Dynamics in Affinity-Weighted Preferential Attachment Networks

Abstract

During the formation process of stochastic networks, nodes tend to establish edges based on selective linkage mechanisms. In general, these mechanisms involve probability distributions that underlie the selection of target nodes. In social networks, edges are often associated to relationships that are homophilic with respect to individual traits. Such traits include, for example, gender and age, and are referred to as node types. Our work considers an affinity-weighted preferential attachment model that characterizes the tendency of two group of nodes to connect to other nodes of the same type. We derive mathematical expressions for the dynamics and convergence of homophily measures at the node, group, and network level. Furthermore, we characterize the convergence of network modularity and show that the formation of community structures can be expressed as a function of network homophily.

Appendix

Note that $\beta _1+\beta _2=2m$. In particular, according to Eq. (8), we know that

$$\begin{aligned} \beta _1&=mp_1+\frac{mp_1q}{2q-1+\frac{2(1-q)}{\alpha _1}}+\frac{mp_2(1-q)}{1-2q+\frac{2q}{\alpha _1}},\\ \beta _2&=mp_2+\frac{mp_2q}{2q-1+\frac{2(1-q)}{\alpha _2}}+\frac{mp_1(1-q)}{1-2q+\frac{2q}{\alpha _2}}, \end{aligned}$$

where

$$\begin{aligned} \alpha _1&=p_1+\frac{p_1q}{2q-1+\frac{1-q}{p_1}}+\frac{p_2(1-q)}{1-2q+\frac{q}{p_1}},\\ \alpha _2&=p_2+\frac{p_2q}{2q-1+\frac{1-q}{p_2}}+\frac{p_1(1-q)}{1-2q+\frac{q}{p_2}}. \end{aligned}$$

First, note that $\alpha _1+\alpha _2=2$. In particular, because $p_2=1-p_1$, we know that

$$\begin{aligned} \frac{q}{2q-1+\frac{1-q}{p_1}}+\frac{1-q}{1-2q+\frac{q}{p_2}}&=\frac{p_1q}{(2q-1)p_1+1-q}+\frac{(1-p_1)(1-q)}{(1-2q)(1-p_1)+q}=1.\\ \end{aligned}$$

Similarly, we can show that

$$\begin{aligned} \frac{q}{2q-1+\frac{1-q}{p_2}}+\frac{1-q}{1-2q+\frac{q}{p_1}}=1, \end{aligned}$$

and so $\alpha _1+\alpha _2=2(p_1+p_2)=2$. Second, since $\alpha _2=2-\alpha _1$, we have

$$\begin{aligned} \frac{q}{2q-1+\frac{2(1-q)}{\alpha _2}}+\frac{1-q}{1-2q+\frac{2q}{\alpha _1}}&=\frac{q}{2q-1+\frac{2(1-q)}{2-\alpha _1}}+\frac{1-q}{1-2q+\frac{2q}{\alpha _1}}\\&=\frac{(2-\alpha _1)q}{(2q-1)(2-\alpha _1)+2-2q}+\frac{\alpha _1(1-q)}{(1-2q)\alpha _1+2q}\\&=1. \end{aligned}$$

Similarly

$$\begin{aligned} \frac{q}{2q-1+\frac{2(1-q)}{\alpha _1}}+\frac{1-q}{1-2q+\frac{2q}{\alpha _2}}=1. \end{aligned}$$

So

$$\begin{aligned} \frac{mp_1q}{2q-1+\frac{2(1-q)}{\alpha _1}}+\frac{mp_1(1-q)}{1-2q+\frac{2q}{\alpha _2}}&=mp_1,\\ \frac{mp_2q}{2q-1+\frac{2(1-q)}{\alpha _2}}+\frac{mp_2(1-q)}{1-2q+\frac{2q}{\alpha _1}}&=mp_2, \end{aligned}$$

and thus $\beta _1+\beta _2=2m(p_1+p_2)=2m$.