Correlation clustering and community detection in graphs and networks

New relationships and algorithms
for correlation clustering and
community detection
David F. Gleich
Purdue University
With Nate Veldt (Purdue), Tony Wirth (Melbourne),
and James Saunderson (Monash)
Paper arXiv:1806.01678, 1712.05825 Code github.com/nveldt/LamCC, github.com/nveldt/MetricOptimization
UIUC 1David Gleich · Purdue

Graph clustering seeks“communities”of nodes in a network
Objective
functions
All seek to
balance
High internal densityLow external connectivity
modularity, densest subgraph, maximum
clique, conductance, sparsest cut, etc.
David Gleich · Purdue 2UIUC

Two objectives at opposite ends of the spectrum
min
cut(S)
`S`
+
cut(S)
`¯S`
Sparsest cut

Sparsest cut
Minimize number of edges removed
to partition graph into cliques
Two objectives at opposite ends of the spectrum
Cluster Deletion
min
cut(S)
`S`
+
cut(S)
`¯S`

We show sparsest cut and cluster deletion are two special
cases of the same new clustering framework:
LAMBDACC = λ Correlation Clustering
This framework also leads to
- new connections to other objectives (including modularity!)
- new approximation algorithms (2-approx for cluster deletion)
- several experiments/applications (social network analysis)
- (aside) fast method for LPs w/ metric constraints (for approx. algs)

6
Our framework is
based on correlation
clustering
Edges in a signed
graph indicate
similarity (+)
or dissimilarity (-)
UIUCDavid Gleich · Purdue

i
j
k
Edges can be weighted, but problems
become harder.
w+
ij wjk
w+
ij wjk
7
Our framework is
clustering
Edges in a signed
graph indicate
similarity (+)

Our framework is
clustering
Edges in a signed
graph indicate
similarity (+)
i
j
k
Mistake Mistake
Objective: Minimize the weight of “mistakes”
w+
ij wjk
w+
ij wjk
8UIUCDavid Gleich · Purdue

Given G = (V,E), construct signed
graph G’ = (V,E+,E- ), an instance
of correlation clustering
You can use correlation clustering to cluster unsigned graphs
David Gleich · Purdue 9
+
++
–
–
–
+
+ –
To model sparsest cut or cluster
deletion, set resolution parameter
λ ∈ (0,1)
LAMBDACC
1
1
1
1
1
Without weights, unweighted
correlation clustering is the same
as cluster editing
UIUC

Consider a restriction to two clusters
Positive mistakes: (1 – λ) cut(S)
Negative mistakes: λ |E–| – λ [ |S| |S| – cut(S) ]
Total weight of mistakes =
S S
cut(S)– λ |S| |S| + λ |E–|
UIUC

This is a scaled version of sparsest cut!
minimize cut(S) `S``¯S` + `E `
constantTwo-cluster LAMBDACC can be written
cut(S) `S``S` < 0 ()
cut(S)
`S``S`
<Note
cut(S)
`S`
+
cut(S)
`S`
= `V`
cut(S)
`S``S`
UIUC

We can write the objective in terms of cuts to get a
relationship with sparsest cut.
The general LAMBDACC objective can be written
THEOREM
Minimizing this objective produces clusters with scaled sparsest
cut at most λ (if they exist). There exists some λ’ such that
minimizing LAMBDACC will return the minimum sparsest cut
partition.
minimize
1
2
kX
i=1
cut(Si)
2
kX
i=1
`Si``Si` + `E `

We show this is
equivalent to LAMBDACC
for the right choice of
λ ≫ (1-λ)
1
1
1
1
1
cluster deletion correlation clustering with infinite
penalties on negative edges
1
1
1
1
1
For large λ,LAMBDACC generalizes cluster deletion
UIUC

1 2 1 4
3
2
4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 6
1 6
4
2
1 6
Degree-weighted LAMBDACC is related to Modularity
Though this does not preserve approximations…
LAMBDACC is a linear function of Modularity
Positive weight: 1 – λdidj
Negative weight: λdidj

