Networks in economics and finance Lecture 1 - Measuring networks
What are networks and why study them? A network is a set of items (nodes) connected by edges or links. Units (nodes) Individuals Firms Banks Airports Websites Interaction Friendship Trade Credit flow Transportation Web links Networks are also found in biological (genetic and metabolic networks) and ecological systems (food webs).
The Connected States of America - senseable.mit.edu/csa
Ahn, Y.Y., et al (2011), Flavor network and the principles of food pairing. Scientific Reports 1, 196
Course objectives Describe properties / measure networks. Mathematical models of network formation. Non-cooperative games on networks. Diffusion of information and learning in networks. Financial crises / development and networks.
What do networks offer us? Questions we can address within a networks paradigm: Does network structure influence system performance and fragility? How does behavior and learning change with network structure? Are there contagious risks (bank failures, spread of disease)?
Structure of course Lectures ~ Monday, 14.00-18.00, MA 142. First lecture: 23 April;; Last lecture: 28 May. Office hours: By appointment (kartik.anand@tu-berlin.de). Course grade ~ 50% final exam + 30% presentation/essay + 20% class participation. Either give a 15-20 minute presentation or write a critical review on a paper (to be assigned).
Outline Types of networks Graphs - notation and terminology Properties of graphs Basic random graph model Why your friends have more friends than you do
The history of the study of networks Graph theory ~ begins with Euler s 1735 solution to the Königsberg Bridge problem. Networks in sociology ~ Focus on structure and interaction in small networks that were constructed from survey data. Typical question Which node in this network would prove most crucial to the network's [property] if it were removed? More recently ~ lots of data (networks with 1000 s of nodes) is available and handled by computers. Statistical tools have been deployed to investigate these networks. Now the question is What percentage of nodes need to be removed to substantially affect network [property]? Modern theory ~ (i) statistical tools to characterize real networks, (ii) build models of networks and make predictions.
Graph theory A graph G =(V, E) consists of a set of nodes (vertices) V = {1,..., N} and edges describing which pair of vertices are connected, E = {(i, j) i, j V }. Graph may be weighted [(i, j) R] or not [(i, j) {0, 1}]. Graph is undirected if (i, j) = (j, i). Adjacency matrix A {0, 1} N N where A ij =1 (i, j) E.
Walks, Paths and Cycles - 1 Walk ~ sequence of edges: (i,c),(c,k),(k,j),(j,p). Path ~ walk between two specific nodes (i and j, for example): (i,c),(c,k),(k,j). Cycle ~ path starting and ending at the same node: (i,c),(c,k),(k,j),(j,i). Geodesic ~ shortest path between two nodes: (i,j). A path s (walk s) length is the number of edges involved.
Walks, Paths and Cycles - 2 Using the adjacency matrix, length 3 in the graph. A 3 = i A ij A jk A ki j,k gives us all the cycles of (A A) ij gives all paths of length 2 between nodes i and j.
Components An undirected graph is connected if every two nodes in the network are joined together in a path. A component in a graph is a connected subgraph, i.e., G =(V,E ), s.t. V V, E E. i, j V a path contains in E A directed graph is strongly connected if there is a directed path in between every pair of nodes, i.e., i -> j and j -> i.
Types of graphs
Maximal independent set Given an undirected graph such that. G =(V, E) i, j U, (i, j) / E an independent set is a subset U V An independent set is maximal if no new node can be added to the set without violating independence.
Neighborhood and degree of a node The neighborhood of a node i is the set of other node that i is adjacent with, N i = {j (i, j) E}. For an undirected graph, the degree of node i is the cardinality of it s neighborhood, d i = N i. For an undirected graph: In degree : d in i = j A ij Out degree : d out i = j A ji
Properties of networks Small networks are easy to visualize. But once we increase the number of nodes, we need to resort to other descriptive summary statistics. Examples of such measures: (i) Average path length, (ii) Clustering, (iii) Centrality and (iv) Degree distribution.
Diameter and average path length Define G(i,j) as the geodesic (shortest) path length between nodes i and j in an undirected graph. The diameter is the largest geodesic path, i.e., diameter = max i,j G(i, j) The average path length is the average distance between any two nodes in the graph. i<j G(i, j) average path length = N(N 1) 2 The average path length is bounded above by the diameter. If a graph is not connected, take the diameter of the largest component.
Clustering Measure for how tightly linked nodes are by counting the number of triangles. Ratio of the number of triangles in the graph over the number of triples (subgraphs of 3 nodes with either 2 or 3 edges between themselves). CI = 3 number of triangles in network number of connected triples of vertices The clustering coefficient is bounded above and below by 1 and 0, respectively.
Clustering One can also measure the clustering of individual nodes. For an undirected graph: j,k CI i = A ij A jk A ki d i (d i 1) 2 The average clustering coefficient: CI Avg = 1 N CI i i
Centrality A measure of how important a particular node is in the network. Examples include: d i degree centrality : (N 1) N 1 closeness centrality : G(i, j) j i eigenvector centrality : x = 1 λ Ax
Degree distribution P(d) ~ frequency of different nodes having degree = d. For a given graph, P(d) is a histogram of fraction of nodes with degree d. For random graphs, P(d) is a well defined probability distribution. Examples of degree distributions include: P (d) =ce α d, c > 0, α > 0 P (d) =cd γ, c > 0, γ > 0
Assortativity A measure for how different nodes prefer to attach themselves to each other. Case 1: High degree nodes typically attach themselves to other high degree nodes. This tendency is referred to as assortativity. Case 2: The opposing case where High degree nodes are typically linked to low degree nodes is referred to as dissortativity. e ij Define as the fraction of edges connecting nodes of degree i with nodes of degree j. e ij =1, ij r = e ij = a i, j e ij = b j. i e ii i a i b i 1 i a. i b i i
Basic random graph model Erdos Renyi Random graph ~ N nodes; Link between two nodes occurs with probability p. Average degree per node? Degree distribution? Clustering coefficient? Diameter? Largest Component?
Why your friends have more friends than you Paper by S. Feld published in the American Journal of Sociology in 1991. Viewing friendship as a way of evaluating ones self, how do people determine what is an adequate number of friends? If people look at the number of friends their friends have, it is likely they will feel relatively inadequate. Specifically: The mean number of friends friends is always greater than the mean number of friends of an individual.
Why your friends have more friends than you Full network consisted of 146 girls.
Why your friends have more friends than you
Why your friends have more friends than you
Why your friends have more friends than you Network structure matters ~ Assortativity influences the mean