Social network Analysis - Community detection

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INTRODUCTION COMPLEX NETWORK ANALYSIS TASKS Community detection Challenges Conclusion

Social network analysis: Community detection

Rushed Kanawati LIPN, CNRS UMR 7030; USPC

http://lipn.fr/ ∼ kanawati

[email protected]

1 / 104

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P LAN

1

I

NTRODUCTION

2

C

OMPLEX NETWORK ANALYSIS TASKS

3

Community detection

Local community detection Global communities detection Community evaluation approaches

4

Challenges

5

Conclusion

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C OMPLEX NETWORK

Definition

Graphs modeling (direct/indirect) interactions among actors.

Basic topological features

I Low Density I Small Diameter I Heterogeneous degree

distribution.

I High Clustering coefficient

I Community structure

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E XAMPLE I : SCIENTIFIC COLLABORATION NETWORK

Density : 10⁻⁴ Diameter : 24 Clustering coeff.: 0.67

Co-authorship network - DBLP 1980-1984 (for authors active for more than 10 years)

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E XAMPLE II: SPATIAL NETWORK

Public sites accessibility in the Bourget district

projet FUI UrbanD 2009-2012

Densiy : 0.052 Diameter : 7 clustering Coef.: 0.87

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E XAMPLE III: P RODUCT PURCHASE NETWORK

projet ANR CADI 2008-2010

Density : 0.02 Diameter : 8

Clustering coef. : 0.84

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E XAMPLE IV: C O - RATING NETWORK

Movie Lens

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E XAMPLE V: PRODUCT NETWORK

MovieLens

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A NOTHER EXAMPLE

I Graph of the Internet I Nodes = service

providers

I Edges : Connections

[CHK

⁺

07]

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A NOTHER EXAMPLE

I Mubarak’s resignation announcement.

I Nodes = Twitter user

I Edges : retweet on

#jan25 hashtag

http://gephi.org/2011/the-egyptian-revolution-on-twitter/

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L AST EXAMPLE !

I Ingredients network extracted fom cooking receipts

I Nodes = Ingredients

I Edges : usage in the same receipt.

[Lada Adamic Course on Network Analysis: Coursera]

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N OTATIONS

A graph G =< V, E ⊆ V × V >

I V is the set of nodes (a.k.a. vertices, actors, sites)

I E is the set of edges (a.k.a. ties, links, bonds)

Notations

I A

_G

Adjacency Matrix d : a

_ij

6= 0 si les nœuds (v

_i

, v

_j

) ∈ E, 0 otherwise.

I n = |V|

I m = |E|. We have often m ∼ n

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C OMPLEX NETWORK ANALYSIS : T ASKS

Node oriented

I Nodes ranking in function of their importance, influence, . . . I Applying centrality functions

I Applications: Ranking (ex. researchers ! ), Viral Marekting, Cyper-attacks (prevention); . . .

Network oriented

I Network formation & evolution models I Link prediction, Diffusion models

I Applications: Explanation, Recommendation, Anomalies detection, infirmation diffusion, . . .

Community oriented

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C ENTRALITY : E XAMPLES

Degree centrality

I C

_d

(v) =

_max^k(Γ(v)k

u∈VkΓ(u)k

I Complexity : O(m)

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C ENTRALITY : SOME EXAMPLES

Closeness centrality

I C

c

(v) =

^P ¹

u∈Vsp(v,u)

I sp(u, v) : shortest path length between u, v.

I Complexity =

O(n × m + n

²

logn)

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C ENTRALITY : SOME EXAMPLES

Betweenness centrality

I C

_i

(v) = P

s,t∈V,stv σs,t(v)

σs,t

I σ

_s,t

(v) : number of shortest paths linking s to t that include v

I σ

_s,t

: total number of shortest paths linking s to t

I Complexity = O(n

³

)

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C OMMUNITY ?

Definitions

I A dense subgraph loosely coupled to other modules in the network

I A community is a set of nodes seen as one by nodes outside the community

I A subgraph where almost all nodes are linked to other nodes in the community.

