The Distribution of Second Degrees in the Buckley-Osthus Random Graph Model
نویسندگان
چکیده
We consider a random graph process analogous to one suggested by Barabási and Albert. Their model can be roughly described as follows. At each step we add one vertex and one outgoing edge, and the probability that the new vertex is connected to a vertex v is proportional to the degree of v. These types of processes serve as models of real-world networks. In this paper we consider a well-known generalization of the Barabási and Albert model – the Buckley-Osthus model. Buckley and Osthus proved that in this model the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from this vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrand concentration inequality over product spaces. As far as we know this is the only application of Talagrand inequality to random web graphs, where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial, and requiring certain novelty.
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عنوان ژورنال:
- Internet Mathematics
دوره 9 شماره
صفحات -
تاریخ انتشار 2013