BEYOND ASSORTATIVITY PROCLIVITY INDEX FOR ATTRIBUTED NETWORKS (PRONE) Reihaneh Rabbany Dhivya Eswaran*

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1 PROCLIVITY INDEX FOR ATTRIBUTED NETWORKS (PRONE) Reihaneh Rabbany Artur W. Dubrawski Dhivya Eswaran* Christos Faloutsos

2 PROBLEM BACKGROUND PRONE PROPERTIES PROBLEM DISCOVERIES

3 3 PROBLEM DATA IMPUTATION BOB ALICE CAROL JOHN

4 4 PROBLEM ADS PLACEMENT BOB ALICE CAROL JOHN

5 5 PROBLEM THE CORE PROBLEM GIVEN a graph of nodes & edges attributes for nodes FIND correlation of attributes with network structure

6 6 PROBLEM PROCLIVITY Inclination of nodes with a certain value of attribute to connect to nodes with a certain ALICE other value for the same or a different attribute BOB self cross

7 PROBLEM BACKGROUND PRONE PROPERTIES BACKGROUND DISCOVERIES

8 8 BACKGROUND MIXING MATRIX - DEFINITION [SRC] ATTR 1 [DEST] ATTR Number of edges between CMU graduates and low income people

9 9 BACKGROUND MIXING MATRIX - EXAMPLE GRAPH WITH 2 ATTRIBUTES SHAPE AND COLOR MIXING MATRIX (COLOR, COLOR)

10 10 BACKGROUND MIXING MATRIX - EXAMPLE GRAPH WITH 2 ATTRIBUTES SHAPE AND COLOR MIXING MATRIX (COLOR, SHAPE)

11 11 BACKGROUND NORMALIZED MIXING MATRIX [SRC] ATTR 1 (r values) [DEST] ATTR 2 (k values) =1

12 12 BACKGROUND Q-MODULARITY How unexpected edges between nodes having the same attribute value are

13 13 BACKGROUND R-INDEX A normalized version of Q-modularity Q-modularity

14 14 BACKGROUND LIMITATIONS (1/2) Q and r fail intuition

15 15 BACKGROUND LIMITATIONS (2/2) Q and r do not even apply

16 PROBLEM BACKGROUND PRONE PROPERTIES PRONE DISCOVERIES

17 17 PRONE WHAT WE WANT GOOD OKAY BAD Single spike per row or column

18 18 PRONE QUANTIFYING DIVERGENCES GOOD BAD (1 + 5) = 10 (3 + 3) = 17

19 19 PRONE DIVERGENCES - CHOOSING F 1 5 GOOD 3 3 BAD (1 + 5) = 10 (3 + 3) = 17 (1 + 5) = 90 (3 + 3) = 162 e 1+5 e 1 e e 3+3 e 3 e f(x + y) f(x)+f(y)

20 20 PRIOR WORK RECALL NOTATIONS [SRC] ATTR 1 (r values) [DEST] ATTR 2 (k values) =1

21 21 PRONE PRONE total row divergences total column divergences maximum total divergence assuming marginals are fixed ProNe f =1 D f

22 22 PRONE SPECIAL CASES OF PRONE f(x) =x log x (similar to Normalized Mutual Information) f(x) =x 2 (similar to Adjusted Random Index)

23 PROBLEM BACKGROUND PRONE PROPERTIES PROPERTIES DISCOVERIES

24 24 PROPERTIES KEY PROPERTIES THOROUGHNESS GENERALITY CONSISTENCY SCALABILITY

25 25 PROPERTIES P1. THOROUGHNESS HETEROPHILY RANDOM l 2 3

26 26 PROPERTIES P2. GENERALITY SELF CROSS l 2 3

27 27 PROPERTIES P3. CONSISTENCY CORRELATED RANDOM l 2 3

28 28 PROPERTIES P4. SCALABILITY COMPUTATION TIME (S) linear scaling GRAPH SIZE (NUMBER OF EDGES)

29 29 PROPERTIES KEY PROPERTIES THOROUGHNESS both homophily and heterophily GENERALITY single or a pair of attributes CONSISTENCY zero score for no correlation SCALABILITY computation linear in graph size

30 PROBLEM BACKGROUND PRONE PROPERTIES DISCOVERIES DISCOVERIES

31 31 DISCOVERIES DATA GENDER DORM STATUS CLASS YEAR MAJOR HIGH SCHOOL MINOR A graph per university, e.g., Rice31

32 32 DISCOVERIES VISUALIZATION (1/2) Which attribute has more correlation?

33 33 DISCOVERIES VISUALIZATION (2/2) What is the extent of correlation?

34 34 DISCOVERIES PRONE (FOR VISUALIZED DATA) Similar inferences for other PRONE variations

35 35 DISCOVERIES PRONE (FOR RICE UNIVERSITY) Similar inferences for other Facebook datasets

36 PROBLEM BACKGROUND PRONE PROPERTIES SUMMARY DISCOVERIES

37 37 SUMMARY PRONE PROCLIVITY INDEX 1 5 GOOD Thoroughness Generality self cross 3 3 BAD Consistency Scalability 1 Questions? deswaran@cs.cmu.edu