algorithmic methods for complex network analysis: …i11 · algorithmic methods for complex network...

KARLSRUHE INSTITUTE OF TECHNOLOGY – INSTITUTE OF THEORETICAL INFORMATICS

Algorithmic Methods for ComplexNetwork Analysis: Graph ClusteringSummer School on Algorithm Engineering

Dorothea Wagner | September 19, 2014

KIT – University of the State of Baden-Wuerttemberg andNational Laboratory of the Helmholtz Association

www.kit.edu

http://www.kit.edu

Scenario of Network AnalysisGiven a network . . .

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explore the instancederive its structureidentify its properties

How can we learn about the instance?

Dorothea Wagner –Algorithmic Methods for Complex Network Analysis: Graph Clustering September 19, 2014 2/49

An Archetypal Example“Zachary’s Karate Club”, a real, social network

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2 years of observation34 vertices = members78 edges = social ties

club split up after disputemanager vs. trainersarchon of toy examples

Caused by an “unequal flow of sentiments and information across the ties”a “factional division led to a formal separation of the club”.[Wayne Zachary: An Information Flow Model for Conflict and Fission in Small Groups, ’77]


A Glimpse of Network Analysis

graph clustering / detecting communities

Group 1

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box = cluster


Scaling of Real-World Instances

”‘Zachary’s Karate Club”’)(vertices/edges = 34/78)



”‘US college football”’ teams and matches(vertices/edges = 115/616)




variables of aSAT-instanceedges = direct dep.(electr. components)(vertices/edges ≈ 2K/6K)


sci. collaborations:3-hop neighorhoodvon D. Wagner(DBLP)(vertices/edges ≈ 10k/40k)



physical Internet: autonomous systemes(vertices/edges ≈ 20K/60K)



instance vertices edges

coauthors in DBLP 300K 1M

roads in the USA 24M 60M

WWW: .UK-domain ’02 20M 500M

( neurons in human brain & 1011 ∼ 1017 )


. . . no limit to be expected. . .

Clustering: Intuition to FormalizationTask: partition graphinto natural groupsParadigm:intra-cluster densityvs. inter-clustersparsity

Different approaches exist to formalize this paradigm, usually:

Paradigm of Graph ClusteringIntra-cluster density vs. inter-cluster sparsity

⇓Mathematical Formalization

quality measures for clusterings

Many exist, optimization generally (NP-)hardThere is no single, universally best strategy


Algorithm Engineering

Algorithms

implement

design

experiment

anal

yze

modelling reality is hard

finding optima is hardsatisfying needs ofapplication is hard

still, we do need to cluster⇒ need good foundation



Algorithms

implement

design

experiment

anal

yze

modelling reality is hardfinding optima is hard

satisfying needs ofapplication is hard




Algorithms

implement

design

experiment

anal

yze

modelling reality is hardfinding optima is hardsatisfying needs ofapplication is hard




Algorithms

implement

design

experiment

anal

yze


still, we do need to cluster

⇒ need good foundation



Algorithms

implement

design

experiment

anal

yze




Clustering vs. Partitioning

clustering partitioningpurpose analysis (pred.) handling of instance. . . and then? zoom/abstraction computations on parts

# of parts open predefined (upper bound)size of parts open upper bound (or even fixed)criteria various (later) weighted cutsconstraints often none see above

applications various (later) often: distributed finite elementmethods on 3d-meshes of objects


Bicriterial Formulations

observations:1 clusterings often “nice” if balanced (like partition)2 intra-density vs. inter-sparsity is bicriterial

bicriterial (or multi-) measures for clusterings can help:constrain sparsity within clustersconstrain density between clustersexplicitly formulate desiderata

(more on bicriteria later)


Postulations to a MeasureGiven a graph G and a clustering C, a quality measure should behave asfollows:

more intra-edges⇒ higher quality

less inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher quality

cliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separated

clusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connected

random clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad quality

disjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum quality

locality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)

double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same result

comparable results across instancesfulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instances

fulfill the desiderata of the application. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application

. . .



more intra-edges⇒ higher qualityless inter-edges⇒ higher qualitycliques must never be separatedclusters must be connectedrandom clusterings should have bad qualitydisjoint cliques should approach maximum qualitylocality of the measure (being better/worse in one part does notdepend on what is done in other part of graph)double the instance, what should happen . . . same resultcomparable results across instancesfulfill the desiderata of the application. . .


