arXiv subjects

Ivan Kryven

Publications and source records attributed to Ivan Kryven.

25 records · Page 2Linked to original sources

arXiv2018

Dynamic Networks that Drive the Process of Irreversible Step-Growth Polymerization

Many research fields, reaching from social networks and epidemiology to biology and physics, have experienced great advance from recent developments in random graphs and network theory. In this paper we propose to view percolation on a directed random graph as a generic model for step-growth polymerisation. This polymerisation process is used to manufacture a broad range of polymeric materials, including: polyesters, polyurethanes, polyamides, and many others. We link features of step-growth polymerisation to the properties of the directed configuration model, and in this way, obtain new analytical expressions describing the polymeric microstructure. Thus, the molecular weight distribution is related to the sizes of connected components, gelation to the emergence of the giant component, and the molecular gyration radii to the Wiener index of these components. A model on this level of generality is instrumental in accelerating the design of new materials and optimizing their properties.

cond-mat.soft↗

arXiv2018

How does bond percolation happen in coloured networks?

Percolation in complex networks is viewed as both: a process that mimics network degradation and a tool that reveals peculiarities of the underlying network structure. During the course of percolation, networks undergo non-trivial transformations that include a phase transition in the connectivity, and in some special cases, multiple phase transitions. Here we establish a generic analytic theory that describes how structure and sizes of all connected components in the network are affected by simple and colour-dependant bond percolations. This theory predicts all locations where the phase transitions take place, existence of wide critical windows that do not vanish in the thermodynamic limit, and a peculiar phenomenon of colour switching that occurs in small connected components. These results may be used to design percolation-like processes with desired properties, optimise network response to percolation, and detect subtle signals that provide an early warning of a network collapse.

physics.soc-ph↗

arXiv2017

Analytical results on the polymerisation random graph model

The step-growth polymerisation of a mixture of arbitrary-functional monomers is viewed as a time-continuos random graph process with degree bounds that are not necessarily the same for different vertices. The sequence of degree bounds acts as the only input parameter of the model. This parameter entirely defines the timing of the phase transition. Moreover, the size distribution of connected components features a rich temporal dynamics that includes: switching between exponential and algebraic asymptotes and acquiring oscillations. The results regarding the phase transition and the expected size of a connected component are obtained in a closed form. An exact expression for the size distribution is resolved up to the convolution power and is computable in subquadratic time. The theoretical results are illustrated on a few special cases, including a comparison with Monte Carlo simulations.

math.CO↗

arXiv2017

General expression for the component size distribution in infinite configuration networks

In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of connected components. This equation is suitable for fast and stable numerical computations up to the machine precision. The analytical analysis reveals that the asymptote of the component size distribution is completely defined by only a few parameters of the degree distribution: the first three moments, scale and exponent (if applicable). When the degree distribution features a heavy tail, multiple asymptotic modes are observed in the component size distribution that, in turn, may or may not feature a heavy tail.

math.CO↗

arXiv2016

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 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 pre-defined 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.

math.CO↗

arXiv2016

Random graph approach to multifunctional molecular networks

Formation of a molecular network from multifunctional precursors is modelled with a random graph process. The random graph model favours reactivity for monomers that are positioned close in the network topology, and disfavours reactivity for those that are obscured by the surrounding. The phenomena of conversion-dependant reaction rates, gelation, and micro-gelation are thus naturally predicted by the model and do not have to be imposed. Resulting non-homogeneous network topologies are analysed to extract such descriptors as: size distribution, crosslink distances, and gel-point conversion. Furthermore, new to the molecular simulation community descriptors are invented. These descriptors are especially useful for understanding evolution of pure gel, amongst them: cluster coefficient, network modularity, cluster size distribution.

cond-mat.soft↗

arXiv2013

Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding

A deterministic mathematical model for the polymerization of hyperbranched molecules accounting for substitution, cyclization, and shielding effect has been developed as a system of nonlinear population balances. The solution obtained by a novel approximation method shows perfect agreement with the analytical solution in limiting cases and provides, for the first time in this class of polymerization problems, full multidimensional results.

math-ph↗