# Encyclopedia of Big Data Technologies

2019 Edition
| Editors: Sherif Sakr, Albert Y. Zomaya

# Degrees of Separation and Diameter in Large Graphs

Reference work entry
DOI: https://doi.org/10.1007/978-3-319-77525-8_59

## Definitions

Given a (di)graph G = (V, E) (strongly) connected, where n = |V | and m = |E| (note that, since the graph is connected, we have mn − 1), the distance between two vertices u, vV 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) =maxvVd(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 =maxuVecc(u) (resp. R =minuVecc(u)).

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

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