network theory david lusseau biol4062/5062 [email protected]
TRANSCRIPT
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Outline
Today: basics of graph theory and network statistics
8 March: incorporating uncertainty, network models
13 March: community structure Suggested readings:
Newman M.E.J. 2003. The structure and function of complex networks. SIAM Review 45,167-256
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What is a network
Set of objects (vertices) with connections (edges)
Represented by an adjacency matrix or a list
1 2 3 4
1 0 0 0 1
2 0 0 1 0
3 0 1 0 1
4 1 0 0 1
v1 v2 weight
Hal John 5
John George 10
Liz Hal 2
Beth Liz 1
Beth John 20
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Types of networks
Undirected graph (weighted or not)
Directed graph (weighted or not) Cyclic (contain loops) Acyclic (no loops)
Hypergraph (one edge join more than two vertices)
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Undirected graph
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Directed graph
cyclic
acyclic?
Cycle: <(a,b),(b,c),(c,a)>
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Hypergraph
Meyers et al. 2004 J Th. Bio.
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Some terminology
Component: set of interconnected vertices (s)
(in- and out- components in a directed graph)
Giant component: the largest component in the graph (S)
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Some terminology
Degree: number of edges connected to a vertex (k)
(in- and out- degrees in a directed graph)
Geodesic path: the shortest path through the network from one vertex to another (l)
Diameter: length of the longest geodesic path (d)
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v=7e=9
v=19
e=27
v=3e=2
v=1e=0
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k=0
k=4
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kin =4kout=4
kin =2kout=3
kin =2kout=1
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l(a,b)=2
Component 4
d(4)=5
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Other centrality measures
Betweenness
Eigenvector
Reach
Clustering coefficient
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Betweenness and bottleneck
Number of geodesic path passing through a vertex
A
B
C
D EBetweenness of B =
1 + 1 + 1 = 3
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Betweenness and bottleneck
Number of geodesic path passing through a vertex
A
B
C
D EBetweenness of D =
½ + ½ = 1
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Eigenvector
Eigenvector of the dominant eigenvalue
ei integrates the connectivity of i (its degree) and the connectivity of its neighbours
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Reach
Number of vertices that can be reached in k steps as a proportion of vertices in the network
Typically 2 or fewer steps
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Reach
Centrality measure integrating link redundancy as well (are your friends only talking to your friends?)
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Clustering coefficient
1 triangle, 8 connected triples: C=(3*1)/8=3/8 Each triangle contributes to 3 triples
Local clustering coefficientn triangle connected to i/ n triples conn. to i
3/3=1
3/3=10/1=0
0/1=0
3/6=0.5
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Dealing with weighted matrices First option: do not deal with them
Ignore the weight of the edges
Transform the weighted matrices in binary matrices Meaningful measures wij>expected by chance, Significance and relevance to hypotheses
ww ij
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Extending to weighted matrices Retrieve more information Relevance of binary matrix statistics strength ↔ degree:
N
1jiji as
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Some examples of real world networks Social networks Contact networks Food webs Man-made networks (internet, electricity grid) Metabolite interactions …
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High school dating
Bearman et al. 2004 Am. J. Soc.
Graph by M.E.J. Newman
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High school friendship
Moody 2001 Am. J. Soc.
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Internet
Cheswick, Bell Labs
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Food webCaribbean coral reef system
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Human protein-protein interactions
Chinnaiyan et al. 2005 Nature Biotech
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Tools for network analyses
Ucinet/Netdraw (http://www.analytictech.com/)
Socprog (http://myweb.dal.ca/hwhitehe/social.htm)
Pajek (http://vlado.fmf.uni-lj.si/pub/networks/pajek/)
Jung (JAVA) (http://jung.sourceforge.net)
SNA (R package) (http://erzuli.ss.uci.edu/R.stuff)
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Tools for network analyses
Net.Linux (Linux OS)
(http://pil.phys.uniroma1.it/%7Eservedio/software.html)
Visualising large graphs
Graphviz (http://www.graphviz.org)
Yed (http://www.yworks.com/en/products_yed_about.htm)