Matching patterns in networks with multi-dimensional attributes: a machine learning approach

Abstract

Assortative matching is a network phenomenon that arises when nodes exhibit a bias towards connections to others of similar characteristics. While mixing patterns in networks have been studied in the literature, and there are well-defined metrics that capture the degree of assortativity (e.g., assortativity coefficient), the latter deal only with single-dimensional enumerative or scalar features. Nevertheless, various complex behaviors of network entities—e.g., human behaviors in social networks—are captured through vector attributes. To date, no formal metric able to cope with similar situations has been defined. In this paper, we propose a novel, two-step process that extends the applicability of the assortativity coefficient to multi-dimensional attributes. In brief, we first apply clustering of the vertices on their vector characteristic. After clustering is completed, each network node is assigned a cluster label, which is an enumerative characteristic and we can compute the assortativity coefficient on the cluster labels. We further compare this method with an alternative baseline, which is an immediate extension of the assortativity coefficient, namely, the assortativity vector. The latter treats each element of the node’s attribute vector separately and then combines the independent results in a single value. Finally, we apply our method and the baseline on two different social network datasets. We also use synthetic network data to delve into the details of each metric/method. Our findings indicate that while the baseline of assortativity vector performs satisfactory when the variance of the elements of the vector attribute across the network population is kept low, it provides biased results as this variance increases. On the contrary, our approach appears to be robust in such scenarios.

Appendix: Toy example for biases introduced by assortativity vector

Consider a toy-network in which each node is described by one of the vectors ${\bf u}_z=(0,0)$, ${\bf u}_w=(0,1)$, ${\bf u}_t=(1,0)$ and ${\bf u}_q=(1,1)$. For ease of presentation let us assume that we have $p$ nodes of each type, and that $z$-vertices are connected with $w$-vertices and $t$-vertices with $q$-vertices. If we calculate the assortativity vector for this network is ${\bf r}=(1,-1).$ Hence, we get $r^{\rm mean}=0.$ The latter implies that associations between vertices in this network are made at random, regardless of their vector attribute. However, in the 2-dimensional space the vectors that describe the nodes of the network can form four distinct classes (each of which contains $p$ vertices) and clearly all the connections in this network are among vertices belonging to different groups (i.e., dissimilar nodes). Even if connections were made at random, one might have expected approximately 25 % of the edges to connect vertices with the same vector attribute, which clearly it is not the case in our toy-example. Hence, $r^{\rm mean}$ for this network with regard to ${\bf u}$ should be negative to capture the underlying mixing patterns. Of course, as mentioned in Section negative mixing is closer to random mixing (compared to positive mixing), but still this example illustrates possible biases introduced by considering elements of u independently.