A Correlation Clustering Framework forCommunity Detection

Nate VeldtMathematics Department,

Purdue UniversityWest Lafayette, Indianalveldt@purdue.edu

David F. GleichComputer Science Department,

Purdue UniversityWest Lafayette, Indianadgleich@purdue.edu

Anthony WirthSchool of Computing andInformation Systems,

The University of MelbourneParkville, VIC, Australiaawirth@unimelb.edu.au

ABSTRACTGraph clustering, or community detection, is the task of identifyinggroups of closely related objects in a large network. In this paperwe introduce a new community-detection framework called Lamb-daCC that is based on a specially weighted version of correlationclustering. A key component in our methodology is a clustering res-olution parameter, λ, which implicitly controls the size and structureof clusters formed by our framework. We show that, by increasingthis parameter, our objective effectively interpolates between twodifferent strategies in graph clustering: finding a sparse cut andforming dense subgraphs. Our methodology unifies and generalizesa number of other important clustering quality functions includingmodularity, sparsest cut, and cluster deletion, and places them allwithin the context of an optimization problem that has been wellstudied from the perspective of approximation algorithms. Ourapproach is particularly relevant in the regime of finding denseclusters, as it leads to a 2-approximation for the cluster deletionproblem. We use our approach to cluster several graphs, includinglarge collaboration networks and social networks.

CCS CONCEPTS•Mathematics of computing→ Graph algorithms; Approxi-mation algorithms; Mathematical optimization;

KEYWORDSgraph clustering; community detection; network analysis; correla-tion clustering; cluster deletion; sparsest cutACM Reference Format:Nate Veldt, David F. Gleich, and Anthony Wirth. 2018. A Correlation Clus-tering Framework for Community Detection. In WWW 2018: The 2018 WebConference, April 23–27, 2018, Lyon, France. ACM, New York, NY, USA,10 pages. https://doi.org/10.1145/3178876.3186110

1 INTRODUCTIONIdentifying groups of related entities in a network is a ubiquitoustask across scientific disciplines. This task is often called graphclustering, or community detection, and can be used to find similarproteins in a protein-interaction network, group related organisms

This paper is published under the Creative Commons Attribution 4.0 International(CC BY 4.0) license. Authors reserve their rights to disseminate the work on theirpersonal and corporate Web sites with the appropriate attribution.WWW 2018, April 23–27, 2018, Lyon, France© 2018 IW3C2 (International World Wide Web Conference Committee), publishedunder Creative Commons CC BY 4.0 License.ACM ISBN 978-1-4503-5639-8/18/04.https://doi.org/10.1145/3178876.3186110

in a food web, identify communities in a social network, and classifyweb documents, among numerous other applications.

Defining the right notion of a “good” community in a graph is animportant precursor to developing successful algorithms for graphclustering. In general, a good clustering is one in which nodes insideclusters are more densely connected to each other than to the restof the graph. However, no consensus exists as to the best way todetermine the quality of network clusterings, and recent resultsshow there cannot be such a consensus for the multiple possiblereasons people may cluster data [32]. Common objective functionsstudied by theoretical computer scientists include normalized cut,sparsest cut, conductance, and edge expansion, all of whichmeasuresome version of the cut-to-size ratio for a single cluster in a graph.Other standards of clustering quality put a greater emphasis on theinternal density of clusters, such as the cluster deletion objective,which seeks to partition a graph into completely connected sets ofnodes (cliques) by removing the fewest edges possible.

Arguably the most widely used multi-cluster objective for com-munity detection is modularity, introduced by Newman and Gir-van [31]. Modularity measures the difference between the truenumber of edges inside the clusters of a given partitioning (“inneredges”) minus the expected number of inner edges, where expecta-tion is calculated with respect to a specific random graph model.

There are a limited number of results which have begun to unifydistinct clustering measures by introducing objective functionsthat are closely related to modularity and depend on a tunableclustering resolution parameter [15, 33]. Reichardt and Bornholdtdeveloped an approach based on finding the minimum-energy stateof an infinite-range Potts spin glass. The resulting Hamiltonianfunction they study is viewed as a clustering objective with a reso-lution parameter, γ , which can be used as a heuristic for detectingoverlapping and hierarchical community structure in a network.When γ = 1, the authors prove an equivalence between minimizingthe Hamiltonian and finding the maximum modularity partitioningof a network [33]. Later, Delvenne et al. introduced a measure calledthe stability of a clustering, which generalizes modularity and alsois related to the normalized cut objective and Fiedler’s spectralclustering method for certain values of an input parameter [15].

