Tree networks with causal structure

A geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree graphs, an analytically solvable case. General formulas are derived, describing the degree distribution, the ancestor-descendant correlation, and the probability that 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, in contrast to the maximally random trees where it is generically finite.