I . . .

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C OMMUNITY DETECTION PROBLEM

I Local community identification (ego-centred).

I Network partition computing

I Overlapping community detection

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L OCAL COMMUNITY

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L OCAL COMMUNITY

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L OCAL COMMUNITY

1 C ← {φ} , B ← { n

₀

} S ← Γ(n

₀

)

2 Q ← 0 /* a community quality function */

3 While Q can be enhanced Do 1 n ← argmax

_n∈S

Q

2 S ← S − { n }

3 D ← D + { n }

4 update B, S, C

4 Return D

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Q UALITY FUNCTIONS : E XEMPLES I

Local modularity R [Cla05]

R =

_B ^Bⁱⁿ

in+Bout

Local modularity M [LWP08]

M =

_D^Dⁱⁿ

out

Local modularity L [CZG09]

L =

^L_Lⁱⁿ

ex

where : L

_in

=

P

i∈D

kΓ(i)∩Dk

kDk

, L

_ex

=

P

i∈B

kΓ(i)∩Sk kBk

And many many others . . . [YL12]

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L OCAL MODULARITY LIMITATIONS : A N EXEMPLE

Blank nodes enhance B

_in

and D

_in

without affecting B

_out

and D

_out

Bank nodes will be added if M or R modularities are used

Low precision computed communities

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M ULTI - OBJECTIVE LOCAL COMMUNITY

IDENTIFICATION [?]

Three main approaches

Combine then Rank

Ensemble ranking

Ensemble clustering

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C OMBINE THEN R ANK

Principle

Let Q

_i

(s) be the local modularity value induced by node s ∈ S

Q ]

_i

(s) =



 

 

Q_i(s)−min

u∈SQ_i(u) maxu∈SQi(u)−min

u∈SQi(u)

if max

u∈S

Q

_i

(u) 6= min

u∈S

Q

_i

(u)

1 otherwise

Q

com

(s) =

¹_k

k

P

i=1

Q ]

_i

(s)

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E NSEMBLE RANKING APPROACHES

Principle

I Rank S in function of each local modularity Q

_i

I Select the winner after applying ensemble ranking approach I What stopping criteria to apply ?

Stopping criteria

I Strict policy : All modularities should be enhanced

I Majority policy : Majority of local modularities are enhanced

I Least gain policy : At least one local modularity is enhanced.

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E NSEMBLE RANKING

Problem

I Let S be a set of elements to rank by n rankers I Let σ

_i

be the rank provided by ranker i

I Goal: Compute a consensus rank of S.

D´ej`a Vu: Social choice algorithms, but . . .

I Small number of voters and big number of candidates I Algorithmic efficiency is required

I Output could be a complete rank

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E NSEMBLE RANKING

Problem

I Let S be a set of elements to rank by n rankers I Let σ

_i

be the rank provided by ranker i

I Goal: Compute a consensus rank of S.

D´ej`a Vu: Social choice algorithms, but . . .

I Small number of voters and big number of candidates I Algorithmic efficiency is required

I Output could be a complete rank

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E NSEMBLE RANKING : APPROACHES

Jean-Charles de Borda [1733-1799]

Borda

I Borda’s score of i ∈ σ

_k

:

B

σ_k

(i) = {count(j)|σ

_k

(j) < σ

_k

(i) ; j ∈ σ

_k

} . I Rank elements in function of

B(i) = P

k

t=1

w

t

× B

σt

(i).

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E NSEMBLE RANKING : APPROACHES

Marquis de Condorcet [1743-1794]

Condorcet

I The winner is the candidate who defeats every other candidate in pairwise majority-rule election

I The winner may not exists

Condorcet 6= Borda

I Votes : 6 × A B C, 4 × B C A I Borda winner : B

I Condorcet winner : A

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E NSEMBLE RANKING : APPROACHES

Marquis de Condorcet [1743-1794]

Condorcet

I The winner is the candidate who defeats every other candidate in pairwise majority-rule election

I The winner may not exists

Condorcet 6= Borda

I Votes : 6 × A B C, 4 × B C A I Borda winner : B

I Condorcet winner : A

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E NSEMBLE RANKING : APPROACHES

Extended Condorcet criterion

If for every a ∈ A and b ∈ B, majority prefers a to b the all elements in A should be ranked before any element in B.