The inherent difficulty in obtaining clusterings that are provablyclose to the optimal solution puts these objective functions at a dis-advantage. Although both the stability and the Hamiltonian-Pottsobjectives provide useful interpretations for community detection,there are no approximation guarantees for either: all current algo-rithms are heuristics. Furthermore, it is known that maximizing

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https://doi.org/10.1145/3178876.3186110https://doi.org/10.1145/3178876.3186110

modularity itself is not only NP-hard, but is also NP-hard to ap-proximate to within any constant factor [19].

Our Contributions. In this paper, we introduce a new clusteringframework based on a specially weighted version of correlationclustering [4]. Our partitioning objective for signed networks lies“between” the family of ±1 complete instances and the most generalcorrelation clustering instances. Our framework comes with severalnovel theoretical properties and leads to many connections betweenclustering objectives that were previously not seen to be related. Insummary, we provide:• A novel framework LambdaCC for community detectionthat is related to modularity and the Hamiltonian, but ismore amenable to approximation results.• A proof that our framework interpolates between the spars-est cut objective and the cluster deletion problem, as weincrease a single resolution parameter, λ.• Several successful algorithms for optimizing our new ob-jective function in both theory and practice, including a2-approximation for cluster deletion, which improves uponthe previous best approximation factor of 3.• A demonstration of our methods in a number of clusteringapplications, including social network analysis and miningcliques in collaboration networks.

2 BACKGROUND AND RELATEDWORKLetG be an undirected and unweighted graph on n nodesV , withmedges E. For all v ∈ V , let dv be node v’s degree. Given S ⊆ V ,let S̄ = V \S be the complement of S and vol(S ) = ∑v ∈S dv be itsvolume. For every two disjoint sets of vertices S,T ⊆ V , cut(S,T )indicates the number of edges between S and T . If T = S̄ , wewrite cut(S ) = cut(S, S̄ ). Let ES denote the interior edge set of S .The edge density of a cluster is density(S ) = |ES |/

( |S |2), the ratio

between the number of edges to the number of pairs of nodes in S .By convention, the density of a single node is 1. We now presentbackground and related work that is foundational to our results,including definitions for several common clustering objectives.

2.1 Correlation ClusteringAn instance of correlation clustering is given by a signed graphwhere every pair of nodes i and j possesses two non-negativeweights, w+i j and w

−i j , to indicate how similar and how dissimi-

lar i and j are, respectively. Typically only one of these weightsis nonzero for each pair i, j. The objective can be expressed as aninteger linear program (ILP):

minimize∑i

at a node and following outgoing edges uniformly at random, willend up in the cluster it started in after a random walk of length t .This t serves as a resolution parameter, since the walker will tendto “wander" farther when t is increased, leading to the formationof larger clusters when the stability is maximized. Delvenne et al.showed that objective (3) is equivalent to a linearized version ofthe stability measure for a specific range of time steps t [15].

2.3 Sparsest Cut and Normalized CutOne measure of cluster quality in an unsigned network G is thesparsest cut score, defined for a set S ⊆ V to be ϕ (S ) = cut(S )/|S | +cut(S )/|S̄ | = n · cut(S )/( |S | |S̄ |). Smaller values for ϕ (S ) are desir-able, since they indicate that S , in spite of its size, is only looselyconnected to the rest of the graph. This measure differs by atmost a factor of two from the related edge expansion measure:cut(S )/(min{|S |, |S̄ |}). If we replace |S | with vol(S ) in these twoobjectives, we obtain the normalized cut and the conductance mea-sure respectively. In our work we focus on a multiplicative scalingof the sparsest cut objective that we call the scaled sparsest cut:ψ (S ) = ϕ (S )/n = cut(S )/( |S | |S̄ |), which is identical to sparsest cutin terms of multiplicative approximations. The best known approx-imation for finding the minimum sparsest cut of a graph is anO (

√logn)-approximation algorithm due to Arora et al. [2].

