15 network centrality

• HW3 deadline on Thu, midnight! • We will provide feedback about the Project Milestone ASAP possibly by the end of this week CS224W: Social and Information Network Analysis Jure Leskovec and Baharan Mirzasoleiman, Stanford http://cs224w.stanford.edu Given a social network, which nodes are more important or influential? ¡ Centrality measures were proposed to account for the importance of the nodes of a network ¡ 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu ¡ Centrality is used often for detecting: § How influential a person is in a social network? § How well used a road is in a transportation network? § How important a web page is? [Wu and He’15] 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu Geometric Measures: § Importance is a function of distances to other nodes ¡ Spectral Measures: § Based on the eigen-structure of some graph-related matrix ¡ Path-based Measures: § Take into account all (shortest) paths coming into a node ¡ (Path based) (Geometric) (Geometric) 11/26/18 (Spectral) Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu ¡ Geometric measures § (In-)Degree Centrality: The number of incoming links !"#$ (&) = )*+ (&) § Or equivalently, number of nodes at distance one § Equivalent to majority voting 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu ¡ Geometric measures § Closeness Centrality: Who are the bridges? § Nodes that are more central have smaller distances !"#$% (') = ∑, -(., ') length of the shortest path from y to x § Nodes that are more central have smaller distances, and higher centrality 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu ¡ Geometric measures § Closeness Centrality: !"#$% (') = ∑, -(., ') length of the shortest path from x to y § How much a vertex can communicate without relying on third parties for his messages to be delivered 0.1 0.14 A B 00123 0.16 C = = : 5+7+8+9 0.14 0.1 D E § Problem: The graph must be (strongly) connected! 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu !"## !"## !"## !"## 0.1 0.1 0.16 !"## 0.1 !"## !"## !"## 0.1 0.1 We get null score for all nodes, if the graph is not connected! 11/26/18 $%&'( (*) = ∑/ 0(1, *) Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu ¡ Geometric measures § Harmonic Centrality: Who are the bridges? § Replace the average distance with the harmonic mean of all distances § The !(! − 1) distances between every pair of distinct nodes: &'() (*) = , = 0(1, *) Harmonic mean -./ , -,/ 45,-./ 0(1, *) § Strongly correlated to closeness centrality § Naturally also accounts for nodes that cannot reach * § Can be applied to graphs that are not strongly connected 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10 ¡ Removing nodes in betweenness order causes a very quick disruption of the network A n n n n 11/26/18 B C D E A lies between no two other vertices B lies between A and other vertices: C, D, and E C lies between pairs of vertices (A,D),(A,E),(B,D),(B,E) note that there are no alternate paths for these pairs to take, so C gets full credit Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 49 ¡ 11/26/18 Non-normalized version: Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 50 ¡ Non-normalized version: n C n A E B n Why C and D each have betweenness 1? They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: ½+½ = Can you figure out why B has betweenness 3.5 while E has betweenness 0.5? D 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 51 The three are clearly related, but they each get at something slightly different 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 52 Degree centrality Closeness centrality Eigenvector centrality 11/26/18 Betwenness centrality Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 53 Closer to all the other nodes (Bridges) Hubs Largest number of neighbors High degree bridges Largest number of walks of length ∞ ends up here (Influentials) Lots of paths pass through these nodes Examples of A) Betweenness centrality, B) Closeness centrality, C) Eigenvector centrality, D) Degree centrality, E) Harmonic Centrality and F) Katz centrality of the same graph 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 54 Largest degree Bridges Hubs High-degree bridges Influentials Bridges with normalized importance A) Degree B) Betweenness C) Eigenvector D) Closeness (normalized) E) Harmonic Centrality F) Closeness (not normalized) 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 55 Peer Experience: Common and Unique Features of Number of Friendships, Social Network Centrality, and Sociometric Status Scott D Gest, Sandra A Graham-Bermann, and Willard W Hartup 2001 ¡ Conceptually distinct dimensions of classroom social position § Number of mutual friendships § Social network centrality § Number of times children are named by classmates as members of informal peer groups § Sociometric status were examined in relation to each other and to peer-nominated behavioral reputation among 205 7- and 8- year old children 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 57 ¡ Individual items were analyzed separately as dependent variables in multiple regression analyses ¡ Network centrality was uniquely associated with both prosocial and antisocial behavioral styles § Positively associated with § reputation as a leader § peer nominations for teasing, showing off, picking on, and bossing classmates § Negatively associated with § sadness § feelings getting hurt easily 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 58 Pearson Correlations Among Peer Relations Measures and Peer Nominated Social Behavior Column entries under each measure indicate the increment in R2 associated with adding that peer relations measure to a regression equation that already includes the other four peer relations measures 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 59 Network Centrality, Power, and Innovation Involvement: Determinants of Technical and Administrative Roles Herminia Ibarra, 1993 ¡ This research was conducted in an advertising and public relations agency § The firm contained 94 full-time employees ¡ The goal is to investigated the relative impacts of individual attributes, on the exercise of individual power § Individual power is measured as involvement in technical and administrative innovations 11/26/18 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 61 ¡ Three sets of features § Individual attributes § § § § Education Tenure in the organization Prestige of past work Professional activity § Formal position § Rank § Department § Centrality § Calculated based on a questionnaire ¡ 11/26/18 Centrality was the most significant predictor of administrative innovation roles Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 62 b: Logistic regression coefficients +p < 10 * p < 05 ** p < 01 *** p < 001 Individual attributes Formal position Centrality 11/26/18 We get 30% improvement by adding centrality! Results of the hierarchical logistic regression of administrative innovation involvement on the three sets of variables: individual attributes, formal position, and centrality Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 63