Network theory David Lusseau BIOL4062/5062

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 The structure and function of complex networks. SIAM Review 45,

What is a network Set of objects (vertices) with connections (edges) Represented by an adjacency matrix or a list v1v2weight HalJohn5 JohnGeorge10 LizHal2 BethLiz1 BethJohn20

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:

Hypergraph Meyers et al 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=7 e=9 v=19 e=27 v=3 e=2 v=1 e=0

k=0 k=4

k in =4 k out =4 k in =2 k out =3 k in =2 k out =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 E Betweenness of B = = 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 e i 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 coefficient n triangle connected to i/ n triples conn. to i 3/3=1 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  w ij >expected by chance,  Significance and relevance to hypotheses

Extending to weighted matrices Retrieve more information Relevance of binary matrix statistics strength ↔ degree:

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 Am. J. Soc. Graph by M.E.J. Newman

High school friendship Moody 2001 Am. J. Soc.

Internet Cheswick, Bell Labs

Food web Caribbean coral reef system

Human protein-protein interactions Chinnaiyan et al Nature Biotech

Tools for network analyses Ucinet/Netdraw ( Socprog ( Pajek ( Jung (JAVA) ( SNA (R package) (

Tools for network analyses Net.Linux (Linux OS) ( Visualising large graphs  Graphviz (  Yed (