Kenneth Arrow, 1921-

Arrow’s Theorem

No vote rule can have the following desired proprieties :

I Every result must be achievable somehow.

I Monotonicity.

I Independence of irrelevant attributes.

I Non-dictatorship.

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E NSEMBLE RANKING : APPROACHES

John Kemeny 1926-1992

Optimal Kemeny rank aggregation

I Let d() be distance over rankings σ

_i

(ex.

Kendall τ , Spearman’s footrule) I Find π that minimise P

i

d(π, σ

_i

) I NP-Hard problem

I Approximation : Local Kemeny : two adjacent candidats are in the good order.

I Local Kemeny : Apply Bubble sort using the

majority preference partial order relationship

I Approximate Kemeny : Apply QuickSort

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E NSEMBLE CLUSTERING APPROACHES

Principle

I Let C

^Q_v_qⁱ

be the the local community of v

_q

applying Q

_i

. I We have a natural partition : P

_Q_i

= {C

^Q_v_qⁱ

, C

^Q_v_qⁱ

}

I Apply an ensemble clustering approach.

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E NSEMBLE CLUSTERING : PRINCIPLE

from A. Topchy et. al. Clustering Ensembles: Models of Consensus and Weak Partitions. PAMI, 2005

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E NSEMBLE CLUSTERING : APPROACHES

CSPA: Cluster-based Similarity Partitioning Algorithm

I Let K be the number of basic models, C

_i

(x) be the cluster in model i to which x belongs.

I Define a similarity graph on objects : sim(v, u) =

K

P

i=1

δ(C_i(v),C_i(u)) K

I Cluster the obtained graph :

Isolate connected components after prunning edges Apply community detection approach

I Complexity : O(n

²

kr) : n # objects, k # of clusters, r# of clustering

solutions

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CSPA : E XEMPLE

from Seifi, M. Cœurs stables de communautés dans les graphes de terrain. Thèse de l’université Paris 6, 2012

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E NSEMBLE CLUSTERING : APPROACHES

HGPA: HyperGraph-Partitioning Algorithm

I Construct a hypergraph where nodes are objects and hyperedges are clusters.

I Partition the hypergraph by minimizing the number of cut hyperedges

I Each component forms a meta cluster

I Complexity : O(nkr)

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E NSEMBLE CLUSTERING : APPROACHES

MCLA: Meta-Clustering Algorithm

I Each cluster from a base model is an item

I Similarity is defined as the percentage of shared common objects I Conduct meta-clustering on these clusters

I Assign an object to its most associated meta-cluster

I Complexity : O(nk

²

r

²

)

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E XPERIMENTS

Protocol ([Bag08])

1 Apply the different algorithms on nodes in networks for which a ground-truth community partition is known.

2 For each node compute the distance between the real-partition and the computed one (Ex. NMI [Mei03])

3 Compute average and standard deviation for the network.

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D ATASETS

Zachary’s Karate Club

US Political books network

Football network

Dolphins social network

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R ESULTS : EVALUATING STOPPING CRITERIA (NMI)

Zachary’s Karate Club Football network

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R ESULTS : C OMPARATIVE RESULTS (NMI)

Zachary’s Karate Club

US Political books network

Football network

Dolphins social network

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G LOBAL COMMUNITIES DETECTION

Problem

I Divid the set of nodes in a number of (overlapping) subsets such that induced subgraphs are dense and loosley coupled.

I S. Fortunato. Community detection in graphs. Physics Reports, 2010, 486, 75-174

I L. Tang, H. Liu. Community Detection and Mining in Social Media, Morgan Claypool Publishers, 2010

I R. Kanawati, D´etection de communaut´es dans les grands graphes

d’interactions multiplexes : ´etat de l’art, (RNTI 2014), hal-00881668

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G LOBAL COMMUNITIES DETECTION

Problem

I Divid the set of nodes in a number of (overlapping) subsets such that induced subgraphs are dense and loosley coupled.

