Hyperbolic Communities: Modelling Communities Beyond Cliques Pauli - - PowerPoint PPT Presentation

Hyperbolic Communities: Modelling Communities Beyond Cliques

Pauli Miettinen
 8 June 2017

Pauli Miettinen 8 June 2017

Joint work with

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Saskia Metzler
 MPI-INF Stephan Günnemann T.U. München Stefan Neumann Uni Wien Sanjar Karaev MPI-INF Rainer Gemulla Uni Mannheim

Your 
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Pauli Miettinen 8 June 2017

Communities = cliques

• Communities are often

modelled as (quasi-) cliques

• Dense subgraphs
• All edges equally likely
• In a community,

everybody knows (or should know) everybody else

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Pauli Miettinen 8 June 2017

Communities ≠ cliques

• But many communities are

not cliques

• There is more structure
• Some people know more

people

• Others know just the

central people
 ⇒ Cliques are not a good model

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Pauli Miettinen 8 June 2017

Communities really 
 are not cliques

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≈ 160 stackexchange communities Nodes are users, edges are comments or answers during the last year

Pauli Miettinen 8 June 2017

Why should I care?

• Better understanding of the community

structures

• Better fit to real-world data
• Better prediction power
• More realistic random graphs
• …

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Pauli Miettinen 8 June 2017

The core/periphery model

• Classical model in social

sciences (from 1999)

• Communities are L-shaped
• A core that is a clique
• A periphery that is only

connected to the core

• In real-world, 0s can

appear both in core and in periphery

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Borgatti & Everett, 1999

Core Periphery

Pauli Miettinen 8 June 2017

Core/periphery example

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Borgatti & Everett, 1999

Pauli Miettinen 8 June 2017

Nested matrices

• Matrix is nested if its rows

and columns can be

• rdered so that
• all 1s are consecutive
• no row has more 1s than

the row above it

• Important concept in

ecology

• Core/periphery matrices are

nested

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Pauli Miettinen 8 June 2017

HyCom communities

• Assume ordered (by

degree)

• In HyCom communities,

an edge (i, j) is in the community iff
 iαjα > τ
 for some α ≤ 0 
 and 0 < τ < 1

• N.B. same as ij < τ’ with

τ’ = τ1/α

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10 20 30 40 50 10 20 30 40 50 α = -0.5, τ = 0.1, 380 edges

Araujo et al. ECMLPKDD ’14

Pauli Miettinen 8 June 2017

Some comments 


• n the models
• Core/periphery model seems too restricted
• T

ails taper towards the end

• Nested model is very general
• Perhaps too general…
• HyCom seems like a good compromise
• But with only one free variable, they’re too

quite limited

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Pauli Miettinen 8 June 2017

The hyperbolic model

• Let us assume the nodes

are ordered and i=0, 1, …

• In hyperbolic model edge

(i, j) is in the community if
 (i + p)(j + p) ≤ θ

• (–p, –p) is the centre of

the hyperbola

• θ places the curve in the

gradient

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• 25

• 25

(–p, –p)

Metzler et al. ICDM 2016

Pauli Miettinen 8 June 2017

The core/tail model

• The core/tail model is

parameterized by the size

• f the core, γ, and the

thickness of the tail, H

• T

ail cannot be thicker than the core

• In fact


γ ≤ (n – 1 + H)/2

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γ H

Pauli Miettinen 8 June 2017

The mixture model

• HyCom: i∙j ≤ τ
• Line: i + j ≤ σ
• Mixture: 


(1 – x)(i∙j) + x(i + j) ≤ Σ 
 for 0 ≤ x ≤ 1 and Σ ∈ ℝ

• Actually


(1 – |x|)(ij) + x(i + j) ≤ Σ
 for –1 ≤ x ≤ 1

• Slightly more general

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• del:

Pauli Miettinen 8 June 2017

All the models 
 are the same

• Equivalence Theorem: Given a valid pair of

parameters for one of the three models above, there exists valid pairs of parameters for the other two models that will model exactly the same graph

• Between hyperbola and core/tail, this is straight-

forward (re-parametrization)

• Requires a proof between mixture and hyperbola

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Pauli Miettinen 8 June 2017

Hyperbolic vs.
 cliques and core/tail

• Cliques are a special case
• f the models
• E.g. set core size to the

community size

• Core/periphery is also a

special case of the model

• Or a limit case (needs to

be checked)

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Pauli Miettinen 8 June 2017

Hyperbolic vs.
 power-law

• HyCom is a special case of
• ur models
• Recall mixture: 


(1 – x)(i∙j) + x(i + j) ≤ Σ

• T

echnically HyCom is
 1/2(i∙j) + 1/2(i + j) ≤ Σ

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20 40 60 80 100 20 40 60 80 100

Pauli Miettinen 8 June 2017

Some example communities

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Examples

Pauli Miettinen 8 June 2017

On likelihoods

• Area under the curve should be

dense

• Area above the curve should be

sparse

• Minimize the log-likelihood


|E|log(d) + |∁E|log(1–d)
 +|O|log(s) + |∁O|log(s)

• |E| = edges in comm, |∁E| = non-

edges in comm., d = density of comm., |O| = edges outside of comm., |∁O| = non-edges outside

• f comm., s = density outside

comm.

