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 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, multi-objective optimization.

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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

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.

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 source-sink 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 lower-bounds for monotone maps do not extend to greedy embeddings. We prove a unified O(log n) lower-bound on the dimension of no-stretch greedy embeddings when the host metric is Euclidean or Lobachevsky geometry. The essence of the lower bound entails showing that low-dimensional 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 3-dimensional Lobachevsky spaces using recursive applications of hyperbolic isometries guided by caterpillar-like 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

Random increasing k-trees represent an interesting, useful class of strongly dependent graphs for which analytic-combinatorial tools can be successfully applied. We study in this paper a notion called connectivity-profile and derive asymptotic estimates for it; some interesting consequences will also be given. 1

Abstract. We analyze the structure of random graphs generated by the geographic 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, diameter and clustering coefficient are related to the weight distribution and threshold values. Key words: random graph, geographical threshold graph, giant component, connectivity, clustering coefficient. 1

Abstract. We are calculating the expectation and variance of the number of leaves in the scale-free network model suggested in [CL06]. Afterwards, we prove a Law of Large Numbers and a Central Limit Theorem for the number of leaves using the Martingale Central Limit Theorem. Finally, we check our results by simulation. 1.

Abstract With the recent explosion of popularity of commercial social-networking sites like Facebook and MySpace, the size of social networks that can be studied scientifically has passed from the scale traditionally studied by sociologists and anthropologists to the scale of networks more typically studied by computer scientists. In this chapter, I will highlight a recent line of computational research into the modeling and analysis of the small-world phenomenon—the observation that typical pairs of people in a social network are connected by very short chains of intermediate friends—and the ability of members of a large social network to collectively find efficient routes to reach individuals in the network. I will survey several recent mathematical models of social networks that account for these phenomena, with an emphasis both on provable properties of these social-network models and on the empirical validation of the models against real large-scale social-network data.

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 age-distribution 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 non-superstar 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 signed-up to receive the tweets of the user. Here our focus is on retweets; these are