I S. Fortunato. Community detection in graphs. Physics Reports, 2010, 486, 75-174

I L. Tang, H. Liu. Community Detection and Mining in Social Media, Morgan Claypool Publishers, 2010

I R. Kanawati, D´etection de communaut´es dans les grands graphes

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C OMMUNITIES DETECTION : M ETHODS

Classification

I Group-based

Nodes are grouped in function of shared topological features (ex. clique)

I Network-based

Clustering, Graph-cut, block-models, modularity optimization I Propagation-based

Unstable approaches, good when used with ensemble clustering I Seed-centred

from local community identification to global community detection

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G ROUP - BASED APPROCHES

Principle

Search for special (dense) subgraphs:

I k-clique I n-clique

I γ-dense clique

I K-core

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G ROUP - BASED APPROCHES

from Symeon Papadopoulos, Community Detection in

Social Media, CERTH-ITI, 22 June 2011

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E XAMPLE : C LIQUE PERCOLATION

Suits fairly dense graph.

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N ETWORK - BASED APPROCHES

Clustering approaches

I Apply classical clustering approaches using graph-based diastase function

I Different types of Graph-based distances: neighborhood-based, path-based (Random-walk)

I Usually requires the number of clusters to discover

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M ODULARITY OPTIMIZATION APPROACHES Modularity: a partition quality criteria

Q(P) = 1 2m

X

c∈P

X

i,j∈c

(A

_ij

− d

_i

d

_j

2m ) (1)

Figure: Example : Q =

(15+6)−(11.25+2.56)

25

= 0.275

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M ODULARITY OPTIMIZATION APPROACHES

I Applying classical optimization algorithms (ex. Genetic algorithms [Piz12]).

I Applying hierarchical clustering and select the level with Q

_max

(ex. Walktrap [PL06])

I Divisive approach : Girvan-Newman algorithm [GN02]

I Greedy optimization : Louvain algorithm [BGL08]

I . . .

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E XAMPLE : G IRVAN -N EWMAN A LGORITHM

1 Compute betweenness centrality for each edge.

2 Remove edge with highest score.

3 Re-compute all scores.

4 Repeat 2nd step.

Complexity : O(n

³

)

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E XAMPLE : L OUVAIN APPROACH

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M ODULARITY OPTIMIZATION LIMITATIONS

Hypothesis

The best partition of a graph is the one that maximize the modularity.

If a network has a community structure, then it is popsicle to find a precise partition with maximal modularity

If a network has a community structure, then partitions inducing high modularity values are structurally similar.

All three hypothesis do not hold [GdMC10, LF11].

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M ODULARITY RESOLUTION LIMITE

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M ODULARITY DISTRIBUTION

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P ROPAGATION - BASED APPROACHES

Algorithm 1 Label propagation

Require: G =< V, E > a connected graph,

1:

Initialize each node with unique label l

v 2:

while Labels are not stable do

3:

for v ∈ V do

l

v

=

_l

|Γ

^l

(v)| /* random tie-breaking */

4:

end for

5:

end while

6:

return communities from labels

Γ

^l

(v) : set of neighbors having label l

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L ABEL PROPAGATION

Advantages

I Complexity : O(m) I Highly parallel

Disadvantages

I No convergence guarantee, oscillation phenomena

I Low robustness

Different runs yields very different community

structure due to randomness

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L ABEL PROPAGATION : C ONVERGENCE ENHANCEMENT

I Asynchronous label update, [RAK07]

I Semi-synchronous label update (graph coloring + propagation by color) [CG12].

I Making stability even worse !

I Harden parallel implementations.

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R OBUST L ABEL PROPAGATION

I Label hop attenuation [LHLC09]

I Balanced label propagation [SB11].

