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									    Graph Mining and Social Network Analysis
                                          Outline
• Graphs and networks
• Graph pattern mining [Borgwardt & Yan 2008]
• Graph classification [Borgwardt & Yan 2008]
• Graph clustering
• Graph evolution [Leskovec & Faloutsos 2007]
• Social network analysis [Leskovec & Faloutsos 2007]
• Trust-based recommendation


    [Han and Kamber 2006, sections 9.1 and 9.2]
    References to research papers at the end of this chapter

                           SFU, CMPT 741, Fall 2009, Martin Ester   350
                     Graphs and Networks
                            Basic Definitions
• Graph G = (V,E)
   V: set of vertices / nodes
   E  V x V: set of edges

• Adjacency matrix (sociomatrix)
  alternative representation of a graph

                         1 if (vi , v j )  E
                yi , j  
                          0 otherwise

• Network: used as synonym to graph
    more application-oriented term

                         SFU, CMPT 741, Fall 2009, Martin Ester   351
                     Graphs and Networks

                          Basic Definitions
• Labeled graph
   set of lables L
   f: V  L or f: E  L

  |L| typically small

• Attributed graph
   set of attributes with domains D1, . . ., Dd
   f: V  D1x . . . x Dd

  |Di| typically large, can be continuous domain



                        SFU, CMPT 741, Fall 2009, Martin Ester   352
Graphs and Networks

          Examples




 SFU, CMPT 741, Fall 2009, Martin Ester   353
                     Graphs and Networks
                             More Definitions
Neighbors Ni of node vi :
                   Ni  {v j V | (vi , v j )  E}

Degree deg(v) of node v :
                deg(v) | N i |

Clustering coefficient of node v
       fraction of pairs of neigbors of v that are connected
Betweenness of node v
      number of shortest paths (between any pair of nodes) in G
      that go through v
Betweenness of edge e
      number of shortest paths in G that go through e
                        SFU, CMPT 741, Fall 2009, Martin Ester    354
                    Graphs and Networks
                         More Definitions
Shortest path distance between nodes v1 and v2
       length of shortest path between v1 and v2
       also called minimum geodesic distance

Diameter of graph G
      maximum shortest path distance for any pair of nodes in G

Effective diameter of graph G
  distance at which 90% of all connected pairs of nodes can be
   reached

Mean geodesic distance of graph G
   average minimum geodesic distance for any pair of nodes in G


                      SFU, CMPT 741, Fall 2009, Martin Ester      355
                   Graphs and Networks
                        More Definitions
Small-world network
    network with „small“ mean geodesic distance / effective diameter




                                                              Microsoft
                                                              Messenger
                                                               network




                     SFU, CMPT 741, Fall 2009, Martin Ester               356
                       Graphs and Networks
                              More Definitions
  Scale-free networks
       networks with a power law degree distribution f (cx)  f ( x)

              P(k )  k  l
       l typically between 2 and 3

P(k)




                         degree k
                          SFU, CMPT 741, Fall 2009, Martin Ester       357
                Graphs and Networks
                Data Mining Scenarios
One large graph
• mine dense subgraphs or clusters
• analyze evolution

Many small graphs
• mine frequent subgraphs

Two collections of many small graphs
• classify graphs


                  SFU, CMPT 741, Fall 2009, Martin Ester   358
                  Graph Pattern Mining
                 Frequent Pattern Mining

• Given a graph dataset DB,
    i.e. a set of labeled graphs G1, . . ., Gn
  and a minimum support  , 0    1
• Find the graphs that are contained in at least  n of the
  graphs of DB
•Assumption: the more frequent, the more interesting
   a graph
• G contained in Gi :
    G is isomorph to a subgraph of Gi
                     SFU, CMPT 741, Fall 2009, Martin Ester   359
Graph Pattern Mining

          Example




 SFU, CMPT 741, Fall 2009, Martin Ester   360
                 Graph Pattern Mining