Degree-
weighted
Standard
Sparsest Cut Cluster
Deletion
Correlation
Clustering
(Cluster Editing)
Normalized Cut Modularity
1
2m
0 1 0 1
m
m + 1
⇢⇤
⇤
=
1/2
17
Many other objectives are special cases of LAMBDACC
m = |E|

Algorithms for LAMBDACC are closely related to other
algorithms for correlation clustering,with better bounds
Any weighted correlation clustering objective gives a O(log n)
approximation via LP relax-and-round approaches.
Adapting the approach of van Zuylen and Williamson we obtain new
algorithms for standard LambdaCC based on the same LP relaxations:
• ThreeLP: 3-approximation for LAMBDACC when λ > ½
• TwoLP: 2-approximation for cluster deletion
We also provide scalable heuristic algorithms
• Lambda-Louvain: based on Louvain method for modularity
• GrowCluster: greedy agglomeration technique
16
[A.van Zuylen and D.P.Williamson.Mathematics of
Operations Research,34(3):594–620,2009.]
Best known
approximation for
cluster deletion!
We get a 5-approx. via
Puleo & Milenkovic
2015 when λ > 1/2

The ThreeLP algorithms begins by solving a metric-
constrained LP-relaxation and then rounding it.
In a few minutes, we’ll talk about how to solve metric LPs using some
scalable approaches.
ALG. Solve (2).
Create a graph F where Fij = +1 if Xij >= 1/3, Fij = -1 if Xij < 1/3.
Run Pivot on F.
UIUCDavid Gleich · Purdue 17
minimize
P
ij2E+ (1 )Xij
+
P
ij2E (1 Xij)
subject to Xij  Xik + Xjk
for all i, j, k
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BLP for min-disagree Relaxed MetricLP (2)

Our heuristic algorithms work in a greedy fashion.
GrowCluster.
• Locally produce a cluster by greedily starting from a random seed and
growing out to the best vertex in the boundary.
• Add the best cluster found, remove those vertices from the graph, and
then repeat until the graph is empty
LambdaLouvain.
• Use the generalized Louvain procedure of Jeub, Bazzi, Jutla, Porter
tweaked to optimize the LambdaCC objective.
• (This merges vertices into a cluster greedily starting from singletons)

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
ProbabilityofSharedAttribute
0
0.2
0.4
0.6
0.8
1
24 101 168 247 312 399 485
S/F
Dorm
Year
Cornell University (Facebook100)
We cluster social networks with various λ to understand the
correlation between communities and metadata attributes
Student/faculty status
Dorm
Graduation year
UIUC

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
24 101 168 247 312 399 485
S/F
Dorm
Year
Probability that two people who share a
cluster also share a metadata attribute
Dorm
Graduation year
UIUC

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
24 101 168 247 312 399 485
S/F
Dorm
Year
Probability that they share a related
fake attribute
Dorm
Graduation year
UIUC

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
24 101 168 247 312 399 485
S/F
Dorm
Year
The gap shows that there is a
noticeable correlation between
each attribute and the clustering
Probability that they share a related
fake attribute
Dorm
Graduation year
UIUC

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
17 71 131 225 363 528 711
S/F
Dorm
Year
n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
4 47 136 242 351 452 533
S/F
Dorm
Year
Swarthmore
Yale

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
17 71 131 225 363 528 711
S/F
Dorm
Year
n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
4 47 136 242 351 452 533
S/F
Dorm
Year
S/F status and graduation year peak early
Swarthmore
Yale

n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
17 71 131 225 363 528 711
S/F
Dorm
Year
n*Lambda
0 0.05 0.1 0.15 0.2 0.25
0
0.2
0.4
0.6
0.8
1
4 47 136 242 351 452 533
S/F
Dorm
Year
S/F status and graduation year peak early
Dorm attribute is more
correlated with small,
dense communities
Swarthmore
Yale

And now, an answer to one of the
most frequently asked questions in
clustering.
“What method should I use”?

