Strongly clustered random graphs via triadic closure: Degree correlations and clustering spectrum

TL;DR

This work presents a model for strongly clustered random graphs in which each triad of a random backbone is closed with a certain probability, and finds positive degree assortativity accompanies high transitivity, and non-trivial structure in the clustering spectrum.

Abstract

Real-world networks often exhibit strong transitivity with nontrivial local clustering spectra and degree correlations. Such features are not easily modelled in tractable network models, creating an obstacle to the theoretical understanding of such complex network structures. Here, we address this problem using a model for strongly clustered random graphs in which each triad of a random backbone is closed with a certain probability. Despite the intricate loopy local structure of the graphs obtained, we provide exact expressions for the local clustering spectrum and the degree correlations, filling the gap in the theoretical description of this model for random graphs. In particular, we find positive degree assortativity accompanies high transitivity, and non-trivial structure in the clustering spectrum. Exact asymptotic analytical results are complemented with extensive numerical characterization of finite size effects.

Figures (11)

• Figure 1: A pictorial representation of the STC mechanism. Imagine some people going to a party. Let us place a link between two individuals if they are friends. This social network before the party is our backbone $\mathcal{G}_0$ on the left. In our simplifying assumptions, we consider that people initially don't have many friends in common, i.e. $\mathcal{G}_0$ is treelike. The party begins, and people start talking with their friends. It may happen that two people having a common friend meet and become friends by the end of the party. If that's the case, we put a green dashed link between them. The social network after the party, $\mathcal{G}_f$, is drawn on the right. Despite the simplicity of the process, the network $\mathcal{G}_f$ clearly presents an intricate local structure with overlapping loops.
• Figure 2: Numerical results for the degree distributions $P(K)$ of STC networks with power-law backbones with exponents (a) $\gamma=2.25$, (b) $\gamma=2.5$, (c) $\gamma=2.75$, (d) $\gamma=3$, (e) $\gamma=3.5$, (f) $\gamma=4.5$. Empty symbols are for $f=0.1$, filled symbols are for $f=1$. The dashed lines represent the scaling with exponent $\widetilde{\gamma}=\gamma-1$ predicted by Eq. \ref{['eq:PK_scaling']}, the dotted line is the scaling with exponent $\gamma$. Apart from strong finite-size effects for $\gamma$ close to $2$ and for large values of $\gamma$, the scaling with exponent $\widetilde{\gamma}$ holds only up to $K^{*} \sim fk_{\textrm{max}}$, while for $K>K^{*}$ the degree distributions follow the power-law of the backbones. The insets show that $K_{\textrm{max}} \sim fk_{\textrm{max}}^{4-\gamma}$ for $2< \gamma \leq 3$ (panels (a)-(d)), and that $K_{\textrm{max}} \sim \theta k_{\textrm{max}}$, for $\gamma>3$, where $\theta=1+f \langle r \rangle$ (panels (e) and (f)). Numerical simulations are perfomed sampling $N$ nodes with degrees $k$ from $p(k)$, with $k_{\textrm{min}}=3$ and $k_{\textrm{max}}=\min\{N^{1/(\gamma-1)},N^{1/2}\}$, each of their $k$ neighbors excess degrees are sampled from $q(r)$, and the new connections created by the STC process are sampled from binomial distributions. The histograms of $P(K)$ are obtained for $N=10^8$, the insets are for various values of $N$. Results are average over 10 independent realizations.
• Figure 3: (a) Pearson correlation coefficient in STC networks generated from random regular networks, as a function of the STC parameter $f$, for different values of the degree parameter $c$. Panels (b)-(c): Pearson correlation coefficient in STC networks generated from Erdős-Rényi networks, (b) as a function of $f$, for different values of $c$, and (c) as a function of the mean degree $c$, for different values of the STC parameter $f$. Solid lines in all panels correspond to the exact results [Eq. (\ref{['eq1.77']}) and Eq. (\ref{['eq1.80']}) for panels (a) and (b)-(c), respectively], and squares correspond to simulation results using networks of size $N=10^6$.
• Figure 4: (a)-(c) Pearson correlation coefficient in STC networks generated from random power-law backbones, as a function of the degree distribution exponent $\gamma$, for various values of the STC parameter $f$. Dashed lines correspond to values of the Pearson coefficient evaluated numerically, using Eq. (\ref{['eq1.72']}), for $k_{\textrm{max}}=10^2, 10^3, 10^4, 10^5, 10^6$, darker colors for larger $k_{\textrm{max}}$. The solid line corresponds to $k_{\textrm{max}} \to \infty$ in which case Eqs. (\ref{['eq1.111']}) and (\ref{['eq1.113']}) apply for $2 < \gamma < 5$, and for $\gamma > 5$, where the first four moments of the degree distribution are finite, these moments can be written and evaluated exactly, using the Hurwitz zeta function: $\langle k^n \rangle = \zeta(\gamma-n, k_{\textrm{min}}) / \zeta(\gamma, k_{\textrm{min}})$. (d) Pearson coefficients shown for large $\gamma$; dashed lines indicating the limits predicted by Eqs. (\ref{['eq1.115']}) and (\ref{['eq1.116']}). In all panels $k_{\textrm{min}}=3$.
• Figure 5: Average nearest-neighbor degree $K_{nn}(K)$ of STC random graphs with PL backbones with exponents (a) $\gamma=2.25$, (b) $\gamma=2.75$, (c) $\gamma=3.5$, (d) $\gamma=4.5$. The symbols represent numerical simulations on synthetic networks of size $N=10^4$ (squares), $N=10^5$ (circles) $N=10^6$ (triangles), with $k_{\textrm{min}}=3$ and $k_{\textrm{max}}=\min\{N^{1/(\gamma-1)}, N^{1/2} \}$. Empty symbols are for $f=0.1$, filled symbols are for $f=1$. The numerical simulations are performed using PL backbones created using the Uncorrelated Configuration Model catanzaro2005generation, creating the STC graphs by random closure of the triads. Results are average over 100 realizations for all data except $N=10^6$ in panel (a), where the average is over 10 realizations.