Copula-based analysis of the generalized friendship paradox in clustered networks

TL;DR

This work advances the analytical understanding of the generalized friendship paradox (GFP) in clustered networks by introducing a vine copula framework to capture both focal-neighbor and neighbor-neighbor attribute correlations. Using a C-vine construction with an FGM copula and a first-order expansion in the correlation parameter, the authors derive a tractable expression for the mean-based peer pressure $h(k,x)$ and provide a closed-form solution for the exponential-attribute case, revealing how triangle structure modulates GFP. Their results show that positive attribute correlations among neighbors can increase peer pressure for high-attribute individuals when neighbors are highly interconnected, while negative correlations have the opposite effect; simulations on clustered networks support these insights. This framework paves the way for more general copula-based analyses of GFP and related perception biases in complex networks, including heavy-tailed attributes and alternative copula families.

Abstract

A heterogeneous structure of social networks induces various intriguing phenomena. One of them is the friendship paradox, which states that on average your friends have more friends than you do. Its generalization, called the generalized friendship paradox (GFP), states that on average your friends have higher attributes than yours. Despite successful demonstrations of the GFP by empirical analyses and numerical simulations, analytical, rigorous understanding of the GFP has been largely unexplored. Recently, an analytical solution for the probability that the GFP holds for an individual in a network with correlated attributes was obtained using the copula method but by assuming a locally tree structure of the underlying network [Jo~et~al., Physical Review E~\textbf{104}, 054301 (2021)]. Considering the abundant triangles in most social networks, we employ a vine copula method to incorporate the attribute correlation structure between neighbors of a focal individual in addition to the correlation between the focal individual and its neighbors. Our analytical approach helps us rigorously understand the GFP in more general networks such as clustered networks and other related interesting phenomena in social networks.

Figures (3)

• Figure 1: Schematic diagram for the generalized friendship paradox in clustered networks. An ego with an attribute $x$ has $k$ neighbors whose attributes are $x_1,x_2,\ldots,x_k$, respectively. It is assumed to have the same correlation structure of attributes of neighboring nodes in every link, which is controlled by a single value of the Pearson correlation coefficient $\rho_x$ in Eq. \ref{['eq:PCC']}.
• Figure 2: Analytical results of $h_{\rm nn}(k,x)$ in Eq. \ref{['eq:h2kx']} (a) and $h_{\rm fn}(k,x)$ in Eq. \ref{['eq:h1kx']} (b) as a function of $k$ for several values of $x$. In the panel (a) the results of $k=1$, leading to $n=0$, were omitted, and the black horizontal line denotes $0$ for guiding eyes.
• Figure 3: Simulation results of the mean-based peer pressure $h(k,x)$ as a function of $n$ for values of $\rho_x=0.05$ (triangle), $0$ (circle), and $-0.05$ (inverse triangle), which we denote by $h(n)$ for clear presentation, and their differences from the results for $n=0$, denoted by $h(n)-h(0)$. $50$ clustered networks were generated with $N=5\times 10^4$ and average degree $50$, and $10^5$ different configurations of attributes for each network were randomly generated using $P(x)$ with $\lambda=1$ in Eq. \ref{['eq:expo_Px']} for each value of $\rho_x$. Error bars denote standard errors. For comparison, we plot the analytical solutions of $h(k,x)$ and $rnh_{\rm nn}(k,x)$ in Eq. \ref{['eq:hkx_mean_expo']} using $r=0.2$, $0$, and $-0.2$ [Eq. \ref{['eq:rel_rhox_r']}] as dotted, solid, and dashed lines, respectively.