I Multiplex approach !

adding new neighborhood similarity based relationships between adjacent nodes → multiplex network

I Ensemble clustering → Communities core [OGS10, SG12, LF12].

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P ARALLEL LP

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L ABEL PROPAGATION PREPROCESSING (EPP)

1 Apply an ensemble clustering on results of basic Parallel LP 2 Coarse the graph according to obtained communities

3 Apply a high quality community detection algorithm on coarsed graph.

4 Expand obtained results to the initial graph.

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E VALUATIONS

benchmark networks Pareto front

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S EED - CENTRIC ALGORITHMS

(KANAWATI, SCSM’2014)

Algorithm 2 General seed-centric community detection algorithm Require: G =< V, E > a connected graph,

1:

C ← ∅

2:

S ← compute seeds(G)

3:

for s ∈ S do

4:

C

_s

← compute local com(s,G)

5:

C ← C + C

_s

6:

end for

7:

return compute community( C )

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S EED - CENTRIC APPROACHES

Table: Characteristics of major seed-centric algorithms

Algorithm Seed Nature Seed Number Seed selection Local Com. Com. computation Leaders-Followers [SZ10] Single Computed informed Agglomerative -

Top-Leaders [KCZ10] Single Input Random Expansion -

PapadopoulosKVS12 Subgraph Computed Informed Expansion -

WhangGD13 Single Computed informed Expansion -

BollobasR09 Subgraph Computed informed expansion -

Licod [Kan11] Set Computed Informed Agglomerative -

Yasca [?] Single Computed Informed Expansion Ensemble clustering

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E XEMPLE : L ICOD

(KANAWATI, SOCILACOMP’2011)

Licod : the idea !

1 Compute a set of seeds that are likely to be leaders in their communities

Heuristic : nodes having higher centralities than their neighbors 2 Each node in the graph ranks seeds in function of its own

preference

3 Each node modify its preference vector in function of neighbor’s preferences

4 iterate max times or till convergence.

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L ICOD : SOME RESULTS

Comparative results on benchmark networks : NMI

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T HE YASCA ALGORITHM

(KANAWATI, COCOON’2014)

The algorithm

Yet Another Seed-centric Community detection Algorithm

1 Compute a set of diverse seed nodes

2 For each node compute a bi-partition of the whole graphe based on local community identification.

3 Apply ensemble clustering to obtain a graph partition.

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Y ASCA : S OME RESULTS

Comparative Results on benchmark networks : NMI

Select 15% of high central nodes and 15% of low central nodes

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E VALUATION METHODS

I Topological criteria : is it useful for applications ? I Ground-truth comparaison : hard to find on large-scale

I Task-oriented approaches : Clustering , Recommendation, link

prediction, . . .

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T OPOLOGICAL EVALUATION CRITERIA

Quality of C = {S

₁

, . . . , S

_i

} :

Q(C) = P

i

f(S

_i

)

|C| (2)

f () is a single- community quality metric 4 types of quality functions

Internal connectivity Externeal connectivity Hybrid functions

Model-based functions: the modularity

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I NTERNAL CONNECTIVITY FUNCTIONS

Internal density : f (S) =

_n ²^×^m^S

S×(nS−1)

Average Degree : f (S) =

^2×m_n ^S

S

FOMD: Fraction over Median Degree : f (s) =

^|{^u:u^∈^S,|(u,v),v_n ^∈^S^|>d^m^}|

S

,

d

_m

est le m´edian de degr´es des nœuds dans V

TPR: Triangle Participation Ratio :

^|{^u^∈^S^}:∃^v,w^∈S:(u,v),(w,v),(u,w)∈E| n_S

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E XTERNAL CONNECTIVITY FUNCTIONS

Expansion : f (S) =

^C_n^S

S

Cut : f (S) =

_n ^C^s

S×(N−nS)

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H YBRID FUNCTIONS

I Conductance : f(S) =

_2m^C^S

S+CS

I MAX-ODF : Out Degree Fraction : f (S) = max

_u∈S|{(u,v)∈E,v6∈S}|

d(u)

I AVG-ODF : f(S) =

_n¹

S

× P

u∈S

|{(u,v)∈E,v6∈S}|

d(u)

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G ROUND - TRUTH BASED EVALUATION

I Principle : Computing a similarity between obtained partition and a ground-truth one.

