Generative Algorithm for Molecular Graphs Uncovers Products of Oil Oxidation.

The autoxidation of triglyceride (or triacylglycerol, TAG) is a poorly understood complex system. It is known from mass spectrometry measurements that, although initiated by a single molecule, this system involves an abundance of intermediate species and a complex network of reactions. For this reason, the attribution of the mass peaks to exact molecular structures is difficult without additional information about the system. We provide such information using a graph theory-based algorithm. Our algorithm performs an automatic discovery of the chemical reaction network that is responsible for the complexity of the mass spectra in drying oils. This knowledge is then applied to match experimentally measured mass spectra with computationally predicted molecular graphs. We demonstrate this methodology on the autoxidation of triolein as measured by electrospray ionization-mass spectrometry (ESI-MS). Our protocol can be readily applied to investigate other oils and their mixtures.

Modeling the free-radical polymerization of hexanediol diacrylate (HDDA): a molecular dynamics and graph theory approach.

In the printing, coating and ink industries, photocurable systems are becoming increasingly popular and multi-functional acrylates are one of the most commonly used monomers due to their high reactivity (fast curing). In this paper, we use molecular dynamics and graph theory tools to investigate the thermo-mechanical properties and topology of hexanediol diacrylate (HDDA) polymer networks. The gel point was determined as the point where a giant component was formed. For the conditions of our simulations, we found the gel point to be around 0.18 bond conversion. A detailed analysis of the network topology showed, unexpectedly, that the flexibility of the HDDA molecules plays an important role in increasing the conversion of double bonds, while delaying the gel point. This is due to a back-biting type of reaction mechanism that promotes the formation of small cycles. The glass transition temperature for several degrees of curing was obtained from the change in the thermal expansion coefficient. For a bond conversion close to experimental values we obtained a glass transition temperature around 400 K. For the same bond conversion we estimate a Young's modulus of 3 GPa. Both of these values are in good agreement with experiments.

Percolation in networks with local homeostatic plasticity.

Percolation is a process that impairs network connectedness by deactivating links or nodes. This process features a phase transition that resembles paradigmatic critical transitions in epidemic spreading, biological networks, traffic and transportation systems. Some biological systems, such as networks of neural cells, actively respond to percolation-like damage, which enables these structures to maintain their function after degradation and aging. Here we study percolation in networks that actively respond to link damage by adopting a mechanism resembling synaptic scaling in neurons. We explain critical transitions in such active networks and show that these structures are more resilient to damage as they are able to maintain a stronger connectedness and ability to spread information. Moreover, we uncover the role of local rescaling strategies in biological networks and indicate a possibility of designing smart infrastructures with improved robustness to perturbations.

Learning heterogenous reaction rates from stochastic simulations.

Reaction rate equations are ordinary differential equations that are frequently used to describe deterministic chemical kinetics at the macroscopic scale. At the microscopic scale, the chemical kinetics is stochastic and can be captured by complex dynamical systems reproducing spatial movements of molecules and their collisions. Such molecular dynamics systems may implicitly capture intricate phenomena that affect reaction rates but are not accounted for in the macroscopic models. In this work we present a data assimilation procedure for learning nonhomogeneous kinetic parameters from molecular simulations with many simultaneously reacting species. The learned parameters can then be plugged into the deterministic reaction rate equations to predict long time evolution of the macroscopic system. In this way, our procedure discovers an effective differential equation for reaction kinetics. To demonstrate the procedure, we upscale the kinetics of a molecular system that forms a complex covalently bonded network severely interfering with the reaction rates. Incidentally, we report that the kinetic parameters of this system feature peculiar time and temperature dependences, whereas the probability of a network strand to close a cycle follows a universal distribution.

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks.

Step-growth and chain-growth are two major families of chemical reactions that result in polymer networks with drastically different physical properties, often referred to as hyper-branched and cross-linked networks. In contrast to step-growth polymerisation, chain-growth forms networks that are history-dependent. Such networks are defined not just by the degree distribution, but also by their entire formation history, which entails a modelling and conceptual challenges. We show that the structure of chain-growth polymer networks corresponds to an edge-coloured random graph with a defined multivariate degree distribution, where the colour labels represent the formation times of chemical bonds. The theory quantifies and explains the gelation in free-radical polymerisation of cross-linked polymers and predicts conditions when history dependance has the most significant effect on the global properties of a polymer network. As such, the edge colouring is identified as the key driver behind the difference in the physical properties of step-growth and chain-growth networks. We expect that this findings will stimulate usage of network science tools for discovery and design of cross-linked polymers.

Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution.

In every network, a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes, and therefore one may speak of a ball, its volume, and how it grows as a function of the radius. Spatial networks tend to feature peculiar volume scaling functions, as well as other topological features, including clustering, degree-degree correlation, clique complexes, and heterogeneity. Here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function. We show that such structures fall into the category of m-colored random graphs and study the percolation transition by using this theory. We prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function. Additionally, we show that the size of the giant component and the cyclomatic number are not affected by volume scaling. These findings may explain the surprisingly good performance of network models that neglect volume scaling. Even though this paper focuses on the implications of the volume growth, the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications.

Bond percolation in coloured and multiplex networks.

Percolation in complex networks is a process that mimics network degradation and a tool that reveals peculiarities of the network structure. During the course of percolation, the emergent properties of networks undergo non-trivial transformations, which include a phase transition in the connectivity, and in some special cases, multiple phase transitions. Such global transformations are caused by only subtle changes in the degree distribution, which locally describe the network. 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-dependent bond percolations. This theory predicts locations of the phase transitions, existence of wide critical regimes that do not vanish in the thermodynamic limit, and a phenomenon of colour switching in small components. These results may be used to design percolation-like processes, optimise network response to percolation, and detect subtle signals preceding network collapse.

Enhancing the robustness of a multiplex network leads to multiple discontinuous percolation transitions.

Determining design principles that boost the robustness of interdependent networks is a fundamental question of engineering, economics, and biology. It is known that maximizing the degree correlation between replicas of the same node leads to optimal robustness. Here we show that increased robustness might also come at the expense of introducing multiple phase transitions. These results reveal yet another possible source of fragility of multiplex networks that has to be taken into the account during network optimization and design.

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 a generic model of step-growth polymerisation as a promising application of the percolation on a directed random graph. 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. In this way, we obtain new analytical expressions describing the polymeric microstructure and compare them to data from experiments and computer simulations. 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, as well as it provides a vital link between network science and experimentally observable physics of polymers.

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation.

Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in nonamenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with interlayer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.

Renormalization group for link percolation on planar hyperbolic manifolds.

Network geometry is currently a topic of growing scientific interest, as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However, the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. Boettcher et al. [Nat. Comm. 3, 787 (2012)2041-172310.1038/ncomms1774] have shown that a special type of two-dimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here we investigate using the renormalization group the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing m-polygons to single edges. We show that when the size m of the polygons is drawn from a distribution q_{m} with asymptotic power-law scaling q_{m}≃Cm^{-γ} for m≫1, different universality classes can be observed depending on the value of the power-law exponent γ. Interestingly, the percolation transition is hybrid for γ∈(3,4) and becomes continuous for γ∈(2,3].

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.

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

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 two-layer 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.

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.

Solution of the chemical master equation by radial basis functions approximation with interface tracking.

BACKGROUND: The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference 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 high-dimensional, 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 multi-modal 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. DISCUSSION: In order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional 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 multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability.