Abstract
Given a graph G, there are a number of global statistics besides the number n of nodes and the number m of edges whose values are classically reported to provide readers with a first impression of the structure of the graph. In this chapter various measures are described, such as the average clustering coefficient, reciprocity and transitivity, connectivity, size and the number of connected components, the graph density, its diameter, and the degree distribution as typical statistics of G.
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Notes
- 1.
Note again, that in many papers, the term “size of a graph” does not refer to the number of edges in the graph, but that it is often used synonymously to “number of nodes”, or both, “number of nodes and edges”.
- 2.
Make sure you understand this formula. See Exercise 3.5 and its solution on p. 532 for a hint.
- 3.
Note that, again, different fields have different names for the same concept: Watts and Strogatz called it the characteristic path length. Note that the equation is correct for both, directed and undirected graphs. In the first case, each pair’s distance is counted twice and thus, the average is divided by \(n(n-1)\), the number of ordered pairs of a set of n objects. Computationally, it is faster to sum every pair of nodes only once and then to divide by \(n(n-1)/2\).
- 4.
The data sets are provided by Jure Leskovec in his Stanford Network Analysis Platform (SNAP) which also provides analysis software http://snap.stanford.edu.
- 5.
Cumulative distributions are discussed in Sect. 3.5.1. The distance distribution is defined as the percentage of node pairs in a given distance k. The cumulative distance distribution gives the percentage of node pairs in at most distance k.
- 6.
In the undirected case it is important to note that one cannot simply count the number of pairs which have a neighbor in common. See Problem 4.9.
- 7.
The children in the survey were between 4 and 11 years old.
- 8.
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Zweig, K.A. (2016). Classic Network Analytic Measures. In: Network Analysis Literacy. Lecture Notes in Social Networks. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0741-6_4
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