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155
Symmetry analysis of reversible markov chains
 INTERNET MATHEMATICS
, 2005
"... We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to ..."
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Cited by 51 (14 self)
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We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a maxdegree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.
Convex optimization of graph Laplacian eigenvalues
 IN INTERNATIONAL CONGRESS OF MATHEMATICIANS
"... We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this p ..."
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Cited by 51 (0 self)
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We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. • Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discretetime) Markov chain on the graph. • Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuoustime) Markov process on the graph. • Absolute algebraic connectivity. Find edge weights that maximize the algebraic
A randomized incremental subgradient method for distributed optimization in networked systems
 SIAM Journal on Optimization
"... Abstract. We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedic ́ and Bertsekas. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e ..."
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Cited by 50 (2 self)
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Abstract. We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedic ́ and Bertsekas. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The stochastic component in the algorithm is described by a Markov chain, which can be constructed in a distributed fashion using local information only. We provide a detailed convergence analysis of the proposed algorithm and compare it with existing, both deterministic and randomized, incremental subgradient methods.
Metastable Walking Machines
, 2008
"... Legged robots that operate in the real world are inherently subject to stochasticity in their dynamics and uncertainty about the terrain. Due to limited energy budgets and limited control authority, these “disturbances” cannot always be canceled out with highgain feedback. Minimallyactuated walkin ..."
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Cited by 42 (11 self)
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Legged robots that operate in the real world are inherently subject to stochasticity in their dynamics and uncertainty about the terrain. Due to limited energy budgets and limited control authority, these “disturbances” cannot always be canceled out with highgain feedback. Minimallyactuated walking machines subject to stochastic disturbances no longer satisfy strict conditions for limitcycle stability; however, they can still demonstrate impressively longliving periods of continuous walking. Here, we employ tools from stochastic processes to examine the “stochastic stability” of idealized rimlesswheel and compassgait walking on randomly generated uneven terrain. Furthermore, we employ tools from numerical stochastic optimal control to design a controller for an actuated compass gait model which maximizes a measure of stochastic stability the mean firstpassagetime and compare its performance to a deterministic counterpart. Our results demonstrate that walking is wellcharacterized as a metastable process, and that the stochastic dynamics of walking should be accounted for during control design in order to improve the stability of our machines.
Data Persistence in Largescale Sensor Networks with Decentralized Fountain Codes
"... It may not be feasible for sensor networks monitoring nature and inaccessible geographical regions to include powered sinks with Internet connections. We consider the scenario where sinks are not present in largescale sensor networks, and unreliable sensors have to collectively resort to storing s ..."
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Cited by 41 (2 self)
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It may not be feasible for sensor networks monitoring nature and inaccessible geographical regions to include powered sinks with Internet connections. We consider the scenario where sinks are not present in largescale sensor networks, and unreliable sensors have to collectively resort to storing sensed data over time on themselves. At a time of convenience, such cached data from a small subset of live sensors may be collected by a centralized (possibly mobile) collector. In this paper, we propose a decentralized algorithm using fountain codes to guarantee the persistence and reliability of cached data on unreliable sensors. With fountain codes, the collector is able to recover all data as long as a sufficient number of sensors are alive. We use random walks to disseminate data from a sensor to a random subset of sensors in the network. Our algorithms take advantage of the low decoding complexity of fountain codes, as well as the scalability of the dissemination process via random walks. We have proposed two algorithms based on random walks. Our theoretical analysis and simulationbased studies have shown that, the first algorithm maintains the same level of fault tolerance as the original centralized fountain code, while introducing lower overhead than naive randomwalk based implementation in the dissemination process. Our second algorithm has lower level of fault tolerance than the original centralized fountain code, but consumes much lower dissemination cost.
RaWMS  Random Walk based Lightweight Membership Service for Wireless Ad Hoc Networks
, 2006
"... This paper presents RaWMS, a novel lightweight random membership service for ad hoc networks. The service provides each node with a partial uniformly chosen view of network nodes. Such a membership service is useful, e.g., in data dissemination algorithms, lookup and discovery services, peer samplin ..."
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Cited by 38 (8 self)
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This paper presents RaWMS, a novel lightweight random membership service for ad hoc networks. The service provides each node with a partial uniformly chosen view of network nodes. Such a membership service is useful, e.g., in data dissemination algorithms, lookup and discovery services, peer sampling services, and complete membership construction. The design of RaWMS is based on a random walk (RW) sampling technique. The paper includes a formal analysis of both the RW sampling technique and RaWMS and verifies it through a detailed simulation study. In addition, RaWMS is compared both analytically and by simulations with a number of other known methods such as flooding and gossipbased techniques.
Mixing times for random walks on geometric random graphs
 In Proc. 7th Workshop on Algorithm Eng. and Experiments and 2nd Workshop on Analytic Algorithmics and Combinatorics (ALENEX/ANALCO 2005
, 2005
"... A geometric random graph, Gd (n, r), is formed as follows: place n nodes uniformly at random onto the surface of the ddimensional unit torus and connect nodes which are within a distance r of each other. The Gd (n, r) has been of great interest due to its success as a model for adhoc wireless netw ..."
