Applications of graph theory in complex systems research

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Applications of graph theory in complex systems
research

• Kai Willadsen

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Outline

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

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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

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Bridges of Königsberg

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

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Bridges of Königsberg

• Is it possible to cross all of the bridges in the
  city without crossing a single bridge twice?
• Euler realised thatthis problem couldbe
  represented asa graph

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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

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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 (lt6) of links

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Friends of friends

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

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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

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What is a graph?

• A graph consists of a set of nodes and a set of
  edges that connect the nodes
• Thats (almost) it
• also directedness, parallel edges,
  self-connection, weighted edges, node values

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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

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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

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Examples of complex systems

• Social networks
• Nodes are actors,edges are relationships

The social network for the java IRC channel
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Examples of complex systems

• Genetic regulatory networks
• Nodes are genes orproteins, edges are regulatory
  interactions

The p53 cancer network
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Examples of complex systems

• Transportation networks
• Nodes are cities, transfer points or depots,
  edges are roads or transport routes

The Brisbanetrain network
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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

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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

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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

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Subgraphs

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

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Graph structure clique

• A clique is a complete connected subgraph
• In a clique, every node isconnected to every
  other node
• There are different ways ofrelaxing the
  completeconnection requirement
• n-clique, n-clan, k-plex, k-core

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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

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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

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Global metrics

• Global metrics provide a measurement of a
  structural property of a whole graph
• Designed to characterise
• System dynamics what aspects of the systems
  structure influence its behaviour?
• Structural dynamics how robust is the systems
  structure to change?

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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

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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

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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 tolog
  N
• In scale-free networks it is proportional tolog
  log N

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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?

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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

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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

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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

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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

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Getting more information

• Tutorial handout
• Available at http//www.itee.uq.edu.au/kaiw/grap
  htheory/
• 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