2.4 Cluster DeletionCluster deletion is the problem of finding a minimum number ofedges in G to be deleted in order to convert G into a disjoint set ofcliques. This problem was first studied by Ben-Dor et al. [6], laterformalized in the work of Natanzon et al. [29], who proved it isNP-hard, and Shamir et al. [36], who showed it is APX-hard. Thelatter studied the problem in conjunction with other related edge-modification problems, including cluster completion and clusterediting. Numerous fixed parameter tractability results are known forcluster deletion [8, 14, 23, 24], as well many results regarding specialgraphs for which the problem can be solved in polynomial time [10,11, 17, 21]. Dessmark et al. proved that recursively findingmaximumcliques will return a clustering with a cluster deletion score withina factor 2 of optimal [17], though in general this procedure is NP-hard.

3 THEORETICAL RESULTSOur novel clustering framework takes an unsigned graph G =(V ,E) and converts it into a signed graph G ′ = (V ,E+,E−) on thesame set of nodes, V , for a fixed clustering resolution parameterλ ∈ (0, 1). PartitioningG ′ with respect to the correlation clusteringobjective will then induce a clustering onG . To construct the signedgraph, we first introduce a node weight wv for each v ∈ V . If(i, j ) ∈ E, we place a positive edge between nodes i and j inG ′, withweight (1−λwiw j ). For (i, j ) < E, we place a negative edge between iand j in G ′, with weight λwiw j . We consider two different choicesfor node weightswv : settingwv = 1 for allv (standard) or choosingwv = dv (degree-weighted). In Figure 1 we illustrate the processof converting G into the LambdaCC signed graph, G ′. The goal ofLambdaCC is to find the clustering that minimizes disagreementsin G ′, or equivalently minimizes the following objective function

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3�

2�

4�

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 41��

1���

�

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1��

1�6�1�6�

4�

2�

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Figure 1:We convert a toy graph (left) into a signed graph forstandard (middle) and degree-weighted (right) LambdaCC.Dashed red lines indicate negative edges.

expressed in terms of edges and non-edges in G:

λCC (x ) =∑

(i, j )∈E(1 − λwiw j )xi j +

∑(i, j )

3.3 Connection to Sparsest CutGivenG and λ, the weight of positive-edge mistakes in the standardLambdaCC objective made by a two-clustering C = {S, S̄ } equalsthe weight of edges crossing the cut: (1− λ) cut(S ). To compute theweight of negative-edge mistakes, we take the weight of all nega-tive edges in the entire network, λ

((n2)− |E |

), and then subtract

the weight of negative edges between S and S̄ : λ(|S | |S̄ | − cut(S )

).

Adding together all terms, we find that the LambdaCC objectivefor this clustering is

cut(S, S̄ ) − λ |S | |S̄ | + λ(n

2)− λ |E | . (7)

Note that if we minimize (7) over all 2-clusterings, we solve thedecision version of the minimum scaled sparsest cut problem: afew steps of algebra confirm that there is some set S ⊆ V withψ (S ) = cut(S )/( |S | |S̄ |) < λ if and only if (7) is less than λ

(n2)− λ |E |.

In a similar way we can show that objective (6) is equivalent to

min12

k∑i=1

cut(Si ) −λ

2

k∑i=1|Si | |S̄i | + λ

(n

2

)− λ |E | , (8)

where we minimize over all clusterings of G (note that the num-ber of clusters k is determined automatically by optimizing theobjective). In this case, optimally solving objective (8) will tell uswhether we can find a clustering C = {S1, S2, . . . , Sk } such that∑ki=1 cut(Si , S̄i )/

(∑kj=1 |Sj | |S̄j |

)< λ. Hence LambdaCC can be

viewed as a multi-cluster generalization of the decision versionof minimum sparsest cut. We now prove an even deeper connec-tion between sparsest cut and LambdaCC. Using degree-weightedLambdaCC yields an analogous result for normalized cut.

Theorem 3.2. Let λ∗ be the minimum scaled sparsest cut valuefor graph G.

(a) For all λ > λ∗, optimal solution (8) partitions G into twoor more clusters, each of which has scaled sparsest cut ≤ λ.There exists some λ′ > λ∗ such that the optimal clustering forLambdaCC is the minimum sparsest cut partition.

