Network Science Class 2: Graph Theory (Ch2) Albert-László Barabási with Emma K. Towlson. Sebastian Ruf, Michael Danziger, and Louis Shekhtman www.BarabasiLab.com Section 1 The Bridges of Konigsberg THE BRIDGES OF KONIGSBERG Can one walk across the seven bridges and never cross the same bridge twice? Network Science: Graph Theory THE BRIDGES OF KONIGSBERG Can one walk across the seven bridges and never cross the same bridge twice? 1735: Euler’s theorem: a) If a graph has more than two nodes of odd degree, there is no path. b) If a graph is connected and has no odd degree nodes, it has at least one path. Network Science: Graph Theory Section 2 Networks and graphs COMPONENTS OF A COMPLEX SYSTEM . components: nodes, vertices N . interactions: links, edges L . system: network, graph (N,L) Network Science: Graph Theory NETWORKS OR GRAPHS? network often refers to real systems • www, • social network • metabolic network. Language: (Network, node, link) graph: mathematical representation of a network • web graph, • social graph (a Facebook term) Language: (Graph, vertex, edge) We will try to make this distinction whenever it is appropriate, but in most cases we will use the two terms interchangeably. Network Science: Graph Theory A COMMON LANGUAGE N=4 L=4 Network Science: Graph Theory CHOOSING A PROPER REPRESENTATION The choice of the proper network representation determines our ability to use network theory successfully. In some cases there is a unique, unambiguous representation. In other cases, the representation is by no means unique. For example, the way we assign the links between a group of individuals will determine the nature of the question we can study. Network Science: Graph Theory CHOOSING A PROPER REPRESENTATION If you connect individuals that work with each other, you will explore the professional network. Network Science: Graph Theory CHOOSING A PROPER REPRESENTATION If you connect those that have a romantic and sexual relationship, you will be exploring the sexual networks. Network Science: Graph Theory CHOOSING A PROPER REPRESENTATION If you connect individuals based on their first name (all Peters connected to each other), you will be exploring what? It is a network, nevertheless. Network Science: Graph Theory Section 2.3 Degree, Average Degree and Degree Distribution NODE DEGREES d Node degree: the number of links connected to the node. e A t c e r i kA =1 kB = 4 d B n U In directed networks we can define an in-degree and out-degree. B D d C e The (total) degree is the sum of in- and out-degree. t c e in out r E kC = 2 kC = 1 kC = 3 i G A D F Source: a node with kin= 0; Sink: a node with kout= 0. A BIT OF STATISTICS Network Science: Graph Theory AVERAGE DEGREE d N e 1 2L t k ki c j k N i=1 e N r i i d n N – the number of nodes in the graph U D 1 N 1 N B k in k in , k out k out , k in = k out d C i i N i=1 N i=1 e t c e E r i A L D k F N Network Science: Graph Theory Average Degree Network Science: Graph Theory DEGREE DISTRIBUTION Degree distribution P(k): probability that a randomly chosen node has degree k Nk = # nodes with degree k P(k) = Nk / N ➔ plot DEGREE DISTRIBUTION DEGREE DISTRIBUTION Discrete Representation: pk is the probability that a node has degree k. Continuum Description: p(k) is the pdf of the degrees, where k2 ò p(k)dk k1 represents the probability that a node’s degree is between k1 and k2. Normalization condition: ¥ ¥ pk =1 ò p(k)dk =1 0 K min where Kmin is the minimal degree in the network. Network Science: Graph Theory UNDIRECTED VS. DIRECTED NETWORKS Undirected Directed Links: undirected (symmetrical) Links: directed (arcs). Graph: Digraph = directed graph: L A D F M B An undirected C link is the I superposition of D two opposite directed links. B G E G H A C F Undirected links : Directed links : coauthorship links URLs on the www Actor network phone calls protein interactions metabolic reactions Network Science: Graph Theory Section 2.2 Reference Networks Question 4 Q4: Adjacency Matrices Section 2.4 