Discovering Roles and Anomalies in Graphs: Theory and Applications

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1 Discovering Roles and Anomalies in Graphs: Theory and Applications Part 1: Theory Tina Eliassi-Rad (Rutgers) Christos Faloutsos (CMU) SDM'12 Tutorial

2 Overview Features Roles Anomalies = rare roles Patterns SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 2

3 Overview Features Roles Anomalies = rare roles Patterns SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 3

4 Roadmap What are roles Roles and communities Roles and equivalences (from sociology) Roles (from data mining) Summary SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 4

5 What are roles? Functions of nodes in the network Similar to functional roles of species in ecosystems Measured by structural behaviors Examples centers of stars members of cliques peripheral nodes SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 5

6 Example of Roles centers of stars members of cliques peripheral nodes SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 6

7 Why are roles important? Role Discovery ü Automated discovery ü Behavioral roles ü Roles generalize Task Role query Role outliers Role dynamics Use Case Identify individuals with similar behavior to a known target Identify individuals with unusual behavior Identify unusual changes in behavior Identity resolution Identify known individuals in a new network Role transfer Network comparison Use knowledge of one network to make predictions in another Determine network compatibility for knowledge transfer

8 Roadmap What are roles Roles and communities Roles and equivalences (from sociology) Roles (from data mining) Summary SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 8

9 Roles and Communities Roles group nodes with similar structural properties Communities group nodes that are wellconnected to each other Roles and communities are complementary SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 9

10 Roles and Communities RolX * Fast Modularity * Henderson, et al. 2012; Clauset, et al SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 10

11 Roles and Communities Consider the social network of a CS dept Roles Faculty Staff Students Communities AI lab Database lab Architecture lab SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 11

12 Roadmap What are roles Roles and communities Roles and equivalences (from sociology) Roles (from data mining) Summary SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 12

13 Equivalences Regular Deterministic Automorphic Equivalences Structural Probabilistic Stochastic SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 13

14 Deterministic Equivalences Regular Automorphic Structural SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 14

15 Structural Equivalence [Lorrain & White, 1971] Two nodes u and v are structurally equivalent if they have the same relationships to all other nodes Hypothesis: Structurally equivalent nodes are likely to be similar in other ways i.e., you are your friend Weights & timing issues are not considered Rarely appears in real-world networks a b c u d v e SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 15

16 Structural Equivalence: Algorithms CONCOR (CONvergence of iterated CORrelations) [Breiger et al. 1975] A hierarchical divisive approach 1. Starting with the adjacency matrix, repeatedly calculate Pearson correlations between rows until the resultant correlation matrix consists of +1 and -1 entries 2. Split the last correlation matrix into two structurally equivalent submatrices (a.k.a. blocks): one with +1 entries, another with -1 entries Successive split can be applied to submatrices in order to produce a hierarchy (where every node has a unique position) SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 16

17 Structural Equivalence: Algorithms STRUCUTRE [Burt 1976] A hierarchical agglomerative approach 1. For each node i, create its ID vector by concatenating its row and column vectors from the adjacency matrix 2. For every pair of nodes i, j, measure the square root of sum of squared differences between the corresponding entries in their ID vectors 3. Merge entries in hierarchical fashion as long as their difference is less than some threshold α SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 17

18 Structural Equivalences: Algorithms Combinatorial optimization approaches Numerical optimization with tabu search [UCINET] Local optimization [Pajek] Partition the sociomatrices into blocks based on a cost function that minimizes the sum of within block variances Basically, minimize the sum of code cost within each block SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 18

19 Deterministic Equivalences Regular Automorphic Structural SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 19

20 Automorphic Equivalence [Borgatti, et al. 1992; Sparrow 1993] Two nodes u and v are automorphically equivalent if all the nodes can be relabeled to form an isomorphic graph with the labels of u and v interchanged Swapping u and v (possibly along with their neighbors) does not change graph distances Two nodes that are automorphically equivalent share exactly the same label-independent properties u v SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 20