(b) For λ ≤ λ∗, it is optimal to place all nodes into a single cluster.

Proof. Statement (a) Let S∗ be some subset ofV that induces asparsest cut, i.e.,ψ (S∗) = cut(S∗)/( |S∗ | |S̄∗ |) = λ∗. The LambdaCCobjective corresponding to C = {S∗, S̄∗} is

cut(S∗) − λ |S∗ | |S̄∗ | + λ(n

2

)− λ |E | . (9)

When minimizing objective (8), we can always obtain a score ofλ(n

2)− λ |E | by placing all nodes into a single cluster. Note however

that the the score of clustering {S∗, S̄∗} in expression (9) is strictlyless than λ

(n2)− λ |E | for all λ > λ∗. Even if {S∗, S̄∗} is not optimal,

this means that when λ > λ∗, we can do strictly better than placingall nodes into one cluster. In this case let C∗ be the optimal Lamb-daCC clustering and consider two of its clusters: Si and Sj . Theweight of disagreements between Si and Sj is equal to the numberof positive edges between them times the weight of a positive edge:(1 − λ) cut(Si , Sj ). Should we form a new clustering by merging Siand Sj , these positive disagreements will disappear; in turn, wewould introduce λ |Si | |Sj | − λ cut(Si , Sj ) new mistakes, being nega-tive edges between the clusters. Because we assumed C∗ is optimal,

we know that we cannot decrease the objective by merging two ofthe clusters, implying that

(1 − λ) cut(Si , Sj ) −(λ |Si | |Sj | − λ cut(Si , Sj )

)= cut(Si , Sj ) − λ |Si | |Sj | ≤ 0 .

Given this, we fix an arbitrary cluster Si and perform a sum overall other clusters to see that∑

j,i cut(Si , Sj ) −∑j,iλ |Si | |Sj | ≤ 0

=⇒ cut(Si , S̄i ) − λ |Si | |S̄i | ≤ 0 =⇒ cut(Si , S̄i )/(|Si | |S̄i |

)≤ λ ,

proving the desired upper bound on scaled sparsest cut.Since G is a finite graph, there are a finite number of scaled

sparsest cut scores that can be induced by a subset of V . Let λ̃ bethe second-smallest scaled sparsest cut score achieved, so λ̃ > λ∗.If we set λ′ = (λ∗ + λ̃)/2, then the optimal LambdaCC clusteringproduces at least two clusters, since λ′ > λ∗, and each cluster hasscaled sparsest cut at most λ′ < λ̃. By our selection of λ̃, all clustersreturned must have scaled sparsest cut exactly equal to λ∗, whichis only possible if the clustering returned has two clusters. Hencethis clustering is a minimum sparsest cut partition of the network.

Statement (b) If λ < λ∗, forming a single cluster must be opti-mal, otherwise we could invoke Statement (a) to assert the existenceof some nontrivial cluster with scaled sparsest cut less than or equalto λ < λ∗, contradicting the minimality of λ∗. If λ = λ∗, forming asingle cluster or using the clustering C = {S∗, S̄∗} yield the sameobjective score, which is again optimal for the same reason. □

3.4 Connection to Cluster DeletionFor large λ, our problem becomes more similar to cluster deletion.We can reduce any cluster deletion problem to correlation cluster-ing by taking the input graph G and introducing a negative edgeof weight “∞” between every pair of non-adjacent nodes. Thisguarantees that optimally solving correlation clustering will yieldclusters that all correspond to cliques inG . Furthermore, the weightof disagreements will be the number of edges in G that are cut, i.e.,the cluster deletion score. We can obtain a generalization of clusterdeletion by instead choosing the weight of each negative edge tobe α < ∞. The corresponding objective is∑

(i, j )∈E+xi j +

∑(i, j )∈E−

α (1 − xi j ) . (10)

If we substitute α = λ/(1 − λ) we see this differs from objective (6)only by a multiplicative constant, and is therefore equivalent interms of approximation. When α > 1, putting dissimilar nodestogether will be more expensive than cutting positive edges, so wewould expect that the clustering which optimizes the LambdaCCobjective will separateG into dense clusters that are “nearly” cliques.We formalize this with a simple theorem and corollary.