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50 100 150 200 50 100 150 200

Dense Spars

Pauli Miettinen 8 June 2017

More than one community

• Generalizing to many communities is mostly

straight forward

• Every community has its own parameters
• Likelihoods inside communities sum up
• The area outside all communities should be

sparse

• But overlapping communities add complexity…

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Pauli Miettinen 8 June 2017

Overlapping communities!

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les

This is a good community This is also a good community But now this is a bad community! And now it’s just a big mess!

Pauli Miettinen 8 June 2017

Forms of overlap

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No overlap Node overlap Edge overlap Just find the 
 communities and
 model them The overlapping 
 edges must be
 ignored The overlapping 
 edges must be
 handled Assign every edge to at most 1 community

Pauli Miettinen 8 June 2017

Are these good models?

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LL ratio block model HyCom Amazon 26450.6 30997.1 DBLP (100) 3148.5

• 788.0

DBLP

• 264974.7

17958.1 Friendster 200627.6 17811.7 LiveJournal 154982.4 22705.8 Orkut 11945.3 1598.5 YouTube 75689.6 12660.0

LL ratio spectral clustering BMF HyCom Email 10895.8 3552.0 250.1 Erd˝

• s

1797.0 949.0 256.3 Jazz 3003.8 4435.0 3718.5 PolBooks 648.0 303.3 228.2

Likelihood ratio test 
 the larger the ratio, the more likely our model is better Ground-truth 
 communities Calculated communities

Pauli Miettinen 8 June 2017

Distributions of parameters

24 Amazon DBLP(100) DBLP Friendster LiveJournal Orkut YouTube 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

. relative to community size

How clique-like the communities are?

Pauli Miettinen 8 June 2017

Distribution of parameters

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H relative to community size

• 1
• 0.8
• 0.6
• 0.4
• 0.2

x

How thick the tail is? How similar to the HyCom the models are?

Pauli Miettinen 8 June 2017

How to find the communities?

• We can use any clique-like community finding

algorithm, and re-model

• But the found communities might not be

very good

• We can try to grow the communities from

cores

• But overlapping communities cause issues

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Pauli Miettinen 8 June 2017

Nested matrix redux

• Work in progress: express the adjacency

matrix as a union of nested submatrices

• More general than our models but

potentially easier

• Interesting on its own right
• We find the nested subgraphs by finding a

suitable matrix factorization

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Pauli Miettinen 8 June 2017

Nested matrices as rank-1 structures

• Nested matrices can have a full rank
• Their nonnegative rounding rank is 1
• Every nested matrix A can be expressed

as A = thr(xyT)

• x and y are nonnegative vectors
• thr(a) = 1 if a ≥ 0.5 and 0 otherwise

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Neumann et al. ICDM 2016

Pauli Miettinen 8 June 2017

Example

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 

1 1 1 1 1 1

  @

4 2 1

1 A

1 1/2 1/4

• =

@

4 2 1 2 1 1/2 1 1/2 1/4

1 A

thr1/2+

   

4 2 1 2 1 1/2 1 1/2 1/4

    =  

1 1 1 1 1 1

 

Pauli Miettinen 8 June 2017

Combining nested matrices: problems

• thr([x1 x2][y1 y2]T) is not necessarily a union
• f two nested matrices
• If x < θ and y < θ, it’s still possible that 


x + y ≥ θ

• Higher ranks work only for completely

disjoint communities

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Pauli Miettinen 8 June 2017

Tropical algebra 
 to the rescue

• The subtropical algebra over the

nonnegative reals has

• the summation ⊞ defined as the maximum
• the product ⊠ defined as usual
• If x < θ and y < θ, x ⊞ y = max{x, y} < θ
• The thresholding distributes over ⊞:


thr(x ⊞ y) = thr(x) ⊞ thr(y)

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Pauli Miettinen 8 June 2017

The full model

• Given a symmetric adjacency matrix A, find

nonnegative matrix B with k columns minimizing 
 ||A – thr(B⊠BT)||

• The matrix product is over the subtropical

algebra

• Equivalently, given A, find k nested (up to

permutations) matrices N1, N2, …, Nk that minimize
 ||A – ∪i Ni||

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Karaev et al. submitted

Pauli Miettinen 8 June 2017

Resulting communities

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Hyperbolic Almost 
 hyperbolic Not at all hyperbolic

Pauli Miettinen 8 June 2017

Future work

• Validate that the hyperbolic models are good

with more data

• More work for algorithms to find the

communities

• Time-evolving communities
• Random graph models and graphons

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