Long a matter of folklore, the “small-world 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 small-world 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 non-negligible probability. We then define an infinite family of network models that naturally generalizes the Watts-Strogatz 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.

Lali Barri`ere , Pierre Fraigniaud , Evangelos Kranakis , and Danny Krizanc Dept. de Matem`atica Aplicada i Telem`atica, Universitat Polit`ecnica de Catalunya.

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 n-cycle 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 triangle-free graph by adding a small number of edges while preserving the triangle-free property, has ∗ Keywords: graphs, diameter of graphs, maximum degree.

Part I of this article, in the January--February 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.

The central observation of this paper is that if n random edges are added to any n-node connected graph or digraph then the resulting graph has diameter O(log n). We apply this to smoothed analysis of algorithms and property testing.

We study three augmentations of ring networks that are intended to decrease a ring's diameter significantly while increasing its structural complexity only modestly. Chordal rings enhance a ring network by adding noncrossing "shortcut" edges, which can be viewed as chords of the ring. Express rings are chordal rings whose chords are oriented either clockwise or counterclockwise, allowing them to be viewed as (noncrossing) arcs of the ring. Multi-rings append subsidiary rings to edges of a ring and, recursively, to edges of appended subrings. Important measures of structural complexity are: the cutwidth of an express ring, viz., the maximum number of arcs that cross "above" any ring edge (counting the edge itself); the depth of a multi-ring, viz., the level of recursive appending of subsidiary subrings. Our first result demonstrates the topological equivalence of these three modes of augmentation: for each augmented ring of one type, there are (graph-theoretically) isomorphic ...

Abstract. We consider the problem of covering and packing subsets of δ-hyperbolic 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 2-hyperbolic graphs. In this paper, using a primal-dual 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 δ-hyperbolic 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 δ-hyperbolic graphs with diameter constraints and slackness δ and for the k-center problem in δ-hyperbolic graphs. 1

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.

A graph is vertex-critical (edge-critical) if deleting any vertex (edge) increases its diameter. A conjecture of Simon and Murty stated that `every edge-critical graph of diameter two on vertices contains at most 1 4 2 edges'. This conjecture has been established for sufficiently large . For vertex-critical 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 vertex-critical graph of diameter two on vertices. In this paper, we construct, for each 5 except = 6, a vertex-critical 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 vertex-critical 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 vertex-critical graph of diameter two on vertices with at most 1 2 (5 \Gamma ...

In this paper we study the fault--tolerant properties of the Supercube, a new interconnection network recently introduced by Sen [15]. The Supercube is a generalization of the Hypercube that can be realized for any number of nodes and not only for powers of 2. Moreover, it has the same diameter and connectivity of the Hypercube. We prove that the diameter of the surviving route graph of the N--node Supercube SN , if less than blog 2 Nc nodes or edges fail, is at most 4 for any minimal routing, and exhibit a minimal routing for which the surviving route graph has diameter 2. Then, we show that, when 2 s + 2 s\Gamma1 N ! 2 s+1 and the failures are dlog 2 Ne, the diameter of the surviving route graph is at most 5 for any minimal routing. We also prove that the fault diameter of SN is exactly blog 2 Nc+ 1 when N 62 f2 s+1 \Gamma 1; 2 s+1 \Gamma 2; 2 s + 2 s\Gamma1 + 1g, and blog 2 Nc+ 2 otherwise. Keywords: Supercube, fault tolerance, routing, surviving route graph, broadc...