Milena Mihail

Random walks in peer-to-peer networks

We quantify the effectiveness of random walks for searching and construction of unstructured peer-to-peer (P2P) networks. We have identified two cases where the use of random walks for searching achieves better results than flooding: a) when the overlay topology is clustered, and h) when a client re-issues the same query while its horizon does not change much. For construction, we argue that an expander can he maintained dynamically with constant operations per addition. The key technical ingredient of our approach is a deep result of stochastic processes indicating that samples taken from consecutive steps of a random walk can achieve statistical properties similar to independent sampling (if the second eigenvalue of the transition matrix is hounded away from 1, which translates to good expansion of the network; such connectivity is desired, and believed to hold, in every reasonable network and network model). This property has been previously used in complexity theory for construction of pseudorandom number generators. We reveal another facet of this theory and translate savings in random bits to savings in processing overhead.

Hybrid search schemes for unstructured peer-to-peer networks

We study hybrid search schemes for unstructured peer-to-peer networks. We quantify performance in terms of number of hits, network overhead, and response time. Our schemes combine flooding and random walks, look ahead and replication. We consider both regular topologies and topologies with supernodes. We introduce a general search scheme, of which flooding and random walks are special instances, and show how to use locally maintained network information to improve the performance of searching. Our main findings are: (a) a small number of supernodes in an otherwise regular topology can offer sharp savings in the performance of search, both in the case of search by flooding and search by random walk, particularly when it is combined with 1-step replication. We quantify, analytically and experimentally, that the reason of these savings is that the search is biased towards nodes that yield more information. (b) There is a generalization of search, of which flooding and random walk are special instances, which may take further advantage of locally maintained network information, and yield better performance than both flooding and random walk in clustered topologies. The method determines edge critically and is reminiscent of fundamental heuristics from the area of approximation algorithms.

Approximating betweenness centrality

Betweenness is a centrality measure based on shortest paths, widely used in complex network analysis. It is computationally-expensive to exactly determine betweenness; currently the fastest-known algorithm by Brandes requires O(nm) time for unweighted graphs and O(nm + n2 log n) time for weighted graphs, where n is the number of vertices and m is the number of edges in the network. These are also the worstcase time bounds for computing the betweenness score of a single vertex. In this paper, we present a novel approximation algorithm for computing betweenness centrality of a given vertex, for both weighted and unweighted graphs. Our approximation algorithm is based on an adaptive sampling technique that significantly reduces the number of single-source shortest path computations for vertices with high centrality. We conduct an extensive experimental study on real-world graph instances, and observe that our random sampling algorithm gives very good betweenness approximations for biological networks, road networks and web crawls.

A primal-dual approximation algorithm for generalized Steiner network problems

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. Ifk is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.

Spectral analysis of Internet topologies

Spectral analysis of the Internet topology at the autonomous system (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 is performed. We observe that the method suggests clusters of ASs 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 ASs.

On certain connectivity properties of the internet topology

We show that random graphs in the preferential connectivity model have constant conductance, and hence have worst-case routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the Vickrey-Clarke-Groves mechanism for shortest paths) of a sparse Erdos-Renyi random graph is bounded by a small constant.

Balanced matroids

Dalancea

On the Eigenvalue Power Law

We show that the largest eigenvalues of graphs whose highest degrees are Zipf-like distributed with slope a are distributed according to a power law with slope α/2. This follows as a direct and almost certain corollary of the degree power law. Our result has implications for the singular value decomposition method in information retrieval.

Strategyproof cost-sharing mechanisms for set cover and facility location games

Strategyproof cost-sharing mechanisms, lying in the core, that recover 1/@a fraction of the cost, are presented for the set cover and facility location games: @a=O(log n) for the former and 1:861 for the latter. Our mechanisms utilize approximation algorithms for these problems based on the method of dual-fitting.

On the complexity of the view-selection problem

A commonly used and powerful technique for improving query response time over very large databases is to precompute (‘Lmaterialize”) frequently’ asked queries (“views”). The problem is to select an appropriate set of views, given a limited amount of resources. Harinarayan, Rajaraman and Ullman formalized this technique by proposing a framework in which queries are modeled by a weighted partial order, and selecting a set of views whose materialization minimizes the average query response time is equivalent to selecting a subset of nodes of the partial order that minimizes a suitably defined cost function. Because this problem is NPHard, the focus is on approximability and heuristics. Harinarayan, Rajaraman and Ullman proposed a greedy heuristic together with a “benefit” criterion to measure its performance; this heuristic and performance measure are used in several subsequent papers which generalize their work. We prove the following lower bounds: (a) The greedy heuristic of Harinarayan, Rajaraman and Ullman has query response time at least n/12 times optimal for infinitely many n. (Compare this to the fact that no algorithm, regardless of how naive it is, ever has query response time exceeding n times optimal.) (b) If PfNP, then for every e > 0, every polynomialtime approximation algorithm for the view-selection problem will output solutions with query response time at least n’-’ times optimal, for infinitely many n, even for partial orders with bounded degrees and *College of Computing, Georgia Institute of Technology, howardQcc.gatech.edu. Research supported in part by NSF grant CCR-9732746. +CoIlege of Computing and School of Industrial and Systems Engineering, Georgia Institute of Technology, mihailQcc.gatech.edu. Work done in part while the author was at Bellcore. Permission to make digital or hard copies of all OT part of this work fbl personal or classroom USC is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this nhx and the full citation on the tirst page. TO COj>y otllerwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission andior a fee. PODS ‘99 Philadelphia PA Copyright ACM 1999 l-58113-062-7/99/05...