We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.
Consider a finite irreducible Markov chain with invariant probability $π$. Define its inverse communication speed as the expectation to go from x to y, when x, y are sampled independently according to $π$. In the discrete time setting and when $π$ is the uniform distribution $\upsilon$, Litvak and Ejov have shown that the permutation matrices associated to Hamiltonian cycles are the fastest Markov chains. Here we prove (A) that the above optimality is with respect to all processes compatible with a fixed graph of permitted transitions (assuming that it does contain a Hamiltonian cycle), not on…
Let $G = (V,E)$ be an undirected graph with maximum degree $Δ$ and vertex conductance $Ψ^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap $γ^*(P)$ satisfies \[Ψ^*(G)^{2}/\logΔ\lesssim γ^*(P) \lesssim Ψ^*(G).\] Our bound is optimal under the Small Set Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who obtained such a result with $\logΔ$ replaced by $\log|V|$. In order to obtain our result, we show how to embed a negative-type semi-metric $d$ defined on $V$ into a negative-type semi-me…
Motivated by applications of distributed linear estimation, distributed control and distributed optimization, we consider the question of designing linear iterative algorithms for computing the average of numbers in a network. Specifically, our interest is in designing such an algorithm with the fastest rate of convergence given the topological constraints of the network. As the main result of this paper, we design an algorithm with the fastest possible rate of convergence using a non-reversible Markov chain on the given network graph. We construct such a Markov chain by transforming the stand…
In a series of recent works, Boyd, Diaconis, and their co-authors have introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixing-time analysis techniques--variational characterizations, conductance, canonical paths--can be used to give simple, nontrivial lower and upper bounds on the fastest mixing time. To test the applicability of this idea, we consider several detailed examples including the Glauber dynamics of the Ising model--and get sha…
We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution $π$ on a given finite partially ordered state space $\mathcal{X}$. When $K\preceq L$ in this partial order we say that $K$ and $L$ satisfy a comparison inequality. We establish that if $K_1,\ldots,K_t$ and $L_1,\ldots,L_t$ are reversible and $K_s\preceq L_s$ for $s=1,\ldots,t$, then $K_1\cdots K_t\preceq L_1\cdots L_t$. In particular, in the time-homogeneous case we have $K^t\preceq L^t$ for every $t$ if $K$ and $L$ are reversible and $K\preceq L$, and using this we sh…
In probability theory, equalities are much less than inequalities. In this paper, we find a series of equalities which characterize the symmetry of the forming times of a family of similar cycles for discrete-time and continuous-time Markov chains. Moreover, we use these cycle symmetries to study the circulation fluctuations for Markov chains. We prove that the empirical circulations of a family of cycles passing through a common state satisfy a large deviation principle with a rate function which has an highly non-obvious symmetry. Finally, we discuss the applications of our work in statistic…
For a given absorbing Markov chain $X^*$ on a finite state space, a chain $X$ is a sharp antidual of $X^*$ if the fastest strong stationary time of $X$ is equal, in distribution, to the absorption time of $X^*$. In this paper we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence - utilizing known results on a limiting di…
We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [Simon et al. 2011], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order…
We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functi…
An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues nu_j of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates nu_j. We give another proof of their theorem. In the case o…
Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced C${}^*$-algebra of a compact quantum group which are translation invariant (w.r.t. to the coproduct) are in one-to-one correspondence with Lévy processes on its $*$-Hopf algebra. We use the theory of Lévy processes on involutive bialgebras to characterize symmetry properties of the associated quantum Markov semigroup. It turns out that the quantum Markov semigr…
The matrix of a permutation is a partial case of Markov transition matrices. In the same way, a measure preserving bijection of a space A with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) A to A, which spreads points of A into measures on A. In this paper, we discuss R-polymorphisms and $\vee$-polymorphisms, who are analogues of the Markov transition operators for the groups of bijections A to A leaving the measure quasiinvariant; two types of the polymorphisms correspond to the cases, when A has fi…
We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We d…
In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the "hidden" symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associat…
In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature $β$ ranges from 0 to $β_0=\frac12\mathrm{arctanh}(\frac13)$. Moreover…
In this article we introduce a dynamic Erdős-Rényi graph model, in which, independently for each vertex pair, edges appear and disappear according to a Markov on-off process. In studying the dynamic graph we present two main results. The first being on how long it takes for the graph to reach stationarity. We give an explicit expression for this time, as well as proving that this is the fastest time to reach stationarity among all strong stationary times. The second result concerns the time it takes for the dynamic graph to reach a certain number of edges. We give an explicit expression for th…
We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions ($x$) and centered values ($h$) of the extreme local maxima in lattice versions of a bounded domain $D\subset\mathbb C$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $Z^D(dx)\otimes e^{-αh}d h$, where $α$ is a constant and $Z^D$ is a random a.s.-finite measure on $D$. The random measures $\{Z^…
We introduce a Markov Chain Monte Carlo algorithm which samples from the space of spanning trees of complete graphs using local rewiring operations only. The probability distribution of graphs of this kind is shown to depend on the symmetries of these graphs, which are reflected in the equilibrium distribution of the Markov chain. We prove that the algorithm is ergodic and proceed to estimate the probability distribution for small graph ensembles with exactly known probabilities. The autocorrelation time of the graph diameter demonstrates that the algorithm generates independent configurations…
We introduce a notion of $k$th order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put - call symmetries in option pricing. Our general $k$th order duality can be financially interpreted as put - call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of $k$th order duality of Markov proce…
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold $\mathcal{M}$. Our method exploits the symmetries of the algorithms' isotropic random search-direction subspaces to analytically average out the variance in the intersection volume caused by the orientation of the search-subspace with respect to the manifold $\mathcal{M}$ it int…
