1. Introduction to Network Theory: Modern Concepts, Algorithms
and Applications
Ernesto Estrada
Department of Mathematics, Department of Physics
Institute of Complex Systems at Strathclyde
University of Strathclyde

2. Types of graphs Weighted graphs Multigraphs Pseudographs Digraphs
Simple graphs

3. Weighted graph is a graph for which each edge has an associated
weight, usually given by a weight function
w: E  R, generally positive

4. Adjacency Matrix of Weighted graphs

5. Degree of Weighted graphs
The sum of the weights associated to every edge
incident to the corresponding node
The sum of the corresponding row or column of
the adjacency matrix
Degree
1.5
4.9 6
2.8
3.3

6. Multigraph or pseudograph
is a graph which is permitted to have multiple
edges. Is an ordered pair G:=(V,E) with
V a set of nodes
E a multiset of unordered pairs of vertices.

7. Adjacency Matrix of Multigraphs

8. Directed Graph (digraph)
Edges have directions
The adjacency matrix is not symmetric

9. Simple Graphs
Simple graphs are graphs without multiple edges or self-loops. They are weighted graphs with all edge weights equal to one. B E D C A

10. Local metrics
Local metrics provide a measurement of a structural property of a single node
Designed to characterise
Functional role – what part does this node play in system dynamics?
Structural importance – how important is this node to the structural characteristics of the system?

11. Degree Centrality
degree B E D C A 1 4 3

12. Betweenness centrality
The number of shortest paths in the graph that pass through the node divided by the total number of shortest paths.

13. Betweenness centrality
Shortest paths are:
AB, AC, ABD, ABE, BC, BD, BE, CBD, CBE, DBE
B has a BC of 5 A B C E D

14. Betweenness centrality
Nodes with a high betweenness centrality are interesting because they
control information flow in a network
may be required to carry more information
And therefore, such nodes
may be the subject of targeted attack

15. Closeness centrality
The normalised inverse of the sum of topological distances in the graph.

16. Closeness centrality B E D C A 6 4 7

17. Closeness centrality B E D C A
Closeness
0.67
1.00
0.57

18. Closeness centrality Node B is the most central one in spreading
information from it to the other nodes in the
network.

19. Local metrics Node B is the most central one according to the degree,
betweenness and closeness
centralities.

20. and the winner is… A is the most central according to the degree A
B is the most central
according to closeness
and betweenness A B
Which is the most central node?

21. Degree: Difficulties

22. Extending the Concept of Degree
Make xi proportional to the average of the centralities
of its i’s network neighbors
where l is a constant. In matrix-vector notation we
can write
In order to make the centralities non-negative we select
the eigenvector corresponding to the principal eigenvalue
(Perron-Frobenius theorem).

23. Eigenvalues and Eigenvectors
The value λ is an eigenvalue of matrix A if there exists a non-zero vector x, such that Ax=λx. Vector x is an eigenvector of matrix A
The largest eigenvalue is called the principal eigenvalue
The corresponding eigenvector is the principal eigenvector
Corresponds to the direction of maximum change

24. Eigenvector Centrality
The corresponding entry of the principal eigenvector of the adjacency matrix of the network.
It assigns relative scores to all nodes in the network based on the principle that connections to high-scoring nodes contribute more.

25. Eigenvector Centrality
Node EC

26. Eigenvector Centrality: Difficulties
In regular graphs all the nodes have exactly the same value of the eigenvector centrality, which is equal to

27. Subgraph Centrality
A closed walk of length k in a graph is a succession of k (not necessarily different) edges starting and ending at the same node, e.g.
1,2,8,1 (length 3)
4,5,6,7,4 (length 4)
2,8,7,6,3,2 (length 5)

28. Subgraph Centrality
The number of closed walk of length k starting at the same node i is given by the ii-entry of the kth power of the adjacency matrix

29. Subgraph Centrality
We are interested in giving weights in decreasing order of the length of the closed walks. Then, visiting the closest neighbors receive more weight that visiting very distant ones.
The subgraph centrality is then defined as the following weighted sum

30. Subgraph Centrality By selecting cl=1/l! we obtain
where eA is the exponential of the adjacency matrix.
For simple graphs we have

31. Subgraph Centrality
Nodes EE(i)
1,2, 4,
3,5,

32. Subgraph Centrality: Comparsions
Nodes BC(i)
1,2, 4,
3,5,
Nodes EE(i)
1,2, 4,
3,5,

33. Subgraph Centrality: Comparisons
Nodes EE(i)
45.696
45.651

34. Communicability
Path of length 6
Walk of length 8
Shortest path

35. Communicability Let be the number of shortest paths of length s
between p and q.
Let be the number of walks of length k>s
between p and q.
DEFINITION (Communicability):
and must be selected such as the communicability converges.

36. Communicability By selecting bl=1/l! and cl=1/l! we obtain
where eA is the exponential of the adjacency matrix.
For simple graphs we have

37. Communicability

38. Communicability q q p p

39. Communicability
intracluster
intercluster

40. Communicability & Communities
A community is a group of nodes for wich the intra-cluster communicability is larger than the inter-cluster one
These nodes communicates better among them than with the rest of extra-community nodes.

41. Communicability Graph
Let
The communicability graph Q(G) is the graph whose adjacency matrix is given by Q(D(G)) results from the elementwise application of the function Q(G) to the matrix D(G).

42. Communicability Graph

43. Communicability Graph
A community is defined as a clique
in the communicability graph.
Identifying communities is reduced
to the “all cliques problem” in the
communicability graph.

44. Social (Friendship) Network
Communities: Example
Social (Friendship) Network

45. Communities: Example
The Network
Its Communicability
Graph

46. Communities
Social Networks Metabolic Networks

47. References
Aldous & Wilson, Graphs and Applications. An Introductory Approach, Springer, 2000.
Wasserman & Faust, Social Network Analysis, Cambridge University Press, 2008.
Estrada & Rodríguez-Velázquez, Phys. Rev. E 2005, 71,
Estrada & Hatano, Phys. Rev. E. 2008, 77,

48. Exercise 1 Identify the most central node according to the following
criteria:
(a) the largest chance of receiving information from closest
neighbors;
(b) spreading information to the rest of nodes in the
network;
(c) passing information from some nodes to others.

49. Exercise 2 T.M.Y. Chan collaborates with 9 scientists in
computational geometry. S.L. Abrams also collaborates with
other 9 (different) scientists in the same network. However,
Chan has a subgraph centrality of 109, while Abrams has 103.
The eigenvector centrality also shows the same trend,
EC(Chan) = 10-2; EC(Abrams) = 10-8.
Which scientist has more chances of being informed about
the new trends in computational geometry?
(b) What are the possible causes of the observed differences
in the subgraph centrality and eigenvector centrality?

50. Exercise 2. Illustration.