Quantifying Network Topology

1. Quantifying Network Topology Jennifer Hallinan ARC Centre for Bioinformatics, Institute for Molecular Biosciences & School of ITEE j.hallinan@imb.uq.edu.au

2. Networks www.surrey.ac.uk/ SBMS/Fgenomics/ http://radio.weblogs.com/0114726/2003/01/02.html http://www.life.uiuc.edu/bio100/lectures/s02lects/foodweb.gif Network Analysis

3. What is a Network? • Relational dataset • One or more sets with explicit relations between their members • Nodes (Vertices) • People, actors, agents, entities, (genes, proteins…) • Edges • Links, ties, interactions Network Analysis

4. Network Data Representation • Matrix • Not necessarily square • Closely related to graph theory • Useful for direct computation (matrix algebra) • Not very efficient for large sparse networks • Linked list • Generally a list of nodes denoting their attributes • Followed by a list of edges • Less standardised • More efficient data storage for large networks • Less efficient computation Network Analysis

5. a b c d e f g h i a 0 1 0 1 1 1 0 0 0 b 1 0 1 0 0 0 0 0 0 c 0 1 0 0 0 1 0 0 1 d 1 0 0 0 1 1 0 0 0 e 1 0 0 1 0 1 0 0 0 f 1 0 1 1 1 0 0 0 0 g 0 0 0 0 0 0 0 1 1 h 0 0 0 0 0 0 1 0 1 i 0 0 0 0 0 0 1 1 0 A Simple Network Network Analysis

6. Network structure Random Small world www.discover.com/dec_issue/smallworld.html Scale free Network Analysis www.21cmagazine.com

7. Network structure:connectivity distribution Network Analysis

8. Network Metrics • Distance: minimum number of links between a pair of nodes, may be weighted • Diameter: longest shortest path between all pairs of nodes • Density: number of edges as a fraction of all possible edges • Cluster coefficient: measure of attachment amongst neighbours Network Analysis

9. Small World Networks Network Analysis

10. Properties • Robust to random error • Most nodes are sparsely connected • Deleting nodes at random tends to leave connectivity of rest of network unchanged • Up to 5% tolerance • Vulnerable to targeted error • Selective removal of most highly connected nodes leads to rapid breakdown • Internet • Enhanced signal propagation speed, computational power and synchronizability • Selective advantage? Network Analysis

11. Network Generation Algorithms • Networks can be generated computationally • Specified distribution • Specified algorithm • Allows large numbers of networks to be generated and studied • Draw conclusions about classes of network Network Analysis

12. Preferential Attachment • R. Albert & A. –L. Barabasi, “Topology of evolving networks: Local events and universality” Physics Review Letters vol. 85, pp. 5234 – 5246, 2000. • Start with a small number of nodes connected in a ring • Network grows by adding nodes, which link to other nodes with probability • Scale free, not small world Network Analysis

13. Preferential Attachment Network Network Analysis

14. Gene Duplication • R. Pastor-Satorras, E. Smith & R. V. Sole, “Evolving protein interaction networks through gene duplication”, Santa Fe Institute Working Paper 02-02-008, 2002 • Start with a small ring of nodes • At each step a node is selected at random and duplicated, with all its links • The links to the new node are deleted with probability  and added with probability  • Scale free network • Small world network Network Analysis

15. Gene Duplication Network Network Analysis

16. Random Network Network Analysis

17. Modularity in Biological Networks • Module: “a biological entity characterized by more internal than external integration” (Bolker, 2000) • Biological systems are inherently modular • Cascades of gene activation • Developmental modularity • Functional modules • The behaviour or function of a module reflects the integration of its parts, not simply the arithmetic sum of those parts • Modules are units of selection • Modular organization provides flexibility: modules can be combined in different ways during development to give different outcomes • This permits complex anatomies without excessive demand on genomic complexity Network Analysis

18. Detecting Modularity • Component: a maximal connected subgraph • Strong component: arcs that make up the paths in the subgraph are in the same direction • Weak component: component in a digraph whose arcs do not form a path in the same direction • Core: most cohesive or highly connected members of a component • k-core: degree-based measure • m-core: multiplicity-based measure (weights on edges) • Clique: subgraph in which every possible pair of points is directly connected and the clique is not contained in any other clique • n-clique: clique with maximal path length of n Network Analysis

19. Detecting Modularity • Analysis of flux modes (the smallest sub-networks enabling the metabolic system to operate in steady state) • find the maximum flow that can be routed from a source node, to a sink node, while obeying all capacity constraints • Identify and remove linking nodes or edges • orthologous groups with mutually exclusive associations • nodes which have more than a threshold number of links • Betweenness • Clustering • Calculate “distance” between each pair of nodes • Cluster according to distance Network Analysis

20. Cancer genes Network Analysis

21. Clustered Network Analysis

22. Clusters Cluster 2: Handling of epidermal growth factor two platelet-derived growth factors (PDGFA and B), two platelet-derived growth factor receptors (PDGFRA and B), a protein which binds the epidermal growth factor receptor (GRB), and one which is involved in the regulation of epidermal growth factor receptor activity (SHC1) Cluster 4: Mitogen activated protein kinases two mitogen activated protein (MAP) kinase kinases and their targets, three MAP kinases. Network Analysis

23. Hierarchical modularity Network Analysis

24. Hierarchical modularity Network Analysis

25. Network Motifs • What are network motifs? • “The simplest units of commonly used transcriptional regulatory network architecture” (Lee et al., 2002). • “Recurring, significant patterns of interconnections.” (Milo et al., 2002). • Motifs with meaning occur in many different network contexts: Network Analysis

26. GRN Motifs Network Analysis

27. Motif Detection • Working on an adjacency matrix representation • Look for all possible two- or three-node configurations • Eg 13 possible 3-node subsets: • look for patterns which occur significantly more frequently in real than in equivalent randomized networks Network Analysis

28. Motif Detection • Two matrices • The overall matrix D consists of binary entries Dij, where a 1 indicates binding of regulator j to intergenic region i with a p-value of less than or equal to 0.001, a 0 indicates a p-value greather than 0.001. • The regulator matrix R is a subset of D, containing only the rows corresponding to the intergenic region assigned to each regulator, in the same order as the columns of regulators • Autoregulatory motif: Find each non-zero entry on the diagonal of R. • Feedforward loop: For each master regulator (column of R), find non-zero entries, which correspond to regulators bound. For each master regulator / secondary regulator pair, find all rows in D bound by both regulators. • Etc. Network Analysis

29. Conclusions • Many biological systems can be modelled as networks of interactions • The dynamics of these networks represent phenomena such as changing gene expression over time, spread of information / disease, etc. • Network dynamics are affected by topology • Topological analysis is interesting in its own right, as giving us more information about the global properties of the system • Topological features of interest include connectivity patterns, cluster coefficient, modularity, and motifs Network Analysis