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- Title
Computational Properties of General Indices on Random Networks.
- Authors
Aguilar-Sánchez, R.; Herrera-González, I. F.; Méndez-Bermúdez, J. A.; Sigarreta, José M.
- Abstract
We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M 1 α (G) and M 2 α (G) , and the general sum-connectivity index, χ α (G) ) as well as of general versions of indices of interest: the general inverse sum indeg index I S I α (G) and the general first geometric-arithmetic index G A α (G) (with α ∈ R ). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks G ER (n ER , p) and random geometric (RG) graphs G RG (n RG , r) . The ER random networks are formed by n ER vertices connected independently with probability p ∈ [ 0 , 1 ] ; while the RG graphs consist of n RG 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 k ER = (n ER − 1) p and k RG = (n RG − 1) (π r 2 − 8 r 3 / 3 + r 4 / 2) . That is, X (G ER) / n ER ≈ X (G RG) / n RG if k ER = k RG , 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.
- Subjects
ZAGREB (Croatia); RANDOM graphs; RANDOM measures; MOLECULAR connectivity index; RANDOM matrices; EUCLIDEAN distance; STATISTICS
- Publication
Symmetry (20738994), 2020, Vol 12, Issue 8, p1341
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym12081341