A comparative analysis of new graph distance measures and graph edit distance
Introduction
To measure the distance or similarity between distinct networks has been an interesting and useful task in applied mathematics and related disciplines. However finding the right measure/method for practical problems has been intricate as structural distance or similarity is in the eye of a beholder, see [13]. A similar situation occurs when studying the problem of determining the structural complexity of graphs, see [14], [19], [28]. Determining network similarity is a classical problem and its early treatment goes back to Sussenguth [36] who investigated the classical graph matching problem by using graph isomorphism. After that, Vizing tackled this problem as well [38] and the Zelinka distance was defined [42]. Note that numerous applications using graph similarity have been examined, see [1], [5], [17], [22], [25], [34].
Exploring methods for network comparisons has been a current research item as various classes of networks exist and, hence, not every measure is generally applicable. For instance, graph measures based on isomorphic relations [35], [42] are only applicable to deterministic graphs, i.e., the graph structure is fixed and structural noise does not exist here. In contrast, graph isomorphism does not apply for matching non-deterministic graphs because they need a statistical treatment as the number of edges and vertices may be altered during measuring the structural similarity or distance.
In [19], Emmert-Streib et al. surveyed more recent and classical results on network similarity. Note that the graph edit distance [7] is one of the most popular measures. The reason for this stems from the fact that GED is quite generic and easily interpretable. For more details on GED, see also [1], [3], [4], [21].
In [11], Dehmer et al. introduced a new distance measure based on graph invariants, see [20], [37]. Let I(G) be a graph invariant or a graph parameter of G. Then We emphasize that can be also regarded as a distance measure for real numbers, see [32]. The initial idea of [11] was to use similarity or distance measures for real numbers and then to use graph parameters as topological indices to obtain comparative graph measures.
To motivate our paper properly, we recall that GED is quite efficient when dealing with various practical problems, e.g., measuring the structural distance of coloured graphs with unique vertex labels [16]. However, in general, it has been difficult to compute GED for graphs generally. If one could show that for general graphs a new measure can be computed in polynomial time which approximates GED well, this apparatus would be powerful. Note that Dehmer and Varmuza [15] already explored the approximation behaviour between GED and the Tanimoto index. In this paper, we will continue to study this topic and turn to GED and our new class of measures.
Section snippets
Used topological indices
Topological indices [37] for graphs have been well-studied. In theoretical graph theory, we use the term “graph invariant” instead of “topological index”, that describe properties of graphs. Actually, researchers found more useful applications of topological index like investigating molecular or biological networks [18] rather than only studying chemical graphs.
For a given connected graph we use d(x, y) to denote the distance between two vertices x and y of G. The famous Wiener index
Main results
This section contributes to the results obtained by applying GED and dI to certain classes of graphs. Observe that the computation of dI is much more easier for small graphs. So, if we could find some topological index I such that dI approximates GED well, then this would be a very useful point.
Summary and conclusion
In this paper, we have investigated the following problem: We compared distance measures for graphs and examined the approximation behaviour of the two used distance measures. As a benchmark, we have chosen GED and approximated this graph distance with others whose input rely on topological indices. On the one hand, it is known that GED has been quite efficient when dealing with special practical problems, whose computation is NP-hard [16]. Then we have shown that one of our measures can
Acknowledgements
Tao Li is partially supported by the Natural Science Foundation of Tianjin (No. 16JCYBJC15200). Yongtang Shi thanks the Natural Science Foundation of Tianjin (No. 17JCQNJC00300) and the National Natural Science Foundation of China. Matthias Dehmer thanks the Austrian Science Funds for supporting this work (project P26142). All authors are supported by Nankai University.
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