               Anti-Monotonicity Property

•If a graph is frequent, all of its subgraphs are frequent.
•Can prune all candidate patterns that have an infrequent
 subgraph, i.e. disregard them from further consideration.
• The higher  , the more effective the pruning




                    SFU, CMPT 741, Fall 2009, Martin Ester    361
Graph Pattern Mining

 Algorithmic Schemes




 SFU, CMPT 741, Fall 2009, Martin Ester   362
                      Graph Pattern Mining
                         Duplicate Elimination
• Given existing patterns G1, . . ., Gm and newly discovered pattern G
     Is G a duplicate?
• Method 1(slow)
  check graph isomorphism of G with each of the Gi
  graph isomorphism test is a very expensive operation
• Method 2 (faster)
  transform each graph Gi into a canonical form and hash it
     into a hash table
  transform G in the same way and check whether there is already
     a graph Gi with the same hash value
  test for graph isomorphism only if such Gi already exists

                          SFU, CMPT 741, Fall 2009, Martin Ester         363
                       Graph Pattern Mining
                       Duplicate Elimination
• Method 3 (fastest)
  define a canonical order of subgraphs and explore them in that order


  e.g., graphs in same equivalence class, if they have the same canonical
       spanning tree




      and define order on the spanning trees


     does not need isomorhism tests

                        SFU, CMPT 741, Fall 2009, Martin Ester         364
                  Graph Pattern Mining
                            Conclusion

• Lots of sophisticated algorithms for mining frequent
   graph patterns: MoFa, gSpan, FFSM, Gaston, . . .
• But: number of frequent patterns is exponential
• This implies three related problems:
  - very high runtimes
  - resulting sets of patterns hard to interpret
  - minimum support threshold hard to set.



                    SFU, CMPT 741, Fall 2009, Martin Ester   365
                 Graph Pattern Mining
                     Research Directions

• Mine only closed or maximal frequent graphs
  i.e. frequent graphs so that no supergraph has the same
     (has at least  ) support
• Summarize graph patterns
   e.g., find the top k most representative graphs
• Constraint-based graph pattern mining
   find only patterns that satisfy certain conditions on their
   size, density, diameter . . .
                    SFU, CMPT 741, Fall 2009, Martin Ester   366
                 Graph Pattern Mining
                   Dense Graph Mining
•Assumption: the denser a graph, the more interesting
• Can add density constraint to frequent graph mining
•In the scenario of one large graph, just want to find
  the dense subgraphs
•Density of graph G                                 2| E |
                          density(G ) 
                                              | V | (| V | 1)
•Want to find all subgraphs with density at least a
•Problem is notoriously hard, even to solve approximately

                   SFU, CMPT 741, Fall 2009, Martin Ester         367
                   Graph Pattern Mining
            Weak Anti-Monotonicity Property
• If a graph of size k is dense, (at least) one of its
        subgraphs of size k-1 is dense.
• Cannot prune all candidate patterns that have a
  subgraph which is not dense.

                                                               G’ denser than
                                                                subgraph G

          density = 8/12         density = 14/20
• But can still enumerate patterns in a level-wise manner,
 extending only dense patterns by another node
                      SFU, CMPT 741, Fall 2009, Martin Ester                    368
                Graph Pattern Mining
                        Quasi-Cliques
• graph G is -quasi-clique if every node v  G
  has at least  (| V | 1) edges




                  SFU, CMPT 741, Fall 2009, Martin Ester   369
                 Graph Pattern Mining
        Mining Quasi-Cliques [Pei, Jiang & Zhang 05]

• for <1, the -quasi-clique property is not anti-monotone,
  not even weakly anti-monotone




         G is 0.8-quasi-clique
         none of the size 5 subgraphs of G is an 0.8-quasi-clique
         since they all have a node with degree 3 < 0.8(5-1) = 3.2

