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Graph Theory in 50 minutes

Graph Theory in 50 minutes. This Graph has 6 nodes (also called vertices) and 7 edges (also called links). Graphs are really useful in biology. Vertices (or nodes) usually represent things Edges (or links) represent interactions Examples: social interactions food webs

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Graph Theory in 50 minutes

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  1. Graph Theory in 50 minutes

  2. This Graph has 6 nodes (also called vertices) and 7 edges (also called links)

  3. Graphs are really useful in biology • Vertices (or nodes) usually represent things • Edges (or links) represent interactions • Examples: • social interactions • food webs • protein-protein interactions • genetic regulatory interactions • epidemic spread

  4. Basic Terminology Edges may be directed or not Edges may be weighted or not The degree of a node is the number of edges it has Distance is counted as number of links between 2 nodes The diameter is the number of links between the two furthest nodes A component is a set of nodes that are connected by links

  5. Tuesday Lake Food web: a directed network

  6. The degree distribution gives the degree of each node in the graph • Random networks are made by taking a set of nodes and connecting them randomly so that the degree is normally distributed • A scale freenetwork has a degree with a power law distribution. Informally, there are many nodes with a few edges, and only a few nodes with many edges.

  7. Random and Power Law Degree Distributions

  8. A small world network can be made by making a locally connected network, and then adding a few random long distance links. These networks have small diameter 6 degrees Milgram experiments The Kevin Bacon game The world wide web? Food webs?

  9. There are many measures of community structure in networks • Example: clustering coefficient or, how often are you friends with your friend’s friends? • Calculated by the percentage of pairs that are connected in triangles A clique is a fully connected graph

  10. Many biological networks are small world, have power law distributed degree and/or have high clustering coefficients. • Profound effects on epidemic spread and robustness of biological networks

  11. Representing Graphs in Matlab % Create a directed graph with 6 nodes and 11 edges W = [.41 .99 .51 .32 .15 .45 .38 .32 .36 .29 .21]; DG = sparse([6 1 2 2 3 4 4 5 5 6 1],… [2 6 3 5 4 1 6 3 4 3 5],W) h = view(biograph(DG,[],'ShowWeights','on'))

  12. % Create a directed graph with 10 nodes and 17 edges g = sparse([1 1 1 2 2 3 3 4 5 6 7 7 8 9 9 9 9], ... [2 6 8 3 1 4 2 5 4 7 6 4 9 8 10 5 3],true,10,10) h = view(biograph(g)); % Find the strongly connected components [S,C] = graphconncomp(g) % Mark the nodes for each component with different color colors = jet(S); for i = 1:numel(h.nodes) h.Nodes(i).Color = colors(C(i),:); end

  13. For more… • # Milgram, Stanley (1967). "The Small World Problem". Psychology Today 2: 60-67. • Watts, Duncan J.; Strogatz, Steven H. (June 1998). "Collective dynamics of 'small-world' networks". Nature 393: 440-442. • Newman, Mark (2003). "The Structure and Function of Complex Networks". SIAM Review 45: 167-256. • Matlab code for analyzing complex networks http://www.levmuchnik.net/Content/Networks/ComplexNetworksPackage.html -graph tools in the Bioinformatics toolkit UCINET software: http://www.analytictech.com/downloaduc6.htm Excel complex network addon

  14. NEW COURSE ! This interdisciplinary course will provide a general background on network theory and analysis. The topics include graph theory, small-world and scale-free networks, fractals, network scaling, and community structure. The course will address the analysis of various real world networks: human social networks, neural and cognitive networks, food webs, the Internet, metabolic networks, transportation systems, and others. Advanced undergraduate and graduate students from diverse disciplines such as computer science, biology, anthropology, sociology, math, and physics are encouraged to register! Networks: an introduction to network theory and its applications in social, biological, and technological systems Instructors: George Bezerra: gbezerra@gmail.com Paul Hooper: phooper@unm.edu Wenyun Zuo: wyzuo@unm.edu Supervisor: Jim H. Brown, distinguished professor in Biology Department Bio 402/502, CS 491/591, Stat/Math 479/579, Phyc 480/580, ECE495/595 or Anth 460/560 Spring 2010, Mon & Wed, 10:00-11:15am. Room 57, Cast.

  15. 1) UCINET 6 software exercise. After an in-class introduction to the UCINET software in week 7, students should download the free software (http://www.analytictech.com/downloaduc6.htm) and use a sampled dataset (provided) to perform a series of exercises specified by the instructors. The results of these exercises should be written up and turned in at the beginning of the Wednesday class in week 8. Students may work alone or in groups of two or three, but each student should turn in his or her own write-up. (20% of grade)

  16. Random • Small world: friends with my friends friends; connectors • Scale free: power law, there is no mean • Fractal • Food webs: Tuesday Lake • social networks: Newman highschool, sexual contacts distribution • Metabolic (2 kinds) • Computational • Neural • Popular books: Strogatz, newmann; linked barabasi, Gladwell

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