Random Ancestor Trees

TL;DR

This paper introduces a network growth model based on genealogical selection, revealing a shallow tree structure with a dominant root hub and highly connected nodes, characterized by algebraic degree distributions and specific growth exponents.

Contribution

The study presents a novel genealogically driven network growth model with analytical results on degree distributions and growth exponents, expanding understanding of hierarchical network structures.

Findings

The network forms a very shallow tree with super-exponentially decreasing node fractions at higher distances.

A macroscopic hub at the root coexists with highly connected nodes at higher generations.

Degree distribution follows a power law with an exponent approximately 2.35.

Abstract

We investigate a network growth model in which the genealogy controls the evolution. In this model, a new node selects a random target node and links either to this target node, or to its parent, or to its grandparent, etc; all nodes from the target node to its most ancient ancestor are equiprobable destinations. The emerging random ancestor tree is very shallow: the fraction g_n of nodes at distance n from the root decreases super-exponentially with n, g_n=e^{-1}/(n-1)!. We find that a macroscopic hub at the root coexists with highly connected nodes at higher generations. The maximal degree of a node at the nth generation grows algebraically as N^{1/beta_n} where N is the system size. We obtain the series of nontrivial exponents which are roots of transcendental equations: beta_1= 1.351746, beta_2=1.682201, etc. As a consequence, the fraction p_k of nodes with degree k has algebraic tail, p_k ~ k^{-gamma}, with gamma=beta_1+1=2.351746.