Network/Graph Network/Graph Informally a graph is a set of nodes - PDF document

What is a Network? What is a Network? • Network = graph Network/Graph Network/Graph • Informally a graph is a set of nodes Theory Theory joined by a set of lines or arrows. 1 2 3 1 3 2 4 5 6 4 5 6 What is network theory? Graph-based representations  Network theory provides a set of  Representing a problem as a graph can techniques for analysing graphs provide a different point of view  Complex systems network theory provides  Representing a problem as a graph can techniques for analysing structure in a make a problem much simpler system of interacting agents, represented as a network  More accurately, it can provide the appropriate tools for solving the problem  Applying network theory to a system means using a graph-theoretic representation What makes a problem graph-like? Friendship Network  There are two components to a graph  Nodes and edges  In graph-like problems, these components have natural correspondences to problem elements  Entities are nodes and interactions between entities are edges  Most complex systems are graph-like

Business ties in US biotech- Scientific collaboration network industry Protein-Protein Interaction Genetic interaction network Networks Transportation Networks Internet

Graph Theory - History Graph Theory - History Ecological Networks Leonhard Euler's paper on “ Seven Bridges of Königsberg” , published in 1736. Graph Theory - History Graph Theory - History Graph Theory - History Graph Theory - History Cycles in Polyhedra Trees in Electric Circuits Thomas P. Kirkman William R. Hamilton Gustav Kirchhoff Hamiltonian cycles in Platonic graphs Graph Theory - History Graph Theory - History Graph Theory - History Graph Theory - History Enumeration of Chemical Isomers Four Colors of Maps Arthur Cayley James J. Sylvester George Polya Francis Guthrie Auguste DeMorgan

Definition: Graph Definitions Definition: Graph Definitions • Vertex • G is an ordered triple G:=(V, E, f) – Basic Element – V is a set of nodes, points, or vertices. – Drawn as a node or a dot . – E is a set, whose elements are known as – V ertex set of G is usually denoted by V ( G ), or V edges or lines. • Edge – f is a function – A set of two elements • maps each element of E – Drawn as a line connecting two vertices, called end vertices, or endpoints. • to an unordered pair of vertices in V. – The edge set of G is usually denoted by E(G), or E. Simple Graphs Example Simple graphs are graphs without multiple edges or self-loops. • V:={1,2,3,4,5,6} • E:={{1,2},{1,5},{2,3},{2,5},{3,4},{4,5},{4,6}} Directed Graph (digraph) Directed Graph (digraph) Weighted graphs • Edges have directions • is a graph for which each edge has an – An edge is an ordered pair of nodes associated weight , usually given by a weight function w: E → R . loop multiple arc 2 1.2 1 2 3 1 2 3 .2 1.5 5 .5 3 .3 1 arc node 4 5 6 4 5 6 .5

Structures and structural Graph structures metrics  Identify interesting sections of a graph  Graph structures are used to isolate  Interesting because they form a significant interesting or important sections of a domain-specific structure, or because they graph significantly contribute to graph properties  Structural metrics provide a measurement  A subset of the nodes and edges in a of a structural property of a graph graph that possess certain characteristics,  Global metrics refer to a whole graph or relate to each other in particular ways  Local metrics refer to a single node in a graph Component Connectivity Component • a graph is connected if • Every disconnected graph can be split – you can get from any node to any other by up into a number of connected following a sequence of edges OR components . – any two nodes are connected by a path. • A directed graph is strongly connected if there is a directed path from any node to any other node. Degree Degree Degree (Directed Graphs) Degree (Directed Graphs) • In-degree: Number of edges entering • Number of edges incident on a node • Out-degree: Number of edges leaving • Degree = indeg + outdeg outdeg(1)=2 indeg(1)=0 outdeg(2)=2 indeg(2)=2 The degree of 5 is 3 outdeg(3)=1 indeg(3)=4

