Memory effect in growing trees

TL;DR

This paper demonstrates that growing trees retain a memory of their initial structure, with the effect diminishing as nodes connect with multiple links, supported by theoretical equations and numerical simulations.

Contribution

It introduces a mathematical framework showing how initial graph structures influence the evolution of exponential and scale-free trees.

Findings

Memory effect is evident in exponential trees through iterative equations.

Numerical simulations confirm the persistence of initial structure influence.

Memory effect diminishes when nodes connect with multiple links.

Abstract

We show that the structure of a growing tree preserves an information on the shape of an initial graph. For the exponential trees, evidence of this kind of memory is provided by means of the iterative equations, derived for the moments of the node-node distance distribution. Numerical calculations confirm the result and allow to extend the conclusion to the Barabasi--Albert scale-free trees. The memory effect almost disappears, if subsequent nodes are connected to the network with more than one link.