I Ground-truth : Expert defined.

Model-generated I Two types of metrics :

Agreement-based metrics: purity, rand, ARI

Information theory metrics: MI, NMI, . . . , etc.

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P URITY

I P = { p

₁

, p

₂

, . . . , p

_k

} : a computed partition I R = { r

₁

, r

2

, . . . , r

n

} : ground-truth partition I purity(P , R) =

_|¹

V|

P

_k

j=1

max

_i

(| p

_k

∩ r

_i

|)

(78)

R AND INDEX

I P = {p

₁

, p

₂

, . . . , p

_k

} : a computed partition I R = { r

₁

, r

₂

, . . . , r

n

} : ground-truth partition

I a : # pairs of nodes in a same community according to P and R I b : # pairs of nodes in a same community according to P and in

different communities in R

I c : # pairs of nodes in a different communities according to P and in same community in R

I d : # pairs of nodes in a different communities in P and R I

I rand(P , R) =

_a+b+c+d^a+d

I ARI =

^C²ⁿ(a+d)−[(a+b)(a+c)+(c+d)(b+d)]

(C²_n)²−[(a+b)(a+c)+(c+d)(b+d)]

I E(ARI)=0

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M UTUAL INFORMATION

I(X, Y) = H(X) + H(Y) − H(X, Y) H(X) = P

_n_x

i=1

p(x

_i

) × log(

_p(x¹

i)

)

Normalisation : NMI(U, V) = √

^I(U,V)

H(U)H(V)

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C HALLENGES

I Real-world networks are Dynamic I Real-world networks are Heterogeneous I Nodes have usually semantic attributes I Scale problem

I Towards multiplex network mining

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M ULTIPLEX N ETWORK

Definition

A set of nodes related by different types of relations

Source: muxviz

Motivation

I Real networks are dynamic.

I Real networks are heterogeneous.

I Nodes are usually qualified by a set

of attributes.

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M ULTIPLEX N ETWORK : N OTATIONS G =< V, E

₁

, . . . , E

α

: E

_k

⊆ V × V ∀k ∈ {1, . . . , α} >

I V: set of nodes (a.k.a. vertices, actors, sites)

I E

_k

: set of edges of type k (a.k.a. ties, links, bonds)

Notations

I A^[k]Adjacency Matrix of slicek:a^[k]_ij 6=0 si les nœuds(vi,vj)∈E_k, 0 otherwise.

I m^[k]=|Ek|. We have oftenm∼n

I Neighbor’s ofvin slicek:Γ(v)^[k]={x∈V: (x,v)∈E_k}. I Node degree in slicek:d^k_v=kΓ(v)^[k]k

I Total degree of nodei:d^tot=Pα

d^[s]

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C OMMUNITY ?

Some definitions

I A dense subgraph loosely coupled to other modules in the network I A community is a set of nodes seen as one by nodes outside the

community

I A subgraph where almost all nodes are linked to other nodes in the community.

What is a dense subgraph in a multiplex network ?

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C OMMUNITY DETECTION IN MULTIPLEX NETWORKS

Approaches

1 Transformation into a monoplex community detection problem

I Layer aggregation approaches.

I Ensemble clustering approaches

I Hypergraph transformation based approaches

2 Generalization of monoplex oriented algorithms to multiplex networks.

I Generalized-modularity optimization

I Seed-centric approaches

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L AYER AGGREGATION

Aggregation functions

A_ij=

(1 ∃1≤l≤α:A^[l]_ij 6=0 0 otherwise

A_ij=k {d:A^[d]_ij 6=0} k

A_ij= 1 α

α

X

k=1

w_kA^[k]_ij A_ij=sim(v_i,v_j)

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L AYER AGGREGATION

Aggregation functions

A_ij=

(1 ∃1≤l≤α:A^[l]_ij 6=0

0 otherwise A_ij= 1

α

X

k=1

w_kA^[k]_ij

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E NSEMBLE CLUSTERING APPROACHES

Ensemble Clustering Strehl2003

I CSPA: Cluster-based Similarity Partitioning Algorithm I HGPA: HyperGraph-Partitioning Algorithm

I MCLA: Meta-Clustering Algorithm

I . . .

