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Applications of graph
theory in complexsystems researchKai Willadsen
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Graph-based representations
Representing a problem as a graph canprovide a different point of viewRepresenting a problem as a graph canmake a problem much simpler
More accurately, it can provide the
appropriate tools for solving the problem
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Bridges of Knigsberg
Is it possible to crossall of the bridges in thecity without crossing asingle bridge twice?
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Bridges of Knigsberg
Is it possible to crossall of the bridges in thecity without crossing asingle bridge twice?Euler realised that
this problem couldbe represented asa graph
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Friends of friends
Social experiments have demonstratedthat the world is a small place after all
There is a high probability of you having anindirect connection, through a small number of friends, to a total stranger
In fact, it is postulated that a connection canbe drawn between two random people in avery small number (
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Friends of friends
In a social network, acommon defaultassumption was thatconnections werelocalised
Distant nodes takemany links to reach
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What is a graph?
A graph consists of a set of nodes and a set of edgesthat connect the nodesThats (almost) it
also directedness, parallel
edges, self-connection,weighted edges, nodevalues
Node Node
Edge
Graph
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What is graph theory?
Graph theory provides a set of techniquesfor analysing graphsComplex systems graph theory providestechniques for analysing structure in asystem of interacting agents, represented
as a graph Applying graph theory to a system meansusing a graph-theoretic representation
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What makes a problem graph-like?
There are two components to a graphNodes and edges
In graph-like problems, these componentshave natural correspondences to problemelements
Entities are nodes and interactions betweenentities are edges
Most complex systems are graph-like
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Examples of complex systems
Genetic regulatory networksNodes are genes or
proteins, edges areregulatory interactions
The p53 cancer network
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Examples of complex systems
Transportation networksNodes are cities, transfer
points or depots, edgesare roads or transportroutes
The Brisbanetrain network
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Structures and structural metrics
Graph structures are used to isolateinteresting or important sections of a graph
Structural metrics provide a measurementof 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 graphInteresting because they form a significant
domain-specific structure, or because theysignificantly 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 waysi.e., a subgraph
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Subgraphs
A subgraph consists of a subset of the nodes
and edges of a graphspanning, induced,complete
Subgraphs are alsographs
Graph
Subgraph
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Graph structure: clique
A clique is a complete connectedsubgraph
In a clique, every node isconnected to every other node
There are different ways of
relaxing the completeconnection requirement
n-clique, n-clan, k-plex, k-core
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Graph structure: clique
B, C, E and F form aclique of size 4
E, F and H form aclique of size 3
A, D, G and I are notpart of any clique
A B
D
H
FE
C
IG
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Graph structure: clique
Subgraphs identified as cliques areinteresting because they
are as tightly connected as possibleare modules in the graph indicate through exclusion sections of the
graph that are not so tightly connected
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Global metric: average path length
The average path length of a graph is the average of
the shortest path lengthsbetween all pairs of nodesin a graph
Also known as diameter or average shortest pathlength
Average path length = 1
Average path length = 1.66
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Global metric: average path length
Shortest paths are AB, AC, ABD, ABE, BC,
BD, BE, CBD, CBE, DBELengths
1, 1, 2, 2, 1, 1, 1, 2, 2, 2
Average path length1.5
B
ED
C
A
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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 generallyproportional 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 astructural property of a single node
Designed to characteriseFunctional role what part does this nodeplay in system dynamics?
Structural importance how important is thisnode to the structural characteristics of thesystem?
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Local metric: betweenness centrality
The number of shortestpaths in the graph that
pass through the nodeOne measure of nodecentrality
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, DBEFive paths go through B
B has a betweenness
centrality of 5
B
ED
C
A
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Local metric: betweenness centrality
Nodes with a high betweenness centralityare interesting because they
control information flow in a networkmay be required to carry more information
And therefore, such nodesmay be the subject of targeted attack
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Graph theory in complex systems
Using complex systems graph theory toisolate interesting system properties
Structural propertiesGlobal and local metrics
Obtaining a better understanding of the
pattern of interactions in a system
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