On the degeneracy of the Randić entropy and related graph measures
Introduction
Exploring the structure of complex networks has been a challenging problem over the last two decades [37], [42], [50]. Analysis of complex networks began, naturally enough, with descriptions of network properties. Random networks were distinguished from deterministic ones. Other considerations such as whether a network satisfies the “small world” hypothesis [50] or is scale free were also examined. Yet, these properties are relevant mainly to graphs representing the World Wide Web and various biological networks [9]. Another approach to analysis has been the quantification of structural properties based on measurement [15]. So-called graph measures can be divided into several categories such as global, local, information-theoretic, non-information-theoretic, distance-based and so forth [22], [28]. Finding a measure that performs well for a particular problem continues to be challenging [15]. Some progress has been made, however, as this approach has been pursued extensively in problem areas such as drug design, QSAR/QSPR, see, e.g., [3]. A critical aspect of this approach involves the definition of measures designed to characterize the structure of complex networks. Many different paths have been followed in quantitative graph analysis. Paths that has been followed for some time are the study of graph complexity [40] and comparative network analysis [23]. A more recent development involves network classification and clustering, see [46]. Critical to the use of quantitative methods to analyze networks are identifying the structural feature or features captured by a graph measure and determining the degeneracy of the measure [7], [19].
In this paper, we continue the work on the degeneracy of structural graph measures. Somewhat related to this problem is the search for complete graph invariants as attempted in [7], [16], [33], [47]. A complete set of graph invariants is a collection of properties which discriminate between all non-isomorphic graphs uniquely. In this paper, we focus on studying the degeneracy, i.e., the failure to discriminate non-isomorphic graphs uniquely utilizing information-theoretic and non-information-theoretic techniques. The work reported here builds on an earlier paper, namely, Dehmer et al. [20]. The point of departure here is Randić entropy [12], since the degeneracy of this measure has not yet been explored. To explore this measure we define a random polynomial using the so-called Randić weights [12], and exploit its roots as the basis for defining novel graph measures. The degeneracy of these graph measures is then investigated by computing numerical values for graphs and trees in an exhaustively set.
Section snippets
Information-theoretic graph measures
Note that a comprehensive review of the many extant graph measures (e.g., [6], [34], [48]) is beyond the scope of this paper. In any case, information-theoretic graph measures have been discussed extensively, see [40]. Here we just sketch two important classes of information-theoretic graph measures as they have been proven useful for network classification [22], [23] and network characterization [23], [40], [48].
A classical entropy based graph measure elaborated by Mowshowitz [40] built on the
Computational procedure
In this section, we present the details of a procedure designed to generate graph classes and to compute the novel graph measures.
- Step 1:
Classes of graphs Ti and Ni are generated exhaustively by means of Nauty [39]. Their sizes can be seen in Table 1. Ti is the class of all pairwise non-isomorphic trees with i vertices, and Ni is the class of all pairwise non-isomorphic graphs with i vertices. For computational details, see also [21]. The reason for using exhaustively generated trees and graphs is
Numerical results
The results of the computation of Randić entropy IR according to Eq. 9 are summarized in Table 1.
From Table 1 it is clear that the bigger the graph class, the higher the degeneracy rate. For example, 241842 of the graphs ∈ N9 could not be distinguished structurally using the Randić entropy IR. In this case 99.92% (almost all) of the exhaustively generated graphs with 9 vertices cannot be discriminated by IR. The situation is quite similar for the classes of exhaustively generated trees. Even
Summary and conclusion
In this paper, we have examined the degeneracy of the Randić entropy and some related graph measures. Our starting point was the Randić entropy [12] which is based on the so-called Randić weights [12] for defining the probabilities on a given edge set. Also, we have defined additional graph measures based on the Randić polynomial as well as on the random Randić roots determined by Eqs. 12 and 14.
This approach is decidedly different from others dealing with ordinary graph measures used to
Acknowledgments
Matthias Dehmer thanks the Austrian Science Funds for supporting this work (project P 30031). Zengqiang Chen was supported by the National Natural Science Foundation of China (NSFC), see Grant No. 61573199. Yongtang Shi was partially supported by the National Natural Science Foundation of China (No. 11771221), the Natural Science Foundation of Tianjin (No. 17JCQNJC00300), the Natural Science Foundation of the Education Department of Shannxi ROC (No. 16JK1456 ). Chengyi Xia was supported by the
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