A random mapping with preferential attachment (original) (raw)
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Random Graphs Associated to some Discrete and Continuous Time Preferential Attachment Models
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We give a common description of Simon, Barab\'asi--Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barab\'asi--Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter alpha\alphaalpha) goes to infinity, a portion of them behave as a Yule model with parameters (lambda,beta)=(1−alpha,1)(\lambda,\beta) = (1-\alpha,1)(lambda,beta)=(1−alpha,1), and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in \cite{Newman2005}. References to traditional and recent applications of the these models are also discussed.
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A preferential attachment model with random initial degrees
Arkiv för Matematik, 2009
In this paper, a random graph process {G(t)} t≥1 is studied and its degree sequence is analyzed. Let {W t } t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex, with W t 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 G(t − 1), the probability that a given edge is connected to vertex i is proportional to d i (t − 1) + δ, where d i (t − 1) is the degree of vertex i at time t − 1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ = min{τ W , τ P }, where τ W is the power-law exponent of the initial degrees {W t } t≥1 and τ P the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.
Random mappings with exchangeable in-degrees
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In this paper we introduce a new random mapping model, TD n , which maps the set {1, 2, ..., n} into itself. The random mapping TD n is constructed using a collection of exchangeable random variableŝ D 1 , ....,D n which satisfy n i=1D i = n. In the random digraph, GD n , which represents the mapping TD n , the in-degree sequence for the vertices is given by the variablesD 1 ,D 2 , ...,D n , and, in some sense, GD n can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions ofD 1 ,D 2 , ...,D n . We also consider two special examples of TD n which correspond to random mappings with preferential and anti-preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above.
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