Mass distribution exponents for growing trees

Abstract

We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that measures the repartition of the mass of large trees between their different subtrees. This distribution is shown to be a broad distribution and we derive explicit expressions for scaling exponents that characterize its behavior when one subtree is much smaller than the others. We show in particular the existence of various regimes with different values of these mass distribution exponents. Our results are corroborated by a number of exact solutions for particular solvable cases, as well as by numerical simulations.

Random tree growth by vertex splitting

“…Firstly, it contains as a very special subclass the so called preferential attachment models, which have been extensively studied in the literature (see e.g. [1,12]). This model consists in allowing only growth by attaching a new vertex by an edge to an already existing k-vertex, with an attachment rate w k depending only on the degree k of the initial vertex.…”

“…we do not allow splittings of vertices of degree d that produce vertices of degree d + 1. If the partitioning weights are chosen such that w i,j = 0 for i = 1 or j = 1, then the vertex-splitting model is equivalent to the preferential attachment model discussed in [13].…”

“…In this paper we introduce and study a general model of growing trees, where growth occurs by random vertex splitting, with very general rules. This model includes as special limit cases several models of growing trees already studied in the physics and the mathematics literature: the well-known preferential attachment model (which models for example the growth of branched polymers out of a soup of monomers or the growth of the Internet [1,12]); the α-model [15] which is a stochastic model of cladograms (binary leaf-labelled trees studied in relation to phylogenetic trees and biological systematics) and its extensions [11]. It is also related to a tree growth model that arises in the theory of RNA secondary structures [13].…”

“…This growth process becomes the preferential attachment growth model [1,13] when we take w j,k = 0 unless j or k is equal to 1. As already mentioned, similar processes of random vertex splitting and random vertex merging are used as ergodic moves in Monte Carlo simulations of triangulations in quantum gravity, see Figure 2.…”

Markov branching in the vertex splitting model

“…When α = 0 there is no grafting and the model reduces to the model of preferential attachment (see e.g. [2,10]) with linear attachment kernel w k . When α = 1 there is no attaching and we simply have a growing linear graph.…”