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

Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

This work presents exact expressions for size distributions of weak/multilayer connected components in two generalisations of the configuration model: networks with directed edges and multiplex networks with 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 two-layer connected components in multiplex networks exhibits exponent $-\frac{3}{2}$ in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents, $-\frac{1}{2}$ and $-\frac{3}{2}$

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

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.Comment: 19 pages, 7 figure

How to Upscale The Kinetics of Complex Microsystems

The rate constants of chemical reactions are typically inferred from slopes and intersection points of observed concentration curves. In small systems that operate far below the thermodynamic limit, these concentration profiles become stochastic and such an inference is less straightforward. By using elements of queuing theory, we introduce a procedure for inferring (time dependent) kinetic parameters from microscopic observations that are given by molecular simulations of many simultaneously reacting species. We demonstrate that with this procedure it is possible to assimilate the results of molecular simulations in such a way that the latter become descriptive on the macroscopic scale. As an example, we upscale the kinetics of a molecular dynamics system that forms a complex molecular network. Incidentally, we report that the kinetic parameters of this system feature a peculiar time and temperature dependences, whereas the probability of a network strand to close a cycle follows a universal distribution

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.Comment: 38 pages, 22 figure

Statistical Modelling of Pre-Impact Velocities in Car Crashes

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 deter- mine probabilities on pre-impact 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 two-car collision model