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

9/25/19 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 Real-World Networks And their common properties 1. Macro-level (graph-level) 2. Micro-level (node-level) 1

9/25/19 Macro-level properties 1. Giant component 2. Small-world 3. Degree distribution 4. Clustering 1. Giant component u Intuitive example– world acquaintance network u Questions u Is it connected? u How many giant components are there? 2

9/25/19 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) 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? 3

9/25/19 What is the implication? High school relationships (1993-95) 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 4

9/25/19 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 Milgram’s experiment (1963) 5

9/25/19 Milgram’s experiment (cont…) Critiques u Only 64 out of 296 cases were successful u How useful? What is the implication? u Milgram: “six worlds apart” 6

9/25/19 Contagion of TB (Valdis Krebs, Oklahoma, 2002) 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 7

9/25/19 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/ Shortest path algorithm Breadth-First Search (BFS) 8

9/25/19 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 Exercise: Draw BFS from MIT ARPANET (1970) 9

9/25/19 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 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 10

9/25/19 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 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 . 11

9/25/19 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 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 12

9/25/19 Example u What is the clustering coefficient of this network? 5 2 1 4 3 Political blogs (2004) u “High” clustering coefficient is observed in real-world networks 13

9/25/19 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 14

9/25/19 Graph Visualization Gephi 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 15

9/25/19 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 Show Labels: Turn it on! Refresh: Must click this button! Otherwise, nothing will be shown. To save the visualization as a pdf file: File à Save 16

9/25/19 Gephi Vocabulary Red: Graph level Black: Node/edge level Term Meaning betweeness centrality how often the node appears on the shortest path between nodes in the network of a node closeness centrality of a average distance from that node to all other nodes in the network node degree of a node the number of edges connected to the node (also connectedness); in a directed graph a node can have in- degree and out-degree measures diameter of a graph the longest shortest path between any two nodes in the graph directed graph this means relationships occur one way only (I follow you, but you do not follow me on Twitter); opposite of undirected (we are friends with each other on Facebook) eccentricity of a node the distance (shortest-path length) from the node to the farthest node from it in the network edge a representation of the connection between two nodes, expresses a relationship (a line) eigenvector centrality in social network analysis, a measure of influence (a node is very influential if it is connected to other of a node influential nodes) layout algorithms also known as graph drawing algorithm; e.g., force-directed drawing where linked nodes attract and non- linked nodes repel leaf node node with a single edge in a “tree-structured” graph modularity a measure of connectedness among groups of nodes (greater than 0.4 is usually considered meaningful) node also called a vertex by mathematicians; a person in a social network graph (a dot or bubble) distance from one node the length of the shortest path (counted in the number of edges) from one node to another to another path length the number of edges in a path singleton node or node with no edge/connection isolated node Micro-level properties Centrality Notation: n = # of nodes Reading: Jackson (Ch 2) 17

9/25/19 Centrality measures Degree centrality 1. Closeness centrality 2. Betweenness centrality 3. Prestige/eigenvector centrality 4. Idea Math Example 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 18