25,955 research outputs found
Twitter event networks and the Superstar model
Condensation phenomenon is 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 that 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 nonsuperstar degree
asymptotics and height of these random trees in the large network limit.Comment: Published at http://dx.doi.org/10.1214/14-AAP1053 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Trees of self-avoiding walks
We consider the biased random walk on a tree constructed from the set of
finite self-avoiding walks on a lattice, and use it to construct probability
measures on infinite self-avoiding walks. The limit measure (if it exists)
obtained when the bias converges to its critical value is conjectured to
coincide with the weak limit of the uniform SAW. Along the way, we obtain a
criterion for the continuity of the escape probability of a biased random walk
on a tree as a function of the bias, and show that the collection of escape
probability functions for spherically symmetric trees of bounded degree is
stable under uniform convergence
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