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

Abstract

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 − .

Connectivity of a general class of inhomogeneous random digraphs

“…constitutes a series of classical results within the random graph literature. Closely related to the results presented here for directed graphs, are the existence of a giant strongly connected component and giant weak-component in the directed configuration model [11,18,19], the existence of a giant strongly connected component in the deterministic directed kernel model with a finite number of types [3], the scale-free property on a directed preferential attachment model [28,31], and the limiting degree distributions in the directed configuration model [7]. 1 From a computational point of view, the work in [33] provides numerical algorithms to identify secondary structures on directed graphs.…”

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

“…In our previous work 39 we have demonstrated on the case of an acrylate polymer featuring predominantly symmetric covalent bonds, that many of the MD-generated network properties can be also reproduced by the configuration model for undirected random networks 41,42 . Furthermore, the recent developments in directed configuration models 29,43 present an opportunity to develop a generic polymerisation framework that will cover asymmetrical bonds as well. The latter, despite posing a more complex mathematical problem, are also more ubiquitous in polymerisation chemistry, and especially in that of hyperbranched and super-molecular polymers 44–46 .…”

Bond percolation in coloured and multiplex networks

“…Interesting developments of the configuration model can be obtained by constraining the configurations with extra requirements. For example, by fixing the clustering coefficient 31 , number of triangles 32 or adding directionality to edges 33 . In edge-coloured configuration model, edges are labelled with an arbitrary number of colours, see Fig.…”

“…Note, there are several other ways to define a connected component, all of which lead to a different asymptotic theory and percolation properties, for example: strong, in-, and out-components 19 , colour-avoiding components 38 and mutually dependent components 21,22,35,36 . In our previous work 33 , the formal expression for the size distribution of connected components w ( n ) is derived by applying Joyal’s theory of species,where operations f g and f denote, respectively, the N -dimensional convolution product and the convolution power 33 . The interpretation of Eqs.…”

“…(39) was derived, ref. 33 applies Joyal’s theory of combinatorial species 53 to write the exact expression for the size distribution of weakly connected components:where the N -dimensional convolution d ( n ) = f ( n ) g ( n ), is defined as…”