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21
Heuristically optimized tradeoffs: a new paradigm for power laws in the internet
, 2002
"... Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: "last mile " connection costs, and transmissi ..."
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Cited by 160 (2 self)
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Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: &quot;last mile &quot; connection costs, and transmission delays measured in hops. We also point out a similar phenomenon, anticipated in [2], in the distribution of file sizes. Our results seem to suggest that power laws tend to arise as a result of complex, multiobjective optimization.
An Algorithmic Approach to Social Networks
 PhD thesis at MIT References 118 Science and Artificial Intelligence Laboratory
, 2005
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Coagulationfragmentation duality, Poisson–Dirichlet distributions and random recursive trees
 Ann. Appl. Probab
, 2006
"... In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in ..."
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Cited by 10 (1 self)
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In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ, respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ +1) distribution, while the reverse is true for Coag α,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ). Repeated application of the Frag α operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality. 1. Introduction. The
The Structure of Geographical Threshold Graphs
 9 M. Bradonjić and Joseph Kong, Wireless Ad Hoc Networks with Tunable Topology, Proceedings of the 45th Annual Allerton Conference on Communication, Control and Computing
, 2007
"... Abstract. We analyze the structure of random graphs generated by the geographical threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as random ..."
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Cited by 8 (3 self)
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Abstract. We analyze the structure of random graphs generated by the geographical threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, clustering coefficient, and diameter relate to the threshold value and weight distribution. We give bounds on the threshold value guaranteeing the presence or absence of a giant component, connectivity and disconnectivity of the graph, and small diameter. Finally, we consider the clustering coefficient for nodes with a given degree l, finding that its scaling is very close to 1/l when the node weights are exponentially distributed. 1.
The connectivityprofile of random increasing ktrees
"... Random increasing ktrees represent an interesting, useful class of strongly dependent graphs for which analyticcombinatorial tools can be successfully applied. We study in this paper a notion called connectivityprofile and derive asymptotic estimates for it; some interesting consequences will als ..."
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Cited by 7 (2 self)
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Random increasing ktrees represent an interesting, useful class of strongly dependent graphs for which analyticcombinatorial tools can be successfully applied. We study in this paper a notion called connectivityprofile and derive asymptotic estimates for it; some interesting consequences will also be given. 1
Greedy embeddings, trees, and euclidean vs. lobachevsky geometry
, 2006
"... A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f: V → X such that for every sourcesink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connec ..."
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Cited by 6 (0 self)
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A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f: V → X such that for every sourcesink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectivity graphs helps to build distributed routing schemes with compact routing tables. In this paper we take a refined look at greedy embeddings, previously addressed in [1, 2], by examining their description complexity as a key parameter in conjunction with their dimensionality. We give arguments showing that the dimensionality lowerbounds for monotone maps do not extend to greedy embeddings. We prove a unified O(log n) lowerbound on the dimension of nostretch greedy embeddings when the host metric is Euclidean or Lobachevsky geometry. The essence of the lower bound entails showing that lowdimensional spaces lack the topological capacity to realize the embeddings of certain graphs with “hard crossroads. ” This technique might be of independent interest. We develop new methods for building concise embeddings of trees (and some other graphs) in 3dimensional Lobachevsky spaces using recursive applications of hyperbolic isometries guided by caterpillarlike decompositions. Our embeddings improve over prior work [1] by achieving O(κ(T) · log n) description complexity, where κ(T) is the caterpillar dimension. We further demonstrate concise O(log n)dimensional greedy embeddings of trees into Euclidean space using techniques inspired by [3], thereby strengthening our belief and intuition that all O(log n) graphs can be embedded with no stretch in ℓ. ∗ PhD candidate. 2
Kernels for the vertex cover problem on the preferred attachment model
 Proc. 5th WEA, Lecture Notes in Computer Science
, 2006
"... Abstract. We examine the behavior of two kernelization techniques for the vertex cover problem viewed as preprocessing algorithms. Specifically, we deal with the kernelization algorithms of Buss and of Nemhauser & Trotter. Our evaluation is applied to random graphs generated under the preferred ..."
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Cited by 1 (0 self)
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Abstract. We examine the behavior of two kernelization techniques for the vertex cover problem viewed as preprocessing algorithms. Specifically, we deal with the kernelization algorithms of Buss and of Nemhauser & Trotter. Our evaluation is applied to random graphs generated under the preferred attachment model, which is usually met in real word applications such as web graphs and others. Our experiments indicate that, in this model, both kernelization algorithms (and, specially, the Nemhauser & Trotter algorithm) reduce considerably the input size of the problem and can serve as very good preprocessing algorithms for vertex cover, on the preferential attachment graphs. 1
On the influence of the seed graph in the preferential attachment model, Preprint available on arxiv
, 1401
"... We study the influence of the seed graph in the preferential attachment model, focusing on the case of trees. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from ..."
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We study the influence of the seed graph in the preferential attachment model, focusing on the case of trees. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from a total variation point of view. We take a first step in proving this conjecture by showing that seeds with different degree profiles lead to different limiting distributions for the (appropriately normalized) maximum degree, implying that such seeds lead to different (in total variation) limiting trees. 1
TWITTER EVENT NETWORKS AND THE SUPERSTAR MODEL
, 1211
"... Abstract. Motivated by “condensation ” phenomena often observed in social networks such as Twitter where one “superstar ” vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail, we formulate a mathematically tractable model fo ..."
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Abstract. Motivated by “condensation ” phenomena often observed in social networks such as Twitter where one “superstar ” vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail, we formulate a mathematically tractable model for this phenomenon which provides a better fit to empirical data than the standard preferential attachment model across an array of networks observed in Twitter. Using embeddings in an equivalent continuous time version of the process, and adapting techniques from the stable agedistribution theory of branching processes, we prove limit results for the proportion of edges that condense around the superstar, the degree distribution of the remaining vertices, maximal nonsuperstar degree asymptotics, and height of these random trees in the large network limit. 1. Retweet Graphs and a mathematically tractable Model Our goal here is to provide a simple model that captures the most salient features of a natural graph that is determined by the Twitter traffic generated by public events. In the Twitter world (or Twitterverse), each user has a set of followers; these are people who have signedup to receive the tweets of the user. Here our focus is on retweets; these are
DISTINGUISHING VERTICES OF INHOMOGENEOUS RANDOM GRAPHS
, 2013
"... We explore under what conditions simple combinatorial attributes and algorithms such as the distance sequence and degreebased partitioning and refinement can be used to distinguish vertices of inhomogeneous random graphs. In the classical setting of ErdősRenyi graphs and random regular graphs it h ..."
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We explore under what conditions simple combinatorial attributes and algorithms such as the distance sequence and degreebased partitioning and refinement can be used to distinguish vertices of inhomogeneous random graphs. In the classical setting of ErdősRenyi graphs and random regular graphs it has been proven that vertices can be distinguished in a constant number of rounds of degreebased refinement or the distance sequence at a logarithmic distance. This yields a highprobability canonical labeling algorithm, and hence an efficient highprobability isomorphism test. In this paper we analyze the same attributes in the context of random graphs that come from distributions that more closely model realworld networks. We first prove a technical result about the effects of one refinement step in the general setting were edges are chosen independently at random. This allows us to prove that an algorithm based on distinguishing vertices yields a canonical labeling for graphs with scalefree degree distribution where edges are added independently, and for Stochastic Kronecker Product Graphs with certain settings of the parameters. Along the way we prove results on the degree distribution, connectedness and diameter of Stochastic Kronecker Graphs with generating matrices of arbitrary size.