2 Networks: Concepts and Empirics m aally. !ffing idual 2.1 Introduction The first aim of this chapter is to present the principal concepts relating to net­ works which will be used in this book. This presentation in section 2.2 will draw upon work carried out in the theory of graphs, in mathematical sociology, and in statistical physics. The study of networks has a rich and distinguished tradition in each of these subjects and there are a number of excellent books available. The exposition below borrows heavily from Harary (1969), Bollobas (1998), and Wasserman and Faust (1994). iould The second aim of this chapter is to discuss these concepts in relation to empiri­ cally observed networks. This discussion is presented in section 2.3. The empirical study of networks has been a very active field of study in the last decade. The discussion highlights some of the key features that have been identified by this research. In doing so, it directs our attention to specific networks which are then taken up in the theoretical investigations of the subsequent chapters. ment 2.2 Concepts eco­ es There is a set of nodes, N = { 1, 2, 3, . .. , n} , where n is a finite number. Rela­ tionships between nodes are conceptualized in terms of binary variables, so that a relationship either exists or does not exist. Denote by gii E {0, 1} a relationship between two nodes i and j. The variable g iJ takes on a value of 1 if there exists a link between i and j and 0 otherwise. 1 The set of nodes taken along with the links between them defines the network; this network is denoted by g and the collection of all possible networks on n nodes is denoted by g, . Given a network g, g + gij and g- gij have the natural interpretation. When gij = Oing, g+ gij adds thelinkgij = 1, while if gij = 1 in g , theng+ gij =g. 1 In this chapter, we consider links which are undirected, i.e., g iJ = gJ i. In some parts of the book (chapters 5, 7, and 8), we will study networks of directed links; most of the concepts defined here can be adopted in a natural way to cover directed links. See chapters 5 and 7 for definitions and additional notation relating to directed networks. 10 2. Networks: Concepts and Empirics 2.2. Corn 0 0 0 0 (a) The complete network (b) Empty network (c) Degree 1 network (d) Degree 2 network Figure 2.1. Regular networks for n = 4. Similarly, if gij l in g, g- gij deletes the link gij, while if gij = 0 in g, then g -gij= g. Neighbors and degrees. Let Ni (g) = {j E N I gij = 1} denote the nodes with which node i has a link; this set will be referred to as the neighbors of i. Let r)i (g) = INi (g) I denote the number of neighbors of node i in network g. Moreover, for any integer d ~ 1, let»/ (g) be the d -neighborhood of i in g: this is defined inductively, 1 Nk (g: »/(g) = Ni (g) and »/(g) = »/- (g) U ( U Nj (g)). other and jE.N;k-1 such an a core-pc Let N 1 (g), N 2 (g), ... , Nn - 1 (g) be a division of nodes into distinct groups, where nodes belong to the same group if and only if they have the same number of links, i.e., i , j E Nk(g), k = 1, 2, ... , n- 1, if and only if 1Ji (g) = 1Jj (g). With satisfy this notation in hand we can now describe a number of well-known networks. A network is said to be regular if every node has the same number of links, INJ (g i.e., r)i(g) = 1], Vi E N (and so all nodes belong to one group in the above are refe partition).2 The complete network, gc, is a regular network in which 1J = n - l, are refen while the empty network, ge, is a regular network in which 17 = 0. Figure 2.1 star \\; presents regular networks for n = 4. An ~ nodes D 2 If the number of nodes is even, then a regular graph of every degree is possible; this is not true when Dm +J(g the number of nodes is odd. For instance, in a network with n = 5, there exist no networks in which Dx(g - ~ every node has degree I or in which every node has degree 3. In some parts of the book, interest will center on how the degree of a regular network matters and then it will be (implicitly) assumed that the gro up ne number of nodes is even. whilen- 'llfi Empirics 2.2. Concepts 11 (a) Star network (b) Interlinked star (two centers) = 0 in g, then (c) Interlinked star (three centers) (d) Core-periphery network l)(e the nodes Figure 2.2. Star and variants. ~ghb ors of i. !II network g. A core-periphery network structure describes the following situation. There JOd of i in g: are two groups of nodes, N,(g) and Nk(g), with k > INk(g)l. Nodes in N 1 (g) constitute the periphery and have a single link each and this link is with a node in Nk(g); nodes in the set Nk(g) constitute the core and are fully linked with each other and with a subset of nodes in N 1 (g). The star network is a special case of such an architecture in which the core contains a single node. Figure 2.2 presents sri:ocl groups, a core-periphery network. me number of An interlinked stars network consists of two groups Nk(g) and Nn-I (g) which =1/j (g). With satisfy the following condition: Ni(g) = Nn- 1 (g) fori E Nk(g). The star networks. network is again a special case of such an architecture with INn-! (g)l = 1 and !llber of links, IN1 (g) I = n - 1. In an interlinked star network, nodes which have n - 1 links 1 in the above are referred to as central nodes or as hubs, while the complementary set of nodes rh '7 = n- 1, are referred to as peripheral nodes or as spokes. Figure 2.2 presents interlinked 0. Figure 2.1 stars with two and three central players. An exclusive groups network consists of m + 1 groups, a group of isolated nodes D, (g), and m ~ 1 distinct groups of completely linked nodes, D2 (g), ... , is· (}(){ true when Dm+!(g). Thus IJi(g) = 0 fori E D,(g), while 1Jj(g) = IDx(g)l-1 for j E avoods in which Dx(g), x E {2, 3, ... , m + 1}. A special case of this architecture is the dominant interest will l assumed that the group network in which there is one complete component with 1 < k < n nodes while n -k > 0 nodes are isolated. Figure 2.3 illustrates exclusive group networks. 12 2. Networks: Concepts and Empirics 2.2. Concepts one of the primary n ® issue of how nodes e 0 a number of diffe o 0· ® ~ The degree variance (a) Exclusive groups (b) Dominant group network network is 0 for all Figure 2.3. Exclusive group networks. The range has a maxi Figure 2.4. Line network. in a star is n - 2, wbl The description o A line network consists of two groups of nodes, N1(g) and N2(g), with elegant way to study IN1 (g)l = 2 and IN2(g)l = n - 2. As the name suggests, the network has links is captured in tti the form of a line; the two nodes with one link each are at the two ends of the line, idea of redistributin while the nodes with two links are in between. Figure 2.4 presents a line network with n = 6 nodes. These figures give us a first impression of how different networks can look! where The examples all contain relatively few nodes; in some economic and social contexts of interest, such as the diffusion of information or the evolution of social coordination and cooperation it is more natural to consider networks with a large number of nodes. When analyzing large networks it is more convenient to work Let P and P ' be I with distributions of links. /P and /P' the two Thedegreeofnodei is thenumberofi 's direct connections; so 1'/i (g) = INi (g)l Definition 2.1. P fu denotes the degree of node i in network g. The degree distribution in a network /P(k) ~ :P'(k) for is a vector P , where P(k) = INk(g)l/n is the frequency/fraction of nodes with degree k; thus P(k) ~ 0 for each k, and z:=~:,~ P(k) = 1. This degree Definition 2.2. P se distribution has support on /D = {0, 1, 2, ... , n - 1}. The average degree in network g is defined as n-1 f,(g) = L P(k)k = L T/i(g). (2.1) for every x E { 0 I _ k=O iEN n Definition 2.3. pi i In a star network the degree distribution has support on degrees 1 and n - 1, have the same mean with n - 1 nodes having degree 1, and 1 node having degree n - 1. The average degree in a star is 2- 2/ n. A simple examplt An important concern in the study of networks is the variation in the degrees. move from a regulaJ This variation is interesting in its own right but it also has an important instru­ mental aspect: degree may be related to node behavior and well-being.