COURSE Introduction to Biological Networks Graph Theory Graph ...

Graph Theory COURSE: Introduction to Biological Networks LECTURE 1: INTRODUCTION TO NETWORKS Arun Krishnan HISTORY HISTORY KOENIGSBERG HISTORY • Koenigsberg, Russia • Is it possible to walk with a route that crosses each bridge exactly once, and return to the starting point? Graph Theory HISTORY KOENIGSBERG HISTORY Euler’s Solution KOENIGSBERG KOENIGSBERG • Euler ( 1783 1707 - Graph Representation 1 Graph Theory Graph & Edge CONCEPTS KOENIGSBERG HISTORY Graoh Theory - Concepts Directed Graph Graph Theory CONCEPTS KOENIGSBERG HISTORY Graph Theory Size & Order CONCEPTS KOENIGSBERG HISTORY 2 Graph Theory Density CONCEPTS KOENIGSBERG HISTORY DegreeGraph Theory & Degree Sequence CONCEPTS KOENIGSBERG HISTORY Indegree & Theory Graph Outdegree CONCEPTS KOENIGSBERG HISTORY Degree Distribution Graph Theory CONCEPTS KOENIGSBERG HISTORY 3 Graph Theory Subgraph CONCEPTS KOENIGSBERG HISTORY Path & Theory Graph Length CONCEPTS KOENIGSBERG HISTORY Euler’s Solution KOENIGSBERG KOENIGSBERG 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 Graph Theory Geodesic CONCEPTS KOENIGSBERG HISTORY Graph Theory - Concepts Cont.. Connectedness & Component Graph Theory CONCEPTS KOENIGSBERG HISTORY Empty Graph Theory & Complete Graphs CONCEPTS KOENIGSBERG HISTORY 5 Star & Cyclic Graphs Graph Theory CONCEPTS KOENIGSBERG HISTORY Graph Theory Tree & Forest CONCEPTS KOENIGSBERG HISTORY BipartiteTheory Graph Graphs CONCEPTS KOENIGSBERG HISTORY Cutpoint CONCEPTS KOENIGSBERG HISTORY 6 Graph Theory Bridge CONCEPTS KOENIGSBERG HISTORY Graph Theory Connectivity CONCEPTS KOENIGSBERG HISTORY Edge Connectivity Graph Theory CONCEPTS KOENIGSBERG HISTORY Vertex Centrality - 1 Graph Theory CONCEPTS KOENIGSBERG HISTORY 7 Vertex Centrality - 2 Graph Theory CONCEPTS KOENIGSBERG HISTORY Vertex Centrality - 3 Graph Theory CONCEPTS KOENIGSBERG HISTORY Shortest GraphMean Path Length Path & Theory CONCEPTS KOENIGSBERG HISTORY Clustering Theory Graph Coefficient • CI = 2nI /k(k-1) CONCEPTS KOENIGSBERG HISTORY • Path Length: Number of links to pass through to go between two nodes • Example – 1-2:1 – 1 - 7: • 1-3-5-7: 3 • 1-2-4-5-7:4 • 1-2-4-5-6-7:5 – k = # of nodes that link to I – nI = 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 – CA = 2/(5*4) = 0.1 • Mean Path Length: – Average of shortest paths between all pairs of nodes – Measure of a network’s overall navigability 8 Is this Graph Theory a connected graph? 2 3 Cyclic or Acyclic? QUIZ BREAK KOENIGSBERG HISTORY QUIZ BREAK 1 Directed or Undirected? 4 6 5 DirectedGraph Theory Graph (Unconnected) 2 3 Cyclic or Acyclic? 4 1 6 5 QUIZ BREAK KOENIGSBERG HISTORY Representing Graphs 9 Adjacency Matrix & List Graph Theory • Adjacency matrix – 2-dimensional array – For each edge (u,v), set A[u][v] to true; otherwise false GRAPH KOENIGSBERG HISTORY REPRESENTATION Laplacian Matrix Graph Theory GRAPH KOENIGSBERG HISTORY REPRESENTATION 1 1 2 3 4 5 6 7 1 7 2 3 x 4 x x 5 x 6 7 x x x x • Adjacency lists – For each vertex, keep a list of adjacent vertices 1 2 3 4 5 6 7 x x 3 4 6 3 4 4 5 6 7 3 2 4 5 6 7 Choosing A Representation Graph Theory • Size of V relative to size of E is a primary factor. GRAPH KOENIGSBERG HISTORY REPRESENTATION – 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 Graphs as Models Graph Theory • The Internet – Communication pathways – DNS hierarchy – The WWW EXAMPLES KOENIGSBERG HISTORY Graphs as Models Graph Theory EXAMPLES KOENIGSBERG HISTORY • Physical objects are often modeled by meshes, which are a particular kind of graph structure. • The physical world – – – – Road topology and maps Airline routes and fares Electrical circuits Job and manufacturing scheduling By Jonathan Shewchuk Graphs as Models Graph Theory EXAMPLES KOENIGSBERG HISTORY World Wide Web Graph Theory • Nodes: EXAMPLES KOENIGSBERG HISTORY – Internet Servers • Connections – Links NASA CFD labs By Paul Heckbert and David Garland See also http://java.sun.com/applets/jdk/1.1/demo/WireFrame/index.html and http://www.mapquest.com 11 Food Web Graph Atlantic North Theory EXAMPLES KOENIGSBERG HISTORY Social Co-Authorship Network Max Plank Institute Graph Theory EXAMPLES KOENIGSBERG HISTORY • Nodes: – Species • Nodes: – Researchers • Connections – Predator-prey • Connections – Co-authorship Economic Network Graph Theory World Trade 1992 EXAMPLES KOENIGSBERG HISTORY Yeast Protein Interaction Network Graph Theory EXAMPLES KOENIGSBERG HISTORY • Nodes: – Countries • Connections – Trade Volume • Nodes: – Proteins • Connections – Protein-Protein Interactions 12 Protein Network Graph Theory EXAMPLES KOENIGSBERG HISTORY Network Types Graph Theory RANDOM KOENIGSBERG HISTORY NETWORK • Erdos-Renyi model – Start with N nodes – Connect each pair of node with probability p – Graph has ~pN(N1)/2 randomly placed links • Nodes: – Residues • Connections – Interactions • Van der Waals, h-bonds, hydrophobic etc etc Network Types Graph Theory RANDOM KOENIGSBERG HISTORY NETWORK