21 Automorphic Equivalence: Algorithms Sparrow (1993) proposed an algorithm that scales linearly to the number of edges Use numerical signatures on degree sequences of neighborhoods Numerical signatures use a unique transcendental number like π, which is independent of any permutation of nodes Suppose node i has the following degree sequence: 1, 1, 5, 6, and 9. Then its signature is S i,1 = (1 + π)(1 + π) (5 + π) (6 + π) (9 + π) The signature for node i at k+1 hops is S i,(k+1) = Π(S i,k + π) To find automorphic equivalence, simply compare numerical signatures of nodes SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 21

22 Deterministic Equivalences Regular Automorphic Structural SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 22

23 Regular Equivalence [Everett & Borgatti, 1992] Two nodes u and v are regularly equivalent if they are equally related to equivalent others President Motes Faculty Graduate Students Hanneman, Robert A. and Mark Riddle Introduction to social network methods. Riverside, CA: University of California, Riverside ( published in digital form at ) SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 23

24 Regular Equivalence (continued) Basic roles of nodes source repeater sink isolate SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 24

25 Regular Equivalence (continued) Based solely on the social roles of neighbors Interested in Which nodes fall in which social roles? How do social roles relate to each other? Hard partitioning of the graph into social roles A given graph can have more than one valid regular equivalence set Exact regular equivalences can be rare in large graphs SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 25

26 Regular Equivalence: Algorithms Many algorithms exist here Basic notion Profile each node s neighborhood by the presence of nodes of other "types" Nodes are regularly equivalent to the extent that they have similar "types" of other nodes at similar distances in their neighborhoods SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 26

27 Equivalences Regular Deterministic Automorphic Equivalences Structural Probabilistic Stochastic SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 27

28 Stochastic Equivalence [Holland, et al. 1983; Wasserman & Anderson, 1987] Two nodes are stochastically equivalent if they are exchangeable w.r.t. a probability distribution u p (a,u) a p (a,u) = p (a,v) p (u,b) = p (v,b) p (a,v) v Similar to structural equivalence but probabilistic p (u,b) b p (v,b) SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 28

29 Stochastic Equivalence: Algorithms Many algorithms exist here Most recent approaches are generative [Airoldi, et al 2008] Some choice points Single [Kemp, et al 2006] vs. mixed-membership [Koutsourelakis & Eliassi-Rad, 2008] equivalences (a.k.a. positions ) Parametric vs. non-parametric models SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 29

30 Roadmap What are roles Roles and communities Roles and equivalences (from sociology) Roles (from data mining) Summary SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 30

31 RolX: Role extraction Introduced by Henderson, et al. 2011b Automatically extracts the underlying roles in a network No prior knowledge required Assigns a mixed-membership of roles to each node Scales linearly on the number of edges SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 31

32 RolX: Flowchart Node Node Matrix Recursive Feature Extraction Node Feature Matrix Role Extraction Node Role Matrix Role Feature Matrix SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 32

neighbors, etc Role Extraction Node Role

33 RolX: Flowchart Node Node Matrix Recursive Feature Extraction Node Feature Matrix Example: degree, avg weight, # of edges in egonet, mean clustering coefficient of neighbors, etc Role Extraction Node Role Matrix Role Feature Matrix SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 33

