Computational Properties of General Indices on Random Networks

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.

Non-uniform random graphs on the plane: A scaling study

“…Since in previous studies of RGGs (embedded in the unit square) the average degree k was shown to play a prominent role in the scaling of topological properties [25,44], before performing the scaling analysis of our random graph model, in the next section we focus on its average degree.…”

“…5, agrees with the scaling of several (normalized) topological indices with k as reported in Refs. [25,44] for Erdös-Renyi graphs and RGGs in the unit square. However, here we are providing an explicit expression for the scaling, see Eq.…”

“…For recent exceptions see Refs. [25,43,44], where the average Randić index has been used to probe the percolation transition in Erdös-Rényi graphs and RGGs.…”

Stolarsky-Puebla index

“…This statistical approach, well known in random matrix theory studies, has been recently applied to random networks by means of topological indices, see e.g. [18][19][20]. Moreover, it has been shown that average topological indices may serve as complexity measures equivalent to standard random matrix theory measures [21,22].…”

Adaptation of Random Binomial Graphs for Testing Network Flow Problems Algorithms

“…Therefore, r and n give the way a RGG is generated. Very recently, RGGs have been successfully applied in nanomaterials [8]. Waxman generalized RGGs by considering a probabilistic connection function [9].…”