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15
The SmallWorld Phenomenon: An Algorithmic Perspective
 in Proceedings of the 32nd ACM Symposium on Theory of Computing
, 2000
"... Long a matter of folklore, the “smallworld phenomenon ” — the principle that we are all linked by short chains of acquaintances — was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960’s. This work was among the first to m ..."
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Cited by 621 (6 self)
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Long a matter of folklore, the “smallworld phenomenon ” — the principle that we are all linked by short chains of acquaintances — was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960’s. This work was among the first to make the phenomenon quantitative, allowing people to speak of the “six degrees of separation ” between any two people in the United States. Since then, a number of network models have been proposed as frameworks in which to study the problem analytically. One of the most refined of these models was formulated in recent work of Watts and Strogatz; their framework provided compelling evidence that the smallworld phenomenon is pervasive in a range of networks arising in nature and technology, and a fundamental ingredient in the evolution of the World Wide Web. But existing models are insufficient to explain the striking algorithmic component of Milgram’s original findings: that individuals using local information are collectively very effective at actually constructing short paths between two points in a social network. Although recently proposed network models are rich in short paths, we prove that no decentralized algorithm, operating with local information only, can construct short paths in these networks with nonnegligible probability. We then define an infinite family of network models that naturally generalizes the WattsStrogatz model, and show that for one of these models, there is a decentralized algorithm capable of finding short paths with high probability. More generally, we provide a strong characterization of this family of network models, showing that there is in fact a unique model within the family for which decentralized algorithms are effective.
Efficient Routing in Networks with Long Range Contacts (Extended Abstract)
, 2001
"... Lali Barri`ere , Pierre Fraigniaud , Evangelos Kranakis , and Danny Krizanc Dept. de Matem`atica Aplicada i Telem`atica, Universitat Polit`ecnica de Catalunya. ..."
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Cited by 80 (14 self)
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Lali Barri`ere , Pierre Fraigniaud , Evangelos Kranakis , and Danny Krizanc Dept. de Matem`atica Aplicada i Telem`atica, Universitat Polit`ecnica de Catalunya.
Decreasing the diameter of bounded degree graphs
 J. Graph Theory
, 1999
"... To the memory of Paul Erdős Let fd(G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n> n0(D) vertices, f2(G) = n − D − 1 and f3(G) ≥ n − O(D 3). For d ≥ 4, fd(G) depe ..."
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Cited by 12 (0 self)
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To the memory of Paul Erdős Let fd(G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n> n0(D) vertices, f2(G) = n − D − 1 and f3(G) ≥ n − O(D 3). For d ≥ 4, fd(G) depends strongly on the actual structure of G, not only on the maximum degree of G. We prove that the maximum of fd(G) over all connected graphs on n vertices is n/⌊d/2 ⌋ − O(1). As a byproduct, we show that for the ncycle Cn, fd(Cn) = n/(2⌊d/2 ⌋ − 1) − O(1) for every d and n, improving earlier estimates of Chung and Garey in certain ranges. 1 Preliminaries and results Extremal problems concerning the diameter of graphs have been initiated by Erdős, Rényi and Sós in [4] and [5]. Problems concerning the change of diameter if edges are added or deleted have been initiated by Chung and Garey in [2], followed by a survey of Chung [1] which contains further references, e.g. the paper by Schoone, Bodlaender and Leeuwen [6]. A related problem, decreasing the diameter of a trianglefree graph by adding a small number of edges while preserving the trianglefree property, has ∗ Keywords: graphs, diameter of graphs, maximum degree.
The Diameter of Randomly Perturbed Digraphs and Some Applications
"... The central observation of this paper is that if n random edges are added to any nnode connected graph or digraph then the resulting graph has diameter O(log n). We apply this to smoothed analysis of algorithms and property testing. ..."
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Cited by 9 (2 self)
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The central observation of this paper is that if n random edges are added to any nnode connected graph or digraph then the resulting graph has diameter O(log n). We apply this to smoothed analysis of algorithms and property testing.
Augmented Ring Networks
, 1999
"... We study four augmentations of ring networks which are intended to enhance a ring's efficiency as a communication medium significantly, while increasing its structural complexity ..."
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Cited by 6 (1 self)
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We study four augmentations of ring networks which are intended to enhance a ring's efficiency as a communication medium significantly, while increasing its structural complexity
Graph Theory in Practice: Part II
 American Scientist
, 2000
"... Part I of this article, in the JanuaryFebruary issue, discussed some very large structures that can usefully be looked upon as mathematical graphs. In this context a graph is a set of vertices (which are usually represented as dots) and a set of edges (lines between the dots). One large object tha ..."