Formalization via Bottleneck

Raul

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Robyn Els

Ken

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KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

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Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

Marc

Toby

Frank

Lee

Raul

Susan

Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

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Toby

Frank

Lee

Quality of the clustering, upper cluster:

inter-cluster sparsity: 2 edges for cutting off 7 nodes (cheap)intra-cluster density: best addit. cut:intra-cluster density: 3 edges for cutting off 4 nodes (expensive)



Raul

Susan

Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

Marc

Toby

Frank

Lee

Raul

Susan

Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

Marc

Toby

Frank

Lee

Quality of the clustering, upper cluster:inter-cluster sparsity: 2 edges for cutting off 7 nodes (cheap)

intra-cluster density: best addit. cut:intra-cluster density: 3 edges for cutting off 4 nodes (expensive)



Raul

Susan

Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

Marc

Toby

Frank

Lee

Raul

Susan

Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

Marc

Toby

Frank

Lee

Quality of the clustering, upper cluster:inter-cluster sparsity: 2 edges for cutting off 7 nodes (cheap)intra-cluster density: best addit. cut:intra-cluster density: 3 edges for cutting off 4 nodes (expensive)


Examples: Conductance, Expansion

conductance of a cut (C,V \ C):

ϕ(C,V \ C) :=ω(E(C,V \ C))

min{∑

v∈C

ω(v),∑

v∈V\Cω(v)

}(i.e.: thickness of bottleneck which cuts off C)

inter-cluster conductance (C) := 1−maxC∈C ϕ(C,V \ C)(i.e.: 1− worst bottleneck induced by some C ∈ C)

intra-cluster conductance (C) := minC∈C minP]Q=C ϕ|C(P,Q)(i.e.: best bottleneck still left uncut inside some C ∈ C)

expansion of a cut (C,V \ C):

ψ(C,V \ C) :=ω(E(C,V \ C))

min{|C|, |V \ C|

}(i.e.: in ϕ, replace ω(v) by 1; intra- and inter-cluster expansion analogously)


Formalization: Counting Edges

Raul

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Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

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Toby

Frank

Lee

Measuring clustering quality by counting edges:inter-cluster sparsity: 6 edges of ca. 800 node pairs (few)

intra-cluster density: 53 edges of 99 node pairs (many)example: quality measure coverage = # intra-cluster edges

# edges

≈ 0.9


Formalization: Counting Edges

Raul

Susan

Phil

Robyn Els

Ken

Alice

KadkaChrisViolaine

Holly

DaveDoro Bob

Yoan

HelenCain

KateSue

Ron

Ralph

Tess

Mandy

Didi

Diane

Elaine

Richard

Clair

Marc

Toby

Frank

Lee

Measuring clustering quality by counting edges:inter-cluster sparsity: 6 edges of ca. 800 node pairs (few)intra-cluster density: 53 edges of 99 node pairs (many)

example: quality measure coverage = # intra-cluster edges# edges

≈ 0.9


Example Counting Measures

coverage: cov(C) := # intra-cluster edges# edges

(i.e.: fraction of covered edges)

performance: perf(C) := # intra-cluster edges+# absent inter-cluster edges12 n(n−1)

(i.e.: fraction of correctly classified pairs of nodes)

density: den(C) := 12

# intra-cluster edges# possible intra-cluster edges + 1

2# absent inter-cluster edges

# possible inter-cluster edges(i.e.: fractions of correct intra- and inter-edges)

modularity: mod(C) := cov(C)− E[cov(C)](i.e.: how clear is the clustering, compared to random network?)


Motivation for Modularity

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coverage = # intra-cluster edges# edges ≈ 0.9

only one cluster⇒ coverage = 1.0


A Promising Remedy[Girvan and Newman: Finding and evaluating community structure in networks,’04]:”. . . if we subtract from [coverage] the expected value [. . . ],we do get a useful measure.”

