Computational Properties of General Indices on Random Networks

• Main Authors: R. Aguilar-Sánchez, I. F. Herrera-González, J. A. Méndez-Bermúdez, José M. Sigarreta
• Format: Article
• Language: English
• Published: MDPI AG 2020-08-01
• Series: Symmetry
• Subjects: computational analysis of networks; general topological indices; Erdös–Rényi networks; random geometric graphs
• Online Access: https://www.mdpi.com/2073-8994/12/8/1341

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>M</mi><mn>1</mn><mi>α</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>M</mi><mn>2</mn><mi>α</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and the general sum-connectivity index, <inline-formula><math display="inline"><semantics><mrow><msub><mi>χ</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>) as well as of general versions of indices of interest: the general inverse sum indeg index <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><msub><mi>I</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and the general first geometric-arithmetic index <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> (with <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks <inline-formula><math display="inline"><semantics><mrow><msub><mi>G</mi><mrow><mi>ER</mi><mi></mi></mrow></msub><mrow><mo>(</mo><msub><mi>n</mi><mi>ER</mi></msub><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and random geometric (RG) graphs <inline-formula><math display="inline"><semantics><mrow><msub><mi>G</mi><mrow><mi>RG</mi></mrow></msub><mrow><mo>(</mo><msub><mi>n</mi><mi>RG</mi></msub><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The ER random networks are formed by <inline-formula><math display="inline"><semantics><msub><mi>n</mi><mrow><mi>ER</mi></mrow></msub></semantics></math></inline-formula> vertices connected independently with probability <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; while the RG graphs consist of <inline-formula><math display="inline"><semantics><msub><mi>n</mi><mrow><mi>RG</mi></mrow></msub></semantics></math></inline-formula> 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 <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><msqrt><mn>2</mn></msqrt><mo>]</mo></mrow></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mfenced open="〈" close="〉"><mi>k</mi></mfenced></semantics></math></inline-formula> of the corresponding random network models, where <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="〈" close="〉"><msub><mi>k</mi><mrow><mi>ER</mi></mrow></msub></mfenced><mo>=</mo><mrow><mo>(</mo><msub><mi>n</mi><mi>ER</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>p</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="〈" close="〉"><msub><mi>k</mi><mrow><mi>RG</mi></mrow></msub></mfenced><mo>=</mo><mrow><mo>(</mo><msub><mi>n</mi><mi>RG</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><msup><mi>r</mi><mn>3</mn></msup><mo>/</mo><mn>3</mn><mo>+</mo><msup><mi>r</mi><mn>4</mn></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. That is, <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="〈" close="〉"><mi>X</mi><mo>(</mo><msub><mi>G</mi><mrow><mi>ER</mi></mrow></msub><mo>)</mo></mfenced><mo>/</mo><msub><mi>n</mi><mi>ER</mi></msub><mo>≈</mo><mfenced separators="" open="〈" close="〉"><mi>X</mi><mo>(</mo><msub><mi>G</mi><mrow><mi>RG</mi></mrow></msub><mo>)</mo></mfenced><mo>/</mo><msub><mi>n</mi><mi>RG</mi></msub></mrow></semantics></math></inline-formula> if <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="〈" close="〉"><msub><mi>k</mi><mrow><mi>ER</mi></mrow></msub></mfenced><mo>=</mo><mfenced separators="" open="〈" close="〉"><msub><mi>k</mi><mi>RG</mi></msub></mfenced></mrow></semantics></math></inline-formula>, with <i>X</i> 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.