Network Types Graph Theory RANDOM KOENIGSBERG HISTORY NETWORK • Degree distribution P(k) – Follows Poisson distribution • ==> Most nodes have same number of nodes = 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 – Tail of distribution decreases exponentially • ==> very few nodes have degree different from average 13 Network Types Graph Theory • Characterized by power-law distribution SCALE FREE KOENIGSBERG HISTORY NETWORK Network Types Graph Theory SCALE FREE KOENIGSBERG HISTORY NETWORK – 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 shorter than logN γγ for random networks Network Types Graph Theory HIERARCHICAL KOENIGSBERG HISTORY NETWORK Network Types Graph Theory HIERARCHICAL KOENIGSBERG HISTORY 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 Origin ofGraph Theory Scale-Free Networks • Two basic mechanisms – Growth BIOLOGICAL KOENIGSBERG HISTORY SYSTEMS Origin ofGraph Theory Scale-Free Networks BIOLOGICAL KOENIGSBERG HISTORY 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 • Network emerges through the subsequent addition of new nodes • 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…. – Preferential Attachment Origin ofGraph Theory Scale-Free Networks BIOLOGICAL KOENIGSBERG HISTORY SYSTEMS World Wide Web Graph Theory • Nodes: EXAMPLES KOENIGSBERG HISTORY • 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 • Connections – Links 15 Food Web Graph Atlantic North Theory EXAMPLES KOENIGSBERG HISTORY Social Co-Authorship Network Max Plank Institute Graph Theory EXAMPLES KOENIGSBERG HISTORY • Nodes: – Species • Nodes: – Researchers • Connections – Predator-prey • Connections – Co-authorship Economic Network Graph Theory World Trade 1992 EXAMPLES KOENIGSBERG HISTORY Yeast Protein Network Graph Theory • Nodes: – Proteins EXAMPLES KOENIGSBERG HISTORY • Nodes: – Countries • Connections – Trade Volume • Connections – Protein-Protein Interactions 16 Protein Network Graph Theory EXAMPLES KOENIGSBERG HISTORY Graph Theory Modularity BIOLOGICAL KOENIGSBERG HISTORY 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 proteinRNA 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 Theory Motifs BIOLOGICAL KOENIGSBERG HISTORY SYSTEMS Graph Theory Motifs • How to identify motifs? BIOLOGICAL KOENIGSBERG HISTORY 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. – 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 Graph Theory Motif Clusters • Motifs also form clusters • Example BIOLOGICAL KOENIGSBERG HISTORY SYSTEMS Network Robustness Graph Theory BIOLOGICAL KOENIGSBERG HISTORY SYSTEMS – Transcriptional network from E. Coli – Shows only the bi-fan motifs (motif with four nodes) • 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 • Such clustering of motifs seems to be a general property of networks • Complex systems can be surprisingly resilient (from internet to the cell) • Topology has a major role to play in this resilience Network Robustness Graph Theory • 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 BIOLOGICAL KOENIGSBERG HISTORY SYSTEMS Graph Theory Conclusions • Studied the basics of graph theory • Examples of networks • Analyzed types of Networks Introduction to KOENIGSBERG HISTORY Networks • 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 – Random, scale-free, hierarchical – We saw how most networks were small-scale in nature • Studied occurrence of motifs, motif clusters • Looked at network robustness. • Hubs tend to be better evolutionarily conserved!! 18 Software Setup Graph Theory • Recurrence Quantification Anaysis – Download from Introduction to KOENIGSBERG HISTORY Networks Software Setup Graph Theory • Installation Instructions Introduction to KOENIGSBERG HISTORY Networks • http://www.iab.keio.ac.jp/~krishnan/downloads/Course_Materia ls/Eval3DStruct – a) Do you want to install a parallel version of the program? [Y/N] • Y • Protein Modularity Detection – Download from • http://www.iab.keio.ac.jp/~krishnan/downloads/Course_Materia ls/GANDivA.tar.gz – b) Do you have MPI Installed? [Y/N] • Y – Copy this to account on cacao.bioinfo.ttck.keio.ac.jp – 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 – Installation Instructions • Tar and unzip the package using – tar zxf GANDivA.tar.gz – d) Please enter the path of the MPI library file • /usr/local/lib/libmpich.a • Change directory to the main GANDivA directory – cd GANDivA_v1.0 • Run the install script – perl install_gandiva.pl – e) Please enter the path of the MPI include directory • /usr/local/include • You will be asked a series of questions – This should automatically install the GANDivA binary in • /path/to/GANDivA_v1.0/bin Software Setup Graph Theory Introduction to KOENIGSBERG HISTORY Networks Graph Theory References Introduction to KOENIGSBERG HISTORY 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 email at: – krishnan@ttck.keio.ac.jp Albert-László Barabási & Zoltán N. Oltvai, Network, Biology, understanding the cell’s functional organization, Nature Reviews, 5, 101-114, 2004 19