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Network theory

Network theory. David Lusseau BIOL4062/5062 d.lusseau@dal.ca. Outline . Today: basics of graph theory and network statistics 8 March: incorporating uncertainty, network models 13 March: community structure Suggested readings:

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Network theory

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  1. Network theory David Lusseau BIOL4062/5062 d.lusseau@dal.ca

  2. 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

  3. What is a network • Set of objects (vertices) with connections (edges) • Represented by an adjacency matrix or a list v1 v2 weight Hal John 5 John George 10 Liz Hal 2 Beth Liz 1 Beth John 20

  4. 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)

  5. Undirected graph

  6. Directed graph cyclic acyclic? Cycle: <(a,b),(b,c),(c,a)>

  7. Hypergraph Meyers et al. 2004 J Th. Bio.

  8. Some terminology • Component: set of interconnected vertices (s) (in- and out- components in a directed graph) • Giantcomponent: the largest component in the graph (S)

  9. Some terminology • Degree: number of edges connected to a vertex (k) (in- and out- degrees in a directed graph) • Geodesicpath: the shortest path through the network from one vertex to another (l) • Diameter: length of the longest geodesic path (d)

  10. v=1 e=0 v=7 e=9 v=19 e=27 v=3 e=2

  11. k=0 k=4

  12. kin =4 kin =2 kin =2 kout=3 kout=1 kout=4

  13. Component 4 l(a,b)=2 d(4)=5

  14. Other centrality measures • Betweenness • Eigenvector • Reach • Clustering coefficient

  15. Betweenness and bottleneck • Number of geodesic path passing through a vertex E D Betweenness of B = B 1 + 1 + 1 = 3 C A

  16. Betweenness and bottleneck • Number of geodesic path passing through a vertex Betweenness of D = E D ½ + ½ = 1 B C A

  17. Eigenvector • Eigenvector of the dominant eigenvalue • ei integrates the connectivity of i (its degree) and the connectivity of its neighbours

  18. Reach • Number of vertices that can be reached in k steps as a proportion of vertices in the network • Typically 2 or fewer steps

  19. Reach • Centrality measure integrating link redundancy as well (are your friends only talking to your friends?)

  20. Clustering coefficient 0/1=0 3/3=1 0/1=0 3/6=0.5 3/3=1 • 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

  21. 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

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

  23. Some examples of real world networks • Social networks • Contact networks • Food webs • Man-made networks (internet, electricity grid) • Metabolite interactions • …

  24. High school dating Bearman et al. 2004 Am. J. Soc. Graph by M.E.J. Newman

  25. High school friendship Moody 2001 Am. J. Soc.

  26. Internet Cheswick, Bell Labs

  27. Food web Caribbean coral reef system

  28. Human protein-protein interactions Chinnaiyan et al. 2005 Nature Biotech

  29. 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)

  30. 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)

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