Randomised Reproducing Graphs

Abstract

We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, α, β and γ, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If (1 + α)(1 + γ) < 1 then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on α and γ. If (1 + α)(1 + γ) > 1 then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree d tends to zero. We also give some results on the number of edges and on the spectral gap.

Chris Cannings: A Life in Games

“…The resulting systems could be considered to be a type of graphical generalization of the Leslie population model [97], where not only the number of individuals of different ages is modelled, but also the network structure connecting them. In Jordan [83] a generalization of the reproducing graph model was considered where links are formed at random.…”

Further properties of a random graph with duplications and deletions

“…If, in addition to (17), sequence (k n ) satisfies (18) ∞ n=1 exp −ε 2 k n < ∞, then the Borel-Cantelli lemma gives min M (t) : √ n ≤ t ≤ √ n + 1 > (1 − ε)k n if n is sufficiently large. Consequently, with n = ⌊t 2 ⌋ we have a.s. M (t) ≥ (1 − ε)k n for all sufficiently large t.…”

“…Recently, Hermann and Pfaffelhuber [17, 2014+] have proved several results on the frequency of isolated vertices and cliques, and also on the evolution of the degree of a fixed vertex in the initial graph. Various other models were also introduced, where the choice of the duplicated vertex is not uniform but depends on the degrees (Jordan [18,2011], Cohen, Jordan and Voliotis [10,2010], Farczadi and Wormald [13, 2014+]) or on the state of a hidden Markov chain (Hamdi, Krishnamurthy and Yin [16, 2013+]).…”

Approximation to the moments of ratios of cumulative sums

“…The inverse moment of interest in their application was an index of posterior uncertainty, of the form , where was a randomly weighted sum of random variables. In Jordan () an approximation to the mean of an inverse moment was used to study the behaviour of Markov chains on randomized graphs.…”