Ensemble of causal trees

2004/03/27 by P. Białas, P. Bialas, Bialas, P.
Environmental Science · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Sustainability and Ecological Systems Analysis #Theoretical and Computational Physics #cond-mat.stat-mech

paper · pdf · doi:10.48550/arxiv.cond-mat/0403669

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

arxiv created 2004/03/27 · arxiv updated 2009/12/01

Abstract

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.

Related

• Causal Models for Growing Networks
• The Recovery of Causal Poly-Trees from Statistical Data
• Geometric and spectral properties of causal maps
• Co-evolving dynamic networks
• Condensation phenomena in preferential attachment trees with neighbourhood influence