Graph Theory. Graph Theory. COURSE: Introduction to Biological Networks. Euler s Solution LECTURE 1: INTRODUCTION TO NETWORKS.
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1 Graph Theory COURSE: Introduction to Biological Networks LECTURE 1: INTRODUCTION TO NETWORKS Arun Krishnan Koenigsberg, Russia Is it possible to walk with a route that crosses each bridge exactly once, and return to the starting point? Graph Theory Euler s Solution Euler ( Graph Representation 1
2 Graph Theory & Edge CONCEPTS Graoh Theory - Concepts Directed Graph Theory Graph Graph Size & Theory Order CONCEPTS CONCEPTS 2
3 Graph Density Theory CONCEPTS Degree Graph & Degree Theory Sequence CONCEPTS Indegree Graph & Theory Outdegree Degree Graph Distribution Theory CONCEPTS CONCEPTS 3
4 Graph Subgraph Theory Path Graph & Theory Length CONCEPTS CONCEPTS Euler s Solution Back to Euler s Solution An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. A connected undirected graph is Eulerian if and only if every graph vertex has even degree. The first Theorem in Graph Theory!!! EVERY edge here has an odd degree 4
5 Graph Geodesic Theory CONCEPTS Graph Theory - Concepts Cont.. Connectedness Graph Theory & Component Empty Graph & Complete TheoryGraphs CONCEPTS 5
6 Star Graph & Cyclic Theory Graphs Graph Tree & Theory Forest CONCEPTS CONCEPTS Bipartite Graph Theory Graphs CONCEPTS Cutpoint CONCEPTS 6
7 Graph Bridge Theory Graph Connectivity Theory CONCEPTS CONCEPTS Edge Graph Connectivity Theory Vertex Graph Centrality Theory - 1 CONCEPTS CONCEPTS 7
8 Vertex Graph Centrality Theory - 2 Vertex Graph Centrality Theory - 3 CONCEPTS CONCEPTS Graph Theory Shortest Path & Mean Path Length CONCEPTS Graph Theory Clustering Coefficient CONCEPTS Path Length: Number of links to pass through to go between two nodes Example 1-2 : 1 1-7: : : :5 Mean Path Length: Average of shortest paths between all pairs of nodes Measure of a network s overall navigability C I = 2n I /k(k-1) k = # of nodes that link to I n I = number of links between the nodes that link to I Essentially find out the number of triangles that pass through node I. Example Node A has only 1 triangle passing through it C A = 2/(5*4) = 0.1 8
9 Is this Graph a connected Theorygraph? QUIZ BREAK 2 3 Cyclic or Acyclic? QUIZ BREAK 1 4 Directed or Undirected? 6 5 Directed Graph Theory (Unconnected) QUIZ BREAK Cyclic or Acyclic? Representing Graphs 6 5 9
10 Adjacency Graph Matrix Theory& List Laplacian Graph Theory Matrix GRAPH REPRESENTATION GRAPH REPRESENTATION Adjacency matrix 2-dimensional array For each edge (u,v), set A[u][v] to true; otherwise false Adjacency lists For each vertex, keep a list of adjacent vertices x x 2 x x 3 x 4 x x x 5 x x Choosing Graph A Representation Theory GRAPH REPRESENTATION Size of V relative to size of E is a primary factor. Dense: E / V is large Sparse: E / V is small Adjacency matrix is expensive in terms of space if the graph is sparse (O( V 2 > O( E + V )). Adjacency list is expensive in terms of checking edges if the graph is dense. Graphs are EVERYWHERE! 10
11 Graphs as Theory Models EXAMPLES Graphs as Theory Models EXAMPLES The Internet Communication pathways DNS hierarchy The WWW The physical world Road topology and maps Airline routes and fares Electrical circuits Job and manufacturing scheduling Physical objects are often modeled by meshes, which are a particular kind of graph structure. By Jonathan Shewchuk Graphs as Theory Models World Graph Wide Theory Web EXAMPLES EXAMPLES NASA CFD labs By Paul Heckbert and David Garland Internet Servers Links See also and 11
12 Food Web Graph Theory North Atlantic Social Co-Authorship Network Max Plank Institute Graph Theory EXAMPLES Species Researchers Predator-prey Economic Network Graph Theory World Trade 1992 EXAMPLES Co-authorship EXAMPLES Yeast Protein Graph Interaction Theory Network EXAMPLES Countries Trade Volume Proteins Protein-Protein Interactions 12
13 Protein Graph Network Theory EXAMPLES Network Graph Theory Types RANDOM NETWORK Residues Interactions Van der Waals, h-bonds, hydrophobic etc etc Erdos-Renyi model Start with N nodes Connect each pair of node with probability p Graph has ~pn(n- 1)/2 randomly placed links Network Graph Theory Types RANDOM NETWORK Network Graph Theory Types RANDOM NETWORK Degree distribution P(k) Follows Poisson distribution ==> Most nodes have same number of nodes = average Tail of distribution decreases exponentially ==> very few nodes have degree different from average Clustering Coefficient C(k) Independent of Node degree Horizontal line Mean path length l ~ logn, N = network size Indicates characteristic small world property Small World property implies any two nodes can be connected by just a few links 13
14 Network Graph Theory Types SCALE FREE NETWORK Network Graph Theory Types SCALE FREE NETWORK Characterized by power-law distribution P(k) ~k -γ Small number of highly connected hubs Power law distributions show up as a straight line on a log-log plot C(k) is independent of k Most biological networks have 2<γ <3 Average path length l ~ log log N which is much γγ shorter than logn for random networks Network Graph Theory Types HIERARCHICAL NETWORK Network Graph Theory Types HIERARCHICAL NETWORK Accounts for coexistance of modularity, local clustering and scale-free topology in many real systems Integrates scale-free topology with an inherent modular structure Power law degree distribution P(k) very similar to scale free network C(k) ~ k -1 has a straight line slope on a log-log plot Implies sparsely connected nodes are part of highly clustered areas Connection between highly connected neighborhoods maintained by a few hubs 14
