Vertex similarity in networks

We consider methods for quantifying the similarity of vertices in networks. We propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar. This leads to a self-consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network. We test our similarity measure on computer-generated networks for which the expected results are known, and on a number of real-world networks.

Figure 1

A vertex $j$ is similar to vertex $i$ (dashed line) if $i$ has a network neighbor $v$ (solid line) that is itself similar to $j$.

Figure 2

(a) A density plot of the similarities of all vertex pairs not directly connected by an edge in our stratified network model. The points give the average similarity as a function of age difference and the line is a least-squares fit to a straight line. (b) A density plot of the cosine similarity values for the same network.

Figure 3

(Color online) The correlation coefficient $r(\sigma ,{\sigma }_{\text{age}})$ for correlation between our similarity measure and the probability of connection, Eq. (20), in our stratified model, for a range of values of $\alpha$. The values given are averaged over an ensemble of graphs generated from the model. The maximum value is found to occur for $\alpha \simeq 0.97$.

Figure 4

(Color online) The correlation coefficient for correlation between our similarity measure and the age difference of all vertex pairs in a single network, as a function of $\alpha$. This plot is typical for the school networks studied.

Figure 5

(Color online) There is only one possible topology for paths of length 1 between distinct vertices, and only one for paths of length 2, but there are four possible topologies for paths of length 3.

Figure 6

(Color online) (a) The actual number of paths of length 2 between vertex pairs in a configuration model versus the expected number of paths, as determined from Eq. (A2). (b) Same as in plot (a), but for paths of length 3.