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Cited by 38 (2 self)
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A geometric random graph, Gd (n, r), is formed as follows: place n nodes uniformly at random onto the surface of the ddimensional unit torus and connect nodes which are within a distance r of each other. The Gd (n, r) has been of great interest due to its success as a model for adhoc wireless networks. It is well known that the connectivity of Gd (n, r) exhibits a threshold property: there exists a constant αd such that for any ɛ> 0, for rd < αd(1 − ɛ) log n/n the Gd (n, r) is not connected with high probability1 and for rd> αd(1 + ɛ) log n/n the Gd (n, r) is connected w.h.p.. In this paper, we study mixing properties of random walks on G d (n, r) for r d (n) = ω(log n/n). Specifically, we study the scaling of mixing times of the fastestmixing reversible random walk, and the natural random walk. We find that the mixing time of both of these random walks have the same scaling laws and scale proportional to r −2 (for all d). These results hold for G d (n, r) when distance is defined using any Lp norm. Though the results of this paper are not so surprising, they are nontrivial and require new methods. To obtain the scaling law for the fastestmixing reversible random walk, we first explicitly characterize the fastestmixing reversible random walk on a regular (gridtype) graph in d dimensions. We subsequently use this to bound the mixing time of the fastestmixing random walk on Gd (n, r). In the course of our analysis, we obtain a tight relation between the mixing time of the fastestmixing symmetric random walk and the fastestmixing reversible random walk with a specified equilibrium distribution on an arbitrary graph.
Further relaxations of the SDP approach to sensor network localization
 SIAM J. Optim
"... Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we prop ..."
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Cited by 34 (0 self)
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Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we propose methods to further relax the SDP relaxation; more precisely, to decompose the single semidefinite matrix cone into a set of smallsize semidefinite matrix cones, which we call the smaller SDP (SSDP) approach. We present two such relaxations or decompositions; and they are, although weaker than SDP relaxation, tested to be both efficient and accurate in practical computations. The speed of the SSDP is much faster than that of the SDP approach as well as other approaches. We also prove several theoretical properties of the new SSDP relaxations.
Multigraph Sampling of Online Social Networks
 IEEE J. SEL. AREAS COMMUN. ON MEASUREMENT OF INTERNET TOPOLOGIES
, 2011
"... Stateoftheart techniques for probability sampling of users of online social networks (OSNs) are based on random walks on a single social relation (typically friendship). While powerful, these methods rely on the social graph being fully connected. Furthermore, the mixing time of the sampling pro ..."
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Cited by 26 (8 self)
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Stateoftheart techniques for probability sampling of users of online social networks (OSNs) are based on random walks on a single social relation (typically friendship). While powerful, these methods rely on the social graph being fully connected. Furthermore, the mixing time of the sampling process strongly depends on the characteristics of this graph. In this paper, we observe that there often exist other relations between OSN users, such as membership in the same group or participation in the same event. We propose to exploit the graphs these relations induce, by performing a random walk on their union multigraph. We design a computationally efficient way to perform multigraph sampling by randomly selecting the graph on which to walk at each iteration. We demonstrate the benefits of our approach through (i) simulation in synthetic graphs, and (ii) measurements of Last.fm an Internet website for music with social networking features. More specifically, we show that multigraph sampling can obtain a representative sample and faster convergence, even when the individual graphs fail, i.e., are disconnected or highly clustered.
Walking on a Graph with a Magnifying Glass: Stratified Sampling via Weighted Random Walks
 in Proc. ACM SIGMETRICS
, 2011
"... Our objective is to sample the node set of a large unknown graph via crawling, to accurately estimate a given metric of interest. We design a random walk on an appropriately defined weighted graph that achieves high efficiency by preferentially crawling those nodes and edges that convey greater info ..."
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Cited by 23 (7 self)
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Our objective is to sample the node set of a large unknown graph via crawling, to accurately estimate a given metric of interest. We design a random walk on an appropriately defined weighted graph that achieves high efficiency by preferentially crawling those nodes and edges that convey greater information regarding the target metric. Our approach begins by employing the theory of stratification to find optimal node weights, for a given estimation problem, under an independence sampler. While optimal under independence sampling, these weights may be impractical under graph crawling due to constraints arising from the structure of the graph. Therefore, the edge weights for our random walk should be chosen so as to lead to an equilibrium distribution that strikes a balance between approximating the optimal weights under an independence sampler and achieving fast convergence. We propose a heuristic approach (stratified weighted random walk, or SWRW) that achieves this goal, while using only limited information about the graph structure and the node properties. We evaluate our technique in simulation, and experimentally, by collecting a sample of Facebook college users. We show that SWRW requires 1315 times fewer samples than the simple reweighted random walk (RW) to achieve the same estimation accuracy for a range of metrics.