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

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Titel
On the asymptotic behavior of the average geodesic distance L and the compactness CB of simple connected undirected graphs whose order approaches infinity
Verfasser
Lokot, Tatiana ; Abramov, Olga ; Mehler, Alexander
Enthalten in
PLOS ONE, Jg. 16 H. 11
Erschienen
2021
Sprache
Englisch
Dokumenttyp
Aufsatz in einer Zeitschrift
URN
urn:nbn:de:0070-pub-29590644
DOI
10.1371/journal.pone.0259776

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Abstract

The average geodesic distance L Newman (2003) and the compactness CB 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 CB(G), subject to the condition that their order |V(G)| approaches infinity. We prove that the limit of L(G)/n and CB(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 (CB(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.

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