Measurement of graphs similarity using graph centralities

Abstract

In this paper, a method to measure similarity between two graphs is proposed, which is based on centralities of the graphs. The similarity between two graphs $G_1$ and $G_2$ is defined by the difference of distance($G_1$, $G_{R_1}$) and distance($G_2$, $G_{R_2}$), where $G_{R_1}$ and $G_{R_2}$ are set of random graphs that have the same number of nodes and edges as $G_1$ and $G_2$, respectively. Each distance ($G_*$, $G_{R_*}$) is obtained by comparing centralities of $G_*$ and $G_{R_*}$. Through the computational experiments, we show that it is possible to compare graphs regardless of the number of vertices or edges of the graphs. Also, it is possible to identify and classify the properties of the graphs by measuring and comparing similarities between two graphs.

Fig. 1. Centrality example graph

Fig. 2. Graphs with specific properties

Fig. 3. Centralities of graphs with specific properties

Fig. 4. The process of making a random tree

Fig. 5. calculation of distance using centralities

Fig. 6. Flowchart for calculating distances

Fig. 7. Distances of [Experiment 1]

Fig. 8. Distances of each specific property graphs

Fig. 9. Distances of [Experiment 2]

Table 1. Each vertex centrality in Fig. 1

Table 2. Distance of Fig. 8

Table 3. Similarity with Ring graph

Table 4. Target graph data sets

Table 5. The distances between random graph set

Table 6. Similarity with $G^{T}_{K}$