Topological versus spectral properties of random geometric graphs

Aguilar-Sanchez, R.; Mendez-Bermudez, J. A.; Rodrigues, Francisco A.; Sigarreta, Jose M.

doi:10.48550/arxiv.2007.02453

Cited by 1 publication

(2 citation statements)

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“…Moreover, recently, it has been shown for RG graphs that there is a a huge correlation between the average-scaled Shannon entropy (of the adjacency matrix eigenvectors) and two average-scaled topological indices [32]: the Randić index R(G) and the harmonic index H(G). We believe that this is a remarkable result because it validates the use of average topological indices as RMT complexity measures; already suggested in Refs.…”

Section: Discussionmentioning

confidence: 99%

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Computational Properties of General Indices on Random Networks

et al. 2020

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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.

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Section: Discussionmentioning

confidence: 99%

“…(We should remark that the behavior reported in Figure 13 for M −1/2 2 (G) /n and χ −1 (G) /n was already expected since they coincide, respectively, with R(G) /n and H(G) /2n; already studied in Refs. [24,25,32] and used there as measures equivalent to RMT complexity measures. )…”

Section: Discussionmentioning

confidence: 99%