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E NSEMBLE CLUSTERING APPROACHES

Ensemble Clustering Strehl2003

I CSPA: Cluster-based Similarity Partitioning Algorithm I HGPA: HyperGraph-Partitioning Algorithm

I MCLA: Meta-Clustering Algorithm

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K- UNIFORM HYPERGRAPH TRANSFORMATION KIVELA 2013 MULTILAYER

Principle

I A k-uniform hypergraph is a hypergraph in which the cardinality of each hyperedge is exactly k

I Mapping a multiplex to a 3-uniform hypergraph H = (V, E) such that :

V = V ∪ { 1, . . . , α}

(u, v, i) ∈ E if ∃ l : A

^[l]_uv

6= 0, u, v ∈ V, i ∈ { 1, . . . , α}

I Apply community detection approaches in Hypergraphs (Ex.

tensor factorization approaches)

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M ULTIPLEX M ODULARITY

Generalized modularity mucha2010community

I

Q

_multiplex

(P) = 1 2µ

X

c∈P

X

i,j∈c k,l:1→α







 A

^[s]_ij

− λ

_k

d

^[k]_i

d

^[k]_j

2m

^[k]



 δ

_kl

+ δ

_ij

C

^kl_ij





I µ = P

j∈V k,l:1→α

m

^[k]

+ C

^l_jk

I C

^kl_ij

Inter slice coupling = 0 ∀i 6= j

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M ODULARITY OPTIMIZATION LIMITATIONS

Hypothesis

I The best partition of a graph is the one that maximize the modularity.

I If a network has a community structure, then it is possible to find a precise partition with maximal modularity.

I If a network has a community structure, then partitions having high modularity values are structurally similar.

All three hypothesis do not hold Good10, LAN11a.

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M UX L ICOD

Multiplex degree centrality [?]

d

^multiplex_i

= −

α

X

k=1

d

^[k]_i

d

^[tot]_i

log d

^[k]_i

d

^[tot]_i

!

Multiplex shortest path

SP(u, v)

^multiplex

=

α

P

k=1

SP(u, v)

^[k]

α Multiplex neighborhood

Γ(v)

^tot

∩ Γ(x)

^tot

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E XPERIMENTS

Datasets

1 European airports network

2 Bibliographical network (DBLP) : Three layer network :

co-authorship, co-venue, co-citing.

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E VALUATION CRITERIA

1 Multiplex modularity 2 Redundancy [?]

ρ(c) = X

(u,v)∈P¯¯c

k {k : ∃A

^[k]_uv

6= 0} k α× k P

_c

k

¯ ¯

P the set of couple (u, v) which are directly connected in at least

two layers

(95)

R ESULTS : L AYER AGGREGATION APPROACHES

Figure: European airports multiplex network

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R ESULTS : P ARTITION AGGREGATION APPROACHES

Figure: European airports multiplex network

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R ESULTS : M UX L ICOD

Figure: European airports multiplex network

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R ESULTS : M UX L ICOD

Figure: DBLP 3-layer multiplex

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C ONCLUSIONS I

I Networks are everywhere (ou presque)

I Network analysis can be applied to different kind of tasks I Community detection can help in different analysis and

application tasks

I Challenges : Scale-problems, multiplex network mining

Problems : Evaluation and interoperation of computed

communities.

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B IBLIOGRAPHY I

J. P. Bagrow,Evaluating local community methods in networks, J. Stat. Mech.2008(2008), no. 5, P05001.

Vincent D Blondel, Jean-loup Guillaume, and Etienne Lefebvre,Fast unfolding of communities in large networks, Journal of Statistical Mechanics: Theory and Experiment2008(2008), P10008.

