On the asymptotic behavior of the average geodesic distance L and the compactness CB of simple connected undirected graphs whose order approaches infinity

• Published in: PloS one Vol. 16; no. 11; p. e0259776
• Main Authors: Lokot, Tatiana, Abramov, Olga, Mehler, Alexander
• Format: Journal Article
• Language: English
• Published: United States Public Library of Science 01.11.2021; Public Library of Science (PLoS)
• Subjects: Algorithms; Asymptotic properties; Biology and Life Sciences; Classification; Complexity; Computer and Information Sciences; Graphs; Humans; Infinity; Information theory; Models, Theoretical; Modems; Physical Sciences; Social Sciences; World problems; Germany
• ISSN: 1932-6203; 1932-6203
• DOI: 10.1371/journal.pone.0259776

The average geodesic distance L Newman (2003) and the compactness C B Botafogo (1992) are important graph indices in applications of complex network theory to real-world problems. Here, for simple connected undirected graphs G of order n , we study the behavior of L ( G ) and C B ( G ), subject to the condition that their order | V ( G )| approaches infinity. We prove that the limit of L ( G )/ n and C B ( G ) lies within the interval [0;1/3] and [2/3;1], respectively. Moreover, for any not necessarily rational number β ∈ [0;1/3] ( α ∈ [2/3;1]) we show how to construct the sequence of graphs { G }, | V ( G )| = n → ∞, for which the limit of L ( G )/ n ( C B ( G )) is exactly β ( α ) (Theorems 1 and 2). Based on these results, our work points to novel classification possibilities of graphs at the node level as well as to the information-theoretic classification of the structural complexity of graph indices.