34 Recursive Feature Extraction ReFeX [Henderson, et al. 2011a] turns network connectivity into recursive structural features Regional Neighborhood ReFeX Nodes Local Egonet Recursive 1411# 0# 1# 2# 1# 0# 0# 0# 1# 1# 0# 1# 0# 0# 1# 1# 2# 2# 1410# 0# 1# 1# 1# 0# 1# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 338# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 1# 0# 0# 0# 339# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 1415# 0# 1# 1# 2# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 1# 1# 1# 941# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1414# 0# 1# 1# 1# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 942# 0# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1413# 0# 1# 1# 1# 0# 1# 1# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 1412# 0# 0# 0# 0# 0# 0# 0# 1# 2# 0# 1# 1# 0# 0# 1# 2# 0# 940# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 1# 0# 1# 1# 1# 1419# 0# 0# 1# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 1# 945# 0# 1# 4# 3# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 1# 3# 1# 332# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1418# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 2# 0# 1# 0# 1# 946# 0# 1# 1# 0# 0# 1# 0# 1# 0# 0# 0# 1# 4# 0# 1# 1# 2# 333# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1417# 0# 1# 1# 1# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 943# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 330# 1# 3# 2# 0# 1# 2# 2# 0# 2# 2# 2# 0# 3# 1# 0# 2# 5# 1416# 0# 1# 1# 1# 1# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 1# 1# 944# 0# 1# 4# 2# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 3# 1# 331# 0# 3# 2# 1# 0# 1# 0# 0# 2# 0# 2# 0# 2# 0# 1# 2# 5# 949# 0# 0# 0# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 0# 0# 0# 336# 0# 0# 0# 0# 2# 0# 0# 1# 1# 1# 1# 1# 0# 0# 0# 1# 0# 337# 1# 1# 1# 0# 0# 1# 2# 0# 1# 1# 1# 0# 1# 1# 1# 1# 1# 947# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 334# 0# 0# 0# 1# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 948# 0# 0# 0# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 0# 335# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 531# 1# 0# 0# 0# 1# 0# 2# 0# 0# 2# 0# 0# 0# 2# 0# 0# 0# Neighborhood features: What is your connectivity pattern? Recursive Features: To what kinds of nodes are you connected? SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 34

35 Propositionalisation (PROP) [Knobbe, et al. 2001; Neville, et al. 2003; Krogel, et al. 2003] From multi-relational data mining with roots in Inductive Logic Programming (ILP) Summarizes a multi-relational dataset (stored in multiple tables) into a propositional dataset (stored in a single target table) Derived attribute-value features describe properties of individuals Related more to recursive structural features than structural roles SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 35

36 Role Extraction Features 1411# 0# 1# 2# 1# 0# 0# 0# 1# 1# 0# 1# 0# 0# 1# 1# 2# 2# Recursively extract roles Nodes 1410# 0# 1# 1# 1# 0# 1# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 338# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 1# 0# 0# 0# 339# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 1415# 0# 1# 1# 2# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 1# 1# 1# 941# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1414# 0# 1# 1# 1# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 942# 0# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1413# 0# 1# 1# 1# 0# 1# 1# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 1412# 0# 0# 0# 0# 0# 0# 0# 1# 2# 0# 1# 1# 0# 0# 1# 2# 0# 940# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 1# 0# 1# 1# 1# 1419# 0# 0# 1# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 1# 945# 0# 1# 4# 3# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 1# 3# 1# 332# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1418# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 2# 0# 1# 0# 1# 946# 0# 1# 1# 0# 0# 1# 0# 1# 0# 0# 0# 1# 4# 0# 1# 1# 2# 333# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1417# 0# 1# 1# 1# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 943# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 330# 1# 3# 2# 0# 1# 2# 2# 0# 2# 2# 2# 0# 3# 1# 0# 2# 5# 1416# 0# 1# 1# 1# 1# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 1# 1# 944# 0# 1# 4# 2# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 3# 1# 331# 0# 3# 2# 1# 0# 1# 0# 0# 2# 0# 2# 0# 2# 0# 1# 2# 5# 949# 0# 0# 0# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 0# 0# 0# 336# 0# 0# 0# 0# 2# 0# 0# 1# 1# 1# 1# 1# 0# 0# 0# 1# 0# 337# 1# 1# 1# 0# 0# 1# 2# 0# 1# 1# 1# 0# 1# 1# 1# 1# 1# 947# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 334# 0# 0# 0# 1# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 948# 0# 0# 0# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 0# 335# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 531# 1# 0# 0# 0# 1# 0# 2# 0# 0# 2# 0# 0# 0# 2# 0# 0# 0# Automatically factorize roles SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 36

37 Automatically Discovered Roles Network Science Co-authorship Graph [Newman 2006] SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 37

