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

Verena Schamboeck, Ivan Kryven, and Piet D. Iedema

Phys. Rev. E 101, 012303 – Published 14 January 2020

Abstract

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.

Received 19 June 2019

DOI:https://doi.org/10.1103/PhysRevE.101.012303

Physics Subject Headings (PhySH)

Percolation phase transition

Directed networks Multilayer & multiplex networks Random graphs

Networks

Authors & Affiliations

Verena Schamboeck ^1,*, Ivan Kryven ², and Piet D. Iedema¹

¹Van 't Hoff Institute for Molecular Sciences, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
²Mathematical Institute, Utrecht University, PO Box 80010, 3508 TA Utrecht, Netherlands

^*v.schamboeck@uva.nl

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Issue

Vol. 101, Iss. 1 — January 2020

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Images

Figure 1
Illustration of the generation model. The directed layered network is illustrated, with the edge orientation in accordance with the generation sequence. Networks with two different volume growth functions are shown: (a) monomial growth and (b) inverted hourglass.
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Figure 2
Illustration of the generation model with intrageneration edges. The intrageneration edges are represented by undirected edges. Two networks with different fractions of intrageneration edges are presented: (a) $0 < a < 1$ , leading to the coexistence of directed edges that connect consecutive generations and undirected edges within one generation, and (b) $a = 1$ , leading to a network with the generations being disconnected.
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Figure 3
Various volume growth functions $P (g)$ : (a) monomial growth with dimensionality $d = 1, 2, 3, 4$ and (b) exponential growth with $α = 2$ and inverted hourglass with $d = 2$ .
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Figure 4
Regular networks with degree $F = 3$ (no cycles) with various types of growth behavior: (i) exponential growth, (ii) one-dimensional monomial growth, (iii) two-dimensional monomial growth, and (iv) three-dimensional monomial growth. (a) Spanning trees (no cycles). The color gradient indicates the generations. (b) and (c) Node neighborhood size $V (l)$ with shortest path $l$ for (b) nodes in first generations and (c) nodes in higher generations. The data (dots) are obtained from stochastic generation networks of size $N = 20 000$ . The dash-dotted lines in (b ii)–(b iv) illustrate the expected neighborhood size for monomial growth and in (b i) and (c i)–(c iv) the expected neighborhood size for exponential growth. In (c ii)–(c iv) the neighborhood first growths exponentially until it switches to monomial growth independent of the volume growth function.
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Figure 5
Average shortest path as a function of system size for regular networks with degree $F = 3$ . Networks with different volume growth functions are studied: one dimensional (circles), two dimensional (stars), and three dimensional (crosses). The dash-dotted lines indicate the expected growth of the average path.
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Figure 6
Percolation on regular graphs with various types of monomial growth behavior with functionalities $F = 3, 4, 5$ : (a) size of the largest or giant component and (b) cyclomatic number. The different types of growth behavior are indicated by markers: one dimensional (circles), two dimensional (stars), and three dimensional (crosses). The percolation points for the different functionalities are indicated by the dashed, dash-dotted, and dotted line for $F = 3, 4, 5$ , respectively. The data points are obtained from networks of size $N = 20 000$ and compared to the percolation on a random regular graph (solid lines).
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