1. Network analysis

2. A network/graph consists of nodes/vertices connected by links/edges
Network: definition
A network/graph consists of nodes/vertices connected by links/edges
node/
vertex
edge
Notation:

3. Multigraph vs. simple graph
loop
multiple edge
Multigraph
can contain loops and multiple edges
Simple graph

4. Directed vs. undirected graph
In a directed graph, an edge has a beginning and an end, i.e.
In a simple graph, an edge is an unordered pair of vertices, i.e.
Asymmetric relationship
Symmetric relationship

5. Weighted vs. unweighted graph
Each edge has an associated ‘strength’:
Carries only information about presence / absence of an edge:
0.5 1 1
-0.7
0.2
0.3
0.9

6. Use a network type according to what is relevant in your dataset
Databases are noisy, so check whether you need to get rid of loops, multiple/directed edges, etc., when constructing a network
We present tools to analyze simple undirected unweighted graphs, but many of them can be extended to deal with other types of networks as well.

7. Storing a graph: adjacency matrix
Two vertices are adjacent/neighbours if they are connected by an edge
Adjacency matrix of a graph G is defined as
where
if vertices i, j are adjacent
otherwise
Example:
Graph
Other possibilities: incidence matrix, edge list

8. Network analysis Comparing different networks to each other OR
e.g. by network properties that assign a single number to each network OR
2. Comparing different nodes within the same network
e.g. by node centrality measures that assign a number to every node in the network

9. 1.a Connectivity: ‘Can I get there?’b
A walk from vertex a to b is a ‘sequence of edges connecting a to b’
A path is a walk that does not intersect itself
A distance between a and b is the length of a shortest path between a and b
A component is a part of a graph in which any two vertices are connected by a path
Is the graph dis/connected?
Number of components c d e
Walk: dcbae
Path: dcab
d(a,b) =1
# of components: 2

10. 1.a Connectivity: ‘Can I get there?’
Edge/vertex connectivity = the minimum number of edges/vertices that need to be removed to disconnect a graph a b
Edge connectivity: 2
Vertex connectivity: 1 c d e

11. 1.b Shortest paths: ‘How far is it?’
Diameter = the length of a longest shortest path between a pair of vertices in a graph
‘at most how far are two vertices from each other?’
Average shortest path length = average of the lengths of shortest paths between all pairs of vertices a b
Diameter: 2 c
Average shortest path length: 15/10 = 1.5 d e

12. 1.c Edge-layout properties
Edge density =
‘Proportion of edges out of all possible edges’
where n = number of vertices in the graph
and a b c d

13. 1.c Edge-layout properties
Degree of vertex v = number of vertices adjacent to v in the graph
Average degree
where n = number of vertices in
the graph a b c
Average
Degree d

14. 1.c Edge-layout properties
where
# triangles(v) = number of triangles in the graph having v as its vertex
Clustering coefficient =
‘if vertex i has neighbours j and k, what is the probability that also j and k are neighbours?’ a b c d

15. 2.a Degree centrality For every vertex v in a graph,
Degree centrality (v) = deg(v)
‘The most important in the graph are the vertices with highest degrees’ a b
a 2
b 2
c 3
d 1 c d

16. 2.b Betweenness centrality
For every vertex v in the graph,
Betweenness centrality (v) =
‘Proportion of shortest paths between vertices i and j that pass through vertex v, out of all shortest paths between i and j; summing over all pairs of vertices in the graph ’ a b
a 1/2
b 0
c 0
d 1/2 c d

17. 2.c ‘Closeness’ centralities
‘The closer a vertex is to other vertices, the more important it is’
For every vertex v in the graph, compute
Geometric centrality (v) =
where d(i,v) is the distance between i and v
Closeness centrality (v) =
The above only work for connected graphs
Harmonic centrality (v) =

18. 2.c ‘Closeness’ centralities
Vertex a has distance 1, 1 and 2 from b, c and d, respectively. Therefore, a b c d
Geometric
Closenness
Harmonic a
4/3
3/4
5/6 b c 1 d
5/3
3/5
2/3

19. 2.d Eigenvector centrality
‘The more important neighbours a vertex has, the more important it is’
Eigenvalue centrality (i) =
where A is the adjacency matrix
Rewriting in a vector form, get
i.e.
The ith entry of the eigenvector corresponding to the (unique) largest eigenvalue give the centrality of vertex i. By Perron-Frobenius theorem, all centralities are positive.
Provides a relative measure. For an absolute measure, need to normalize the eigenvector, e.g. so that its entries sum up to 1.

20. 2. Exercise
Find a small network in which the maximum for 3 different centrality measures is attained by 3 different vertices.