                    SFU, CMPT 741, Fall 2009, Martin Ester     370
                Graph Pattern Mining
                  Mining Quasi-Cliques
• enumerate (all) the subgraphs




•prune based on maximum diameter of -quasi-clique G




                  SFU, CMPT 741, Fall 2009, Martin Ester   371
                 Graph Pattern Mining
    Mining Cohesive Patterns [Moser, Colak and Ester 2009]
•  Cohesive pattern:
  subgraph G‟ satisfying three conditions:
  (1) subspace homogeneity, i.e. attribute values are within a
      range of at most w in at least d dimensions,
  (2) density, i.e. has at least a of all possible edges, and
  (3) connectedness, i.e. each pair of nodes has a connecting
      path in G‟.
• Task
  Find all maximal cohesive patterns.


                   SFU, CMPT 741, Fall 2009, Martin Ester    372
Graph Pattern Mining
          Example

              density = 7/10




 cohesive                                 a = 0.7
 patterns                                  =3
                                          w = 0.0



               density = 8/10




 SFU, CMPT 741, Fall 2009, Martin Ester         373
                 Graph Pattern Mining
                              Algorithm
• Cohesive Pattern Mining problem is NP-hard
  decision version reduceable from Max-Clique problem
• A constraint is anti-monotone:
  if for each network G of size n that satisfies the constraint,
  all induced subnetworks G‟ of G of size n - 1 satisfy the
  constraint
• Can prune all candidate networks that have a subnetwork
  not satisfying the constraint
    cohesive pattern constraints are not anti-monotone


                    SFU, CMPT 741, Fall 2009, Martin Ester    374
                 Graph Pattern Mining
                      Algorithm CoPaM
•  A constraint is loose anti-monotone:
  if for each network G of size n that satisfies the constraint,
  there is at least one induced subnetwork G‟ of G of size n - 1
  satisfying the constraint.
• For a >= 0.5, the cohesive pattern constraints are loose
  anti-monotone
• CoPaM algorithm performs level-wise search of the lattice
  structure in a bottom-up manner
        construct only connected subgraphs
• Prune all candidates that do not satisfy the constraints of
  density and homogeniety

                   SFU, CMPT 741, Fall 2009, Martin Ester   375
Graph Pattern Mining
            Example




                                          a = 0.8
                                           =2
                                          w = 0.5




 SFU, CMPT 741, Fall 2009, Martin Ester         376
                 Graph Classification
                           Introduction
• given two (or more) collections of (labeled) graphs
  one for each of the relevant classes
• e.g., collections of program flow graphs
  to distinguish faulty graphs from correct ones




                   SFU, CMPT 741, Fall 2009, Martin Ester   377
                 Graph Classification
            Feature-based Graph Classification
• define set of graph features
       global features such as diameter, degree distribution
       local features such as occurence of certain subgraphs
• choice of relevant subgraphs
       based on domain knowledge
               domain expert
       based on frequency
               pattern mining algorithm [Huan et al 04]



                   SFU, CMPT 741, Fall 2009, Martin Ester   378
                 Graph Classification
             Kernel-based Graph Classification
•kernel-based
  map two graphs x and x„ into feature space via function 
  compute similarity (inner product) in feature space
                       ( x), ( x' ) 
  kernel k avoids actual mapping to feature space
                k ( x, x' )  ( x), ( x' ) 
•many graph kernels have been proposed
            e.g. [Kashima et al 2003]
•graph kernels should capture relevant graph features
      and be efficient to compute [Borgwardt & Kriegel 2005]

                   SFU, CMPT 741, Fall 2009, Martin Ester      379
                   Graph Clustering
                           Introduction
• group nodes into clusters such that nodes within a cluster
  have similar relationships (edges) while nodes in different
  clusters have dissimilar relationships
•compared to graph classification: unsupervised
•compared to graph pattern mining: global patterns,
       typically every node belongs to exactly one cluster
•main approaches
       - hierarchical graph clustering
       - graph cuts
       - block models

