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

2017

Phys. Rev. E

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Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions Kryven, I. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. 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 − .

Section: Asymptote For Weakly Connected Componentssupporting

confidence: 83%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

2017

Phys. Rev. E

Self Cite

“…A random graph refers to a probability distribution over all possible realizations of a graph and allows us to draw results that are typical to MC simulations with an analytical probabilistic consideration. One of the most attractive features is that results on random graphs are often obtainable as an explicit exact equation bypassing any simulations at all . Thus, the topology becomes accessible without the need of balancing expensive computational time against statistical errors.…”

Section: Introductionmentioning

confidence: 99%

Acrylate Network Formation by Free‐Radical Polymerization Modeled Using Random Graphs

Schamboeck

Macro Theory & Simulations

Iedema

2017

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A novel technique is developed to predict the evolving topology of a diacrylate polymer network under photocuring conditions, covering the low‐viscous 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 (<2%). Typically, this low‐conversion network is very sparse, as becomes apparent from the predicted crosslink distribution.

“…Various values of k c / k p considered. The dashed line represents the gel fraction as obtained from the mean field model for reference.…”

Section: Discussion/application To Linseed Oil Networkmentioning

confidence: 99%

Random Graph Approach to Multifunctional Molecular Networks

Macromol. Theory Simul.

Duivenvoorden

Hermans

et al. 2016

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one must realize that such simulations imply an enormous spread of length and time scales: even the simple case of a linear polymer exhibits geometrical structures from the scale of a chemical bond to the scale of the gyration radius, to the collective length scales that might be much larger in dense materials. [ 1 ] Moreover, our drive for modelling is not limited to linear polymers but also is directed at tree-like branched structures, cross-linked polymers, and even infinite molecular networks, i.e., gels. In this paper we propose a strategy that draws a consensus between what can be computed on the one hand and pursuing fundamental roots of the matter at hand on the other.The process of assembly or polymerization of molecular networks usually goes through a few temporal stages: disconnected monomers, linear polymer, branched polymer, and gel (an infi nite network). Furthermore, the gel itself should not be perceived as a single, fi nal state of polymer topology. After the gel point the connectivity patterns of the network continue to evolve. [2][3][4] In fact, in many cases a major part of the whole polymerization process may occur in the gel regime. [ 5 ] In view of extremely long time scales associated with polymerization, namely, the time scales that might well exceed hours for some industrial processes, [ 6 ] or even years (drying of oil paint [ 7 ] ), direct atomistic simulation 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 self-avoiding 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 conversion-dependent 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 gel-point 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 coeffi cient, network modularity, cluster size distribution.