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On Cubical Graphs
 JOURNAL OF COMBINATORIAL THEORY (B) 18, 86 % (1975)
, 1975
"... It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of ndimensional cubes. This has found applications, for example, in coding theory, data transmission, and lin ..."
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Cited by 81 (6 self)
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It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of ndimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of those graphs 6, called cubical graphs (not to be confused with cubic graphs, those graphs for which all vertices have degree 3), which can be embedded into an ndimensional cube. A basic technique used is the investigation of graphs which are critically nonembeddable, i.e., which can not be embedded but all of whose subgrapbs can be embedded.
A Geometric Preferential Attachment Model of Networks
 In Algorithms and Models for the WebGraph: Third International Workshop, WAW 2004
, 2004
"... We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with powerlaw degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generat ..."
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Cited by 64 (4 self)
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We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with powerlaw degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2,..., xn chosen uniformly at random from the unit sphere in R 3. After generating xt, we randomly connect it to m points from those points in x1, x2,..., xt−1. 1
A survey of models of the web graph
 In Combinatorial and algorithmic aspects of networking
, 2005
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The volume of the giant component of a random graph with given expected degrees
 SIAM J. Discrete Math
"... Abstract. We consider the random graph model G(w) for a given expected degree sequence w =(w1,w2,...,wn). If the expected average degree is strictly greater than 1, then almost surely the giant component in G of G(w) has volume (i.e., sum of weights of vertices in the giant component) equal to λ0Vol ..."
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Cited by 33 (4 self)
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Abstract. We consider the random graph model G(w) for a given expected degree sequence w =(w1,w2,...,wn). If the expected average degree is strictly greater than 1, then almost surely the giant component in G of G(w) has volume (i.e., sum of weights of vertices in the giant component) equal to λ0Vol(G)+O ( √ n log3.5 n), where λ0 is the unique nonzero root of the equation n∑ wie i=1 −w n∑ iλ =(1−λ) wi, i=1 and where Vol(G) = ∑ i wi.
The many facets of Internet topology and traffic
 Networks and Heterogeneous Media
"... ABSTRACT. The Internet’s layered architecture and organizational structure give rise to a number of different topologies, with the lower layers defining more physical and the higher layers more virtual/logical types of connectivity structures. These structures are very different, and successful Inte ..."
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Cited by 24 (12 self)
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ABSTRACT. The Internet’s layered architecture and organizational structure give rise to a number of different topologies, with the lower layers defining more physical and the higher layers more virtual/logical types of connectivity structures. These structures are very different, and successful Internet topology modeling requires annotating the nodes and edges of the corresponding graphs with information that reflects their networkintrinsic meaning. These structures also give rise to different representations of the traffic that traverses the heterogeneous Internet, and a traffic matrix is a compact and succinct description of the traffic exchanges between the nodes in a given connectivity structure. In this paper, we summarize recent advances in Internet research related to (i) inferring and modeling the routerlevel topologies of individual service providers (i.e., the physical connectivity structure of an ISP, where nodes are routers/switches and links represent physical connections), (ii) estimating the intraAS traffic matrix when the AS’s routerlevel topology and routing configuration are known, (iii) inferring and modeling the Internet’s ASlevel topology, and (iv) estimating the interAS traffic matrix. We will also discuss recent work on Internet connectivity structures that arise at the higher layers in the TCP/IP protocol stack and are more virtual and dynamic; e.g., overlay networks like the WWW graph, where nodes are web pages and edges represent existing hyperlinks, or P2P networks like Gnutella, where nodes represent peers and two peers are connected if they have an active network connection. 1. Introduction. The
ShortestPath Queries for Complex Networks: Exploiting Low Treewidth Outside the Core
"... We present new and improved methods for efficient shortestpath query processing. Our methods are tailored to work for two specific classes of graphs: graphs with small treewidth and complex networks. Seemingly unrelated at first glance, these two classes of graphs have some commonalities: complex ne ..."
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Cited by 5 (3 self)
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We present new and improved methods for efficient shortestpath query processing. Our methods are tailored to work for two specific classes of graphs: graphs with small treewidth and complex networks. Seemingly unrelated at first glance, these two classes of graphs have some commonalities: complex networks are known to have a core–fringe structure with a dense core and a treelike fringe. Our main contributions are efficient algorithms and data structures on three different levels. First, we provide two new methods for graphs with small but not necessarily constant treewidth. Our methods achieve new tradeoffs between space and query time. Second, we present an improved treedecompositionbased method for complex networks, utilizing the methods for graphs with small treewidth. Third, we extend our method to handle the highly interconnected core with existing exact and approximate methods. We evaluate our algorithms both analytically and experimentally. We prove that our algorithms for lowtreewidth graphs achieve improved tradeoffs between space and query time. Our experiments on several realworld complex networks further confirm the efficiency of our methods: Both the exact and the hybrid method have faster preprocessing and query times than existing methods. The hybrid method in particular provides an improved tradeoff between space and accuracy.
Robust MultiChannel Wireless Networks
"... Abstract—One wants to deploy an nnode multichannel wireless network in an environment that is inaccessible for repairs and/or that contains malicious adversaries. One wants to design the network to be robust in the following strong sense. Even if any set of m < n nodes is disabled, one still wa ..."
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Cited by 1 (0 self)
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Abstract—One wants to deploy an nnode multichannel wireless network in an environment that is inaccessible for repairs and/or that contains malicious adversaries. One wants to design the network to be robust in the following strong sense. Even if any set of m < n nodes is disabled, one still wants all of the surviving n − m nodes to be able to communicate with one another. We present a mathematical model for multichannel wireless network that facilitates the problem of designing such networks with desired properties. We then present a scalable, deterministic design strategy that produces robust networks that are: (a) within a factor of 2 of optimal in size and complexity; (b) powerefficient, in that nodes attempt to communicate with nearer neighbors before trying more remote ones. I. THE ROBUSTNETWORK PROBLEM Advances in technology allow us to deploy increasingly
AN EDGE DELETION MODEL FOR COMPLEX NETWORKS
"... Abstract. We propose a new random graph model—Edge Popularity—for the web graph and other complex networks, where edges are deleted over time and an edge is chosen to be deleted with probability inversely proportional to the indegree of the destination. We show that with probability tending to one ..."
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Abstract. We propose a new random graph model—Edge Popularity—for the web graph and other complex networks, where edges are deleted over time and an edge is chosen to be deleted with probability inversely proportional to the indegree of the destination. We show that with probability tending to one as time tends to infinity, the model generates graphs whose degree distribution follows a power law. Depending on the parameters of the model, the exponent of the power law can be any number in (2,∞). 1.
Infinite Limits of Stochastic Graph Models
"... We present new generalized copying models for massive selforganizing networks such as the web graph, and analyze their limit behaviour. Our model is motivated by a desire to unify common design elements of the copying models of the web graph, and the partial duplication model for biological net ..."
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We present new generalized copying models for massive selforganizing networks such as the web graph, and analyze their limit behaviour. Our model is motivated by a desire to unify common design elements of the copying models of the web graph, and the partial duplication model for biological networks. In these models, new nodes copy (with some error) the link structure of existing nodes, and a certain number of random links may be added to the new node that can link to any of the existing nodes. In our new models, a function # parameterizes the number of random links, and thereby allows for the analysis of threshold behaviour.
Averagecase analysis for combinatorial problems
, 2006
"... This thesis considers the average case analysis of algorithms, focusing primarily on NPhard combinatorial optimization problems. It includes a catalog of distributions frequently used in averagecase analysis and a collection of mathematical tools that have been useful in studying these distributio ..."
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This thesis considers the average case analysis of algorithms, focusing primarily on NPhard combinatorial optimization problems. It includes a catalog of distributions frequently used in averagecase analysis and a collection of mathematical tools that have been useful in studying these distributions. The bulk of the thesis consists of casestudies in averagecase analysis of algorithms. Algorithms for 3SAT, Subset Sum, Strong Connectivity, Stochastic Minimum Spanning Tree, and Uncapacitated Facility Location are analyzed on random instances.