Modularity

mod(C) := cov(C) − E(cov(C))

=# intra-cluster edges

|#edges| − 14|#edges|2

∑C∈C

(∑v∈C

deg(v)

)2

first: stopping criterion for cuttingthen: optimization criterion


Modularity in Practiceeasy to use & implementreasonable behavior on many practical instances; heavily used in various fields

ecosystem explorationcollaboration analysesbiochemistrystructure of the internet (AS-graph, www, routers)

close to human intuition of quality[Gorke et al.: Comp. aspects of lucidity-driven clustering, 2010]

scaling behavior (double instance, result differs) [folklore]non-locality of optimal clustering [folklore]resolution limit (no tiny and large clusters at the same time)[Fortunato and Barthelemy ’07]large sparse graph ; high values, balanced clusters [Good et al.: Theperformance of modularity maximization in practical contexts, 2009]


Modularity, Algorithmic Theory

The complexity of modularity optimization:

finding C with maximum modularity is NP-hard; reduction from 3-PARTITION

restriction to |C| = 2 also hard⇒ not FPT wrt. |C|greedy maximization (later) does not approximatevery limited families combinatorially solvableILP-formulation, feasible for ≈ |V | ≤ 200

[Brandes et al.: On modularity clustering, 2008]

diverse results on approximability on specific classes of graphs

[DasGupta, Devine: On the complexity of newman’s community finding approachfor biological and social networks, 2011]


How to Cluster?Optimization of quality function:

Bottom-up: start with singletons

⇒ merge clusters

Top-down: start with the one-cluster

⇒ split clusters

Local Opt.: start with random clustering

⇒ migrate nodes

Variants of recursive min-cutting

Percolation of network by removal of highly central edges

Spectral methods using eigenanalysis of adjacency Laplacian

Direct identification of dense substructures

Random walks

Geometric approaches

. . .



Bottom-up: start with singletons

⇒ merge clustersTop-down: start with the one-cluster

⇒ split clusters


⇒ migrate nodes





Random walks


. . .



Bottom-up: start with singletons⇒ merge clusters

Top-down: start with the one-cluster

⇒ split clusters


⇒ migrate nodes





Random walks


. . .



Bottom-up: start with singletons⇒ merge clustersTop-down: start with the one-cluster

⇒ split clustersLocal Opt.: start with random clustering

⇒ migrate nodes





Random walks


. . .



Bottom-up: start with singletons⇒ merge clustersTop-down: start with the one-cluster⇒ split clusters


⇒ migrate nodes





Random walks


. . .



Bottom-up: start with singletons⇒ merge clustersTop-down: start with the one-cluster⇒ split clustersLocal Opt.: start with random clustering

⇒ migrate nodes





Random walks


. . .



Bottom-up: start with singletons⇒ merge clustersTop-down: start with the one-cluster⇒ split clustersLocal Opt.: start with random clustering⇒ migrate nodes





Random walks


. . .


Density-Constrained Clustering: Overview

New Optimization Problem:Find clusterings with guaranteed intra-cluster density and goodinter-cluster sparsity


Density-Constrained Clustering: Overview

New Optimization Problem:Find clusterings with guaranteed intra-cluster density and goodinter-cluster sparsity

This talk:Systematic collection of sparsity and density measuresClassification of measures with respect to their behaviorExperimental evaluation of greedy merge vs. greedy movesQualitative comparison of clusterings obtained by optimizing differentmeasures

See also:[Schumm et al.: Density-Constrained Graph Clustering, WADS’2011][Kappes et al.: Experiments on Density-Constrained Graph Clustering, to appearin ACM JEA]


Inter-cluster-sparsity: Cut-based

Isolated View: Each cluster induces a cutPairwise View: Each pair of clusters induces a cut in their subgraphGlobal View: A clustering with k clusters induces a k-way cut



Isolated View: Each cluster induces a cut

Pairwise View: Each pair of clusters induces a cut in their subgraphGlobal View: A clustering with k clusters induces a k-way cut



Isolated View: Each cluster induces a cutPairwise View: Each pair of clusters induces a cut in their subgraph

