requires that the factor contain a safe symbol.

math.DS↗