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Cited by 5 (0 self)
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Part I of this article, in the JanuaryFebruary issue, discussed some very large structures that can usefully be looked upon as mathematical graphs. In this context a graph is a set of vertices (which are usually represented as dots) and a set of edges (lines between the dots). One large object that can be described in this way is the World Wide Web; its 800 million pages are the vertices of a graph, and links from one page to another are the edges. A second example comes out of Hollywood: The vertices are 225,000 actors, and an edge connects any two actors who have appeared in a feature film together. Although graph theory has a history of two centuries and more, only in recent years has it been applied routinely to structures like these, with many thousands or millions of vertices and edges. Studying such enormous graphs is by no means easy. The Hollywood collaboration graph just barely fits in the memory of a large computer. The Web, a few orders of magnitude larger, requires all the resources of the Internet to keep track of its tentacles. Certain other graphs are even bigger. The human acquaintanceship graph, with a vertex for every person on earth and edges linking all those who know each other, may never be recorded beyond a few small, sampled regions. Even when the vertices and edges of a large graph can be catalogued in full detail, gaining a deeper understanding of the graph's structure still calls for something more. What's needed is a mathematical theory or model. Typically this takes the form of an algorithm for generating new graphs that share certain properties of the graph under examination: You understand the original graph by building structures that resemble it. Part II of this article looks at a few such models.
Packing and covering δhyperbolic spaces by balls
 In APPROXRANDOM 2007
"... Abstract. We consider the problem of covering and packing subsets of δhyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic gr ..."
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Cited by 5 (2 self)
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Abstract. We consider the problem of covering and packing subsets of δhyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic graph G and a positive number R, let γ(S, R) be the minimum number of balls of radius R covering S. It is known that computing γ(S, R) or approximating this number within a constant factor is hard even for 2hyperbolic graphs. In this paper, using a primaldual approach, we show how to construct in polynomial time a covering of S with at most γ(S, R) balls of (slightly larger) radius R + δ. This result is established in the general framework of δhyperbolic geodesic metric spaces and is extended to some other set families derived from balls. This covering algorithm is used to design better than in general case approximation algorithms for the augmentation problem of δhyperbolic graphs with diameter constraints and slackness δ and for the kcenter problem in δhyperbolic graphs. 1
Mixed covering of trees and the augmentation problem with odd diameter constraints
 Algorithmica
, 2006
"... Abstract. In this paper we present a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R2 (R1 < R2) to maximize the total number of covered vertices. The solutions provided by this algorithm in the particular case R1 = R ..."
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Cited by 2 (2 self)
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Abstract. In this paper we present a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R2 (R1 < R2) to maximize the total number of covered vertices. The solutions provided by this algorithm in the particular case R1 = R − 1, R2 = R can be used to obtain for any integer δ>0 a factor (2 + 1/δ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2R + 1: Given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤ D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.
Maximal and Minimal Vertexcritical Graphs of Diameter Two
, 1998
"... A graph is vertexcritical (edgecritical) if deleting any vertex (edge) increases its diameter. A conjecture of Simon and Murty stated that `every edgecritical graph of diameter two on vertices contains at most 1 4 2 edges'. This conjecture has been established for sufficiently large . For v ..."
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Cited by 1 (0 self)
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A graph is vertexcritical (edgecritical) if deleting any vertex (edge) increases its diameter. A conjecture of Simon and Murty stated that `every edgecritical graph of diameter two on vertices contains at most 1 4 2 edges'. This conjecture has been established for sufficiently large . For vertexcritical graphs, little is known about the number of edges. Plesn ' ik implicitly asked whether it is also true that 1 4 2 is an upper bound for the number of edges in a vertexcritical graph of diameter two on vertices. In this paper, we construct, for each 5 except = 6, a vertexcritical graph of diameter two on vertices with at least 1 2 2 \Gamma p 2 3 2 + c 1 edges, where c 1 is some constant. We show that each vertexcritical graph contains at most 1 2 2 \Gamma p 2 2 3 2 + c 2 edges, where c 2 is some constant. In addition, we also construct, for each 5 except = 6, a vertexcritical graph of diameter two on vertices with at most 1 2 (5 \Gamma ...
Minimum VertexDiameter2Critical Graphs
, 1999
"... . We prove that the minimum number of edges in a vertexdiameter2critical graph on n 37 vertices is (5n 17)=2 if n is odd, and is 5n=2 7 if n is even. x 1. Introduction. For a graph G, let V (G); E(G); n(G) and e(G) denote its vertex set, edge set, number of vertices and number of edges, respect ..."
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. We prove that the minimum number of edges in a vertexdiameter2critical graph on n 37 vertices is (5n 17)=2 if n is odd, and is 5n=2 7 if n is even. x 1. Introduction. For a graph G, let V (G); E(G); n(G) and e(G) denote its vertex set, edge set, number of vertices and number of edges, respectively. For a vertex x 2 V (G), we dene the open neighborhood of x to be the set of the vertices adjacent to x, that is, N(x) := f y 2 V (G): y is adjacent to x g; and the closed neighborhood to be N [x] := N(x) [ x. We dene the degree deg(x) of x to be the number of its neighbors, deg(x) :=j N(x) j. We denote the minimum degree of G by (G) = minf deg(x): x 2 V (G) g. If A; B V (G), we dene G[A; B] to be the subgraph with vertex set A [ B and edge set E(G[A; B]) consisting of the edges of G between A and B, that is, E(G[A; B]) = fab 2 E(G) : a 2 A; b 2 Bg: If A = B, then we abbreviate the subgraph G[A; A] as G[A] and its edge set E(G[A; A]) as E(G[A]). Also, jE(G[A; B])j = e(G[A; B])...