Global View: A clustering with k clusters induces a k-way cut



Isolated View: Each cluster induces a cutPairwise View: Each pair of clusters induces a cut in their subgraphGlobal View: A clustering with k clusters induces a k-way cut


Inter-cluster Sparsity:Degrees of Freedom


Set of cutsisolated (one for eachcluster)pairwise (one for eachpair of clusters)global (k -way cut)



Measuresnumber of cut-edgesdensityconductanceexpansion






Combinationsaverage sparsityminimum sparsity


⇒ 14 (reasonable) inter-cluster sparsity measures




Combinationsaverage sparsityminimum sparsity

Intra-cluster density

Definitions analoguous to inter-cluster sparsity possibleFinding cut with optimal density/conductance/expansion is NP-hard

Practical approach: evaluate |intra-cluster edges||possible intra-cluster edges|



Definitions analoguous to inter-cluster sparsity possible

Finding cut with optimal density/conductance/expansion is NP-hard




610





610



⇒ minimum/average/global intra-cluster density


Problem Statement

Density-Constrained ClusteringGiven a graph G = (V ,E), among all clusterings with an intra-clusterdensity of no less than α, find a clustering C with optimum inter-clustersparsity.

3 possible intra-cluster density measure14 possible inter-cluster sparsity measures⇒ Family of 42 optimization problems


Complexity (Example)

...

Kn

...

Kn

S1

Sm

VX

vx1

vxn

Reduction from Exact Cover by 3-Sets

TheoremDensity-Constrained Clustering combining any intra-cluster densitymeasure with the number of inter-cluster edges is NP-hard.


Complexity (Example)

...

Kn

...

Kn

S1

Sm

VX

vx1

vxn

Reduction from Exact Cover by 3-Sets

TheoremDensity-Constrained Clustering combining any intra-cluster densitymeasure with the number of inter-cluster edges is NP-hard.

⇒ motivates use of heuristic greedy algorithms


Greedy Algorithms

Greedy Merge (GM)Popular for modularity-based clusteringIdea: Merge clusters iteratively

Greedy Vertex Moving (GVM)Closely related to algorithms for graph partitioningVery successfull for optimizing modularity [Rotta et al. ‘11]


Generic Greedy Merge Algorithm

Idea: Merge clusters greedilyObjective: Increase inter-cluster sparsityConstraint: Intra-cluster density must not drop below given threshold


Generic Greedy Merge AlgorithmExample: Minimize number of inter-cluster edges such that the density ofeach cluster is at least 3

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Influence of Measures on Algorithm:Coarseness

inter-cluster sparsityintra-cluster density

Rough Intuition

QuestionWithout constraints, is there always a merge that improves the objectivefunction?


(Un-)Boundedness

DefinitionAn objective function measure f is unbounded if for any clustering C with|C| > 1 there exists a merge that does not deteriorate f .

Max. pw. inter-cluster conductanceis bounded

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e.g., modularity is bounded


(Un-)Boundedness



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(Un-)Boundedness



28


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(Un-)Boundedness



28


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boundedmpxcmpxe

apxdapxc

aixd

(Un-)Boundedness



28


28



boundedmpxcmpxe

apxdapxc

aixd

unboundednxegxdmixc

mixdmixeaixc

aixemixdmpxd

Influence of Measures on Algorithm

Feasiblemerges



Update Feasiblemerges?

QuestionDoes feasibility of a merge only depend on involved clusters?



Update Feasiblemerges?

QuestionDoes feasibility of a merge only depend on involved clusters?

⇒ Context insensitivity of an intracluster measure


Context Insensitivity

DefinitionA constraint is context insensitive, if the feasibility of a merge does notdepend on the remainder of the clustering.

E.g., minimum intra-cluster density is context insensitive




E.g., global intra-cluster density is context sensitive

Constraint: |intra-cluster edges||possible intra-cluster edges| =

11 ≥ 0.7







23 < 0.7







11 ≥ 0.7







22 ≥ 0.7







34 ≥ 0.7



Context Insensitivity: Classification


context insensitiveminimumintra-cluster density

context sensitiveaverageintra-cluster densityglobalintra-cluster density


Optimum

Heap

?Feasiblemerges

QuestionGiven context insensitivity, can the set of feasible merges be efficientlymaintained in a heap?



