
I. Kryven. Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions, Physical Review E 96 (5), 052304, 2017
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This work presents exact expressions for size distributions of weak and multilayer connected components in two generalizations of the configuration model: networks with directed edges and multiplex networks with an arbitrary number of layers. The expressions are computable in a polynomial time and, under some restrictions, are tractable from the asymptotic theory point of view. If first partial moments of the degree distribution are finite, the size distribution for twolayer connected components in multiplex networks exhibits an exponent 3/2
in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents
−1/2 and −3/2.

I. Kryven. Analytic results on the polymerisation random graph model, J Math Chem, 2017, DOI:10.1007/s1091001707851,
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The stepgrowth polymerisation of a mixture of arbitraryfunctional
monomers is viewed as a timecontinuos 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.

V. Schamboeck, I. Kryven, P. Iedema. Acrylate Network Formation by FreeRadical Polymerization Modeled Using Random Graphs, Macromolecular Theory and Simulations 2017, DOI:10.1002/mats.201700047
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A novel technique is developed to predict the evolving topology of a diacrylate
polymer network under photocuring conditions, covering the lowviscous
initial state to full transition into polymer gel. The model is based on a new
graph theoretical concept being introduced in the framework of population
balance equations (PBEs) for monomer states (mPBEs). A trivariate degree
distribution that describes the topology of the network locally is obtained
from the mPBE, which serves as an input for a directional random graph
model. Thus, access is granted to global properties of the acrylate network
which include molecular size distribution, distributions of molecules with a
specific number of crosslinks/radicals, gelation time/conversion, and gel/sol
weight fraction. Furthermore, an analytic criterion for gelation is derived. This
criterion connects weight fractions of converted monomers and the transition
into the gel regime. Valid results in both sol and gel regimes are obtained by
the new model, which is confirmed by a comparison with a “classical” macromolecular
PBE model. The model predicts full transition of polymer into gel at
very low vinyl conversion. Typically, this lowconversion network is very
sparse, as becomes apparent from the predicted crosslink distribution.

I. Kryven. General expression for componentsize distribution in infinite configuration networks, Physical Review E 95, 052303, 2017
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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.

I. Kryven. Emergence of the giant weakcomponent in directed random graphs with arbitrary degree distributions, Physical Review, E 94, 012315, 2016
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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 weakcomponent criterion to
obtain the exact time of the phase transition. The phasetransition time is obtained as a function of the distribution
that bounds the degrees. Remarkably, when viewed from the steppolymerization formalism, the new results
yield FloryStockmayer gelation theory and generalize it to a broader scope.

I. Kryven, J. Duivenvoorden, J. Hermans, P. Iedema. Random graph approach to multifunctional molecular networks, Macromolecular Theory and Simulations 25 (5), 2016, 449465.
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Formation of a molecular network from multifunctional precursors is modelled with a random graph process. The process does not account for spatial positions of the monomers explicitly, yet the Euclidean distances between the monomers are derived from the topological information by applying selfavoiding random walks. This allows favoring reactivity of monomers that are close to each other, and to disfavor the reactivity for monomers obscured by the surrounding. As a result, the model is applicable to large time scales. The phenomena of conversiondependent reaction rates, gelation, microgelation, and structural inhomogeneity are predicted by the model. Resulting nonhomogeneous network topologies are analyzed to extract such descriptors as: size distribution, crosslink distances, and gelpoint conversion. Furthermore, new to the molecular simulation community descriptors are suggested that are especially useful when explaining evolution of the gel as being a single molecule: local clustering coefficient, network modularity, cluster size distribution.
 I. Kryven, S. Rüoblitz, Ch. Schütte.
Solution of the chemical master equation by
radial basis functions approximation with
interface tracking,
BMC System Biology 2015
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Background: The chemical master equation is the fundamental equation of
stochastic chemical kinetics. This differentialdifference equation describes
temporal evolution of the probability density function for states of a chemical
system. A state of the system, usually encoded as a vector, represents the
number of entities or copy numbers of interacting species, which are changing
according to a list of possible reactions. It is often the case, especially when the
state vector is highdimensional, that the number of possible states the system
may occupy is too large to be handled computationally. One way to get around
this problem is to consider only those states that are associated with probabilities
that are greater than a certain threshold level.
