We perform spectral analysis of the Internet topology at the AS level, by adapting the standard spectral filtering method of examining the eigenvectors corresponding to the largest eigenvalues of matrices related to the adjacency matrix of the topology. We observe that the method suggests clusters of ASes with natural semantic proximity, such as geography or business interests. We examine how these clustering properties vary in the core and in the edge of the network, as well as across geographic areas, over time, and between real and synthetic data. We observe that these clustering properties may be suggestive of traffic patterns and thus have direct impact on the link stress of the network. Finally, we use the weights of the eigenvector corresponding to the first eigenvalue to obtain an alternative hierarchical ranking of the ASes.

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...

We show that a number of graph-theoretic counting problems remain NP-hard, indeed #P-complete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. To achieve these results, a new interpolationbased reduction technique which preserves properties such as constant degree is introduced. In addition, the problem of approximately counting minimum cardinality vertex covers is shown to remain NP-hard even when restricted to graphs of maximal degree 3. Previously, restrictedcase complexity results for counting problems were elusive; we believe our techniques may help obtain similar results for many other counting problems. 1 Introduction Ever since the introduction of NP-completeness in the early 1970's, the primary focus of complexity theory has been on decision ...

While decision theory provides an appealing normative framework for representing rich preference structures, eliciting utility or value functions typically incurs a large cost. For many applications involving interactive systems this overhead precludes the use of formal decision-theoretic models of preference. Instead of performing elicitation in a vacuum, it would be useful if we could augment directly elicited preferences with some appropriate default information. In this paper we propose a case-based approach to alleviating the preference elicitation bottleneck. Assuming the existence of a population of users from whom we have elicited complete or incomplete preference structures, we propose eliciting the preferences of a new user interactively and incrementally, using the closest existing preference structures as potential defaults. Since a notion of closeness demands a measure of distance among preference structures, this paper takes the first step of studying various distance mea...

It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size of the network? We study routing in families of sparse random graphs whose degrees follow heavy tailed distributions. Instantiations of such random graphs have been proposed as models for the topology of the Internet at the level of Autonomous Systems as well as at the level of routers. Let n be the number of nodes. Suppose that for each pair of nodes with degrees du and dv we have O(dudv ) units of demand. Thus the total demand is O(n ). We argue analytically and experimentally that in the considered random graph model such demand patterns can be routed so that the flow through each link is at most O . This is to be compared with a bound # that holds for arbitrary graphs. Similar results were previously known for sparse random regular graphs, a.k.a. "expander graphs." The significance is that Internet-like topologies, which grow in a dynamic, decentralized fashion and appear highly inhomogeneous, can support routing with performance characteristics comparable to those of their regular counterparts, at least under the assumption of uniform demand and capacities. Our proof uses approximation algorithms for multicommodity flow and establishes strong bounds of a generalization of "expansion," namely "conductance." Besides routing, our bounds on conductance have further implications, most notably on the gap between first and second eigenvalues of the stochastic normalization of the adjacency matrix of the graph.

The learning process in Boltzmann Machines is computationally very expensive. The computational complexity of the exact algorithm is exponential in the number of neurons. We present a new approximate learning algorithm for Boltzmann Machines, which is based on mean field theory and the linear response theorem. The computational complexity of the algorithm is cubic in the number of neurons. In the absence of hidden units, we show how the weights can be directly computed from the fixed point equation of the learning rules. Thus, in this case we do not need to use a gradient descent procedure for the learning process. We show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role. Finally, we apply the method to a pattern completion task and show good performance for networks up to 100 neurons. 1 Introduction Boltzmann Machines (BMs) (Ackley et al., 1985), are networks of binary neurons with a stoc...

This work is a continuation of [4]. The focus is on the problem of sampling independent sets of a graph with maximum degree ffi. The weight of each independent set is expressed in terms of a fixed positive parameter 2 ffi\Gamma2 , where the weight of an indepednent set oe is joej . The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. In [4], we showed fast convergence of this dynamics for triangle-free graphs. This paper proves fast convergence for arbitrary graphs. Computer Science Division, University of California at Berkeley, and International Computer Science Institute. Supported in part by National Science Foundation Fellowship. 1 Introduction For a more general introduction and a discussion of related work we refer the reader to the companion work [4]. The aim of this work is given a graph G = (V; E) to efficiently sample from the probability measure ¯G defined on the set of indepedent sets\Omega =\Omega G of G weight...

We analyse the convergence to stationarity of a simple non-reversible Markov chain that serves as a model for several non-reversible Markov chain sampling methods that are used in practice. Our theoretical and numerical results show that non-reversibility can indeed lead to improvements over the diffusive behavior of simple Markov chain sampling schemes. The analysis uses both probabilistic techniques and an explicit diagonalisation. Keywords: Non-reversible Markov chain, Markov chain Monte Carlo, Metropolis algorithm. 5 June 1997 Acknowledgments We thank David Aldous, Martin Hildebrand, Brad Mann, and Laurent Saloff-Coste for their help. 1 Introduction Markov chain sampling methods are commonly used in statistics [30, 29], computer science [28], statistical mechanics [2], and quantum field theory [31, 21]. In all these fields, distributions are encountered that are difficult to sample from directly, but for which a Markov chain that converges to the distribution can easily be con...

Staircase walks are lattice paths from (0; 0) to (2n; 0) which take diagonal steps and which never fall below the x-axis. A path hitting the x-axis k times is assigned a weight of k ; where ? 0 . A simple local Markov chain which connects the state space and converges to the Gibbs measure (which normalizes these weights) is known to be rapidly mixing when = 1 , and can easily be shown to be rapidly mixing when ! 1 . We give the first proof that this Markov chain is also mixing in the more interesting case of ? 1 , known in the statistical physics community as adsorbing staircase walks. The main new ingredient is a decomposition technique which allows us to analyze the Markov chain in pieces, applying different arguments to analyze each piece. 1. Introduction 1.1. The model Staircase walks (also called Dyck paths) are walks in ZZ 2 from (0; 0) to (n; n) which stay above the diagonal x = y . Rotating by 45 o , they correspond to walks from (0; 0) to (2n; 0) which take diagon...

Graph models for real-world complex networks such as the Internet, the WWW and biological networks are necessary for analytic and simulation-based studies of network protocols, algorithms, engineering and evolution. To date, all available data for such networks suggest heavy tailed statistics, most notably on the degrees of the underlying graphs. A practical way to generate network topologies that meet the observed data is the following degree-driven approach: First predict the degrees of the graph by extrapolation from the available data, and then construct a graph meeting the degree sequence and additional constraints, such as connectivity and randomness. Within the networking community, this is currently accepted as the most successful approach for modeling the inter-domain topology of the Internet.