Walks Degree: Simple Facts • If G is a graph with m edges, then A walk of length k in a graph is a succession of k Σ deg( v ) = 2 m = 2 | E | (not necessarily different) edges of the form • If G is a digraph then uv,vw,wx,…,yz. Σ indeg( v )= Σ outdeg( v ) = | E | This walk is denote by uvwx…xz, and is referred to as a walk between u and z . • Number of Odd degree Nodes is even A walk is closed is u=z. Path Path Cycle • A path is a walk in which all the edges and all • A cycle is a closed path in which all the the nodes are different. edges are different. Walks and Paths 1,2,5,2,3,4 1,2,5,2,3,2,1 1,2,3,4,6 1,2,5,1 2,3,4,5,2 walk of length 5 CW of length 6 path of length 4 3-cycle 4-cycle Special Types of Graphs Trees Trees • Empty Graph / Edgeless graph • Connected Acyclic Graph – No edge • Two nodes have exactly one path between them • Null graph – No nodes – Obviously no edge

Special Trees Special Trees Regular Connected Graph Paths All nodes have the same degree Stars Bipartite Bipartite graph graph Special Regular Graphs: Cycles • V can be partitioned into 2 sets V 1 and V 2 such that ( u , v ) ∈ E implies – either u ∈ V 1 and v ∈ V 2 C 3 C 4 C 5 – OR v ∈ V 1 and u ∈ V 2. Complete Graph Complete Graph Complete Bipartite Graph Complete Bipartite Graph • Every pair of vertices are adjacent • Bipartite Variation of Complete Graph • Has n(n-1)/2 edges • Every node of one set is connected to every other node on the other set Stars

Planar Graphs Subgraph Planar Graphs Subgraph • Can be drawn on a plane such that no two edges • Vertex and edge sets are subsets of intersect those of G • K 4 is the largest complete graph that is planar – a supergraph of a graph G is a graph that contains G as a subgraph. Special Subgraphs Special Subgraphs: Cliques : Cliques Spanning subgraph subgraph Spanning • Subgraph H has the same vertex set as A clique is a maximum complete connected subgraph . G. . – Possibly not all the edges A B C – “H spans G”. D E F G H I Isomorphism Isomorphism Spanning tree Spanning tree  Let G be a connected graph. Then a Let G be a connected graph. Then a • Bijection, i.e., a one-to-one mapping: spanning tree in G is a spanning tree in G is a subgraph subgraph of G of G f : V(G) -> V(H) that includes every node and is also a that includes every node and is also a u and v from G are adjacent if and only tree. tree. if f(u) and f(v) are adjacent in H. • If an isomorphism can be constructed between two graphs, then we say those graphs are isomorphic .

Representation (Matrix) Representation (Matrix) Isomorphism Problem Isomorphism Problem • Determining whether two • Incidence Matrix graphs are isomorphic – V x E • Although these graphs look – [vertex, edges] contains the edge's data very different, they are isomorphic; one isomorphism • Adjacency Matrix between them is – V x V f(a)=1 f(b)=6 f(c)=8 f(d)=3 – Boolean values (adjacent or not) f(g)=5 f(h)=2 f(i)=4 f(j)=7 – Or Edge Weights Representation (List) Representation (List) Matrices Matrices • Edge List 1 , 2 1 , 5 2 , 3 2 , 5 3 , 4 4 , 5 4 , 6 1 1 1 0 0 0 0 0 – pairs (ordered if directed) of vertices 2 1 0 1 1 0 0 0 3 0 0 1 0 1 0 0 – Optionally weight and other data 4 0 0 0 0 1 1 1 5 0 1 0 1 0 1 0 • Adjacency List (node list) 6 0 0 0 0 0 0 1 1 2 3 4 5 6 1 0 1 0 0 1 0 2 1 0 1 0 1 0 3 0 1 0 1 0 0 4 0 0 1 0 1 1 5 1 1 0 1 0 0 6 0 0 0 1 0 0 Edge and Node Lists Edge and Node Lists Implementation of a Graph. Implementation of a Graph. • Adjacency-list representation Node List Edge List 1 2 2 1 2 – an array of | V | lists, one for each vertex in 2 3 5 1 2 V . 2 3 3 3 – For each u ∈ V , ADJ [ u ] points to all its 4 3 5 2 5 adjacent vertices. 5 3 4 3 3 4 3 4 5 5 3 5 4