Tree Networks with Causal Structure

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.

arXiv comment9 pages, 2-column revtex format, 1 eps figure, misprints corrected

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• Fractality in complex networks: critical and supercritical skeletons
• Complex Quantum Network Geometries: Evolution and Phase Transitions
• Homogeneous complex networks
• Duality between equilibrium and growing networks
• Transition from small to large world in growing networks
• Statistical mechanics of random graphs
• Power laws in zero-range processes on random networks
• Average distance in growing trees
• Ferromagnetic Ising spin systems on the growing random tree
• Ageing of complex networks
• Approaching the thermodynamic limit in equilibrated scale-free networks
• Is Kinetic Growth Walk equivalent to canonical Self Avoiding Walk?
• A simple stochastic model for the evolution of protein lengths
• Memory effect in growing trees