Applications of graph theory in complex systems research Kai Willadsen

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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?

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

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

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

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

Getting more information Tutorial handout Available at: http://www.itee.uq.edu.au/~kaiw/graphtheory/ Reference material Available at: http://130.102.66.173/wiki/index.php/Main_Page Try looking up node centrality, degree distribution, scale-free topology, diameter, girth, edge-connectivity, robustness