Gennaro Cordasco and Luisa Gargano,Label propagation algorithm: a semi-synchronous approach, IJSNM1(2012), no. 1, 3–26.

S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, and E. Shir,A model of internet topology using k-shell decomposition, PNAS104 (2007), no. 27, 11150–11154.

Aaron Clauset,Finding local community structure in networks, Physical Review E (2005).

Jiyang Chen, Osmar R. Za¨ıane, and Randy Goebel,Local community identification in social networks, ASONAM, 2009, pp. 237–242.

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B IBLIOGRAPHY II

B. H. Good, Y.-A. de Montjoye, and A. Clauset.,The performance of modularity maximization in practical contexts., Physical ReviewE(2010), no. 81, 046106.

M. Girvan and M. E. J. Newman,Community structure in social and biological networks, PNAS99(2002), no. 12, 7821–7826.

Rushed Kanawati,Licod: Leaders identification for community detection in complex networks, SocialCom/PASSAT, 2011, pp. 577–582.

Reihaneh Rabbany Khorasgani, Jiyang Chen, and Osmar R Zaiane,Top leaders community detection approach in information networks, 4th SNA-KDD Workshop on Social Network Mining and Analysis (Washington D.C.), 2010.

Andrea Lancichinetti and Santo Fortunato,Limits of modularity maximization in community detection, CoRRabs/1107.1 (2011).

,Consensus clustering in complex networks, Sci. Rep.2(2012).

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B IBLIOGRAPHY III

Ian X.Y. Leung, Pan Hui, Pietro Lio, and Jon Crowcroft,Towards real-time community detection in large networks, Phys. Phy. E.

79(2009), no. 6, 066107.

Feng Luo, James Zijun Wang, and Eric Promislow,Exploring local community structures in large networks, Web Intelligence and Agent Systems6(2008), no. 4, 387–400.

Marina Meila,Comparing clusterings by the variation of information, COLT (Bernhard Sch ¨olkopf and Manfred K. Warmuth, eds.), Lecture Notes in Computer Science, vol. 2777, Springer, 2003, pp. 173–187.

Michael Ovelg ¨onne and Andreas Geyer-Schulz,Cluster cores and modularity maximization, ICDM Workshops, 2010, pp. 1204–1213.

Clara Pizzuti,A multiobjective genetic algorithm to find communities in complex networks, IEEE Trans. Evolutionary Computation16(2012), no. 3, 418–430.

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B IBLIOGRAPHY IV

Pascal Pons and Matthieu Latapy,Computing communities in large networks using random walks, J. Graph Algorithms Appl.

10(2006), no. 2, 191–218.

Usha N. Raghavan, Roka Albert, and Soundar Kumara,Near linear time algorithm to detect community structures in large-scale networks, Physical Review E76(2007), 1–12.

Lovro Subelj and Marko Bajec,Robust network community detection using balanced propagation, European Physics Journal B 81(2011), no. 3, 353–362.

Massoud Seifi and Jean-Loup Guillaume,Community cores in evolving networks, WWW (Companion Volume) (Alain Mille, Fabien L. Gandon, Jacques Misselis, Michael Rabinovich, and Steffen Staab, eds.), ACM, 2012, pp. 1173–1180.

D Shah and T Zaman,Community detection in networks: The leader-follower algorithm, Workshop on Networks Across Disciplines in Theory and Applications, NIPS, 2010.

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B IBLIOGRAPHY V

Zied Yakoubi and Rushed Kanawati,Interactions in complex systems, ch. Community detection in complex interaction networks: Seed-based approaches, Cambridge Scholar Publishing, 2014.

Jaewon Yang and Jure Leskovec,Defining and evaluating network communities based on ground-truth, ICDM (Mohammed Javeed Zaki, Arno Siebes, Jeffrey Xu Yu, Bart Goethals, Geoffrey I. Webb, and Xindong Wu, eds.), IEEE Computer Society, 2012, pp. 745–754.