Real-World Networks And their common properties 1. Macro-level - PDF document

9/22/20 DCS/CSCI 2350 Social & Economic Networks What does a real-world network look like? Reading: Ch 2 of EK, Ch 2 & 3 of Jackson Graph visualization using Gephi Mohammad T . Irfan Email: mirfan@bowdoin.edu 1 Real-World Networks And their common properties 1. Macro-level (graph-level) 2. Micro-level (node-level) 3 1

9/22/20 Macro-level properties 1. Giant component 2. Small-world 3. Degree distribution 4. Clustering 4 1. Giant component u Intuitive example– world acquaintance network u Questions u Is it connected? u How many giant components are there? 5 2

9/22/20 Examples u Actor network u Edge between two actors iff they appear together in a movie u 98% of 449,913 actors belong to the giant component (IMDB, May 2000) 6 More examples u Instant messaging u Microsoft IM: one giant component in a network of 240 million users (2008) u Co-author network u Email u Biological networks (neural networks) u Technology networks (power grid) u The Internet (web of links) Can you think of a network that doesn’t have any giant component? 8 3

9/22/20 What is the implication? High school relationships (1993-95) 9 2. Small-world property Also known as distance u Proposition u The average shortest path between any two nodes in a connected component is “small” u Intuition 11 4

9/22/20 Six degrees of separation u Hungarian author Frigyes Karinthy (1929 short story “Chain-Links”) “A fascinating game grew out of this discussion. One of us suggested performing the following experiment to prove that the population of the Earth is closer together now than they have ever been before. We should select any person from the 1.5 billion inhabitants of the Earth – anyone, anywhere at all. He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances.” u John Guare’s play (1990) & later movie 12 Milgram’s experiment (1963) 14 5

9/22/20 Milgram’s experiment (cont…) 15 Critiques u Only 64 out of 296 cases were successful u How useful? What is the implication? u Milgram: “six worlds apart” 16 6

9/22/20 Contagion of TB (Valdis Krebs, Oklahoma, 2002) 17 Another example u Microsoft instant messenger (2008) u 240M node network u Edge: Two-way conversation at some point during a month-long observation period u Average distance: 6.6 Fraction of pairs of nodes having this distance 18 7

9/22/20 Computational question u How to find the “right 6 people?” u Breadth-first search (BFS) algorithm to find the shortest path u Fun application– Bacon number u Bacon number of an actor = distance from Kevin Bacon u Average Bacon number: 2.9 u https://oracleofbacon.org/ 19 Shortest path algorithm Breadth-First Search (BFS) 20 8

9/22/20 BFS algorithm u Resulting graph: BFS tree AKA "root" Other existing edges within a Yo layer are not drawn here. Distance = 0 u Draw only the edges explored. Distance = 1 Your friends Distance = 2 Friends of friends Friends of friends Distance = 3 of friends Nodes whose distance have not yet been calculated and who have edges to nodes in the previous layer 22 Exercise: Draw BFS from MIT ARPANET (1970) 23 9

9/22/20 Network among Patrick Dial M for Murder Allen movie actors Grace Kelly High Noon The Eagle Has Landed Nicole Cold Mountain Lloyd Kidman Bridges Donald Sutherland Kathleen Joe vs. Volcano Quinlan Animal House John Tom Kevin Apollo 13 Belushi Hanks Bacon Da Vinci Yves Code Portrait of a Lady Bill Aubert The Wild River Paxton Paul Meryl Titanic Herbert Streep John Kate Holiday Winslet Gielgud Hamlet Cameron Diaz Bill Charlie’s Angels Murray 24 When does BFS give shortest paths? u When all the edges have the same "weight"/dist. u Negative example: Frankfurt—Kassel—Munchen not shortest path 25 10

9/22/20 Some special types of graphs u Tree u Connected, acyclic graph u Example: BFS tree u Bipartite graph u Two sets of nodes with no edge within the same set of nodes u Example: Network between movies and actors Actors Movies 26 3. Degree distribution u What’s the probability of finding a node with degree k? u What fraction of nodes have degree k? Call it P k . 27 11

