11,495 research outputs found
Weighted distances in scale-free preferential attachment models
We study three preferential attachment models where the parameters are such
that the asymptotic degree distribution has infinite variance. Every edge is
equipped with a non-negative i.i.d. weight. We study the weighted distance
between two vertices chosen uniformly at random, the typical weighted distance,
and the number of edges on this path, the typical hopcount. We prove that there
are precisely two universality classes of weight distributions, called the
explosive and conservative class. In the explosive class, we show that the
typical weighted distance converges in distribution to the sum of two i.i.d.
finite random variables. In the conservative class, we prove that the typical
weighted distance tends to infinity, and we give an explicit expression for the
main growth term, as well as for the hopcount. Under a mild assumption on the
weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear
in Random Structures and Algorithm
A preferential attachment model with random initial degrees
In this paper, a random graph process is studied and its
degree sequence is analyzed. Let be an i.i.d. sequence. The
graph process is defined so that, at each integer time , a new vertex, with
edges attached to it, is added to the graph. The new edges added at time
t are then preferentially connected to older vertices, i.e., conditionally on
, the probability that a given edge is connected to vertex i is
proportional to , where is the degree of vertex
at time , independently of the other edges. The main result is that the
asymptotical degree sequence for this process is a power law with exponent
, where is the power-law exponent
of the initial degrees and the exponent predicted
by pure preferential attachment. This result extends previous work by Cooper
and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is
incomplete. This version contains the complete proo
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