SFU, CMPT 741, Fall 2009, Martin Ester 1

Graph Mining and Social Network AnalysisOutline

• 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 2

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

• Network: used as synonym to graphmore application-oriented term

otherwise0

),(if1,

Evvy ji

ji

SFU, CMPT 741, Fall 2009, Martin Ester 3

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 4

Graphs and Networks

Examples

SFU, CMPT 741, Fall 2009, Martin Ester 5

Graphs and Networks

More DefinitionsNeighbors

Degree

Clustering coefficient of node vfraction of pairs of neigbors of v that are connected

Betweenness of node vnumber of shortest paths (between any pair of nodes) in Gthat go through v

Betweenness of edge enumber of shortest paths in G that go through e

}),(|{

:nodeof

EvvVvN

vN

jiji

ii

||)deg(

:nodeof)deg(

iNv

vv

SFU, CMPT 741, Fall 2009, Martin Ester 6

Graphs and Networks

More DefinitionsShortest path distance between nodes v1 and v2

length of shortest path between v1 and v2 also called minimum geodesic distance

Diameter of graph Gmaximum 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 7

Graphs and Networks

More DefinitionsSmall-world network

network with „small“ mean geodesic distance / effective diameter

MicrosoftMessenger

network

SFU, CMPT 741, Fall 2009, Martin Ester 8

Graphs and Networks

More Definitions

Scale-free networksnetworks with a power law degree distribution

typically between 2 and 3

)()( xfcxf

kkP )(

degree k

P(k)

SFU, CMPT 741, Fall 2009, Martin Ester 9

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 10

Graph Pattern Mining

Frequent Pattern Mining

• Given a graph dataset DB,

i.e. a set of labeled graphs G1, . . ., Gn

and a minimum support

• Find the graphs that are contained in at least of the graphs of DB

•Assumption: the more frequent, the more interestinga graph

• G contained in Gi :

G is isomorph to a subgraph of Gi

10,

n

SFU, CMPT 741, Fall 2009, Martin Ester 11

Graph Pattern Mining

Example

SFU, CMPT 741, Fall 2009, Martin Ester 12

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 13

Graph Pattern Mining

Algorithmic Schemes

SFU, CMPT 741, Fall 2009, Martin Ester 14

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 15

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 16

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 17

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 18

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

•Want to find all subgraphs with density at least

•Problem is notoriously hard, even to solve

approximately

)1|(|||

||2)(

VV

EGdensity

SFU, CMPT 741, Fall 2009, Martin Ester 19

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.

• But can still enumerate patterns in a level-wise manner, extending only dense patterns by another node

G’ denser than subgraph G

density = 8/12 density = 14/20

SFU, CMPT 741, Fall 2009, Martin Ester 20

Graph Pattern Mining

edges)1|(| V

Quasi-Cliques

• graph G is -quasi-clique if every node has at least

Gv

SFU, CMPT 741, Fall 2009, Martin Ester 21

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-cliquenone 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 22

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 23

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 24

Graph Pattern Mining

= 0.7 = 3

= 0.0

density = 7/10

density = 8/10

cohesivepatterns

Example

SFU, CMPT 741, Fall 2009, Martin Ester 25

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 26

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 27

Graph Pattern MiningExample

= 0.8 = 2

= 0.5

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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

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Graph Classification

Feature-based Graph Classification

• define set of graph featuresglobal features such as diameter, degree

distributionlocal features such as occurence of certain

subgraphs• choice of relevant subgraphs

based on domain knowledgedomain expert

based on frequencypattern mining algorithm [Huan et al 04]

SFU, CMPT 741, Fall 2009, Martin Ester 30

Graph Classification

)'(),()',( xxxxk

Kernel-based Graph Classification•kernel-based map two graphs x and x‘ into feature space via function compute similarity (inner product) in feature space

kernel k avoids actual mapping to feature space

•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]

)'(),( xx

SFU, CMPT 741, Fall 2009, Martin Ester 31

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 32

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 33

Graph Clustering

Example

friendship network from Zachary’s karate club

hierarchical clustering(dendrogram)

shapes denote the true community

SFU, CMPT 741, Fall 2009, Martin Ester 34

Graph Clustering

j

iji ea

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

• modularity Q

)( 2 i

iii ae

SFU, CMPT 741, Fall 2009, Martin Ester 35

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 36

Graph Clustering

college football network, shapes denote conferences (true communities)

SFU, CMPT 741, Fall 2009, Martin Ester 37

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 38

Graph Clustering

Graph Cuts [Shi & Malik 2000]

• minimum cut tends to cut off very small, isolated components

• normalized cut

where assoc(A, V) = sum of weights of all edges in V that touch A

),(

),(

),(

),(

VBassoc

BAcut

VAassoc

BAcut

SFU, CMPT 741, Fall 2009, Martin Ester 39

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 40

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 41

Graph Clustering

Example

graph(sociomatrix)

block model(permuted and

partitionedsociomatrix)

SFU, CMPT 741, Fall 2009, Martin Ester 42

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 43

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]

CMPT 884, SFU, Martin Ester, 1-09 44

Graph Clustering

Generative Model •assign nodes to clusters

•determine link (edge) probability between clusters

•determine edges between nodes

SFU, CMPT 741, Fall 2009, Martin Ester 45

Graph Clustering

)()|()|( zPzRPRzP

ab

abab

B

mmBzRP

),(

),()|(

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)

function Beta theB(.,.) and

b and a clustersbetween edges missing ofnumber theis m and

b and a clustersbetween edges ofnumber theis m where

ab

ab

SFU, CMPT 741, Fall 2009, Martin Ester 46

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 be recoveredab

SFU, CMPT 741, Fall 2009, Martin Ester 47

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 48

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 49

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 50

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)

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Graph Evolution

Diffusion Models

• Threshold model [Granovetter 78]

- each  node  has  a  threshold  t - node u is activated when 

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

)(uactivevuv tw

SFU, CMPT 741, Fall 2009, Martin Ester 52

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 53

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 54

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 55

Social Network Analysis

Viral Marketing

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

DVD purchases

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Social Network Analysis

Viral Marketing

Probability of joining community increases sharply with the first 10 friends in the community Absolute values of probabilities are very small

LiveJournalcommunitymembership

SFU, CMPT 741, Fall 2009, Martin Ester 57

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 58

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 59

Trust-Based Recommendation

Introduction

• Collaborative filteringgiven a user-item rating matrixpredict missing ratings by aggregating ratingsof users with similar rating profiles

Standard method for recommender systems• Online social networks

• Trust-based recommendationgiven additionally a trust (social) networkaggregate ratings of trusted neighbors

SFU, CMPT 741, Fall 2009, Martin Ester 60

Trust-Based Recommendation

Introduction

• Explore the trust network to find raters.• Aggregate their ratings.

• Advantages:can better deal with cold start users

• Challengethe larger the distance, the noisier the ratingsbut low probability of finding rater at small distances

SFU, CMPT 741, Fall 2009, Martin Ester 61

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 62

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 63

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

•

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Trust-Based Recommendation

TrustWalker

• Prediction = expected value returned by random walk.

SFU, CMPT 741, Fall 2009, Martin Ester 65

Trust-Based Recommendation

TrustWalker

• Special cases of TrustWalkerΦu,i,k = 1

Random walk never starts.Item-based Collaborative Filtering.

Φu,i,k = 0Pure trust-based recommendation.Continues until finding the exact target item.Aggregates the ratings weighted by probability of reaching them.Existing methods approximate this.

• ConfidenceHow confident is the prediction?

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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

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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

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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