                  SFU, CMPT 741, Fall 2009, Martin Ester   380
                   Graph Clustering
 Divisive Hierarchical Clustering [Girvan and Newman 2002]

• for every edge, compute its betweenness
• remove the edge with the highest betweenness
• recompute the edge betweenness
• repeat until no more edge exists
 or until specified number of clusters produced
• runtime O(m2n) where m = |E| and n = |V|
 produces meaningful communities,
  but does not scale to large networks

                  SFU, CMPT 741, Fall 2009, Martin Ester   381
                      Graph Clustering
                                  Example




friendship network from                          hierarchical clustering
  Zachary’s karate club                              (dendrogram)
                                            shapes denote the true community

                     SFU, CMPT 741, Fall 2009, Martin Ester                382
                   Graph Clustering
          Agglomerative Hierarchical Clustering
                             [Newman 2004]

•divisive hierarchical algorithm always produces a clustering,
 whether there is some natural cluster structure or not
•define the modularity of a partitioning to measure its
 meaningfulness (deviation from randomness)
• eij: percentage of edges between partitions i and j
      ai   eij
            j

• modularity Q     (eii  ai )
                                     2

                     i
                   SFU, CMPT 741, Fall 2009, Martin Ester   383
                    Graph Clustering
          Agglomerative Hierarchical Clustering
• start with singleton clusters
• in each step, perform the merge of two clusters
  that leads to the largest increase of the modularity
• terminate when no more merges improve modularity
  or when specified number of clusters reached
• need to consider only connected pairs of clusters
• runtime O((m+n) n) where m = |E| and n = |V|
 scales much better than divisive algorithm
  clustering quality quite comparable
                   SFU, CMPT 741, Fall 2009, Martin Ester   384
                      Graph Clustering




college football network, shapes denote conferences (true communities)

                     SFU, CMPT 741, Fall 2009, Martin Ester              385
                   Graph Clustering
                            Graph Cuts
• graph cut is a set of edges whose removal partitions the
  set of vertices V into two (disconnected) sets S and T




• cost of a cut is the sum of the weights of the cut edges
• edge weights can be derived from node attributes,
  e.g. similarity of attributes (attribute vectors)
• minimum cut is a cut with minimum cost

                   SFU, CMPT 741, Fall 2009, Martin Ester    386
                        Graph Clustering
                     Graph Cuts [Shi & Malik 2000]
• minimum cut tends to cut off very small, isolated components




• normalized cut             cut ( A, B)    cut ( A, B)
                                          
                            assoc ( A, V ) assoc ( B, V )
    where assoc(A, V) = sum of weights of all edges in V that touch A

                        SFU, CMPT 741, Fall 2009, Martin Ester          387
                 Graph Clustering
                          Graph Cuts
• minimum normalized cut problem is NP-hard
• but approximation can be computed by solving
  generalized eigenvalue problem




                 SFU, CMPT 741, Fall 2009, Martin Ester   388
                    Graph Clustering
              Block Models [Faust &Wasserman 1992]
• actors in a social network are structurally equivalent if they
   have identical relational ties to and from all the actors in a
   network
•partition V into subsets of nodes that have the same
  relationships
        i.e., edges to the same subset of V
• graph represented as sociomatrix
• partitions are called blocks



                   SFU, CMPT 741, Fall 2009, Martin Ester    389
                Graph Clustering
                           Example


    graph
(sociomatrix)




 block model
(permuted and
  partitioned
 sociomatrix)


                SFU, CMPT 741, Fall 2009, Martin Ester   390
                   Graph Clustering
                              Algorithms
• agglomerative hierarchical clustering
• CONCOR algorithm
  repeated calculations of correlations between rows
  (or columns) will eventually result in a correlation matrix
  consisting of only +1and -1
        - calculate correlation matrix C1 from sociomatrix
        - calculate correlation matrix C2 from C1
        - iterate until the entries are either +1 or -1



