Generalized friendship paradox in networks with tunable degree-attribute correlation

One of the interesting phenomena due to topological heterogeneities in complex networks is the friendship paradox: Your friends have on average more friends than you do. Recently, this paradox has been generalized for arbitrary node attributes, called the generalized friendship paradox (GFP). The origin of GFP at the network level has been shown to be rooted in positive correlations between degrees and attributes. However, how the GFP holds for individual nodes needs to be understood in more detail. For this, we first analyze a solvable model to characterize the paradox holding probability of nodes for the uncorrelated case. Then we numerically study the correlated model of networks with tunable degree-degree and degree-attribute correlations. In contrast to the network level, we find at the individual level that the relevance of degree-attribute correlation to the paradox holding probability may depend on whether the network is assortative or dissortative. These findings help us to understand the interplay between topological structure and node attributes in complex networks.

Figure 1

Analytic results of the uncorrelated model with Gamma distributions for $x$ and $k$ in Eq. (9). (a) Heat map of the paradox holding probability $h_k\left(x\right)$ in Eq. (10) as a function of $\alpha k$ and $x/\langle x\rangle$. (b) $h_k\left(x\right)$ as a function of $\alpha k$ for values of $x/\langle x\rangle =0.9$, 1, and $1.1$ (curves), which are compared to the numerical results (circles) from the uncorrelated network of size $N={10}^5$ and of $\langle x\rangle =50$ using the same Gamma distribution in Eq. (9).

Figure 2

Paradox holding probability $h_k\left(x\right)$ of the correlated networks of size $N={10}^5$ for values of $r_{kk}=-0.2$, 0, and $0.4$ (from left to right) and of ${\rho }_{kx}=-0.5$, 0, and $0.5$ (from bottom to top). Degrees $k$ follow the Gamma distribution in Eq. (9) with $\alpha =1$ and $\beta =50$, i.e., $\langle k\rangle =50$, and attributes $x$ are assigned to nodes using Eq. (14). For comparison to the uncorrelated case, $x$ has been regularized by $\langle k\rangle$ that has the same value as $\langle x\rangle$ for ${\rho }_{kx}=0$. Blue horizontal lines correspond to $\langle x\rangle /\langle k\rangle$ for each case.

Figure 3

Numerical results for correlated networks of size $N={10}^5$ and of $\langle k\rangle =50$ with the Gamma distribution for degrees (a)–(c): (a) Average paradox holding probability $H$ as a function of $r_{kk}$ and ${\rho }_{kx}$. (b) Scatter plot showing $r_{xx}$ and $r_{kk}\left|{\rho }_{kx}\right|$ for $-0.8\le r_{kk}\le 0.8$. The solid line corresponds to $r_{xx}=r_{kk}\left|{\rho }_{kx}\right|$. (c) Paradox holding probability of the FP for various values of degree-degree correlations. (d) Empirical paradox holding probability of the FP for coauthorship networks of Physical Review (PR) journals and Google Scholar (GS) profiles from Ref. [18].

Figure 4

Paradox holding probability $h_k\left(x\right)$ of the correlated networks of size $N={10}^4$ for values of $r_{kk}=-0.1$, 0, and $0.1$ (from left to right) and of ${\rho }_{kx}=-0.5$, 0, and $0.5$ (from bottom to top). Degrees $k$ follow the power-law distribution in Eq. (15) with $\gamma =2.7$ and $k_{\min }=6$, and attributes $x$ are assigned to nodes using Eq. (14). For comparison to the uncorrelated case, $x$ has been regularized by $\langle k\rangle$ that has the same value as $\langle x\rangle$ for ${\rho }_{kx}=0$. Blue horizontal lines correspond to $\langle x\rangle /\langle k\rangle$ for each case.

Figure 5

Numerical results for correlated networks of size $N={10}^4$ with the power-law distribution for degrees: (a) Average paradox holding probability $H$ as a function of $r_{kk}$ and ${\rho }_{kx}$. (b) Scatter plot showing $r_{xx}$ and $r_{kk}\left|{\rho }_{kx}\right|$ for $-0.1\le r_{kk}\le 0.1$. The solid line corresponds to $r_{xx}=r_{kk}\left|{\rho }_{kx}\right|$. (c) Paradox holding probability of the FP for various values of degree-degree correlations.