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12 Networ k Co mpar ison Michael Baur and Marc Benkert A fundamental question in comparative network analysis is whether two given networks have the same structure. To formalize what to relate to structural equivalence, the following definition was made: Definition 12.0.1. Two undir e cte d simple graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 ,E 2 ) ar e isomorphic (denoted by G 1 G 2 ) if there is an edge-preserving bijective vertex mapping φ : V 1 −→ V 2 , i.e. a bije cti on φ with ∀u, v ∈ V 1 : {u, v} ∈ E 1 ⇐⇒ {φ(u), φ(v )} ∈ E 2 . The graph isomorphism problem (GI) is to determine whether two given graphs are isomorphic. Figure 12.1 shows an example of two – differently em- bedded – isomorphic graphs. However, in practice it will be extremely rare that two graphs are isomorphic. We can deal with this fact, as in most cases it is comparatively easy to recognize two graphs as non-isomorphic. We simply have to check necessary conditions: trivially, the number of vertices and edges has to match. For each degree value the number of vertices having this degree has to match, the two graphs must form the same number of connected components, the diameter has to match, and so on. We can also use more complicated proper- ties like those from other chapters in this book: if it should be possible that two graphs are isomorphic, their spectra should be equal, all centrality indices have to match, etc. One could give an ever increasing list of necessary conditions, but thus far no one has succeeded in giving a sufficient condition that is polynomially computable. More details are given in Section 12.1, especially in the overview and in Section 12.1.3. 1 3 2 4 5 6 8 7 1 2 3 4 5 6 7 8 Fig. 12.1. Two isomorphic graphs. The labeling indicates a possible isomorphism; it is not part of the graph

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