Changing your method (implicitly) changes the value of
λ that you are using.
Lambda
1e-05 0.00022 0.0046 0.1 0.25 0.55 0.85
RatiotoLPbound
1
2
3
4
Graclus
Louvain
InfoMap
RMQC
RMC
Dense subgraph regimeSparse cut regime
This figure shows that if you
use one of these algorithms
(Graclus, Louvain, InfoMap,
recursive max-quasi clique,
or recursive max-clique)
then you implicitly
minimize λ-CC for some
choice of λ.
Turns the question
“what method should I use?”into
“what λ should I use?”

Changing your method (implicitly) changes the value of
λ that you are using.
Lambda
1e-05 0.00022 0.0046 0.1 0.25 0.55 0.85
RatiotoLPbound
1
2
3
4
Graclus
Louvain
InfoMap
RMQC
RMC
Dense subgraph regimeSparse cut regime
This figure shows that if you
use one of these algorithms
(Graclus, Louvain, InfoMap,
recursive max-quasi clique,
or recursive max-clique)
then you implicitly
minimize λ-CC for some
choice of λ.
Turns the question
“what method should I use?”into
“what λ should I use?”
The rest of the talk is an aside on
how we made this figure!
LP bound involves an LP
with 12 billion constraints.

For the rest of the talk, we’ll discuss
solving LPs with up to 700 billion
metric constraints.

Metric constrained LPs show up in a variety of
approximation algorithms for NP-hard problems.
A distance metric X is a matrix that encodes distances between n points.
Paradigm. Formulate NP-hard problem as optimization over {0,1}-metrics.
Then relax to [0,1]-metrics to get a linear program and bound.
Examples. Leighton-Rao sparsest cut, correlation clustering, cluster editing,
modularity optimization, cluster deletion.
Challenge. These LPs are really hard to solve as they have n3 constraints.
· Xij = distance from i to j
· Xii = 0, Xij 0
· Xij = Xji
· Xik  Xij + Xik<latexit sha1_base64="02GYV6F8DvHZ7Ao+7wxUq3E60Uk=">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</latexit><latexit sha1_base64="02GYV6F8DvHZ7Ao+7wxUq3E60Uk=">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</latexit><latexit sha1_base64="02GYV6F8DvHZ7Ao+7wxUq3E60Uk=">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</latexit><latexit sha1_base64="02GYV6F8DvHZ7Ao+7wxUq3E60Uk=">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</latexit>
Note. Some consider our definition to
be a pseudo- or semi-metric because
we allow Xij = 0 for i ≠ j
Xik ≤ Xij + Xik is a triangle constraintLinear
constraints!

