Similarity (network science)

Similarity in network analysis occurs when two nodes (or other more elaborate structures) fall in the same equivalence class. Similarity in network analysis occurs when two nodes (or other more elaborate structures) fall in the same equivalence class. There are three fundamental approaches to constructing measures of network similarity: structural equivalence, automorphic equivalence, and regular equivalence. There is a hierarchy of the three equivalence concepts: any set of structural equivalences are also automorphic and regular equivalences. Any set of automorphic equivalences are also regular equivalences. Not all regular equivalences are necessarily automorphic or structural; and not all automorphic equivalences are necessarily structural. Agglomerative Hierarchical clustering of nodes on the basis of the similarity of their profiles of ties to other nodes provides a joining tree or Dendrogram that visualizes the degree of similarity among cases - and can be used to find approximate equivalence classes. Usually our goal in equivalence analysis is to identify and visualize 'classes' or clusters of cases. In using cluster analysis, we are implicitly assuming that the similarity or distance among cases reflects as single underlying dimension. It is possible, however, that there are multiple 'aspects' or 'dimensions' underlying the observed similarities of cases. Factor or components analysis could be applied to correlations or covariances among cases. Alternatively, multi-dimensional scaling could be used (non-metric for data that are inherently nominal or ordinal; metric for valued). 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.