38 Making Sense of Roles Degree Avg. Weight Clustering Coeff. Eccentricity PageRank GateKeeper GateKeeper-Local Pivot Structural Hole Role 1 Role 2 Role 3 Role 4 NodeSense NeighborSense Role 1 Role 2 Role 3 Role 4 Default cliquey bridge periphery isolated SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 38

39 Mixed-Membership over Roles liberal conservative neutral Bright red nodes are locally central nodes Bright blue nodes are peripheral nodes Amazon Political Books Co-purchasing Network [V. Krebs 2000] SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 39

40 Role Query Node Similarity for J. Rinzel (isolate) Node Similarity for M.E.J. Newman (bridge) Node Similarity for F. Robert (cliquey) SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 40

Roles Generalize across Disjoint Networks Classification Accuracy 1 0.8 0.6 0.4 0.

41 Roles Generalize across Disjoint Networks Classification Accuracy RolX Feat Default A1-B A2-B A3-B TrainSet-TestSet A4-B A-B SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 41

42 Roles Generalize across Networks Discovery Stage Tue Network Wed Network V: (node feature) Thu matrix Network G: (node role) matrix E.g., degree, avg wgt, etc Feature Discovery Feature Extraction F: (role feature) matrix Feature L: List of feature names Extraction C: Class labels V 1 L V 2 M: model V 3 C 1 RolX Regression Regression G 1 F G 2 G 3 Learning Inference Inference M SDM'12 Tutorial C 2 42 C 3

43 Roles Generalize across Networks Tue Network Wed Network Thu Network E.g., degree, avg wgt, etc Feature Discovery Feature Extraction Feature Extraction V: (node feature) matrix V 1 L V 2 V 3 G: (node role) matrix F: (role feature) matrix C 1 RolX Regression Regression L: List of feature names C: Class labels G 1 F G 2 G 3 M: model Learning Inference Inference M Discovery Stage C 2 C 3 Application Stage 43

44 Roles: Regular Equivalence vs. RolX Mixed-membership over roles Fully automatic Uses structural features RolX Regular Equivalence Uses structure Generalizable across disjoint networks Scalable (linear on # of edges)? SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 44

45 Roadmap What are roles Roles and communities Roles and equivalences (from sociology) Roles (from data mining) Summary SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 45

46 Roles Summary Structural behavior ( function ) of nodes Complementary to communities Previous work mostly in sociology under equivalences Recent graph mining work produces mixedmembership roles, is fully automatic and scalable Can be used for many tasks: transfer learning, reidentification, node dynamics, etc SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 46

47 Acknowledgement LLNL: Brian Gallagher, Keith Henderson IBM Watson: Hanghang Tong Google: Sugato Basu CMU: Leman Akoglu, Danai Koutra, Lei Li Thanks to: LLNL, NSF, IARPA. SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 47

48 References Deterministic Equivalences S. Boorman, H.C. White: Social Structure from Multiple Networks: II. Role Structures. American Journal of Sociology, 81: , S.P. Borgatti, M.G. Everett: Notions of Positions in Social Network Analysis. In P. V. Marsden (Ed.): Sociological Methodology, 1992:1 35. S.P. Borgatti, M.G. Everett, L. Freeman: UCINET IV, S.P. Borgatti, M.G. Everett, Regular Blockmodels of Multiway, Multimode Matrices. Social Networks, 14:91-120, R. Breiger, S. Boorman, P. Arabie: An Algorithm for Clustering Relational Data with Applications to Social Network Analysis and Comparison with Multidimensional Scaling. Journal of Mathematical Psychology, 12: , R.S. Burt: Positions in Networks. Social Forces, 55:93-122, SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 48

49 References P. DiMaggio: Structural Analysis of Organizational Fields: A Blockmodel Approach. Research in Organizational Behavior, 8:335-70, P. Doreian, V. Batagelj, A. Ferligoj: Generalized Blockmodeling. Cambridge University Press, M.G. Everett, S. P. Borgatti: Regular Equivalence: General Theory. Journal of Mathematical Sociology, 19(1):29-52, K. Faust, A.K. Romney: Does Structure Find Structure? A critique of Burt's Use of Distance as a Measure of Structural Equivalence. Social Networks, 7:77-103, K. Faust, S. Wasserman: Blockmodels: Interpretation and Evaluation. Social Networks, 14: R.A. Hanneman, M. Riddle: Introduction to Social Network Methods. University of California, Riverside, SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 49

