The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

Sun, Jun; Boyd, Stephen; Xiao, Lin; Diaconis, Persi

doi:10.1137/s0036144504443821

Cited by 133 publications

(144 citation statements)

References 27 publications

(38 reference statements)

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“…In situations where Ω is of large size, such an answer is not satisfactory, since the above mentioned P may not be practical (implementable). To make the above problem more interesting and nontrivial, following the work of [9,10,62,72], let us now further impose that we are given an arbitrary graph G = (V, E), with π a probability measure on V as before, and demand that any kernel P has to be zero on the pairs which are not adjacent in G (i.e., not connected by an edge from E) and such that π is the stationary measure for P . Now given this setting, the main issue is to come up with the fastest such P , in the sense that such a P has as large a spectral gap as possible.…”

Section: The Fastest Mixing Markov Process Problemmentioning

confidence: 99%

“…also investigated [72] the closely related problem of designing a fastest Markov process (meaning, a continuous-time Markov chain) under the above stipulation. Since the rates of a continuous-time Markov process can be arbitrary (up to scaling), a particular normalization such as the sum of all non-diagonal entries being at most 1, was imposed by [72] to make the problem meaningful in continuous-time.…”

Section: The Fastest Mixing Markov Process Problemmentioning

confidence: 99%

“…Since the rates of a continuous-time Markov process can be arbitrary (up to scaling), a particular normalization such as the sum of all non-diagonal entries being at most 1, was imposed by [72] to make the problem meaningful in continuous-time. Perhaps it would have been more common to have made the assumption that the sum in each row (barring the non-diagonal entry) be at most 1, however the above assumption seems rather interesting and powerful (see the remark at the end of this section).…”

Section: The Fastest Mixing Markov Process Problemmentioning

confidence: 99%

See 2 more Smart Citations

Mathematical Aspects of Mixing Times in Markov Chains

Montenegro

Tetali

2006

FNT in Theoretical Computer Science

209

222

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In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques to bounding the convergence to stationarity. This includes functional techniques such as logarithmic Sobolev and Nash inequalities, refined spectral and entropy techniques, and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or less self-contained treatment of some of these modern techniques, after reviewing several preliminaries. We also review classical and modern lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and decomposition methods, which are not included here; our choice was to keep the analytical methods as the theme of this presentation. We illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.

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Section: The Fastest Mixing Markov Process Problemmentioning

confidence: 99%

Section: The Fastest Mixing Markov Process Problemmentioning

confidence: 99%

Section: The Fastest Mixing Markov Process Problemmentioning

confidence: 99%

See 1 more Smart Citation

Mathematical Aspects of Mixing Times in Markov Chains

Montenegro

Tetali

2006

FNT in Theoretical Computer Science

209

222

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“…Boyd et al maximize the mixing rate by assigning optimum link weights in the setting of a single layer (see Refs. [9,10]). …”

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confidence: 99%

Maximizing algebraic connectivity in interconnected networks

Shakeri¹,

Albin²,

Sahneh³

et al. 2016

Phys. Rev. E

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Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with interlayer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these interlayer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one interlayer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be nonuniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically. DOI: 10.1103/PhysRevE.93.030301Real-world networks are often connected together and therefore influence each other [1]. Robust design of interdependent networks is critical to allow uninterrupted flow of information, power, and goods in spite of possible errors and attacks [2][3][4]. The second eigenvalue of the Laplacian matrix, λ 2 (L), is a good measure of network robustness [5]. Fiedler shows that algebraic connectivity increases by adding links [6]. Moreover, it is harder to bisect a network with higher algebraic connectivity [7]. The second eigenvalue of the Laplacian matrix is also a measure of the speed of mixing for a Markov process on a network [8]. Boyd et al. maximize the mixing rate by assigning optimum link weights in the setting of a single layer (see Refs. [9,10]).For multiplex networks (see Fig. 1), a natural question is the following. Given fixed network layers, how should the weights be assigned to interlayer links in order to maximize algebraic connectivity?The behavior of λ 2 , in the case of identical weights, i.e., with a fixed coupling weight p for every interlayer link, has been studied recently. networks by choosing from a set of interlayer links with predefined weights [16].In this Rapid Communication we remove the constraint of identical interlinking weights and pose the problem of finding the maximum algebraic connectivity for a one-toone interconnected structure between different layers in the presence of limited resources. We show that up to the threshold budget p * N (where p * is the same threshold studied in Refs. [11,13,14]) the uniform distribution of identical weights is actually optimal. For larger budgets, the optimal distribution of weights is generally not uniform.Let G = (V,E) represent a network a...

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“…Interest in fastest mixing Markov chains and graph conductivity led Boyd, Diaconis and Xiao [11] to investigate the same object (up to a trivial scaling). There and in [8] it was observed for connected G, that via semidefinite dualityâ(G) may also be expressed as the embedding problem |E| a(G) = maximize i∈N v i 2 subject to i∈N v i = 0, v i − v j ≤ 1 for ij ∈ E, v i ∈ R n for i ∈ N.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The rotational dimension of a graph

2010

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Given a connected graph G = (N, E) with node weights s ∈ R N + and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of G: Find vi ∈ R |N| , i ∈ N so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter of all points is at the origin, and P i∈N si vi 2 is maximized. In the case of a two dimensional optimal solution this corresponds to the equilibrium position of a quickly rotating net consisting of weighted mass points that are linked by massless cables of given lengths. We define the rotational dimension of G to be the minimal dimension k so that for all choices of lengths and weights an optimal solution can be found in R k and show that this is a minor monotone graph parameter. We give forbidden minor characterizations up to rotational dimension 2 and prove that the rotational dimension is always bounded above by the tree-width of G plus one.

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The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

Cited by 133 publications

References 27 publications

Mathematical Aspects of Mixing Times in Markov Chains

Mathematical Aspects of Mixing Times in Markov Chains

Maximizing algebraic connectivity in interconnected networks

The rotational dimension of a graph

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