Abstract
We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1α(G) and M2α(G), and the general sum-connectivity index, χα(G)) as well as of general versions of indices of interest: the general inverse sum indeg index ISIα(G) and the general first geometric-arithmetic index GAα(G) (with α∈R). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks GER(nER,p) and random geometric (RG) graphs GRG(nRG,r). The ER random networks are formed by nER vertices connected independently with probability p∈[0,1]; while the RG graphs consist of nRG vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius r∈[0,2]. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree k of the corresponding random network models, where kER=(nER−1)p and kRG=(nRG−1)(πr2−8r3/3+r4/2). That is, X(GER)/nER≈X(GRG)/nRG if kER=kRG, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.
Highlights
Harold Wiener can be considered the pioneer in the study of the so-called topological indices.His first investigations in this subject appeared in 1947 [1]
Winner’s work did not produce an immediate impact. Note that it was about 30 years later, in 1971, that Haruo Hosoya continued the research on topological indices by introducing the Z index [2]; currently known as the Hosoya Z index, which has been successfully applied in establishing quantitative structure–property relationships (QSPRs) and quantitative structure–activity relationships (QSARs)
In addition to well-known general indices (the first and second variable Zagreb indices, M1α ( G ) and M2α ( G ), and the general sum-connectivity index, χα ( G )) we have introduced general versions of the inverse sum indeg index
Summary
Harold Wiener can be considered the pioneer in the study of the so-called topological indices. Winner’s work did not produce an immediate impact Note that it was about 30 years later, in 1971, that Haruo Hosoya continued the research on topological indices by introducing the Z index [2]; currently known as the Hosoya Z index, which has been successfully applied in establishing quantitative structure–property relationships (QSPRs) and quantitative structure–activity relationships (QSARs). The computation of a given topological index on a large ensemble of random networks, all characterized by the same parameter pair (nER , p). [(nRG , r )], may provide useful average information about the full ensemble This statistical approach, well known in random matrix theory studies, is not widespread in studies involving topological indices, mainly because topological indices are not commonly applied to random networks; for very recent exceptions see [24,25].
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