NETWORK SCIENCE Graphs and Networks Prof. Marcello Pelillo Ca’ Foscari University of Venice a.y. 2016/17 Section 1 The Bridges of Konigsberg Drawing Curves with a Single Stroke… Königsberg (today’s Kaliningrad, Russia) Konigsberg’s People Immanuel Kant (1724 – 1804) Gustav Kirchhoff (1824 – 1887) David Hilbert (1862 – 1943) THE BRIDGES OF KONIGSBERG Can one walk across the seven bridges and never cross the same bridge twice? Network Science: Graph Theory hp://www.numericana.com/answer/graphs.htm 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, or two such vertices, it has at least one path. Euler’s solution is considered to be the first theorem in graph theory. Network Science: Graph Theory hp://www.numericana.com/answer/graphs.htm The Bridges Today A “Local” Variation of Euler’s Problem Graphs and networks after the “bridges” • Laws of electrical circuitry (G. Kirchhoff, 1845) • Molecular structure in chemistry (A. Cayley, 1874) • Network representaon of social interacIons (J. Moreno, 1930) • Power grids (1910) • Telecommunicaons and the Internet (1960) • Google (1997), Facebook (2004), Twi#er (2006), . 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 UNDIRECTED VS. DIRECTED NETWORKS Undirected Directed Links: undirected (symmetrical) Links: directed (arcs). Graph: Digraph = directed graph: L A D M B F An undirected C link is the I superposition of D two opposite directed links. G B 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 NETWORK NODES LINKS DIRECTED N L k UNDIRECTED Internet Routers Internet connections Undirected 192,244 609,066 6.33 WWW Webpages Links Directed 325,729 1,497,134 4.60 Power Grid Power plants, transformers Cables Undirected 4,941 6,594 2.67 Mobile Phone Calls Subscribers Calls Directed 36,595 91,826 2.51 Email Email addresses Emails Directed 57,194 103,731 1.81 Science Collaboration Scientists Co-authorship Undirected 23,133 93,439 8.08 Actor Network Actors Co-acting Undirected 702,388 29,397,908 83.71 Citation Network Paper Citations Directed 449,673 4,689,479 10.43 E. Coli Metabolism Metabolites Chemical reactions Directed 1,039 5,802 5.58 Protein Interactions Proteins Binding interactions Undirected 2,018 2,930 2.90 Section 2.3 Degree, Average Degree and Degree Distribution NODE DEGREES Node degree: the number of links connected to the node. A kA =1 kB = 4 B Undirected In directed networks we can define an in-degree and out-degree. D B C The (total) degree is the sum of in- and out-degree. in out E G kC = 2 kC = 1 kC = 3 A Directed F Source: a node with kin= 0; Sink: a node with kout= 0. SECTION 2.3 DEGREE, AVERAGE DEGREE, AND DEGREE DISTRIBUTION SECTION 2.3 A key property of each node is its degree, representing the number of links it has to other nodes. The degree can represent the number of mobile phone contacts an individual has in the call graph (i.e. the number of dif- BOX 2.2 DEGREE, AVERAGEferent individuals the personDEGREE, has talked to), or the number of citations a BRIEF STATISTICS REVIEW research paper gets in the citation network. AND DEGREE DISTRIBUTION Four key quantities characterize Degree a sample of N values x , ... , x : th 1 N We denote with ki the degree of the i node in the network. For exam- ple, for the undirected networks shown in Figure 2.2 we have k =2, k =3, 1 2 Average (mean): k3=2, k4=1. In an undirected network the total number of links, L, can be expressed as the sum of the node degrees: xx++…+ x 1 N = 12 N = N x ∑ xi NN= 1 (2.1) i 1 Lk = ∑ i . 2 i=1 The nth moment: Here the 1/2 factor corrects for the fact that in the sum (2.1) each link is A key property of each node is its degreecounted, representing twice. For the example, number the of linkA BIT connecting OF STATISTICS the nodes 2 and 4 in Figure nn n N n xx12++…+ x N 1 n links it has to other nodes. The degree can2.2 represent will be counted the number once of in mobile the degree of node 1 and once in the degree of x = = ∑ xi NNi=1 phone contacts an individual has in the nodecall graph 4. (i.e. the number of dif- BOX 2.2 ferent individuals the person has talked to), or the number of citations a BRIEF STATISTICS REVIEW research paper gets in the citation network. Standard deviation: Average Degree An important property of a network is itsFour average key quantities degree (BOX characterize 2.2), which Degree N 1 2 for an undirected network is a sample of N values x1, ... , xN : σ = − We denote with k the degree of the ith node in the network. For exam- xi∑()xx i N = N i 1 ple, for the undirected networks shown in Figure 2.2 we have k =2, k =3, 12L 1 2 ≡ = . (2.2) k ∑ ki Average (mean): k =2, k =1. In an undirected network the total number of links, L, can be N i=1 N 3 4 Distribution of x: expressed as the sum of the node degrees: xx++…+ x 1 N In directed networks we distinguish between= 12 incoming degreeN = , k in, rep- N x ∑ixi 1 NNi=1 1 = (2.1) p = δ . Lk ∑resentingi . the number of links that point to node i, and outgoing degree, x ∑ xx, i 2 = N i i 1 k out, representing the number of links that point from node i to other i The nth moment: Here the 1/2 factor corrects for the fact that in the sum (2.1) each link is where p follows nodes. Finally, a node’s total degree, ki, is given by x counted twice. For example, the link connecting the nodes 2 and 4 in Figure nn n N in nout xx12++…+ x N 1 n 2.2 will be counted once in the degree of node 1 and once in the degree of =+x .= = xi (2.3) kkiiki ∑ NNi=1 node 4. For example, on the WWW the number of pages a given document points to represents its outgoing degree, Standardkout, and deviationthe number: of docu- Average Degree ments that point to it represents its incoming degree, k . The total number An important property of a network is its average degree (BOX 2.2), which in Network Science: Graph Theory N for an undirected network is 1 2 σ xi= ∑()xx− GRAPH THEORY N = 8 N i 1 12L ≡ = . (2.2) k ∑ ki N i=1 N Distribution of x: In directed networks we distinguish between incoming degree, k in, rep- i 1 = δ . resenting the number of links that point to node i, and outgoing degree, px ∑ xx, i N i out ki , representing the number of links that point from node i to other where p follows nodes. Finally, a node’s total degree, ki, is given by x in out. (2.3) kkii=+ki For example, on the WWW the number of pages a given document points to represents its outgoing degree, kout, and the number of docu- ments that point to it represents its incoming degree, kin. The total number GRAPH THEORY 8 AVERAGE DEGREE 1 N 2L k k j ≡ ∑ i k ≡ N i=1 N i N – the number of nodes in the graph Undirected D 1 N 1 N B k in k in , k out k out , k in k out C ≡ ∑ i ≡ ∑ i = N i=1 N i=1 E A L Directed k ≡ F N Network Science: Graph Theory Average Degree NETWORK NODES LINKS DIRECTED N L k UNDIRECTED Internet Routers Internet connections Undirected 192,244 609,066 6.33 WWW Webpages Links Directed 325,729 1,497,134 4.60 Power Grid Power plants, transformers Cables Undirected 4,941 6,594 2.67 Mobile Phone Calls Subscribers Calls Directed 36,595 91,826 2.51 Email Email addresses Emails Directed 57,194 103,731 1.81 Science Collaboration Scientists Co-authorship Undirected 23,133 93,439 8.08 Actor Network Actors Co-acting Undirected 702,388 29,397,908 83.71 Citation Network Paper Citations Directed 449,673 4,689,479 10.43 E.