9/22/20 Real-world degree distributions u Power law distribution (or Pareto distrib.) vs. normal distribution u Mathematical formulation u Scale-free networks Extremely important Please take note 29 4. Clustering coefficients u Clustering coeff = Average probability that two friends of a node are also friends u How to calculate? u Local clustering coeff. of node i, Need to count Actual # of edges among i’s friends C i = Max possible # of edges among i’s friends d i (d i -1) / 2 where d i = degree of i u Clustering coefficient of the whole network = average C i of all the nodes i 30 12

9/22/20 Example u What is the clustering coefficient of this network? 5 2 1 4 3 31 Political blogs (2004) u “High” clustering coefficient is observed in real-world networks 32 13

9/22/20 Empirical study of network properties u Uzzi et al., 2007 u https://www.kellogg.northwestern.edu/facu lty/uzzi/ftp/Uzzi_EuropeanManReview_2007. pdf u N = # of nodes k = Avg degree L = Avg shortest path length CC = Clustering coefficient 33 34 14

9/22/20 Micro-level properties Centrality Notation: n = # of nodes Reading: Jackson (Ch 2) 35 Caution u Six Degrees, pg. 51 36 15

9/22/20 Centrality measures Degree centrality 1. Closeness centrality 2. Betweenness centrality 3. Prestige/eigenvector centrality 4. Idea Math Example 37 1. Degree centrality u A node’s centrality = The node’s degree / (n-1) u Who is the most central here? u How about node 4 in this network? 1 6 4 3 5 2 7 38 16

9/22/20 2. Closeness centrality u Idea: node i is very central if it’s pretty close to the other nodes u Avg distance from node i to all other nodes = Need to do BFS ∑ dist ( i , j ) with root i j ≠ i n − 1 u Closeness centrality of i = 1/Avg distance from i 39 Example u Compute the closeness centralities of nodes 1 and 4 1 6 3 4 5 2 7 40 17

9/22/20 3. Betweenness centrality u Idea: a node i is very central if a lot of shortest paths go through i u Betweenness centrality of i, # of shortest paths between j and k passing through i β i = Σ # of shortest paths between j and k, irrespective of j,k j ≠ k ≠ i passing through i u Florentine marriage: Medici most central 41 Example u Compute the between centrality of nodes 1, 2, and 3 1 u β 1 = 0 6 3 2 u β 2 = 0 u β 3 = ? 4 5 42 18

9/22/20 Matrix algebra u Images from this tutorial: http://www.intmath.com/matrices- determinants/3-matrices.php u 4x1 matrix (AKA vector) u 3x3 matrix 43 Matrix multiplication u 2x3 matrix multiplied by 3x2 matrix must match u Result is a 2x2 matrix 44 19

9/22/20 Transpose of matrix u Transpose operator: superscript T ! $ 1 4 # & A = 2 5 # & # 3 6 & " % ! $ 1 2 3 A T = # & # & 4 5 6 " % u (A B) T = B T A T 45 4. Prestige/Eigenvector/power centrality u Idea (Phillip Bonacich, 1987): A node’s importance is determined by its friends’ importance u Mathematical formulation and example 46 20

9/22/20 Eigenvector calculator 47 More on eigenvector centrality u Tutorial on eigenvector u Jackson’s Section 2.4 (Appendix) 48 21

9/22/20 Comparison of centrality measures 49 Graph Visualization Gephi 50 22

9/22/20 Links u Download u https://gephi.org/ u Windows: Gephi 0.9.2 will only run with Java 7 or 8. Most modern Windows PCs will already have it. u How to find Java version in Windows? https://www.java.com/en/download/help/version_ manual.xml u Where to get Java for Windows? https://www.java.com/en/download/ u Mac OS X: Java is bundled with the application so it doesn't have to be installed separately. u Tutorial: http://bit.ly/gephi_tutorial u Dataset: http://bit.ly/gephi_dataset 51 Ranking: rank and color Data Laboratory: Manipulate Statistics: Computes graph-level properties. Partition: partition nodes, edges, and their the input graph files (e.g., Some of them (e.g., Average Degree) must nodes and edges labels by numeric properties apply labels to nodes) be done before using other features Preview: Produces a nice visualization (next slide) Color palette for coloring schemes Filters: Filter out nodes/ed ges based on their propertie s Useful filter: Topology à Degree Range Layout: Select a "Magnifying T: Toggle T: Toggle Slider: Tune Slider: Tune Network among graph drawing glass": Centers showing node showing edge edge thickness the size of the algorithm the graphics labels labels node labels the characters of Les Miserables 52 23