Tree Networks with Causal Structure

TL;DR

This paper analyzes the geometry and causal structure of tree networks using statistical mechanics, deriving formulas for degree distribution, correlations, and network dimensions, revealing that causal networks typically have infinite Hausdorff dimension.

Contribution

It introduces an analytical framework for causal tree networks, deriving key properties and contrasting their dimensions with random trees.

Findings

Causal networks have infinite Hausdorff dimension.

Derived formulas for degree distribution and ancestor-descendant correlations.

Causal structure influences network geometry significantly.

Abstract

Geometry of networks endowed with a causal structure is discussed using the conventional framework of 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 formulae are derived, 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, in contrast to the maximally random trees, where it is generically finite.