Theorem 3.3. If C minimizes the LambdaCC objective for theunsigned network G = (V ,E), then the edge density of every clusterin C is at least λ.

Proof. Take a cluster S ∈ C and consider what would happen ifwe broke apart S so that each of its nodes were instead placed intoits own singleton cluster. This means we are now making mistakesat every positive edge previously in S , which increases the weight

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Degree-weighted

StandardSparsest Cut Cluster DeletionCorrelation Clustering

Normalized Cut Modularity

1

2m

0 1 0 1m

m + 1

⇢⇤

�⇤� =

1/2

Figure 2: LambdaCC is equivalent to several other objectivesfor specific values of λ ∈ (0, 1). Values λ∗ and ρ∗ are notknown a priori, but can be obtained by solving LambdaCCfor increasingly smaller values of λ.

Table 1: The best approximation factors known for stan-dard LambdaCC, for λ ∈ (0, 1), both for minimizing dis-agreements andmaximizing agreements.We contribute twoconstant-factor approximations for minimizing disagree-ments when λ > 1/2.

λ ∈ (0, 1/2) λ = 1/2 λ ∈ (1/2, 1)Max-Ag. 0.7666 [37] PTAS [4] 0.7666 [37]

Min-Dis. O (logn) [12, 16, 20] 2.06 [13] 3(2 : λ > mm+1

)

of disagreements by (1 − λ) |ES |. On the other hand, there are nolonger negative mistakes between nodes in S , so the LambdaCCobjective would simultaneously decrease by λ

(( |S |2)− |ES |

). The

total change in the objective made by pulverizing S is

(1 − λ) |ES | − λ(( |S |

2)− |ES |

)= |ES | − λ

( |S |2),

which must be nonnegative, since C is optimal, so |ES | −λ( |S |

2)≥ 0,

which implies density(S ) = |ES |/( |S |

2)≥ λ. □

Corollary 3.4. Let G havem edges. For every λ > m/(m + 1),optimizing LambdaCC is equivalent to optimizing cluster deletion.

Proof. All output clusters must have density at leastm/(m + 1),which is only possible if the density is actually 1, sincem is the totalnumber of edges in the graph. Therefore all clusters are cliquesand the LambdaCC and cluster deletion objectives differ only by amultiplicative constant (1 − λ). □

3.5 Equivalences and ApproximationsWe summarize the equivalence relationships between LambdaCCand other objectives in Figure 2. Accompanying this, Table 1 out-lines the best-known approximation results both for maximizingagreements and minimizing disagreements for the standard Lamb-daCC signed graph. For degree-weighted LambdaCC, the best-known approximation factors for all λ are O (logn) [12, 16, 20]for minimizing disagreements, and 0.7666 for maximizing agree-ments [37]. Thus, LambdaCC is more amenable to approximationthan modularity (and relatives) because of additive constants.

Algorithm 1 threeLP

Input: Signed graph G ′ = (V ,E+,E−), λ ∈ (0, 1)Output: Clustering C of G ′Solve the LP relaxation of ILP (6), obtaining distances (xi j )Let G̃ = (V , F+, F−), where F+ = {(i, j ) : xi j < 1/3} and

5: F− = {(i, j ) : xi j ≥ 1/3}Apply Pivot on G̃.

4 ALGORITHMSWe present several new algorithms, tailored specifically to ourLambdaCC framework; some come with approximation guarantees,some are designed for efficiency.1 Our first two methods rely ona key theorem of van Zuylen and Williamson, which can be usedto prove approximation results for different cases of correlationclustering using pivoting algorithms, which operate by selectinga pivot node k , clustering it with all its positive neighbors, andrecursing on the rest of the graph. If pivots are chosen uniformly atrandom this corresponds to the Pivot algorithm of Ailon et al. [1].We state the theorem here for completeness, with minor changesto match our notation and presentation:

Theorem 4.1. (Theorem 3.1 in [38]) Let G = (V ,W +,W −) be asigned, weighted graph where each pair of nodes (i, j ) has positive andnegative weights w+i j ∈W

+ and w−i j ∈W−. Given a set of LP costs

{ci j : i ∈ V , j ∈ V , i , j}, and an unweighted graph G̃ = (V , F+, F−)satisfying the following assumptions:

(i) w−i j ≤ αci j for all (i, j ) ∈ F+ and

w+i j ≤ αci j for all (i, j ) ∈ F−,

(ii) w+i j +w+jk +w

−ik ≤ α

(ci j + c jk + cik

)for every bad triangle in G̃: (i, j ), (j,k ) ∈ F+, (i,k ) ∈ F−,

then applying Pivot on G̃ will return a solution that costs α∑i 1/2.

