Similarity (network science)

Similarity in network analysis occurs when two nodes (or other more elaborate structures) fall in the same equivalence class.

In using cluster analysis, we are implicitly assuming that the similarity or distance among cases reflects a single underlying dimension.

Alternatively, multi-dimensional scaling could be used (non-metric for data that are inherently nominal or ordinal; metric for valued).

[2] MDS represents the patterns of similarity or dissimilarity in the tie profiles among the actors (when applied to adjacency or distances) as a "map" in multi-dimensional space.

This map lets us see how "close" actors are, whether they "cluster" in multi-dimensional space and how much variation there is along each dimension.

[3] While structurally equivalent actors have identical relational patterns or network positions, institutional equivalence captures the similarity of institutional influences that actors experience from being in the same fields, regardless of how similar their network positions are.

For example, two banks in Chicago might have very different patterns of ties (e.g., one may be a central node, and the other may be in a peripheral position) such that they are not structural equivalents, but because they both operate in the field of finance and banking and in the same geographically defined field (Chicago), they will be subject to some of the same institutional influences.

Salton proposed that we regard the i-th and j-th rows/columns of the adjacency matrix as two vectors and use the cosine of the angle between them as a similarity measure.

The cosine similarity of i and j is the number of common neighbors divided by the geometric mean of their degrees.

[1] Pearson product-moment correlation coefficient is an alternative method to normalize the count of common neighbors.

This method compares the number of common neighbors with the expected value that count would take in a network where vertices are connected randomly.

The maximum means that there are no common neighbors, in which case the distance is equal to the sum of the degrees of the vertices.