                   SFU, CMPT 741, Fall 2009, Martin Ester   391
                     Graph Clustering
                    Stochastic Block Models

• requirement of structural equivalence often too strict
• relax to stochastic equivalence:
 two actors are stochastically equivalent if the actors are
 “exchangeable” with respect to the probability distribution
• Infinite Relational Model
       [Kemp et al 2006]




                    SFU, CMPT 741, Fall 2009, Martin Ester    392
                   Graph Clustering
                       Generative Model
•assign nodes to clusters



•determine link (edge) probability between clusters



•determine edges between nodes




                    CMPT 884, SFU, Martin Ester, 1-09   393
                           Graph Clustering
                               Generative Model
• assumption
  edges conditionally independent given cluster assignments
• prior P(z) assigns a probability to all possible partitions
   of the nodes
• find z that maximizes P(z|R)
       P ( z | R )  P ( R | z ) P( z )
                           B(mab  a , mab   )
       P( R | z )  
                      ab        B(a ,  )
     where m ab is the number of edges between clustersa and b
     and mab is the number of missing edges between clustersa and b
     and B(.,.) the Beta function

                           SFU, CMPT 741, Fall 2009, Martin Ester     394
                      Graph Clustering
                                 Inference
• sample from the posterior P(z|R)
       using Markov Chain Monte Carlo
• possible moves:
       - move a node from one cluster to another
       - split a cluster
       - merge two clusters
• at the end, can  ab be recovered




                     SFU, CMPT 741, Fall 2009, Martin Ester   395
                    Graph Evolution
                            Introduction
•so far, have considered only the static structure of networks
•but many real life networks are very dynamic and evolve
 rapidly in the course of time
•two aspects of graph evolution
       - evolution of the structure (edges): generative models
       - evolution of the attributes: diffusion models
•questions, e.g.
       does the graph diameter increase or decrease?
       how does information about a new product spread?
       what nodes should be targeted for viral marketing?

                   SFU, CMPT 741, Fall 2009, Martin Ester   396
                     Graph Evolution
                       Generative Models
• Erdos Renyi model
 - connect each pair of nodes i.i.d. with probability p
  lots of theory, but does not produce power law degree
    distribution
• Preferential attachment model
 - add a new node, create m out-links to existing nodes
 - probability of linking an existing node is proportional to its
   degree
  produces power law in-degree distribution
    but all nodes have the same out-degree

                    SFU, CMPT 741, Fall 2009, Martin Ester   397
                   Graph Evolution
                     Generative Models
• Copy model
 - add a node and choose k, the number of edges to add
 - with probability β select k random vertices and link to
   them
 - with probability 1- β edges are copied from a randomly
   chosen node
 generates power law degree distributions with exponent
    1/(1-β)
    generates communities



                  SFU, CMPT 741, Fall 2009, Martin Ester   398
                       Graph Evolution
                           Diffusion Models
• each edge (u,v) has probability puv / weight wuv




• initially, some nodes are active (e.g., a, d, e, g, i)

                      SFU, CMPT 741, Fall 2009, Martin Ester   399
                    Graph Evolution
                        Diffusion Models
• Threshold model [Granovetter 78]
  - each node has a threshold t
  - node u is activated when  wuv  t
                                   vactive(u )

      where active(u) are the active neighbors of u
  - deterministic activation
• Independent contagion model [Dodds & Watts 2004]
  - when node u becomes active, it activates each of its
    neighbors v with probability puv
  - a node has only one chance to influence its neighbors
  - probabilistic activation
                   SFU, CMPT 741, Fall 2009, Martin Ester   400
              Social Network Analysis
                       Viral Marketing
• Customers becoming less susceptible to mass marketing
• Mass marketing impractical for unprecedented variety of
  products online
•Viral marketing successfully utilizes social networks for
  marketing products and services
• We are more influenced by our friends than strangers
• 68% of consumers consult friends and family before
 purchasing home electronics (Burke 2003)
• E.g., Hotmail gains 18 million users in 12 months,
  spending only $50,000 on traditional advertising