1For the initial submission of our work we presented a 5-approximation for LambdaCCwhen λ > 1/2 and a 4-approximation for cluster deletion by altering the approachof Charikar et al. [12]. Here we show improved approximations that we developedbased on a helpful suggestion from an anonymous reviewer. We include the originalapproximation results in the full version of our paper [39].

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Proof. We begin by stating the correspondence between thenotation of Theorem 4.1 and the edge weights and LP costs forLambdaCC. The graph G ′ = (V ,E+,E−) is made up of positiveedges of weight (1 − λ) and negative edges with weight λ, so

ci j = (1 − λ)xi j and (w+i j ,w−i j ) = (1 − λ, 0) if (i, j ) ∈ E

+

ci j = λ(1 − xi j ) and (w+i j ,w−i j ) = (0, λ) if (i, j ) ∈ E

−.

By construction, if (i, j ) ∈ F− then xi j < 1/3, otherwise (i, j ) ∈ F+and we know xi j ≥ 1/3. The first two inequalities we need to checkfor Theorem 4.1 are

w−i j ≤ αci j for all (i, j ) ∈ F+ (11)

w+i j ≤ αci j for all (i, j ) ∈ F− (12)

where α = 3. If (i, j ) ∈ F+ ∩ E+, then w−i j = 0 and inequality (11)is trivial since the left hand side is zero. Similarly, inequality (12)is trivial if (i, j ) ∈ F− ∩ E−. Assume then that (i, j ) ∈ F+ ∩ E−.Then w−i j = λ and ci j = λ(1 − xi j ), and we know xi j < 1/3 =⇒(1 − xi j ) > 2/3. Therefore:

w−i j = λ < 3λ (2/3) < 3λ(1 − xi j ) = αci j .

On the other hand, if (i, j ) ∈ F− ∩ E+, then w+i j = (1 − λ), ci j =(1 − λ)xi j , and xi j ≥ 1/3, so we see:

w+i j = (1 − λ) = 3(1 − λ) (1/3) ≤ 3(1 − λ)xi j = αci j .

This concludes the proof for inequalities (11) and (12). Next weconsider a triplet of nodes {i, j,k } where (i, j ) ∈ F+, (j,k ) ∈ F+ but(i,k ) ∈ F−. This is called a bad triangle since we will have to violateat least one of these edges when clustering G̃. We must prove that:

w+i j +w+jk +w

−ik ≤ 3

(ci j + c jk + cik

), (13)

which is somewhat tedious to show. The variables in (13) are highlydependent on the types of edges shared among nodes {i, j,k } in theoriginal signed graph G ′; there are two possibilities for each edgefor a total of eight cases. We give a proof here for the first case.

Case 1: Assume (i, j ) ∈ E+, (j,k ) ∈ E+, and (i,k ) ∈ E−. For thiscase, we note that (ci j , c jk , cik ) = ((1−λ)xi j , (1−λ)x jk , λ(1−xik ))and (w+i j ,w

+jk ,w

−ik ) = (1 − λ, 1 − λ, λ). By our construction of F

+

and F− we know that xi j < 1/3, x jk < 1/3, and by the triangleinequality we have xik ≤ xi j + x jk < 2/3. Combining these facts:

3(ci j + c jk + cik ) = 3((1 − λ) (xi j + x jk ) + λ(1 − xik )

)≥ 3 ((1 − λ)xik + λ(1 − xik )) = 3 ((1 − 2λ)xik + λ)> 3 ((1 − 2λ)2/3 + λ) = 2 − λ = w+i j +w

+jk +w

−ik .