50 References F. Lorrain, H.C. White: Structural Equivalence of Individuals in Social Networks. Journal of Mathematical Sociology, 1:49-80, L.D. Sailer: Structural Equivalence: Meaning and Definition, Computation, and Applications. Social Networks, 1:73-90, M.K. Sparrow: A Linear Algorithm for Computing Automorphic Equivalence Classes: The Numerical Signatures Approach. Social Networks, 15: , S. Wasserman, K. Faust: Social Network Analysis: Methods and Applications. Cambridge University Press, H.C. White, S. A. Boorman, R. L. Breiger: Social Structure from Multiple Networks I. Blockmodels of Roles and Positions. American Journal of Sociology, 81: , D.R. White, K. Reitz: Graph and Semi-Group Homomorphism on Networks and Relations. Social Networks, 5: , SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 50

51 References Stochastic blockmodels E.M. Airoldi, D.M. Blei, S.E. Fienberg, E.P. Xing: Mixed Membership Stochastic Blockmodels. Journal of Machine Learning Research, 9: , P.W. Holland, K.B. Laskey, S. Leinhardt: Stochastic Blockmodels: Some First Steps. Social Networks, 5: , C. Kemp, J.B. Tenenbaum, T.L. Griffiths, T. Yamada, N. Ueda: Learning Systems of Concepts with an Infinite Relational Model. AAAI P.S. Koutsourelakis, T. Eliassi-Rad: Finding Mixed-Memberships in Social Networks. AAAI Spring Symposium on Social Information Processing, Stanford, CA, K. Nowicki,T. Snijders: Estimation and Prediction for Stochastic Blockstructures, Journal of the American Statistical Association, 96: , Z. Xu, V. Tresp, K. Yu, H.-P. Kriegel: Infinite Hidden Relational Models. UAI S. Wasserman, C. Anderson: Stochastic a Posteriori Blockmodels: Construction and Assessment, Social Networks, 9:1-36, SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 51

52 References Role Discovery K. Henderson, B. Gallagher, L. Li, L. Akoglu, T. Eliassi-Rad, H. Tong, C. Faloutsos: It's Who Your Know: Graph Mining Using Recursive Structural Features. KDD 2011: K. Henderson, B. Gallagher, T. Eliassi-Rad, H. Tong, S. Basu, L. Akoglu, D. Koutra, L. Li, C. Faloutsos: RolX: Structural Role Extraction & Mining in Large Graphs. Technical Report, Lawrence Livermore National Laboratory, Livermore, CA, Ruoming Jin, Victor E. Lee, Hui Hong: Axiomatic ranking of network role similarity. KDD 2011: SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 52

53 References Community Discovery A. Clauset, M.E.J. Newman, C. Moore: Finding Community Structure in Very Large Networks. Phys. Rev. E., 70:066111, M.E.J. Newman: Finding Community Structure in Networks Using the Eigenvectors of Matrices. Phys. Rev. E., 74:036104, Propositionalisation A.J. Knobbe, M. de Haas, A. Siebes: Propositionalisation and Aggregates. PKDD 2001: M.-A. Krogel, S. Rawles, F. Zelezny, P.A. Flach, N. Lavrac, S. Wrobel: Comparative Evaluation of Approaches to Propositionalization. ILP 2003: J. Neville, D. Jensen, B. Gallagher: Simple Estimators for Relational Bayesian Classifiers. ICDM 2003: SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 53

54 Back to Overview Features Roles Anomalies = rare roles Patterns SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 54

Generalized blockmodeling of sparse networks

Metodološki zvezki, Vol. 10, No. 2, 2013, 99-119 Generalized blockmodeling of sparse networks Aleš Žiberna 1 Abstract The paper starts with an observation that the blockmodeling of relatively sparse binary