                  SFU, CMPT 741, Fall 2009, Martin Ester   401
                  Social Network Analysis
             Most Influential Nodes [Kempe et al 2003]

• S: initial active node set
• f(S): expected size of final active set
•Most influential set of size k:
  the set S of k nodes producing largest f(S), if activated




                      SFU, CMPT 741, Fall 2009, Martin Ester   402
                Social Network Analysis
                    Most Influential Nodes
• Can use various diffusion models
• Diminishing returns: pv(u,S) ≥ pv(u,T) if S ⊆T
  where pv(u,S) denotes the marginal gain of f(S) when adding u to S
• Independent contagion model has diminishing returns
• Greedy algorithm
   repeatedly select node with maximum marginal gain
•Performance guarantee
 solution of greedy algorithm is within (1‐1/e) ~63%
 of optimal solution
• Reason: f is submodular
  f submodular: if S ⊆T then f(S∪{x}) –f(S) ≥ f(T∪{x}) –f(T)

                     SFU, CMPT 741, Fall 2009, Martin Ester    403
                  Social Network Analysis
                           Viral Marketing




                                                               DVD purchases




Probability of buying increases with the first 10 recommendations
Diminishing returns for further recommendations (saturation)

                      SFU, CMPT 741, Fall 2009, Martin Ester                   404
                  Social Network Analysis
                            Viral Marketing



                                                                LiveJournal
                                                                community
                                                                membership




 Probability of joining community increases sharply with the first 10
  friends in the community
 Absolute values of probabilities are very small
                       SFU, CMPT 741, Fall 2009, Martin Ester                 405
               Social Network Analysis
                    Role of Communities
• Consider connectedness of friends
• E.g., x and y have both three friends in the community
        - x‟s friends are independent
        - y‟s friends are all connected




   •Who is more likely to join the community?
                   SFU, CMPT 741, Fall 2009, Martin Ester   406
                Social Network Analysis
                     Role of Communities
• Competing sociological theories
• Information argument [Granovetter 1973]
      unconnected friends give independent support
• Social capital argument [Coleman 1988]
      safety / trust advantage in having friends
       who know each other
• In LiveJournal, community joining probability increases
   with more connections among friends in community
 Independent contagion model too simplistic for real life data

                    SFU, CMPT 741, Fall 2009, Martin Ester   407
             Trust-Based Recommendation
                            Introduction
• Collaborative filtering
   given a user-item rating matrix
   predict missing ratings by aggregating ratings
   of users with similar rating profiles
 Standard method for recommender systems
• Online social networks


• Trust-based recommendation
    given additionally a trust (social) network
    aggregate ratings of trusted neighbors

                    SFU, CMPT 741, Fall 2009, Martin Ester   408
            Trust-Based Recommendation
                            Introduction
• Explore the trust network
  to find raters.
• Aggregate their ratings.


• Advantages:
   can better deal with
   cold start users
• Challenge
   the larger the distance, the noisier the ratings
   but low probability of finding rater at small distances

                    SFU, CMPT 741, Fall 2009, Martin Ester   409
            Trust-Based Recommendation
                            Introduction
• How far to go in the network?
   tradeoff between precision and recall




• Instead of distant neighbors with same item
  use near neighbor with similar item

                    SFU, CMPT 741, Fall 2009, Martin Ester   410
              Trust-Based Recommendation
                              TrustWalker
• Random walk-based method
• Start from source user u0.
• In step k, at node u:
   If u has rated i, return ru,i
   With probability Φu,i,k , random walk stops
       Randomly select item j rated by u and return ru,j .
   With probability 1- Φu,i,k , continue random walk to a direct
   neighbor of u.

