Ensemble of causal trees

Details Ensemble of causal trees Ensemble of causal trees

2004-03-27

arxiv.org/abs/cond-mat/0403669v1

Physics

Condensed Matter

Statistical Mechanics

11 Pages, published in proceedings of Random Geometry Workshop May 15-17, 2003 Krakow

Scientific paper

We discuss the geometry of trees endowed with a causal structure using the conventional framework of equilibrium statistical mechanics. We show how this ensemble is related to popular growing network models. In particular we demonstrate that on a class of afine attachment kernels the two models are identical but they can differ substantially for other choice of weights. We show that causal trees exhibit condensation even for asymptotically linear kernels. We derive general formulae describing the degree distribution, the ancestor-descendant correlation and the probability a randomly chosen node lives at a given geodesic distance from the root. It is shown that the Hausdorff dimension d_H of the causal networks is generically infinite.

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