Exactly solvable model of a coalescing random graph

Kinetic model of two-monomer polymerization

We propose a kinetic gelation model of polymer growth with two monomeric types that have distinct functionalities (reaction sites), and can polymerize using different reaction types. The heterotypic aggregation of two monomer types is modeled using a moment generating function approach by tracking the temporal evolution of a closed system of moment equations up until gelation. We investigate several scenarios of polymerization with two distinct monomers that differ in the types of reactions that can occur. We determine numerical and analytical conditions for finite time blow-up (the emergence of an oligomer of infinite size) that depend on initial conditions, reaction rates, and number of reaction sites per monomer.

Kinetics of aggregation with choice

We generalize the ordinary aggregation process to allow for choice. In ordinary aggregation, two random clusters merge and form a larger aggregate. In our implementation of choice, a target cluster and two candidate clusters are randomly selected and the target cluster merges with the larger of the two candidate clusters. We study the long-time asymptotic behavior and find that as in ordinary aggregation, the size density adheres to the standard scaling form. However, aggregation with choice exhibits a number of different features. First, the density of the smallest clusters exhibits anomalous scaling. Second, both the small-size and the large-size tails of the density are overpopulated, at the expense of the density of moderate-size clusters. We also study the complementary case where the smaller candidate cluster participates in the aggregation process and find an abundance of moderate clusters at the expense of small and large clusters. Additionally, we investigate aggregation processes with choice among multiple candidate clusters and a symmetric implementation where the choice is between two pairs of clusters.

Emergence of the giant weak component in directed random graphs with arbitrary degree distributions

The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for the existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition, we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a predefined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phase-transition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the step-polymerization formalism, the new results yield Flory-Stockmayer gelation theory and generalize it to a broader scope.

Source-enhanced coalescence of trees in a random forest

The time evolution of a random graph with varying number of edges and vertices is considered. The edges and vertices are assumed to be added at random by one at a time with different rates. A fresh edge connects either two linked components and forms a new component of larger order g (coalescence of graphs) or increases (by one) the number of edges in a given linked component (cycling). Assuming the vertices to have a finite valence (the number of edges connected with a given vertex is limited) the kinetic equation for the distribution of linked components of the graph over their orders and valences is formulated and solved exactly by applying the generating function method for the case of coalescence of trees. The evolution process is shown to reveal a phase transition: the emergence of a giant linked component whose order is comparable to the total order of the graph. The time dependencies of the moments of the distribution of linked components over their orders and valences are found explicitly for the pregelation period and the critical behavior of the spectrum is analyzed. It is found that the linked components are γ distributed over g with the algebraic prefactor g-5/2. The coalescence process is shown to terminate by the formation of the steady-state γ spectrum with the same algebraic prefactor.