Graph Theory Presentation

7/28/2019 Graph Theory Presentation

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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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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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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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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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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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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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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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Graph Theory Presentation