Results: We introduce an algorithm that significantly reduces computational
resources and is especially powerful when dealing with multimodal distributions.
The algorithm is built according to two key principles. Firstly, when performing
time integration, the algorithm keeps track of the subset of states with significant
probabilities (essential support). Secondly, the probability distribution that solves
the equation is parametrised with a small number of coefficients using collocation
on Gaussian radial basis functions. The system of basis functions is chosen in
such a way that the solution is approximated only on the essential support instead
of the whole state space. In order to demonstrate the effectiveness of the method,
we consider four application examples: a) the selfregulating gene model, b) the
2dimensional bistable toggle switch, c) a generalisation of the bistable switch to
a 3dimensional tristable problem, and d) a 3dimensional cell differentiation
model that, depending on parameter values, may operate in bistable or tristable
modes. In all multidimensional examples the manifold containing the system
states with significant probabilities undergoes drastic transformations over time.
This fact makes the examples especially challenging for numerical methods.
Conclusions: The proposed method is a new numerical approach permitting to
approximately solve a wide range of problems that have been hard to tackle until
now. A full representation of multidimensional distributions is recovered. The
method is especially attractive when dealing with models
 I. Kryven, PD. Iedema.
Deterministic modelling of copolymer microstructure: composition drift and sequence patterns,
Macromolecular Reaction Engineering [Special Issue: Statistical Modeling Tools and Approaches for Polymerization Reaction Engineering], Volume 9, Issue 3, pages 285–306, 2015;
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A concept of a population balance is applied to study evolving structures of propagating linear copolymer chains. As a result, a general numerical toolbox was developed capable to handle various copolymerization mechanisms utilizing twodimensional distributions as means of system description. The method was applied to free radical copolymerization of styrene–acrylonitrile explaining peculiar bimodality in chain length distribution observed for asymmetrical initial monomer ratios.
 I. Kryven, PD. Iedema.
Transition into the gel regime for crosslinking radical polymerisation in a continuously stirred tank reactor,
Chemical Engineering Science, Volume 126, Issue 14, Pages 296–308, 2015
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Crosslinking radical polymerisation in a continuously stirred tank reactor has been studied by means of a fourdimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiradicals as dimensions. The model covers both pregel and gel regimes in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The comparison with Monte Carlo simulations shows interesting and unexpected features.

X. Cao,
A. van Dam,
B. de Leeuw,
C. Geldhauser,
J. Grasman,
I. Kryven,
D. Lahaye,
L. Morelli,
V. Rottschäfer,
H. Zhou.
Always Nice Weather in Europe.
Proceedings SWI 2015
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Weather forecasting relies on mathematical models that exhibit chaotic behavior. This
renders the solution of these models very sensitive to errors in the model, to choices of the
initial conditions and to rounding errors in the numerical solution procedure. Over the course
of the past decade, various meteorological institutes in Europe have developed different at
mosphere models. Each of these models has its strengths and weaknesses. The principle
behind the socalled Super Modeling approach is to merge these existing models into a single
larger model to combine common strengths while overcoming individual weaknesses. This
approach was initially proposed and developed by the KNMI in the Netherlands to improve
the reliability of its weather forecasts. The task formulated for this Study Group problem was
to reevaluate the Super Modeling approach and to formulate recommendations for its future
development.
