network theory david lusseau biol4062/5062 [email protected]
TRANSCRIPT
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
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
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)
Undirected graph
Directed graph
cyclic
acyclic?
Cycle: <(a,b),(b,c),(c,a)>
Hypergraph
Meyers et al. 2004 J Th. Bio.
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)
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)
v=7e=9
v=19
e=27
v=3e=2
v=1e=0
k=0
k=4
kin =4kout=4
kin =2kout=3
kin =2kout=1
l(a,b)=2
Component 4
d(4)=5
Other centrality measures
Betweenness
Eigenvector
Reach
Clustering coefficient
Betweenness and bottleneck
Number of geodesic path passing through a vertex
A
B
C
D EBetweenness of B =
1 + 1 + 1 = 3
Betweenness and bottleneck
Number of geodesic path passing through a vertex
A
B
C
D EBetweenness of D =
½ + ½ = 1
Eigenvector
Eigenvector of the dominant eigenvalue
ei integrates the connectivity of i (its degree) and the connectivity of its neighbours
Reach
Number of vertices that can be reached in k steps as a proportion of vertices in the network
Typically 2 or fewer steps
Reach
Centrality measure integrating link redundancy as well (are your friends only talking to your friends?)
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
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
Extending to weighted matrices Retrieve more information Relevance of binary matrix statistics strength ↔ degree:
N
1jiji as
Some examples of real world networks Social networks Contact networks Food webs Man-made networks (internet, electricity grid) Metabolite interactions …
High school dating
Bearman et al. 2004 Am. J. Soc.
Graph by M.E.J. Newman
High school friendship
Moody 2001 Am. J. Soc.
Internet
Cheswick, Bell Labs
Food webCaribbean coral reef system
Human protein-protein interactions
Chinnaiyan et al. 2005 Nature Biotech
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)
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)