A Graph Similarity Relation Defined by Graph Transformation

Similarity is an important way of categorizing mathematical objects. In this paper we consider how similarity between undirected graphs, or networks, can be defined. We do this by reducing any undirected graph, G = (V,E) to its irreducible spine, I. First we show that the reduction process, ω, is a unique transformation and thus is a well-defined function. Each irreducible spine, I, thus defines an equivalence class consisting of all graphs {G
_i
} such that G
_i
.ω = I. We then show that reduction by ω preserves many key properties. Specifically it preserves shortest path lengths in I, and (under rather general conditions) preserves the location of distance and betweeness centers. In essence, every member graph of the equivalence class will have much the same path structure. Next, an algorithm ε is presented which, given any irreducible spine I, will generate (randomly) a graph G
_k
in the equivalence class defined by I, that is, we must have G
_k
.ω = I. Since I.ε.ω = I, ε = ω
^-1
. It is of some importance in network analysis to be able to quantify the notion of network similarity; to be able to say that a network G
₁
is "closer" to the network G
₂
than to G
₃
. A popular method of comparing similarity between two networks G
₁
and G
₂
is to calculate the Spearman coefficient based on corresponding node degrees. We do this for 5 random graphs in the equivalence class of our running example. Finally, we show that any irreducible spine consists of a system of cycles. We then note that every spine, I, is topologically equivalent to a cubic graph. Thus individual cubic graphs generate a second, and coarser, equivalence relation on the family of all undirected graphs, which we call its genus.