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Degrees of Separation and Diameter in Large Graphs

Encyclopedia of Big Data Technologies

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Synonyms

Approximated neighborhood function; Computing average distance; Distance distribution; Maximum eccentricity in real-world networks

Definition

Given a (di)graph G = (V, E) (strongly) connected, where n = |V | and m = |E| (note that, since the graph is connected, we have m ≥ n − 1), the distance between two vertices u, v ∈ V is the number of edges along the shortest path from u to v and is denoted as d(u, v). The number of nodes separating u and v, i.e., d(u, v) − 1, is also called degree of separation.

The eccentricity of a node u is ecc(u) =max v ∈ V d(u, v), which measures in how many hops u can reach any other node in the graph. Hence, the diameter D (resp. the radius R) of G is defined as the maximum (resp. minimum) eccentricity among all the nodes, i.e., D =max u ∈ V ecc(u) (resp. R =min u ∈ V ecc(u)).

Given a node u ∈ V and an h ∈ [D] (where, for any positive integer x, [x] denotes the set {1, 2, …, x}), we call B h (u) the set {v : v ∈ V, d(u, v) ≤ h} and, similarly, N...

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Firenze, Florence, Italy

    Pierluigi Crescenzi

  2. Dipartimento di Informatica, Università di Pisa, Pisa, Italy

    Andrea Marino

Authors
  1. Pierluigi Crescenzi
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  2. Andrea Marino
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Corresponding authors

Correspondence to Pierluigi Crescenzi or Andrea Marino .

Editor information

Editors and Affiliations

  1. School of Comp. Sci. and Engineering, University of New South Wales School of Comp. Sci. and Engineering, Eveleigh, New South Wales, Australia

    Sherif Sakr

  2. Sch of Info Techno, Building J12, University of Sydney Sch of Info Techno, Building J12, Sydney, Australia

    Albert Zomaya

Section Editor information

  1. Department of Computer Science, University of Pisa, Largo B. Pontecorvo 3, 56127, Pisa, Italy

    Paolo Ferragina

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Crescenzi, P., Marino, A. (2018). Degrees of Separation and Diameter in Large Graphs. In: Sakr, S., Zomaya, A. (eds) Encyclopedia of Big Data Technologies. Springer, Cham. https://doi.org/10.1007/978-3-319-63962-8_59-1

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  • DOI: https://doi.org/10.1007/978-3-319-63962-8_59-1

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Chapter history

  1. Latest

    Degrees of Separation and Diameter in Large Graphs
    Published:
    15 June 2022

    DOI: https://doi.org/10.1007/978-3-319-63962-8_59-2

  2. Original

    Degrees of Separation and Diameter in Large Graphs
    Published:
    14 February 2018

    DOI: https://doi.org/10.1007/978-3-319-63962-8_59-1

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