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27
Network Topology Generators: DegreeBased vs. Structural
, 2002
"... Following the longheld belief that the Internet is hierarchical, the network topology generators most widely used by the Internet research community, TransitStub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal paper by Faloutsos et al. revealed tha ..."
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Cited by 165 (14 self)
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Following the longheld belief that the Internet is hierarchical, the network topology generators most widely used by the Internet research community, TransitStub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal paper by Faloutsos et al. revealed that the Internet's degree distribution is a powerlaw. Because the degree distributions produced by the TransitStub and Tiers generators are not powerlaws, the research community has largely dismissed them as inadequate and proposed new network generators that attempt to generate graphs with powerlaw degree distributions.
Routing in Distributed Networks: Overview and Open Problems
 ACM SIGACT News  Distributed Computing Column
, 2001
"... This article focuses on routing messages in distributed networks with efficient data structures. After an overview of the various results of the literature, we point some interestingly open problems. ..."
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Cited by 49 (12 self)
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This article focuses on routing messages in distributed networks with efficient data structures. After an overview of the various results of the literature, we point some interestingly open problems.
Complexity classification of some edge modification problems
, 2001
"... In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, c ..."
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Cited by 41 (2 self)
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In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.
Network topologies, power laws, and hierarchy
 Comput. Commun. Rev
"... It has long been thought that the Internet, and its constituent networks, are hierarchical in nature. Consequently, the network topology generators most widely used by the Internet research community, GTITM [7] and Tiers [11], create networks with a deliberately hierarchical structure. However, rec ..."
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Cited by 32 (5 self)
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It has long been thought that the Internet, and its constituent networks, are hierarchical in nature. Consequently, the network topology generators most widely used by the Internet research community, GTITM [7] and Tiers [11], create networks with a deliberately hierarchical structure. However, recent work by Faloutsos et al. [13] revealed that the Internet’s degree distribution — the distribution of the number of connections routers or Autonomous Systems (ASs) have — is a powerlaw. The degree distributions produced by the GTITM and Tiers generators are not powerlaws. To rectify this problem, several new network generators have recently been proposed that produce more realistic degree distributions; these new generators do not attempt to create a hierarchical structure but instead focus solely on the degree distribution. There are thus two families of network generators, structural generators that treat hierarchy as fundamental and degreebased generators that treat the degree distribution as fundamental. In this paper we use several topology metrics to compare the networks produced by these two families of generators to current measurements of the Internet graph. We find that the degreebased generators produce better models, at least according to our topology metrics, of both the ASlevel and routerlevel Internet graphs. We then seek to resolve the seeming paradox that while the Internet certainly has hierarchy, it appears that the Internet graphs are better modeled by generators that do not explicitly construct hierarchies. We conclude our paper with a brief study of other network structures, such as the pointer structure in the web and the set of airline routes, some of which turn out to have metric properties similar to that of the Internet. 1
Reconciling Simplicity and Realism in Parallel Disk Models
 Parallel Computing
, 2001
"... For the design and analysis of algorithms that process huge data sets, a machine model is needed that handles parallel disks. There seems to be a dilemma between simple and flexible use of such a model and accurate modelling of details of the hardware. This paper explains how many aspects of this pr ..."
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Cited by 16 (3 self)
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For the design and analysis of algorithms that process huge data sets, a machine model is needed that handles parallel disks. There seems to be a dilemma between simple and flexible use of such a model and accurate modelling of details of the hardware. This paper explains how many aspects of this problem can be resolved. The programming model implements one large logical disk allowing concurrent access to arbitrary sets of variable size blocks. This model can be implemented efficienctly on multiple independent disks even if zones with different speed, communication bottlenecks and failed disks are allowed. These results not only provide useful algorithmic tools but also imply a theoretical justification for studying external memory algorithms using simple abstract models.