Adjacency matrix ADJACENCY MATRIX 4 4 3 3 2 2 1 1 Aij=1 if there is a link between node i and j Aij=0 if nodes i and j are not connected to each other. Note that for a directed graph (right) the matrix is not symmetric. if there is a link pointing from node j and i Network Science: Graph Theory if there is no link pointing from j to i. ADJACENCY MATRIX AND NODE DEGREES æ 0 1 0 1ö N 4 ki = Aij d ç ÷ 1 0 0 1 j =1 e ç ÷ t A = ij N c ç 0 0 0 1÷ e 3 ç ÷ k j = Aij r 2 i è 1 1 1 0ø i=1 1 d n Aij = A ji 1 N 1 N U L = ki = Aij A = 0 2 2 ii i=1 ij æ 0 0 0 0 ö ç ÷ d 4 1 0 0 1 e A = ç ÷ t ij ç ÷ N c 0 0 0 1 out e ç ÷ k j = Aij r è 1 0 0 0 ø i 3 2 i=1 D N N N 1 in out L = ki = k j = Aij Aij ¹ A ji i=1 j=1 i, j Aii = 0 ADJACENCY MATRIX a e a b c d e f g h a 0 1 0 0 1 0 1 0 b 1 0 1 0 0 0 0 1 c 0 1 0 1 0 1 1 0 d 0 0 1 0 1 0 0 0 h b d e 1 0 0 1 0 0 0 0 f 0 0 1 0 0 0 1 0 g 1 0 1 0 0 0 0 0 f h 0 1 0 0 0 0 0 0 g c Network Science: Graph Theory ADJACENCY MATRICES ARE SPARSE Network Science: Graph Theory Section 4 Real networks are sparse COMPLETE GRAPH The maximum number of links a network æ Nö N(N -1) of N nodes can have is: Lmax = ç ÷ = è 2 ø 2 A graph with degree L=Lmax is called a complete graph, and its average degree is <k>=N-1 Network Science: Graph Theory REAL NETWORKS ARE SPARSE Most networks observed in real systems are sparse: L << Lmax or <k> <<N-1. 6 12 WWW (ND Sample): N=325,729; L=1.4 10 Lmax=10 <k>=4.51 7 Protein (S. Cerevisiae): N= 1,870; L=4,470 Lmax=10 <k>=2.39 5 10 Coauthorship (Math): N= 70,975; L=2 10 Lmax=3 10 <k>=3.9 6 13 Movie Actors: N=212,250; L=6 10 Lmax=1.8 10 <k>=28.78 (Source: Albert, Barabasi, RMP2002) Network Science: Graph Theory ADJACENCY MATRICES ARE SPARSE Network Science: Graph Theory METCALFE’S LAW The maximum number of links a network æ Nö N(N -1) of N nodes can have is: Lmax = ç ÷ = è 2 ø 2 Network Science: Graph Theory Section 2.6 WEIGHTED AND UNWEIGHTED NETWORKS WEIGHTED AND UNWEIGHTED NETWORKS METCALFE’S LAW The maximum number of links a network æ Nö N(N -1) of N nodes can have is: Lmax = ç ÷ = è 2 ø 2 Network Science: Graph Theory Section 2.7 BIPARTITE NETWORKS BIPARTITE GRAPHS bipartite graph (or bigraph) is a graph whose nodes can be divided into two disjoint sets U and V such that every link connects a node in U to one in V; that is, U and V are independent sets. Examples: Hollywood actor network Collaboration networks Disease network (diseasome) Network Science: Graph Theory GENE NETWORK – DISEASE NETWORK DISEASOME PHENOME GENOME Gene network Disease network Goh, Cusick, Valle, Childs, Vidal & Barabási, PNAS (2007) Network Science: Graph Theory HUMAN DISEASE NETWORK Ingredient-Flavor Bipartite Network Y.-Y. Ahn, S. E. Ahnert, J. P. Bagrow, A.-L. Barabási Flavor network and the principles Network Science: Graph Theory of food pairing , Scientific Reports 196, (2011). Section 2.8 PATHOLOGY PATHS A path is a sequence of nodes in which each node is adjacent to the next one Pi0,in of length n between nodes i0 and in is an ordered collection of n+1 nodes and n links Pn = {i0,i1,i2,...,in} Pn = {(i0 ,i1),(i1,i2 ),(i2 ,i3 ),...,(in-1,in )} • In a directed network, the path can follow only the direction of an arrow. Network Science: Graph Theory DISTANCE IN A GRAPH Shortest Path, Geodesic Path B The distance (shortest path, geodesic path) between two A nodes is defined as the number of edges along the shortest path connecting them. C D *If the two nodes are disconnected, the distance is infinity. B In directed graphs each path needs to follow the direction of A the arrows. Thus in a digraph the distance from node A to B (on an AB path) is generally different from the distance from node B to A C D (on a BA path). Network Science: Graph Theory NUMBER OF PATHS BETWEEN TWO NODES Adjacency Matrix Nij, number of paths between any two nodes i and j: Length n=1: If there is a link between i and j, then Aij=1 and Aij=0 otherwise. Length n=2: If there is a path of length two between i and j, then AikAkj=1, and AikAkj=0 otherwise.