The Leighton-Rao problem for sparsest cut.
Sparsest cut. Find a set S that has the smallest boundary-to-size ratio for
itself and its complement.
The metric problem. Encode the set S as a distance metric X where Xij = 0 if
i,j are in S or its complement and Xij = const. if the edge is cut.
The LP.
S S
minimize cut(S)/|S| + cut(¯S)/|¯S|<latexit sha1_base64="6ajYtNpiJP2CFKiADuZROdE/BgY=">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</latexit><latexit sha1_base64="6ajYtNpiJP2CFKiADuZROdE/BgY=">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</latexit><latexit sha1_base64="6ajYtNpiJP2CFKiADuZROdE/BgY=">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</latexit><latexit sha1_base64="6ajYtNpiJP2CFKiADuZROdE/BgY=">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</latexit>
, minimize cut(S)/(|S||¯S|)<latexit sha1_base64="p+UA9CoDFlmfmJF5Walt0YoDbBo=">AAAFgHicbVThb9w0FM/KDsYxWMc+8iWjTOpQ1yUIREEMVWKaNqnTCrluk+pTsZ2XO+vsJLNf6F1d/0P8NXwdfw12krLmWktJXp5/7/38fn42q6UwmCTvb2x8dHP08Se3Ph1/dvvzL+5s3v3ytakazeGIV7LSbxk1IEUJRyhQwttaA1VMwhu2+C3Mv/kLtBFVOcFVDVNFZ6UoBKfoXSebT8kBFKjFbI5U6+o0Ju8amsdEiVIocQb9vyUIS7S8QbedPXy8fZ6dnxNGtc3c+UN3srmV7CbtiK8aaW9sRf04PLl78xuSV7xRUCKX1JjjNKlxaqlGwSW4MWkM1JQv6AyOvVlSBWZq23Jd/MB78riotH9KjFvv+HKIz6PpapDFImWNpHo59LKqWvgZ48ZDSiz2plaUdYNQ8o6xaGSMVRxEjHOhgaNcxUNaXJw9mmlazzsSFIszKZimemVbcc2OmdMazA6nku8UAj2uXb0EtJOmQPgDcmc15Pf3kvtM+rSXETiHmQYonW0/AXM6FwhrGCYbcDa8LyGG9U3SqQ3aheIGFRxOMlp6NYiGEk55pRQtc0sKqoRc5VDQRqKzxBQX9lAAUyiKvvoHl8mMrxbyJ8nuTzvctxX6Iqj0u+kJcGmKkKLlko0qg7r2ubO/eg5AVi2TJ2TuP4TNdNXUjv9iCXTmvnVtWYougPq+RwQ9Jn5dpMRlWMZ+R2zNt8e+BX+Yugts5UUKDfMUfO9pyFaKVfKZl8N2WYyzr14eOFuG5SnhrHK23asM8Dqwd+TrIawP6TlCQNYw409oEw7e9QTrDNmzl0HOC4JJOpDesqWzRn4gCeAu2r7wyKABlfWcug9L/fPF2o7lMwmCzx91+3bdjG8S40/d8Nio/3et7xCViZnyTMSjGw0hnSVMWdL53ZWWUgf+ksqvi+gnfIi/V9L1W+Sq8fq73TTZTX//fmt/r79hbkVfRV9H21Ea/RjtR8+jw+go4tHf0T/R++jf0cZoe/R4lHbQjRt9zL1oMEY//wfqN/Ns</latexit><latexit sha1_base64="p+UA9CoDFlmfmJF5Walt0YoDbBo=">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</latexit><latexit sha1_base64="p+UA9CoDFlmfmJF5Walt0YoDbBo=">AAAFgHicbVThb9w0FM/KDsYxWMc+8iWjTOpQ1yUIREEMVWKaNqnTCrluk+pTsZ2XO+vsJLNf6F1d/0P8NXwdfw12krLmWktJXp5/7/38fn42q6UwmCTvb2x8dHP08Se3Ph1/dvvzL+5s3v3ytakazeGIV7LSbxk1IEUJRyhQwttaA1VMwhu2+C3Mv/kLtBFVOcFVDVNFZ6UoBKfoXSebT8kBFKjFbI5U6+o0Ju8amsdEiVIocQb9vyUIS7S8QbedPXy8fZ6dnxNGtc3c+UN3srmV7CbtiK8aaW9sRf04PLl78xuSV7xRUCKX1JjjNKlxaqlGwSW4MWkM1JQv6AyOvVlSBWZq23Jd/MB78riotH9KjFvv+HKIz6PpapDFImWNpHo59LKqWvgZ48ZDSiz2plaUdYNQ8o6xaGSMVRxEjHOhgaNcxUNaXJw9mmlazzsSFIszKZimemVbcc2OmdMazA6nku8UAj2uXb0EtJOmQPgDcmc15Pf3kvtM+rSXETiHmQYonW0/AXM6FwhrGCYbcDa8LyGG9U3SqQ3aheIGFRxOMlp6NYiGEk55pRQtc0sKqoRc5VDQRqKzxBQX9lAAUyiKvvoHl8mMrxbyJ8nuTzvctxX6Iqj0u+kJcGmKkKLlko0qg7r2ubO/eg5AVi2TJ2TuP4TNdNXUjv9iCXTmvnVtWYougPq+RwQ9Jn5dpMRlWMZ+R2zNt8e+BX+Yugts5UUKDfMUfO9pyFaKVfKZl8N2WYyzr14eOFuG5SnhrHK23asM8Dqwd+TrIawP6TlCQNYw409oEw7e9QTrDNmzl0HOC4JJOpDesqWzRn4gCeAu2r7wyKABlfWcug9L/fPF2o7lMwmCzx91+3bdjG8S40/d8Nio/3et7xCViZnyTMSjGw0hnSVMWdL53ZWWUgf+ksqvi+gnfIi/V9L1W+Sq8fq73TTZTX//fmt/r79hbkVfRV9H21Ea/RjtR8+jw+go4tHf0T/R++jf0cZoe/R4lHbQjRt9zL1oMEY//wfqN/Ns</latexit><latexit sha1_base64="p+UA9CoDFlmfmJF5Walt0YoDbBo=">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</latexit>
minimize
X
P
ij AijXij
subject to
P
ij Xij = n, Xij 0
Xij  Xik + Xjk for all i, j, k<latexit sha1_base64="Jgpj4JELAcDXxwqa2wJZWf20v0M=">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</latexit><latexit sha1_base64="Jgpj4JELAcDXxwqa2wJZWf20v0M=">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</latexit><latexit sha1_base64="Jgpj4JELAcDXxwqa2wJZWf20v0M=">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</latexit><latexit sha1_base64="Jgpj4JELAcDXxwqa2wJZWf20v0M=">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</latexit>
Also. This is related to multi-
commodity flow.
Aij is the adjacency matrix