                      SFU, CMPT 741, Fall 2009, Martin Ester   411
             Trust-Based Recommendation
                             TrustWalker
• Φu,i,k
   sim(i,j): similarity of target item i and item j rated by user u.
   k: the step of random walk

•




                     SFU, CMPT 741, Fall 2009, Martin Ester    412
            Trust-Based Recommendation
                           TrustWalker
• Prediction = expected value returned by random walk.




                   SFU, CMPT 741, Fall 2009, Martin Ester   413
            Trust-Based Recommendation
                           TrustWalker
• Special cases of TrustWalker
   Φu,i,k = 1
       Random walk never starts.
       Item-based Collaborative Filtering.
   Φu,i,k = 0
       Pure trust-based recommendation.
       Continues until finding the exact target item.
       Aggregates the ratings weighted by probability of
       reaching them.
       Existing methods approximate this.
• Confidence
   How confident is the prediction?
                   SFU, CMPT 741, Fall 2009, Martin Ester   414
     Graph Mining and Social Network Analysis
                                   References
• R. Albert and A.L. Barabasi: Emergence of scaling in random networks,
  Science, 1999
• Karsten M. Borgwardt, Hans-Peter Kriegel: Shortest-Path Kernels on Graphs,
  ICDM 2005
• Karsten Borgwardt, Xifeng Yan: Graph Mining and Graph Kernels,
  Tutorial KDD 2008
• Peter Sheridan Dodds and Duncan J.Watts: Universal Behavior in a Generalized
  Model of Contagion, Phys. Rev. Letters, 2004
• P. Erdos and A. Renyi: On the evolution of random graphs, Publication of the
  Mathematical Institute of the Hungarian Acadamy of Science, 1960
• K. Faust and S.Wasserman: Blockmodels: Interpretation and evaluation,
  Social Networks,14, 1992
• M. Girvan and M. E. J. Newman: Community structure in social and biological
  networks, Natl. Acad. Sci. USA, 2002

                         SFU, CMPT 741, Fall 2009, Martin Ester            415
     Graph Mining and Social Network Analysis
                            References (contd.)
• Mark Granovetter: Threshold Models of Collective Behavior, American Journal
  of Sociology, Vol. 83, No. 6, 1978
• M. Jamali, M. Ester: TrustWaker: A Random Walk Model for Combining
  Trust-based and Item-based Recommendation, KDD 2009
• H. Kashima,K. Tsuda, and A. Inokuchi: Marginalized kernels between labeled
  graphs, ICML 2003
• Kemp, C., Tenenbaum, J. B., Griffiths, T. L., Yamada, T. & Ueda, N.: Learning
  systems of concepts with an infinite relational model, AAAI 2006
• D. Kempe, J Kleinberg, É Tardos: Maximizing the spread of influence through a
  social network, KDD 2003
• J.Kleinberg, S. R.Kumar, P.Raghavan, S.Rajagopalan and A.Tomkins:
   The web as a graph: Measurements, models and methods, COCOON 1998
• Jure Leskovec and Christos Faloutsos: Mining Large Graphs, Tutorial
  ECML/PKDD 2007

                         SFU, CMPT 741, Fall 2009, Martin Ester            416
     Graph Mining and Social Network Analysis
                             References (contd.)
• F. Moser, R. Colak, A. Rafiey, and M. Ester: Mining cohesive patterns from
  graphs with feature vectors, SDM 2009
• M. E. J. Newman: Fast algorithm for detecting community structure in networks,
  Phys. Rev. E 69, 2004
• Jian Pei, Daxin Jiang, Aidong Zhang: On Mining CrossGraph QuasiCliques,
   KDD 2005
• Jianbo Shi and Jitendra Malik: Normalized Cuts and Image Segmentation, IEEE
  Transactions on Pattern Analysis and Machine Intelligence, Vol. 22, No. 8, 2000




                          SFU, CMPT 741, Fall 2009, Martin Ester             417

								
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