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Measurement and Analysis of Online Social Networks
 In Proceedings of the 5th ACM/USENIX Internet Measurement Conference (IMC’07
, 2007
"... Online social networking sites like Orkut, YouTube, and Flickr are among the most popular sites on the Internet. Users of these sites form a social network, which provides a powerful means of sharing, organizing, and finding content and contacts. The popularity of these sites provides an opportunity ..."
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Cited by 698 (14 self)
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Online social networking sites like Orkut, YouTube, and Flickr are among the most popular sites on the Internet. Users of these sites form a social network, which provides a powerful means of sharing, organizing, and finding content and contacts. The popularity of these sites provides an opportunity to study the characteristics of online social network graphs at large scale. Understanding these graphs is important, both to improve current systems and to design new applications of online social networks. This paper presents a largescale measurement study and analysis of the structure of multiple online social networks. We examine data gathered from four popular online social networks: Flickr, YouTube, LiveJournal, and Orkut. We crawled the publicly accessible user links on each site, obtaining a large portion of each social network’s graph. Our data set contains over 11.3 million users and 328 million links. We believe that this is the first study to examine multiple online social networks at scale. Our results confirm the powerlaw, smallworld, and scalefree properties of online social networks. We observe that the indegree of user nodes tends to match the outdegree; that the networks contain a densely connected core of highdegree nodes; and that this core links small groups of strongly clustered, lowdegree nodes at the fringes of the network. Finally, we discuss the implications of these structural properties for the design of social network based systems.
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 166 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Conductance and Congestion in Power Law Graphs
, 2003
"... 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 ..."
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Cited by 69 (6 self)
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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 Internetlike 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.
Approximation algorithms for disjoint paths and related routing and packing problems
 Mathematics of Operations Research
, 2000
"... Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems consi ..."
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Cited by 59 (1 self)
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Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems considered; the central theme of this work is the underlying multicommodity flow relaxation. Applications of these techniques to approximating families of packing integer programs are also presented. Key words and phrases. Disjoint paths, approximation algorithms, unsplittable flow, routing, packing, integer programming, multicommodity flow, randomized algorithms, rounding, linear programming. 1
Improved Bounds for the Unsplittable Flow Problem
 In Proceedings of the 13th ACMSIAM Symposium on Discrete Algorithms
, 2002
"... In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for eac ..."
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Cited by 56 (6 self)
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In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity.
Approximation Algorithms for the Unsplittable Flow Problem
"... We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the nonuniform capacity case in which the edge capacities can vary arbitrarily ..."
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Cited by 55 (9 self)
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We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the nonuniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are: For undirected graphs we obtain a O(\Delta ff \Gamma 1 log2 n) approximation ratio, where n is the number of vertices, \Delta the maximum degree, and ff the expansion of the graph. Our ratio is capacity independent and improves upon the earlier O(\Delta ff \Gamma 1(c max=cmin) log n) bound [15] for large values of cmax=cmin. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an O(\Delta ff \Gamma 1 log n) approximation, which matches the performance of the bestknown algorithm [15] for this special case. For certain strong constantdegree expanders considered by Frieze [10] we obtain an O(plog n) approximation for the uniform capacity case, improving upon the current O(log n) approximation. For UFP on the line and the ring, we give the first constantfactor approximation algorithms. Previous results addressed only the uniform capacity case. All of the above results improve if the maximum demand is bounded
On the kSplittable Flow Problem
, 2002
"... In traditional multicommodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, ..."
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Cited by 31 (3 self)
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In traditional multicommodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However,
OnLine Randomized Call Control Revisited
 SIAM J. COMPUTING
, 2001
"... We consider the problem of online call admission and routing on trees and meshes. Previous work gave randomized online algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probabi ..."
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Cited by 29 (5 self)
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We consider the problem of online call admission and routing on trees and meshes. Previous work gave randomized online algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probability. We investigate the question of devising for these problems online competitive algorithms that also guarantee a "good" solution with "good" probability. We give a new
EdgeDisjoint Paths in Expander Graphs
, 2000
"... Given a graph G = (17, E) and a set of t ¢ pairs of vertices in V, we are interested in finding for each pair (hi, b~), a path connecting ai to bi, such that the set of t ¢ paths so found is edgedisjoint. (For arbitrary graphs the problem is AfT~complete, although it is in 7 ~ if n is fixed.) We p ..."
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Cited by 27 (0 self)
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Given a graph G = (17, E) and a set of t ¢ pairs of vertices in V, we are interested in finding for each pair (hi, b~), a path connecting ai to bi, such that the set of t ¢ paths so found is edgedisjoint. (For arbitrary graphs the problem is AfT~complete, although it is in 7 ~ if n is fixed.) We present a polynomial time randomized algorithm for finding edge disjoint paths in an rregular expander graph G. We show that if G has sufficiently strong expansion properties and r is sufficiently large then all sets of n = g~(n/logn) pairs of vertices can be joined. This is within a constant factor of best possible.
Minorembedding in adiabatic quantum computation: I The parameter setting problem
 Quantum Inf. Process 7: pp
, 2008
"... We show that the NPhard quadratic unconstrained binary optimization (QUBO) problem on a graph G can be solved using an adiabatic quantum computer that implements an Ising spin1/2 Hamiltonian, by reduction through minorembedding of G in the quantum hardware graph U. There are two components to thi ..."
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Cited by 21 (3 self)
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We show that the NPhard quadratic unconstrained binary optimization (QUBO) problem on a graph G can be solved using an adiabatic quantum computer that implements an Ising spin1/2 Hamiltonian, by reduction through minorembedding of G in the quantum hardware graph U. There are two components to this reduction: embedding and parameter setting. The embedding problem is to find a minorembeddingGemb of a graph G in U, which is a subgraph of U such that G can be obtained from Gemb by contracting edges. The parameter setting problem is to determine the corresponding parameters, qubit biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper, we focus on the parameter setting problem. As an example, we demonstrate the embedded Ising Hamiltonian for solving the maximum independent set (MIS) problem via adiabatic quantum computation (AQC) using an Ising spin1/2 system. We close by discussing several related algorithmic problems that need to be investigated in order to facilitate the design of adiabatic algorithms and AQC architectures. 1