How well does Gurobi do at solving this LP?
Graph |V| |E| # constraints Gurobi Time
Jazz 198 2742 3.8 ⇥ 106
60
SmallW 233 994 6.2 ⇥ 106
93
C.El-Neural 297 2148 1.2 ⇥ 107
274
USAir97 332 2126 1.8 ⇥ 107
471
Netscience 379 914 2.7 ⇥ 107
887
Erdos991 446 1413 4.4 ⇥ 107
2574
C.El-Meta 453 2025 4.6 ⇥ 107
2497
Harvard500 500 2043 6.2 ⇥ 107
18769
Roget 994 3640 4.9 ⇥ 108
out of memory
SmaGri 1024 4916 5.4 ⇥ 108
out of memory
Email 1133 5451 7.3 ⇥ 108
out of memory
Polblogs 1222 16714 9.1 ⇥ 108
out of memory
Vassar85 3068 119161 1.4 ⇥ 1010
out of memory
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We setup Gurobi to skip the
cross-over step and just use
the minimum time of
SIMPLEX vs. Interior-point.
• Interior point always wins
We limited experiments to
100GB of RAM for
reproducibility. Gurobi used
all 28-cores on our system.

Our DykstraSC solver provides an extra order-of-
magnitude in problem solvablility.
Graph |V| |E| # constraints Gurobi Time Dykstra Time Approx
Jazz 198 2742 3.8 ⇥ 106
60 81 1.003
SmallW 233 994 6.2 ⇥ 106
93 166 1.001
C.El-Neural 297 2148 1.2 ⇥ 107
274 350 1.000
USAir97 332 2126 1.8 ⇥ 107
471 511 1.041
Netscience 379 914 2.7 ⇥ 107
887 1134 1.000
Erdos991 446 1413 4.4 ⇥ 107
2574 1954 1.011
C.El-Meta 453 2025 4.6 ⇥ 107
2497 1138 1.000
Harvard500 500 2043 6.2 ⇥ 107
18769 1427 1.000
Roget 994 3640 4.9 ⇥ 108
out of memory 53449 1.008
SmaGri 1024 4916 5.4 ⇥ 108
Email 1133 5451 7.3 ⇥ 108
Polblogs 1222 16714 9.1 ⇥ 108
Vassar85 3068 119161 1.4 ⇥ 1010
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We only use a single thread. The approximation can be controlled by a parameter.

