Analytical approach to the generalized friendship paradox in networks with correlated attributes

One of the interesting phenomena due to the topological heterogeneities in complex networks is the friendship paradox, stating that your friends have on average more friends than you do. Recently, this paradox has been generalized for arbitrary nodal attributes, called a generalized friendship paradox (GFP). In this paper, we analyze the GFP for the networks in which the attributes of neighboring nodes are correlated with each other. The correlation structure between attributes of neighboring nodes is modeled by the Farlie-Gumbel-Morgenstern copula, enabling us to derive approximate analytical solutions of the GFP for three kinds of methods summarizing the neighborhood of the focal node, i.e., mean-based, median-based, and fraction-based methods. The analytical solutions are comparable to simulation results, while some systematic deviations between them might be attributed to the higher-order correlations between nodal attributes. These results help us get deeper insight into how various summarization methods as well as the correlation structure of nodal attributes affect the GFP behavior, hence better understand various related phenomena in complex networks.

Figure 1

Mean-based GFP: (a–c) Heat maps of the analytical solution of $h_{\mathrm{mn}}(k,x)$ in Eq. (17) in the cases with negatively correlated attributes ($r=-0.16$) (a), uncorrelated attributes ($r=0$) (b), and positively correlated attributes ($r=0.16$) (c). (d–f) Heat maps of the simulation results of $h_{\mathrm{mn}}(k,x)$ for ${\rho }_x=-0.04$ (d), 0 (e), and 0.04 (f). For each value of ${\rho }_x$, $2\times {10}^3$ random configurations of attributes per network were generated using $P\left(x\right)$ with $\lambda =1$ in Eq. (12) for 10 networks of size $N=5\times {10}^4$ with an exponential degree distribution with the average of 50. In panels (a–f) solid lines denote $x_r^{*}\left(k\right)/\langle x\rangle$, where the transition point $x_r^{*}\left(k\right)$ is defined in Eq. (19), and horizontal dotted lines for $x/\langle x\rangle =1$ are included for guiding eyes. (g–i) Comparison of $h_{\mathrm{mn}}(k,x)$ for $k=1$, 10, and 100 between the analytical solution (solid lines) and simulation results (symbols) in the cases with $(r,{\rho }_x)=(-0.16,-0.04)$ (g), (0, 0) (h), and (0.16, 0.04) (i). Error bars for confidence intervals for simulation results at significance level $\alpha =0.05$ are smaller than symbols.

Figure 2

Comparison of the conditional distributions of $P\left(x\right|x^{\prime })$ for various values of $x^{\prime }$, in which $x$ and $x^{\prime }$ are attributes of neighboring nodes, between the simulation results (symbols) and the analytical from in Eq. (23) (solid lines) in the cases with $(r,{\rho }_x)=(-0.16,-0.04)$ (a) and (0.16, 0.04) (b). The simulation results are obtained using the same set of random configurations of attributes on networks used in Fig. 1.

Figure 3

Mean-based GFP: Differences in the peer pressure between the correlated cases (${\rho }_x=-0.04$ and 0.04) and the uncorrelated case (${\rho }_x=0$), i.e., $\mathrm{\Delta }{h}_{\mathrm{mn}}(k,x)$ in Eq. (25), for $k=1$ (a), 10 (b), and 100 (c). For each value of ${\rho }_x$, $5\times {10}^4$ random configurations of attributes per network were generated for 10 networks used in Fig. 1. In each panel, simulation results are denoted by symbols, while the analytical result, i.e., the first-order term of $r$ on the right-hand side of Eq. (17) with $r=0.16$, is denoted by a solid line.

Figure 4

Fraction-based GFP: (a–c) Comparison of the analytical solution of $h_{\mathrm{fr}}(k,x)$ in Eq. (30) (solid lines) to simulation results (symbols) for $k=1$, 10, and 100 in the cases with $(r,{\rho }_x)=(-0.16,-0.04)$ (a), (0, 0) (b), and (0.16, 0.04) (c). The simulation results are obtained using the same set of random configurations of attributes on networks used in Fig. 1. (d–f) Differences in the peer pressure between the correlated cases (${\rho }_x=-0.04$ and 0.04) and the uncorrelated case (${\rho }_x=0$), i.e., $\mathrm{\Delta }{h}_{\mathrm{fr}}(k,x)$, for $k=1$ (d), 10 (e), and 100 (f). Here $\mathrm{\Delta }{h}_{\mathrm{fr}}(k,x)$ is defined similarly to $\mathrm{\Delta }{h}_{\mathrm{mn}}(k,x)$ in Eq. (25). The simulation results are obtained using the same set of random configurations of attributes on networks used in Fig. 3. Simulation results are denoted by symbols, while the analytical result, i.e., the first-order term of $r$ on the right-hand side of Eq. (30) with $r=0.16$, is denoted by a solid line. In all panels, standard errors are smaller than symbols.

Figure 5

Median-based GFP: (a–c) Heat maps of the analytical solution of $h_{\mathrm{md}}(k,x)$ in Eq. (35) for $r=-0.16$ (a), 0 (b), and 0.16 (c). (d–f) Heat maps of the simulation results of $h_{\mathrm{md}}(k,x)$ for ${\rho }_x=-0.04$ (d), 0 (e), and 0.04 (f). The simulation results are obtained using the same set of random configurations of attributes on networks used in Fig. 1. In panels (a–f) solid lines denote $x_r^{*}\left(k\right)/\langle x\rangle$, where the transition point $x_r^{*}\left(k\right)$ is defined similarly to the mean-based case in Eq. (19), and horizontal dotted lines for $x/\langle x\rangle =ln2\approx 0.69$ are included for guiding eyes. (g–i) Comparison of $h_{\mathrm{md}}(k,x)$ for $k=1$, 10, and 100 between the analytical solution (solid lines) and simulation results (symbols) in the cases with $(r,{\rho }_x)=(-0.16,-0.04)$ (g), (0, 0) (h), and (0.16, 0.04) (i). Error bars for confidence intervals for simulation results at significance level $\alpha =0.05$ are smaller than symbols.

Figure 6

Median-based GFP: Differences in the peer pressure between the correlated cases (${\rho }_x=-0.04$ and 0.04) and the uncorrelated case (${\rho }_x=0$), i.e., $\mathrm{\Delta }{h}_{\mathrm{md}}(k,x)$, for $k=1$ (a), 10 (b), and 100 (c). Here $\mathrm{\Delta }{h}_{\mathrm{md}}(k,x)$ is defined similarly to $\mathrm{\Delta }{h}_{\mathrm{mn}}(k,x)$ in Eq. (25). The simulation results are obtained using the same set of random configurations of attributes on networks used in Fig. 3. Simulation results are denoted by symbols, while the analytical results derived from the right-hand side of Eq. (35) with $r=\mp 0.16$ are denoted by solid lines and dashed lines, respectively.