Optimum

Heap

?Feasiblemerges

QuestionGiven context insensitivity, can the set of feasible merges be efficientlymaintained in a heap?

⇒ Locality of an objective function


Locality: Intuition

Example: Maximum isolated inter-cluster conductance

A,B -0.3 C ,D 0 E ,F C, F G ,H G , I 0.30 0 0A,B C ,D

First approach: Use gain in inter-cluster sparsity as key

goodmerges

badmerges


Locality: Intuition




merge G and I

A,B -0.3 C ,D 0 E ,FC, F 0.2-0.2A,B C ,DG ∪ I ,T -0.3

goodmerges

badmerges


Locality: Intuition




merge G and I


goodmerges

badmerges

Clever tie-breaking possible?


Locality: Intuition




merge G and I


goodmerges

badmerges


Needed: Suitable order that does not change if unrelated clusters merge


Locality: Intuition




merge G and I


goodmerges

badmerges


Needed: Suitable order that does not change if unrelated clusters merge

Existence of such an order ≈ Locality of the inter-cluster measure


Example: Max. isolated inter-clusterconductance

0.5

Current sequence of conductance of all clusters (sorted)

A 0.4B 0.3C 0.3D 0.1E

Ordering merges lexicographically is stableTwo merges can be compared in constant time by comparing keysconsisting of three numbers



0.5


A 0.4B 0.3C 0.3D 0.1E

Sequence if A and B are merged

0.45A ∪ B 0.3C 0.3D 0.1E




0.5


A 0.4B 0.3C 0.3D 0.1E


0.45A ∪ B 0.3C 0.3D 0.1E

Sequence if A and D are merged

0.45A ∪ D 0.3C 0.1E0.4B




0.5


A 0.4B 0.3C 0.3D 0.1E


0.45A ∪ B 0.3C 0.3D 0.1E


0.45A ∪ D 0.3C 0.1E0.4B

compare lexicographically:

Merging A and B is better!




0.5


A 0.4B 0.3C 0.3D 0.1E


0.45A ∪ B 0.3C 0.3D 0.1E


0.45A ∪ D 0.3C 0.1E0.4B

compare lexicographically:

Merging A and B is better!


⇒ Maximum isolated inter-cluster conductance is local


Locality: Results

Does such an order exist for all objective functions?

|inter-cluster edges||possible inter-cluster edges| =

1743

global inter-cluster density is not local

localmixdmixcmixe

aixdaixcaixe

nxe

not localmpxdapxdmpxc

mpxegxdapxe

apxc


Locality: Results



1539


localmixdmixcmixe

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Locality: Results



1743


localmixdmixcmixe

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Locality: Results



1642


localmixdmixcmixe

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nxe


mpxegxdapxe

apxc


Locality: Results



1743

better

worse

global inter-cluster density is not localglobal inter-cluster density is not local

localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

apxc


Locality: Results



937


localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

apxc


Locality: Results



733


localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

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Locality: Results



937


localmixdmixcmixe

aixdaixcaixe

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mpxegxdapxe

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Locality: Results



836


localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

apxc


Locality: Results



937


localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

apxc


Locality: Results


better

worse


836


localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

apxc


Locality: Results


better

worse


836

global inter-cluster density is not local localmixdmixcmixe

aixdaixcaixe

nxe


mpxegxdapxe

apxc



sufficient?

Feasiblemerges

connectedmerges

Feasiblemerges

important?

QuestionDo we have to consider pairs of unconnected clusters?



sufficient?

Feasiblemerges

connectedmerges

Feasiblemerges

important?

QuestionDo we have to consider pairs of unconnected clusters?

⇒ Connectedness of an objective function


Disconnectedness

DefinitionAn objective function f is connected if merging unconnected clusters isnever the best option with respect to f .

