1. Applications of graph theory in complex systems research
Kai Willadsen

2. Outline Graph-based representations What makes a problem graph-like?
Applications of graph theory
Measuring graph characteristics
Graph structures
Global metrics
Local metrics

3. Graph-based representations
Representing a problem as a graph can provide a different point of view
Representing a problem as a graph can make a problem much simpler
More accurately, it can provide the appropriate tools for solving the problem

4. Bridges of Königsberg
Is it possible to cross all of the bridges in the city without crossing a single bridge twice?

5. Bridges of Königsberg
Is it possible to cross all of the bridges in the city without crossing a single bridge twice?
Euler realised that this problem could be represented as a graph

6. Bridges of Königsberg
Does this graph have a path covering every edge without duplicates? (a Euler walk)
In order to have such a path, the graph must have either zero or two nodes with an odd number of edges
It has four, therefore no

7. Friends of friends
Social experiments have demonstrated that the world is a small place after all
There is a high probability of you having an indirect connection, through a small number of friends, to a total stranger
In fact, it is postulated that a connection can be drawn between two random people in a very small number (<6) of links

8. Friends of friends
In a social network, a common default assumption was that connections were localised
Distant nodes take many links to reach

9. Friends of friends
Watts and Strogatz showed that randomly rewiring only a few links in such a network dramatically reduced the number of links between distant nodes
Small-world networks

10. What is a graph?
A graph consists of a set of nodes and a set of edges that connect the nodes
That’s (almost) it
also directedness, parallel edges, self-connection, weighted edges, node values…
Node
Edge
Graph

11. What is graph theory?
Graph theory provides a set of techniques for analysing graphs
Complex systems graph theory provides techniques for analysing structure in a system of interacting agents, represented as a graph
Applying graph theory to a system means using a graph-theoretic representation

12. What makes a problem graph-like?
There are two components to a graph
Nodes and edges
In graph-like problems, these components have natural correspondences to problem elements
Entities are nodes and interactions between entities are edges
Most complex systems are graph-like

13. Examples of complex systems
Social networks
Nodes are actors, edges are relationships
The social network
for the #java IRC
channel

14. Examples of complex systems
Genetic regulatory networks
Nodes are genes or proteins, edges are regulatory interactions
The p53 cancer network

15. Examples of complex systems
Transportation networks
Nodes are cities, transfer points or depots, edges are roads or transport routes
The Brisbane train network

16. Why are graphs useful?
The structure of relationships between system elements provides information about system properties
Bridges of Königsberg – the graph structure demonstrated the lack of the property in question
Small world networks – the way in which the desired property was obtained informed understanding of the network structure

17. Structures and structural metrics
Graph structures are used to isolate interesting or important sections of a graph
Structural metrics provide a measurement of a structural property of a graph
Global metrics refer to a whole graph
Local metrics refer to a single node in a graph

18. Graph structures Identify interesting sections of a graph
Interesting because they form a significant domain-specific structure, or because they significantly contribute to graph properties
A subset of the nodes and edges in a graph that possess certain characteristics, or relate to each other in particular ways
i.e., a subgraph

19. Subgraphs
Graph
Subgraph
A subgraph consists of a subset of the nodes and edges of a graph
spanning, induced, complete
Subgraphs are also graphs

20. Graph structure: clique
A clique is a complete connected subgraph
In a clique, every node is connected to every other node
There are different ways of relaxing the complete connection requirement
n-clique, n-clan, k-plex, k-core

21. Graph structure: clique
B, C, E and F form a clique of size 4
E, F and H form a clique of size 3
A, D, G and I are not part of any clique A B D H F E C I G

22. Graph structure: clique
Subgraphs identified as cliques are interesting because they
are as tightly connected as possible
are ‘modules’ in the graph
indicate through exclusion sections of the graph that are not so tightly connected

23. Global metrics
Global metrics provide a measurement of a structural property of a whole graph
Designed to characterise
System dynamics – what aspects of the system’s structure influence its behaviour?
Structural dynamics – how robust is the system’s structure to change?

24. Global metric: average path length
The average path length of a graph is the average of the shortest path lengths between all pairs of nodes in a graph
Also known as diameter or average shortest path length
Average path length = 1
Average path length = 1.66

25. Global metric: average path length
Shortest paths are
AB, AC, ABD, ABE, BC, BD, BE, CBD, CBE, DBE
Lengths
1, 1, 2, 2, 1, 1, 1, 2, 2, 2
Average path length
1.5 B E D C A

26. Global metric: average path length
In graphs with a low average path length, transfer of information between nodes takes place rapidly
Average path length is generally proportional to the size (N) of a network
In small-world networks it is proportional to log N
In scale-free networks it is proportional to log log N

27. 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?

28. Local metric: betweenness centrality
The number of shortest paths in the graph that pass through the node
One measure of node centrality
also closeness centrality, degree centrality

29. Local metric: betweenness centrality
Shortest paths are:
AB, AC, ABD, ABE, BC, BD, BE, CBD, CBE, DBE
Five paths go through B
B has a betweenness centrality of 5 B E D C A

30. Local metric: 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

31. Graph theory in complex systems
Using complex systems graph theory to isolate interesting system properties
Structural properties
Global and local metrics
Obtaining a better understanding of the pattern of interactions in a system

32. Getting more information
Tutorial handout
Available at:
Reference material
Available at:
Try looking up node centrality, degree distribution, scale-free topology, diameter, girth, edge-connectivity, robustness