Basics of Graph Theory

Abstract

A graph is a very important mathematical representation of a network because it can used for varied purposes such as understanding how a particular pecularity of a network behaves, how tweaking certain parts of a network can give rise to expected and sometimes unexpected results, and help visualise the network from different angles. However, this is incumbent upon the ability of the graph to represent all the features of the network. This chapter describes the different types of graphs that can be used to accommodate several specific features of the graph, and some important mathematical properties concerning these graphs. Several common graph theory concepts that are fundamental for social network analysis as well as other important definitions related to properties of the graph will also be discussed.

Problems

Download the email-Eu-core directed network from the SNAP dataset repository available at http://snap.stanford.edu/data/email-Eu-core.html.

For this dataset compute the following network parameters:

1

Number of nodes

2

Number of edges

3

In-degree, out-degree and degree of the first five nodes

4

Number of source nodes

5

Number of sink nodes

6

Number of isolated nodes

7

In-degree distribution

8

Out-degree distribution

9

Average degree, average in-degree and average out-degree

10

Distance between five pairs of random nodes

11

Shortest path length distribution

12

Diameter

13

Is the graph strongly connected? If so, compute the strongly connected component size distribution

14

Is the graph weakly connected? If so, compute the weakly connected component size distribution

15

Number of bridge edges

16

Number of articulation nodes

17

Number of nodes in In(v) for five random nodes

18

Number of nodes in Out(v) for five random nodes

19

Clustering coefficient for five random nodes

20

Clustering coefficient distribution

21

Average clustering coefficient