What does our algorithm do?
We extend the framework of Brickell, Dhillon, Sra, and Tropp
on fast solvers for the metric nearness problem.
1. Convert the LPs into an equivalent QP. (improved)
2. Use Dykstra’s projection method to solve the QP.
3. Implement it carefully with a sparse set of dual variables
(active triangle constraints).
4. (new) Uses a convergence criteria based on the KKT
conditions.
Richard L.Dykstra

We abstract our setup as a LP with an extremely sparse
constraint matrix and use a Dykstra projection method
This is exactly what is proposed in Brickell et al.
The target problem. Is an LP where A has O(n3) rows and x is O(n2) entries
The quadratic program. Is a a smoothed dual with W diagonal, positive. For
! sufficiently large these two are equivalent (a “proximity function”in convex opt)
The dual. Only involves a non-negative vector.
minimize cT
x subject to Ax  b<latexit sha1_base64="RK8zV6OP5umjKF5qt5XqskbhJOU=">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</latexit><latexit sha1_base64="RK8zV6OP5umjKF5qt5XqskbhJOU=">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</latexit><latexit sha1_base64="RK8zV6OP5umjKF5qt5XqskbhJOU=">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</latexit><latexit sha1_base64="RK8zV6OP5umjKF5qt5XqskbhJOU=">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</latexit>
minimize Q(x) = cT
x +
1
2
xT
Wx subject to Ax  b
<latexit sha1_base64="uwTLqCGa66SItr3pYBEk2SN7aYQ=">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</latexit><latexit sha1_base64="uwTLqCGa66SItr3pYBEk2SN7aYQ=">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</latexit><latexit sha1_base64="uwTLqCGa66SItr3pYBEk2SN7aYQ=">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</latexit><latexit sha1_base64="uwTLqCGa66SItr3pYBEk2SN7aYQ=">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</latexit>
maximize D(y) = bT
y
1
2
(AT
y + c)T
W 1
(AT
y + c) subject to y 0
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Dykstra projection algorithm visits constraints cyclically.
This can be made efficient for Metric LPs
Input: A 2 RN⇥M
, b 2 RM
, c 2 RN
, > 0, W 2 RN⇥N
(diagonal, positive deﬁnite)
Output: ˆx = argminx2A Q(x) where A = {x 2 RN
: Ax  b}
y := 0 2 RM
x := W 1
c, k := 0
while not converged do
k := k + 1
(Visit constraints cyclically): i := (k 1) mod M + 1
(Perform correction step): x := x + yi( W 1
ai) where ai is the ith row of A
(Perform projection step): x := x ✓+
i ( W 1
ai) where ✓+
i =
max{aT
i x bi,0}
aT
i W 1ai
(Update dual variables): yi := ✓+
i 0
end while<latexit 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New (kinda). This is a coordinate ascent procedure on the dual for any pd W!
Richard Dykstra (1983),Hildreth (1957),Dax (2003).
Note. These updates only involve the
extremely sparse rows of A
This could be zero!

The method works by visiting all constraints and
updating variables.
This shows the solution
in the matrix, and the
constraint violation in
the cube.
• First it illustrates
what it’s like going
through each triplet
one at a time.
• Then it speeds
things up by only
updating the plot
one time per outer
loop (i.e. the changes
made at all triplets
involving node i)

The final algorithm we use stores y as a sparse vector
with the index of the constraints
Choice 1. Store y as a dictionary.
• Easy to implement, enables random access
• Slow to access dictionary
Choice 2. Store an array of the indices where y is non-zero.
• Just push a constraint on to the new non-zero list when we change it.
• Easy to trace through to know when the next non-zero arises.
For our serial code with a specific loop, we use Choice 2 as it’s
considerably faster.
I’ll return to this point later!

We have a variety of technical results that make this
method robust and easy-to-use for specific LPs
We also have a clear understanding of how !, W can give approx. LP solns.
We have good primal-dual theory to certify optimality or near optimality.
We have approximation-related non-optimality. (e.g. if OPT is a two-
approx. to the NP-hard problem, and we get a theta-approx. to the LP, then
we get an OPT+theta approx)
We get fast aposterori approx. bounds for LP too.

