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Relevance of Massively Distributed Explorations of the Internet Topology: Simulation Results
, 2005
"... Internet maps are generally constructed using the traceroute tool from a few sources to many destinations. It appeared recently that this exploration process gives a partial and biased view of the real topology, which leads to the idea of increasing the number of sources to improve the quality of ..."
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Cited by 42 (14 self)
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Internet maps are generally constructed using the traceroute tool from a few sources to many destinations. It appeared recently that this exploration process gives a partial and biased view of the real topology, which leads to the idea of increasing the number of sources to improve the quality of the maps. In this paper, we present a set of experiments we have conduced to evaluate the relevance of this approach. It appears that the statistical properties of the underlying network have a strong influence on the quality of the obtained maps, which can be improved using massively distributed explorations. Conversely, we show that the exploration process induces some properties on the maps. We validate our analysis using realworld data and experiments and we discuss its implications.
Distances in random graphs with finite mean and infinite variance degrees.
 Electron. J. Probab.,
, 2007
"... Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph wi ..."
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Cited by 38 (13 self)
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Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with N nodes is investigated when N → ∞. When τ ∈ (2, 3), this graph distance grows like 2 log log N  log(τ −2) . In different papers, the cases τ > 3 and τ ∈ (1, 2) have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.
Generating stationary random graphs on Z with prescribed i.i.d
 degrees, Adv. Appl. Probab
, 2006
"... Let F be a probability distribution with support on the nonnegative integers. Two algorithms are described for generating a stationary random graph, with vertex set Z, so that the degrees of the vertices are i.i.d. random variables with distribution F. Focus is on an algorithm where, initially, a r ..."
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Cited by 11 (7 self)
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Let F be a probability distribution with support on the nonnegative integers. Two algorithms are described for generating a stationary random graph, with vertex set Z, so that the degrees of the vertices are i.i.d. random variables with distribution F. Focus is on an algorithm where, initially, a random number of “stubs ” with distribution F is attached to each vertex. Each stub is then randomly assigned a direction, left or right, and the edge configuration is obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a welldefined configuration, but that the expected length of the shortest edge of a vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions gives infinite mean for the total length of the edges of a given vertex. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.
Diameters in preferential attachment models
, 2009
"... In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. There is a substantial amount of literature proving that, in quite generality, PAgraphs possess powerlaw degree sequences with exponent τ> 2. Th ..."
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In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. There is a substantial amount of literature proving that, in quite generality, PAgraphs possess powerlaw degree sequences with exponent τ> 2. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider linear PAmodels. We prove that the diameter is bounded by a constant times log t, where t is the size of the graph. When the powerlaw exponent τ exceeds 3, then we also log t log log t prove a lower bound of the form, while when τ ∈ (2, 3), we improve the upper bound to a constant times log log t. These bounds are consistent with predictions by physicists that the distances in PAgraphs are similar to the ones in other scalefree random graphs, where distances have been shown to be of order log log t, when τ ∈ (2, 3), and of order log t when τ> 3. 1
ELECTRONIC COMMUNICATIONS in PROBABILITY STATIONARY RANDOM GRAPHS ON Z WITH PRE SCRIBED IID DEGREES AND FINITE MEAN CON
, 2006
"... Let F be a probability distribution with support on the nonnegative integers. A model is proposed for generating stationary simple graphs on Z with degree distribution F and it is shown for this model that the expected total length of all edges at a given vertex is finite if F has finite second mom ..."
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Let F be a probability distribution with support on the nonnegative integers. A model is proposed for generating stationary simple graphs on Z with degree distribution F and it is shown for this model that the expected total length of all edges at a given vertex is finite if F has finite second moment. It is not hard to see that any stationary model for generating simple graphs on Z will give infinite mean for the total edge length per vertex if F does not have finite second moment. Hence, finite second moment of F is a necessary and sufficient condition for the existence of a model with finite mean total edge length. 1