Small Worlds  The Structure of Social Networks
, 1999
"... Experimentally it has been found that any two people in the world, chosen at random, are connected to one another by a short chain of intermediate acquaintances, of typical length about six. This phenomenon, colloquially referred to as the six degrees of separation, has been the subject of a conside ..."
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Cited by 14 (2 self)
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Experimentally it has been found that any two people in the world, chosen at random, are connected to one another by a short chain of intermediate acquaintances, of typical length about six. This phenomenon, colloquially referred to as the six degrees of separation, has been the subject of a considerable amount of recent research and modeling, which we review here.
The Compactness of Interval Routing for Almost All Graphs
, 2001
"... Interval routing is a compact way for representing routing tables on a graph. It is based on grouping together, in each node, destination addresses that use the same outgoing edge in the routing table. Such groups of addresses are represented by some intervals of consecutive integers. We show th ..."
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Cited by 10 (4 self)
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Interval routing is a compact way for representing routing tables on a graph. It is based on grouping together, in each node, destination addresses that use the same outgoing edge in the routing table. Such groups of addresses are represented by some intervals of consecutive integers. We show that almost all the graphs, i.e., a fraction of at least 1 \Gamma 1=n 2 of all the nnode graphs, support a shortest path interval routing with three intervals per outgoing edge, even if the addresses of the nodes are arbitrarily fixed in advance and cannot be chosen by the designer of the routing scheme. In case the addresses are initialized randomly, we show that two intervals per outgoing edge suffice, and conversely, that two intervals are required, for almost all graphs. Finally, if the node addresses can be chosen as desired, we show how to design in polynomial time a shortest path interval routing with a single interval per outgoing edge, for all but at most O(log 3 n) outgoing edges in each node. It follows that almost all graphs support a shortest path routing scheme which requires at most n + O(log 4 n) bits of routing information per node, improving on the previous upper bound.
TwoDimensional WeightConstrained Codes through Enumeration Bounds
 IEEE Trans. Inform. Theory
, 2000
"... For a rational ff 2 (0; 1), let A n\Thetam;ff be the set of binary n \Theta m arrays in which each row has Hamming weight ffm and each column has Hamming weight ffn, where ffm and ffn are integers. (The special case of twodimensional balanced arrays corresponds to ff = 1=2 and even values for n ..."
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Cited by 8 (1 self)
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For a rational ff 2 (0; 1), let A n\Thetam;ff be the set of binary n \Theta m arrays in which each row has Hamming weight ffm and each column has Hamming weight ffn, where ffm and ffn are integers. (The special case of twodimensional balanced arrays corresponds to ff = 1=2 and even values for n and m.) The redundancy of A n\Thetam;ff is defined by ae n\Thetam;ff = nmH(ff) \Gamma log 2 jA n\Thetam;ff j, where H(x) = \Gammax log 2 x \Gamma (1\Gammax) log 2 (1\Gammax). Bounds on ae n\Thetam;ff are obtained in terms of the redundancies of the sets A `;ff of all binary `vectors with Hamming weight ff`, ` 2 fn; mg. Specifically, it is shown that ae n\Thetam;ff nae m;ff + mae n;ff ; where ae `;ff = `H(ff) \Gamma log 2 jA `;ff j, and that this bound is tight up to an additive term O(n + log m). A polynomialtime coding algorithm is presented that maps unconstrained input sequences into A n\Thetam;ff at a rate H(ff) \Gamma (ae m;ff =m) \Gamma (ae n;ff =n). Keywords: Twodimensio...
Kolmogorov random graphs and the incompressibility method
 SIAM J. Comput
"... Abstract. We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered l ..."
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Cited by 8 (1 self)
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Abstract. We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of the maximum possible Kolmogorov complexity) and (ii) a new elementary proof for the number of unlabeled graphs. Key words. Kolmogorov complexity, incompressiblity method, random graphs, enumeration of graphs, algorithmic information theory AMS subject classifications. 68Q30, 05C80, 05C35, 05C30 1. Introduction. The