 I. Kryven,
Topology evolution in macromolecular networks,
Univeristy of Amsterdam, PhD Thessis, ISBN: 9789090285191, 2014;
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Governed by various intermolecular forces, molecular networks tend to evolve
from simple to very complex formations that have random structure. This
randomness in the connectivity of the basic units can still be
captured employing distributional description of the state of the system;
the evolution itself by particular stochastic processes, for instance
Smoluchowski coagulation. The Smoluchowski coagulation equation can be
extended to include collisions of orders distinct from 2, which allows
developing a framework that covers various special cases being far beyond
the scope of the original application of the coagulation equation. The
combination of the population balance equation based on a generalization of
the Smoluchowski coagulation and a meshless projection method based on
Gaussian basis functions has been found to be a powerful tool allowing to
solve many problems of prior unmanageable complexity. For instance, in the
case of crosslinking polymerisation in bulk, the population balance model
describing a particular set of reaction mechanisms has been present in
literature for decades before it has been successfully and to a full extent
solved by the numerical method. The other case studies address formation of
various molecular networks in a few important cases: polymerisation of AB2
monomers, modification of linear chains into a branched
polymer, copolymerisation with a composition drift, and coagulation/
coalescence of colloids of variable fractal dimension.
 I. Kryven, PD. Iedema.
Transition into the gel regime for free radical crosslinking polymerisation in a batch reactor,
Polymer, Volume 55, Issue 16, Pages 3475–3489, 2014;
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Crosslinking polymerization has been studied by means of a fourdimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiradicals as dimensions. The model, for the first time and to a full extent resolves the crosslinking problem as formulated by Zhu et al.
[1] and covers both pregel and gel regimes, in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The model has been validated with data from an experimental crosslinking polymerization, Methyl Methacrylate with Ethylene Glycol Dimethacrylate. Nontrivial patterns in the time evolution of average quantities like crosslink densities, partly observed in prior studies [
2–
3], are naturally emerging from the model by computing marginals of the fourdimensional distribution possessing an interesting multimodal structure.
 I. Kryven, S. Lazzari, G. Storti.
Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems,
Macromolecular Theory and Simulations, Volume 55, Issue 16, Pages 3475–3489, 2014;
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A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation (PBE) based on cluster mass and fractal dimension is solved, employing a discretization method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusionlimitedcluster aggregation (DLCA) and reactionlimitedclusteraggregation (RLCA) kernels and different coalescence rates. It turns out that in both DLCA and RLCA regimes, a faster coalescence leads to smaller sized and more compact clusters, whereas a slow coalescence promotes the formation of highly reactive
 I. Kryven, PD. Iedema.
Topology Evolution in Polymer Modification,
Macromolecular Theory and Simulations, Volume 23, Issue 1, pages 7–14, 2014
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[JOURNAL COVER]
A recent numerical method has opened new opportunities in multidimensional population balance modeling. Here, this method is applied to a full threedimensional population balance model (PBM) describing branching topology evolution driven by chain end to backbone coupling. This process is typical for polymer modification reactions, e.g., in polyethylene, where initially linear polymer chains undergo hydrogen abstraction, and subsequent branching or scission. Topologies are distinguished by chain ends, number of branches, and number of reactive ends. The resulting time dependent trivariate distribution is utilized to extract various distributive properties of the polymer. The results exhibit excellent agreement with data from Monte Carlo simulations
 I. Kryven, PD. Iedema.
Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding,
Volume 54, Issue 14, Pages 3472–3484, 2013
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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.
 I. Kryven, A. Berkenbos, P. Melo, DM. Kim. PD. Iedema.
Modeling Crosslinking Polymerization in Batch and Continuous Reactors,
Volume 7, Issue 5, pages 205–220, 2013
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A new pseudodistribution approach is applied to the modeling of crosslinking copolymerization of vinyl and divinyl monomer and compared to Monte Carlo (MC) simulations. With the number of free pending double bonds as the main distribution variable, a rigorous solution of the three leading moments of the molecular size distribution becomes possible. Validation takes place with data of methyl methacrylate with ethylene glycol dimethacrylate. Well within the sol regime perfect agreement is found, but near the gelpoint larger discrepancies do appear. This is probably due to the existence of multiradicals that are not taken into account in the population balance approaches.