On graphs with thousands of nodes,the algorithms for
sparsest cut method take thousands of iterations.
We find sparsest cut is the
hardest problem.
• Total time = 11.5 hrs
• 9.1 billion constraints
• 99.67% are tight.
These are scaled so OPT=0.1
The primal is not always an
upper-bound (iters 1-500)
Number of Iterations
1000 2000 3000 4000 5000
ConstraintTol/QPScores
-0.05
0
0.05
0.1
0.15
0.2
Polblogs (n = 1222), 1 iter = 4.9 s
0.1*Dual/OPT
0.1*Primal/OPT
0.1 = Scaled OPT
Constraint Tol

We use an approach to create correlation clustering
problems from graphs that isn’t λ-CC
• For each pair of nodes (not just edges) , compute Jaccard (i, j) = Ji,j
•
For this problem,lazy constraint generation for Gurobi helped.
Wang et al.ADMA 2013
Set Si,j = log[(1 + (Ji,j ))/(1 + (Ji,j ))] = 0.05<latexit sha1_base64="RBLV+6hb4MlpuA1+naxY4jFhnmI=">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</latexit><latexit sha1_base64="RBLV+6hb4MlpuA1+naxY4jFhnmI=">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</latexit><latexit sha1_base64="RBLV+6hb4MlpuA1+naxY4jFhnmI=">AAAFu3icdVRtbxQ3EF4o18L1LbQf+bI0QgrtJd2tippKBCK1QlCBGrgLIGVPqdc7u2fOXm/9Qu6w/Af7D/gXfIVPHd9uSvaSWtrz3HhmnpnHM84bzrRJkreXLn9yZfDpZ1evDT//4suvvt64/s1zLa2icEgll+plTjRwVsOhYYbDy0YBETmHF/n8t3D+4jUozWQ9McsGpoJUNSsZJQZVxxtFZmBh3BhM7MfHjo1e+XgvzrisjuKtNP4h3vqj027HWQHckNu34x///2gaZ39bUnQKDJXsJHeONzZxW634vJB2wmbUrYPj61fuZ4WkVkBtKCdaH6VJY6aOKMMoBz/MrIaG0Dmp4AjFmgjQU7eiw8e3UFPEpVT41SZeaYdnXTCOIsteFGdIbjlRi742l3KOJ9oP+5Cm3J06VjfWQE1bxNLy2Mg4kBwXTAE1fBn3Yc38zXalSDNrQQybv+EsV0QtQ0byRI/0jDSgR5RwOiqZQbtV9hyMm9jSwDMovFNQ3NxNbuYcw561MDOoFEDt3WoLNiczZmDNJucWvAu/Zyz69U3SqQvcheJ6FRxMxqRGNjIFNZxQKQSpC5eVRDC+LKAklhvvMl2eyn0CdCmIwepvnQXTWC0Ue8nOryMqGIIiRRxvEwHMQpchxAqLW1EHdt1D7+4hBphcLpK9bIZblldK2sbTuy6DVtx3flWWIHMgOBfGgBpiW5ZZbRYhjf0W2Onvj7AF70z9qa1EkkLD/A7YewrGS5FL/gDpcG0U7d2fTx57V4f0BPNOeLe6K5yii4xRUay75J1LhxEcxjbXOME2DObFAOsI4wdPAp2nAJO0R73LF95p/hEkGLfe7hFaBg4Ib2bEf0z1r0drN1ZUHBidbbf3dtEJNonGqeuPjej/J7xqNFgcaVnAfxfaNY8Ys0pgEhkGsgoCksty4bJW7891m3iM71txkUd3gC745KTrD8x54flPO2mykz79eXN/t3t8rkY3ou+irSiNfon2o4fRQXQY0eif6F30Pvow2BvQwasBb00vX+p8vo16a2D/BboaA/Y=</latexit><latexit sha1_base64="RBLV+6hb4MlpuA1+naxY4jFhnmI=">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</latexit>
Graph |V| |E| # constraints Gurobi Time Dykstra Time Approx
power 4941 6594 6.0 ⇥ 1010
549 s 7.6 hrs 1.07
caGrQc 4158 13422 3.6 ⇥ 1010
out of memory 6.6 hrs 1.33
caHepTh 8638 24806 3.2 ⇥ 1011
caHepPh 11204 117619 7.0 ⇥ 1011
<latexit 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Wi,j =
(
Si,j + sign(Si,j)" Si,j 6= 0
" + 2"Ind[(i, j) 2 E] Si,j = 0
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On graphs with thousands of nodes,the algorithms for
correlation clustering take hundreds of iterations.
The correlation clustering
problems tend to be easier.
• Total time = 7.6 hrs
• 60 billion constraints
• 1.07 approx. LP soln
The jumps are because we
only check constraint tol every
20 steps
• We could have made this parallel and faster
Number of Iterations
0 50 100 150
-0.2
0
0.2
0.4
0.6
0.8
1
power (n = 4941), 1 iter = 180s
Constraint Tol
Duality Gap