15 Origin of Graph Scale-Free Theory Networks BIOLOGICAL SYSTEMS Two basic mechanisms Growth Network emerges through the subsequent addition of new nodes Preferential Attachment Origin of Graph Scale-Free Theory Networks BIOLOGICAL SYSTEMS Preferential Attachment New nodes prefer to link to more connected nodes Eg. Red node has 2 times greater probability of connecting to node 1 than node 2 Growth and preferential attachment More connected a hub is, more nodes will link to it which means they get still more links and so on. Origin of Graph Scale-Free Theory Networks BIOLOGICAL SYSTEMS World Graph Wide Theory Web EXAMPLES Scale free model predicts that evolutionarily older nodes are more connected Eg. Metabolic Hubs: Remnants of RNA world like coenzyme A, NAD and GTP are most connected substrates For protein networks: Evolutionarily older proteins have more links to other proteins than their younger counterparts Internet Servers Links 15
16 Food Web Graph Theory North Atlantic EXAMPLES Graph Theory EXAMPLES Species Researchers Predator-prey Economic Network Graph Theory World Trade 1992 Social Co-Authorship Network Max Plank Institute Co-authorship EXAMPLES Yeast Graph Protein Theory Network Countries Proteins Trade Volume EXAMPLES Protein-Protein Interactions 16
17 Protein Graph Network Theory EXAMPLES Graph Modularity Theory BIOLOGICAL SYSTEMS Cellular functions are likely to be carried out in a modular manner Modularity group of physically or functionally linked molecules (nodes) that work together to achieve a (relatively) distinct function. Examples Relatively invariant protein-protein and protein- RNA complexes are at the core of many basic biological functions Temporally regulated groups of molecules Involved in Cell Cycle Convey extracellular signals in bacterial chemotaxis Graph Motifs Theory BIOLOGICAL SYSTEMS Graph Motifs Theory BIOLOGICAL SYSTEMS Some subgraphs are over-represented in motifs compared to a random network Ex: feedforward loops emerge in transcriptional regulatory as well as neural networks Four-node networks occur in electrical circuits but not in biological networks. How to identify motifs? All subgraphs of n nodes are determined Network is randomized keeping number of nodes, links and degree distribution the same Subgraphs that occur significantly more frequently in real networks than in random networks are designated as motifs 17
18 Graph Motif Clusters Theory BIOLOGICAL SYSTEMS Network Graph Robustness Theory BIOLOGICAL SYSTEMS Motifs also form clusters Example Transcriptional network from E. Coli Shows only the bi-fan motifs (motif with four nodes) Such clustering of motifs seems to be a general property of networks Robustness --> ability to respond to external changes while maintaining relatively normal behavior Random networks: If a critical fraction of nodes is removed, then functional disruption takes place Complex systems can be surprisingly resilient (from internet to the cell) Topology has a major role to play in this resilience Network Graph Robustness Theory Graph Conclusions Theory SYSTEMS Networks BIOLOGICAL Introduction to Scale-free networks Amazingly robust Even if > 80% of nodes are disconnected This is due to the presence of few hubs However, this can lead to attack vulnerability Deletion of a few of this hubs can completely disrupt network Strong relationship between hub status of molecule and effect on cell viability Eg: In S. Cerevisiae: ~10% of proteins with < 5 links are essential ~60% of proteins with > 15 links are essential Hubs tend to be better evolutionarily conserved!! Studied the basics of graph theory Examples of networks Analyzed types of Networks Random, scale-free, hierarchical We saw how most networks were small-scale in nature Studied occurrence of motifs, motif clusters Looked at network robustness. 18
19 Software Graph Theory Setup Introduction to Networks Software Graph Theory Setup Introduction to Networks Recurrence Quantification Anaysis Download from ls/eval3dstruct Protein Modularity Detection Download from ls/gandiva.tar.gz Copy this to account on cacao.bioinfo.ttck.keio.ac.jp Installation Instructions Tar and unzip the package using tar zxf GANDivA.tar.gz Change directory to the main GANDivA directory cd GANDivA_v1.0 Run the install script perl install_gandiva.pl You will be asked a series of questions Installation Instructions a) Do you want to install a parallel version of the program? [Y/N] Y b) Do you have MPI Installed? [Y/N] Y c) Where would you like to have PGA installed? Give full path of directory in which to install Eg: /home/krishnan/gandiva_v1.0/pga d) Please enter the path of the MPI library file /usr/local/lib/libmpich.a e) Please enter the path of the MPI include directory /usr/local/include This should automatically install the GANDivA binary in /path/to/gandiva_v1.0/bin Software Graph Theory Setup Introduction to Networks Graph References Theory Introduction to Networks You will be using these two programs (Eval3DStruct) and GANDivA for assignments in the later classes. If you have any problems, please drop me an at: krishnan@ttck.keio.ac.jp Albert-László Barabási & Zoltán N. Oltvai, Network, Biology, understanding the cell s functional organization, Nature Reviews, 5, ,
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