# Types of Nodes and Centrality Measures in Networks

## Abstract

Equilibrium behaviors in games on networks are often defined by centralities of players. We show that centrality measures of a class (degree, eigenvalue centrality, Katz-Bonacich centrality, diffusion centrality, alpha-gamma centrality, and alpha-beta centrality) do characterize not just separate nodes but types of nodes. The typology relates the fact that the nodes in an undirected graph may be colored in a minimal number of colors in such a way that any node of a color has definite numbers of neighbors of definite colors. Networks of the same typology are characterized by a “type adjacency” matrix T, which shows for each type numbers of neighbors of different types. For any typology, if *i* and *j* are nodes of the same type (may be even belonging different networks of this typology), then *c*(*i*) = *c*(*j*), where *c* is any of the above-mentioned centrality measures. Networks of different size but with the same typology have common properties; in particular, game equilibria may be transplanted among networks of the same typology. For calculation of any of these centrality measures, the type adjacency matrix may be used instead of the adjacency matrix. A problem is: for which classes of networks each of a set of several centrality measures defines the same order on the set of nodes of network? We show that any network typology with two types of nodes possesses this property for the above-mentioned class of centrality measures.

## Notes

### Acknowledgements

The research is supported by the Russian Foundation for Basic Research (project 17-06-00618).

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