These are slow algorithms right now.We tried a variety
of ideas to make them faster.
Parallelization!
• The updates can be done in any order over constraints for the method
to guarantee linear convergence.
• So we could get away with sequentially consistent guarantees.
• Need locking and some fancier techniques to store the updates, but it could work.
• Our worry was about lock-contension (3-number compare exchange?)
• Static schedule for complete 3-regular hypergraphs?
• True–parallel versions of Dykstra algorithm exist, but they average over
solutions computed between processors, because of our large numbers
of constraints, these made little progress.
• Could a hog-wild style approach be adapted to work here?
• We lose coordinate-ascent in a simplistic implementation.

The memory required is still O(n3) in the worst case.
O(n2) memory algs exist,but they are really slow!
Bauschke designed methods that avoid dual-variables for the constraints.
Dykstra is a linearly convergent algorithm. Bauschke’s is asymptotically convergent.
We couldn’t make these competitive.
Iterations
500 1000 1500 2000 2500
QPobjectivescore
-0.2
0
0.2
0.4
0.6
0.8
1
BauschkeSC on celegansmetabolic
< = 1/10
< = 1/25
< = 1/50
< = 1/100
< = 1/250
< = 1/500
Iterations
500 1000 1500 2000 2500
QPobjectivescore
-0.2
0
0.2
0.4
0.6
0.8
1
DykstraSC on celegansmetabolic
Dual QP
Primal QP
OPT
Bauschke 1996 J.Math Analysis & Approx.

A quick summary of other work from our research team
on data-driven scientific computing
Our team’s overall goal is to design algorithms and methods tuned to
the evolving needs and nature of scientific data analysis.
Low-rank methods for network alignment – Huda Nassar
• Principled methods that scale to
aligning thousands of networks.
Spectral properties and generation of realistic
networks – Nicole Eikmeier
• “Power-laws” in the top sing. vals of adj matrix are most
robust than degree “power-laws”
• Fast sampling for hypergraph models with higher-order structure.
Local analysis of network data – Meng Liu
• Applications in bioinformatics, software https://github.com/kfoynt/LocalGraphClustering
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Fig. 5. For a Kronecker graph with a 2 ⇥ 2 initi
been “⌦-powered” three times to an 8 ⇥ 8 probability

We have extensively explored principled methods in
terms of higher-order and multi-way data.
The key question. Much of the data now collected and curated has rich
multi-way and higher-order structure. How can we engineer algorithms
with guarantees that capture the structure?
Figure 1: An illustration of Markov chain methods and our
proposed RHOMP model.
surfer had visited a search-query result page and then clicked the
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Multiway structure in sequence
modeling for prediction
Higher-order structures in
networks
Benson,Gleich,Leskovec (2015,2016)
Klymko,Gleich,Kolda (2014)
Mohammadi,Gleich,Kolda,Grama (2017)
Higher-order methods for data
Yu,Gleich,Lim (SIMAX 2015)
Benson,Gleich,Lim (SIAM Review 2017)
Wu,Benson,Gleich (2016) Wu,Gleich (2017)
9
10
8
7
2
0
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3
11
6
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1

UIUC
Papers. arXiv: 1712.05825 (at WWW2018),1806.01678
Software. github: nveldt/LamCC,nveldt/MetricOptimization
47
A different framework for clusters communities in graphs. (LAMBDACC)
An improved procedure to solve LPs with metric constraints
Issues.
• Links with other approaches
such as cut-matching games?
• Would love to solve problems with 100k
node graphs J can we get there with
parallel / distributed settings?
With Nate Veldt (Purdue),
Tony Wirth (Melbourne),
and James Saunderson (Monash)
David Gleich · Purdue

Correlation clustering and community detection in graphs and networks

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