Biological Networks Analysis Dijkstra s algorithm and Degree Distribution

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1 iological Networks nalysis ijkstra s algorithm and egree istribution Genome 373 Genomic Informatics Elhanan orenstein

2 Networks: Networks vs. graphs The Seven ridges of Königsberg collection of nodes and links irected/undirected; weighted/non-weighted, Many types of biological networks Transcriptional regulatory networks Metabolic networks Protein-protein interaction (PPI) networks quick review

3 The Kevin acon Number Game Tropic Thunder (008) Tropic Thunder Iron Man Proof Flatliners Tom ruise Robert owney Jr. Gwyneth Paltrow Hope avis Kevin acon Tropic Thunder Iron Man Frost/Nixon Tom ruise Robert owney Jr. Frank Langella Kevin acon

4 The Paul Erdos Number Game

5 The shortest path problem Find the minimal number of links connecting node to node in an undirected network How many friends between you and someone on F (6 degrees of separation, Erdös number, Kevin acon number) How far apart are two genes in an interaction network What is the shortest (and likely) infection path Find the shortest (cheapest) path between two nodes in a weighted directed graph GPS; Google map

ijkstra s lgorithm Edsger Wybe ijkstra 1930 00 "omputer Science

6 ijkstra s lgorithm Edsger Wybe ijkstra "omputer Science is no more about computers than astronomy is about telescopes."

7 Solves the single-source shortest path problem: Works on both directed and undirected networks Works on both weighted and non-weighted networks Find the shortest path from a single source to LL nodes in the network Greedy algorithm but still guaranteed to provide optimal solution!!! pproach: Iterative ijkstra s algorithm Maintain shortest path to each intermediate node

8 ijkstra s algorithm 1. Initialize: i. ssign a distance value,, to each node. Set to zero for start node and to infinity for all others. ii. iii. Mark all nodes as unvisited. Set start node as current node.. For each of the current node s unvisited neighbors: i. alculate tentative distance, t, through current node. ii. iii. If t smaller than (previously recorded distance): t Mark current node as visited (note: shortest dist. found). 3. Set the unvisited node with the smallest distance as the next "current node" and continue from step. 4. Once all nodes are marked as visited, finish.

9 ijkstra s algorithm simple synthetic network F 3 E 1 1. Initialize: i. ssign a distance value,, to each node. Set to zero for start node and to infinity for all others. ii. Mark all nodes as unvisited. iii. Set start node as current node.. For each of the current node s unvisited neighbors: i. alculate tentative distance, t, through current node. ii. If t smaller than (previously recorded distance): t iii. Mark current node as visited (note: shortest dist. found). 3. Set the unvisited node with the smallest distance as the next "current node" and continue from step. 4. Once all nodes are marked as visited, finish.

10 ijkstra s algorithm Initialization Mark (start) as current node E F : E 1 F

11 ijkstra s algorithm heck unvisited neighbors of E F vs. 9 5 : vs E 1 F

12 ijkstra s algorithm Update Record path E F 0 9, : 0 3 1, E 1 F

13 Mark as visited ijkstra s algorithm E F 0 9, : 0 3 1, E 1 F

14 ijkstra s algorithm Mark as current (unvisited node with smallest ) E F 0 9, : 0 3 1, E 1 F

15 ijkstra s algorithm heck unvisited neighbors of E F vs. 9 9, vs : 0 3 1, E 1 F 3+ vs.

16 ijkstra s algorithm Update distance Record path E F 0 9,9,7, : 0 3 1, E,5 1 F

17 ijkstra s algorithm Mark as visited Note: istance to is final!! E F : 0 9 3,9,7 1, ,6 E,5 5 1 F

18 ijkstra s algorithm Mark E as current node heck unvisited neighbors of E E F : 0 9 3,9,7 1, ,6 E,5 5 1 F

19 ijkstra s algorithm Update Record path E F 0 9,9,7, : 0 3 1, E,5 1 F,17

20 Mark E as visited ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17

21 ijkstra s algorithm Mark as current node heck unvisited neighbors of E F : 0 9 3,9,7 1, ,6 E,5 5 1 F,17

22 ijkstra s algorithm Update Record path (note: path has changed) E F : ,9,7, ,6 E,5 5 1 F,17,11

23 Mark as visited ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

24 ijkstra s algorithm Mark as current node heck neighbors E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

25 ijkstra s algorithm No updates.. Mark as visited E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

26 Mark F as current ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

27 Mark F as visited ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11 11

28 We now have: Shortest path from to each node (both length and path) Minimum spanning tree We are done! 9,9,7,6 5 E F : 0 3 1, E,5 1 Will we always get a tree? an you prove it? F,17,11

29 Measuring Network Topology

31 omparing networks We want to find a way to compare networks. Similar (not identical) topology ommon design principles We seek measures of network topology that are: Simple apture global organization Potentially important (equivalent to, for example, G content for genomes) Summary statistics

32 Node degree / rank egree = Number of neighbors Node degree in PPI networks correlates with: Gene essentiality onservation rate Likelihood to cause human disease

33 egree distribution P(k): probability that a node has a degree of exactly k ommon distributions: Poisson: Exponential: Power-law:

34 The power-law distribution Power-law distribution has a heavy tail! haracterized by a small number of highly connected nodes, known as hubs.k.a. scale-free network Hubs are crucial: ffect error and attack tolerance of complex networks (lbert et al. Nature, 000)

35 Govindan and Tangmunarunkit, 000 The Internet Nodes 150,000 routers Edges physical links P(k) ~ k -.3

36 arabasi and lbert, Science, 1999 Movie actor collaboration network Tropic Thunder (008) Nodes 1,50 actors Edges co-appearance in a movie P(k) ~ k -.3

37 Yook et al, Proteomics, 004 Protein protein interaction networks Nodes Proteins Edges Interactions (yeast) P(k) ~ k -.5

38 Jeong et al., Nature, 000 Nodes Metabolites Edges Reactions Metabolic networks P(k) ~ k -.±.Fulgidus (archae) E. oli (bacterium) Metabolic networks across all kingdoms of life are scale-free.elegans (eukaryote) veraged (43 organisms)

39 Why do so many real-life networks exhibit a power-law degree distribution? Is it selected for? Is it expected by chance? oes it have anything to do with the way networks evolve? oes it have functional implications??

41 omputational Representation of Networks List of edges: (ordered) pairs of nodes [ (,), (,), (,), (,) ] onnectivity Matrix Name: ngr: p1 p Object Oriented Name: ngr: p1 Name: ngr: Name: ngr: p1 Which is the most useful representation?

A quick review. The clustering problem: Hierarchical clustering algorithm: Many possible distance metrics K-mean clustering algorithm:

A quick review. The clustering problem: Hierarchical clustering algorithm: Many possible distance metrics K-mean clustering algorithm: The clustering problem: partition genes into distinct sets with high homogeneity and high separation Hierarchical clustering algorithm: 1. Assign each object to a separate cluster.. Regroup the pair of