 T. van der Aalst, D. Denteneer, H. Döring, MH. Duong, RJ. Kang, M. Keane, J. Kool, I. Kryven, T. Meyfroyt, T. Müller, G. Regts, J. Tomczyk
The random disc thrower problem,
Proceedings SWI 2013, Pages 5978, 2013
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We describe a number of approaches to a question posed by Philips Research, described as the "random disc thrower" problem. Given a square grid of points in the plane, we cover the points by equalsized planar discs according to the following random process. At each step, a random point of the grid is chosen from the set of uncovered points as the centre of a new disc. This is an abstract model of spatial reuse in wireless networks. A question of Philips Research asks what, as a function of the grid length, is the expected number of discs chosen before the process can no longer continue? Our main results concern the onedimensional variant of this problem, which can be solved reasonably well, though we also provide a number of approaches towards an approximate solution of the original twodimensional problem. The twodimensional problem is related to an old, unresolved conjecture ([6]) that has been the object of close study in both probability theory and statistical physics. Keywords: generating functions, Markov random fields, random sequential adsorption, Rényi’s parking problem, wireless networks
 V. Kukharskyy, Ya. Savula, I. Kryven
Modified method of residualfree bubbles for solving the advectiondiffusion problem with high Peclet number,
Series Appl. Math. and Informatics. Visnyk of the Lviv University, Volum 20, Pages 8594, 2013
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The numerical scheme based on the residual free bubbles approach and the hierarchy basis discretization for advectiondiffusion problems with dominated advection is introduced. Among properties of the developed method are better stability to oscillations for hight Peclet numbers, better time performance, and simplicity in parallel computing application. The computational analysis of convergence and time performance on an example of a 2D symmetric advectiondiffusion problem for 10e10 Peclet number is performed.
 I. Kryven, PD. Iedema.
A Novel Approach to Population Balance Modeling of Reactive Polymer Modification Leading to Branching,
Volume 22, Issue 2, pages 89–106, 2013
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[JOURNAL COVER]
The mathematical treatment of polymer modification systems, described by population balances containing convolution is discussed. The twodimensional case (molecular weight vs. number of branch points) was considered by utilizing approximations of distributions, expanding them in terms of Gaussian basis functions. Three branching reactions were addressed: chain backbone to chain end point coupling; threefunctional coupling of chain ends; and crosslinking. The results were compared to those of Monte Carlo (MC) simulations. Good agreement was observed, although the quality of a distribution as generated by the numerical approach is much better in view of the strong scatter in the MC data.
 J.B. van den Berg, R. Castro, J. Draisma, J. Evers, M. Hendriks, O. Krehel, I. Kryven, K. Mora, B. Szabó, P. Zwiernik.
Nonimaging Optics for LEDLighting,
Proceedings of SWI 2012, Editor MAA. Boon, ISBN 9789064646300, Pages 70103, 2013
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In this report, several methods are investigated to rapidly compute the light intensity function, either in the far field or on a finitedistance screen, of light emanating from a light fixture with a given shape. Different shapes are considered, namely polygonal and (piecewise) smooth. In the first case, analytic methods are sought to circumvent the use of Monte Carlo methods and raytracing with large sample size. In the second case, refinements of the Monte Carlo method (notably using a bootstrap procedure) are devised to minimize the number of samples needed for a good approximation of the intensity function.
 W. Kager, I. Kryven, K. Myerscough, T. van Opstal, T. Rot.
Statistical Modelling of PreImpact Velocities in Car Crashes,
Proceedings SWI 2011, Pages 133148, 2013
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The law wants to determine if any party involved in a car crash is guilty. The Dutch court invokes the expertise of the Netherlands Forensic Institute (NFI) to answer this question. We discuss the present method of the NFI to determine probabilities on preimpact car velocities, given the evidence from the crash scene. A disadvantage of this method is that it requires a prior distribution on the velocities of the cars involved in the crash. We suggest a different approach, that of statistical significance testing, which can be carried out without a prior. We explain this method, and apply it to a toy model. Finally, a sensitivity analysis is performed on a simple twocar collision model.