14

max. pw. inter-cluster conductanceis not connected

14

14

14

14

14

14

14

connectednxe

unconnectedgxdmixcmixdmixeaixc

aixemixdmpxdmpxcmpxe

apxd

apxc

aixd


Disconnectedness



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connectednxe



apxd

apxc

aixd


Disconnectedness



Best option!

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apxd

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aixd


Influence of Measures on Efficiency

(Given the necessary data can efficiently be maintained:)

Contextinsensitivity Locality+ =

O(n2 log n)running time


Influence of Measures on Efficiency

(Given the necessary data can efficiently be maintained:)

Contextinsensitivity Locality+ =

O(n2 log n)running time

Contextinsensitivity

Locality+ =

O(md log n)running time

Connectedness+ &

linear space


Example: Email Graph of ourDepartment


chair

Modularity-based algorithm greedy merge (mid + aixc)

Local Moving

Example: Minimize number of intercluster edges such that the density ofeach cluster is at least 3

4

Idea: Move vertices greedilyObjective: Increase intercluster sparsityConstraint: Intracluster density must not drop below given threshold


Local Moving


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Local Moving


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Local Moving


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Local Moving


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Local Moving


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Local Moving


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Local Moving


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Local Moving


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Greedy Vertex Moving

Idea: Use Local Moving on multiple levels



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Effectiveness: Merge vs. Move

Question: Which greedy algorithm is more effective?

Setup:Preliminary Experiments: Pairwise measures behavecounter-intuitively⇒ left out of experimental analysisExperiments on Real-World Networks taken from the benchmark setsof Arenas and Newman

Outcome:Different Configurations

Intracluster density measureIntercluster sparsity measureParameter α

Summary: In 74 percent of all configurations, greedy vertex movingperforms better than greedy merging


Social Network of Dolphins

[Lusseau ’04]



Objectives: average intercluster density

Restriction: global intracluster density > 0.2

maximum intercluster density

global intercluster density

intercluster edges



Objectives: av. intercluster conductanceav. intercluster expansion




Objective: max. intercluster expansionmax. intercluster conductance




Objective: modularity


Planted Partition Graphs: Setup

Planted Partition Graph:

pinpout




pinpout

QuestionWhat is the distance between clustering found by objective function andhidden clustering?




pinpout

QuestionWhat is the distance between clustering found by objective function andhidden clustering?

Parameter α≈

expected intracluster density


Planted Partition Graphs: RoughSummary

ML-M

OD

MOD

NXE

GXD

MIXD

AIXD

MIXC

AIXC

MIXE

AIXE

MOD

NXE

GXD

MIXD

AIXD

MIXC

AIXC

MIXE

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···········global intracluster density minimum intracluster density

Distance to reference clustering



ML-M

OD

MOD

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GXD

MIXD

AIXD

MIXC

AIXC

MIXE

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MOD

NXE

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MIXD

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MIXC

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MIXE

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reference




ML-M

OD

MOD

NXE

GXD

MIXD

AIXD

MIXC

AIXC

MIXE

AIXE

MOD

NXE

GXD

MIXD

AIXD

MIXC

AIXC

MIXE

AIXE

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Planted Partition Graphs: Insights

Investigating different configurations yields further insights:Using average intracluster density as constraint leads to veryunbalanced clusteringsConstraining modularity by maximum intracluster density improves itsresults

. . . especially if expected number of clusters is high

Fine reference clusterings disbalance maximum objectivesAverage intercluster expansion/density identify many clusters


Conclusion

Clustering as bicriterial problemOptimize inter-cluster sparsity respecting intra-cluster densityCollection of new measuresAlgorithm Engineering aspects:Formulation of measuresClassification of measures with respect to greedy merge⇒ Insightsabout behavior of measuresExperimental evaluation of greedy methodsExperimental comparison on planted partition graphs


Conclusion

Clustering as bicriterial problemOptimize inter-cluster sparsity respecting intra-cluster densityCollection of new measuresAlgorithm Engineering aspects:Formulation of measuresClassification of measures with respect to greedy merge⇒ Insightsabout behavior of measuresExperimental evaluation of greedy methodsExperimental comparison on planted partition graphs

Thank you for your attention!


algorithmic methods for complex network analysis: …i11 · algorithmic methods for complex network...

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