We rely above on the fact that (1−2λ) < 0, which restricts our proofto cases where λ > 1/2. Due to space constraints we defer the proofof the other seven cases to the full version of the paper [39]. □

4.2 2-Approximation for Cluster DeletionIn order to obtain an approximation algorithm for cluster deletionwe alter threeLP in two ways:

• Begin by solving the LP relaxation of cluster deletion:minimize

∑(i, j )∈E+ xi j

subject to xi j ≤ xik + x jk for all i, j,kxi j ∈ [0, 1] for all (i, j ) ∈ E+xi j = 1 for all (i, j ) ∈ E−

(14)

• Define F+ = {(i, j ) : xi j < 1/2}, F− = {(i, j ) : xi j ≥ 1/2}.As before, we then run Pivot on G̃ = (V , F+, F−). We name theresulting procedure twoCD, and prove the following result:

Theorem 4.3. Algorithm twoCD returns a 2-approximation forcluster deletion.

Proof. First observe that no negative edge mistakes are madeby performing Pivot on G̃: if k is the pivot and i, j are two positiveneighbors of k in G̃, then xik < 1/2, x jk < 1/2, and xi j ≤ xik +x jk < 1. Since all distances are less than one, all nodes must sharepositive edges inG ′, because xi j = 1 for any (i, j ) ∈ E−. The rest ofthe proof follows from showing that the newly constructed graph G̃satisfies the assumptions of Theorem 4.1 when α = 2. For the sakeof space we defer details to the full version of the paper [39]. □

This result is particularly interesting given that no constant-factor approximation for cluster deletion has been explicitly pre-sented in previous literature.2 Until now, the best approximationsfor correlation clustering were stronger than any result knownfor cluster deletion; our result indicates that the latter problem isperhaps the easier of the two to approximate.

4.3 Scalable Heuristic AlgorithmsAs a counterpart to the previous approximation-driven approaches,we provide fast algorithms for LambdaCC based on greedy localheuristics. The first of these is GrowCluster, which iterativelyselects an unclustered node uniformly at random and forms a clusteraround it by greedily aggregating adjacent nodes, until there is nomore improvement to the LambdaCC objective.

A variant of this, called GrowCliqe, is specifically designedfor cluster deletion. It monotonically improves the LambdaCCobjective, but differs in that at each iteration it uniformly at randomselects k unclustered nodes, and greedily grows cliques around eachof these seeds. The resulting cliques may overlap: at each iterationwe select only the largest of such cliques.

Finally, since the LambdaCC and Hamiltonian objectives areequivalent, we can use previously developed algorithms and soft-ware for modularity-like objectives with a resolution parameter. Inparticular we employ adaptations of the Louvain method, an algo-rithm developed by Blondel et al. [7]. It iteratively visits each nodein the graph and moves it to an adjacent cluster, if such a movegives a locally maximum increase in the modularity score. Thiscontinues until no move increases modularity, at which point theclusters are aggregated into super-nodes and the entire process isrepeated on the aggregated network. By adapting the original Lou-vain method to make greedy local moves based on the LambdaCCobjective, rather than modularity, we obtain a scalable algorithm

2The results of van Zuylen and Williamson for constrained correlation clustering canbe used to obtain a 3-approximation for cluster deletion (Theorem 4.2 in [38]), thoughcluster deletion is not mentioned explicitly in their work.

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that is known to provide good approximations for a related objec-tive, and additionally adapts well to changes in our parameter λ.We refer to this as Lambda-Louvain. Both standard and degree-weighted versions of the algorithm can be achieved by employingexisting generalized Louvain algorithms (e.g., the GenLouvain al-gorithm of Jeub et al. http://netwiki.amath.unc.edu/GenLouvain/).Our heuristic algorithms satisfy the following guarantee:

Theorem 4.4. For every λ, both the algorithms GrowCluster andLambda-Louvain either place all nodes in one cluster or they produceclusters that have scaled sparsest cut bounded above by λ.

An analogous result for normalized cut holds when the algo-rithms greedily optimize degree-weighted LambdaCC. We providea detailed proof in the full version of the paper [39].

5 EXPERIMENTSWe begin by comparing our new methods against existing correla-tion clustering algorithms on several small networks. This showsour algorithms for LambdaCC are superior to common alternatives.We then study how well-known graph partitioning algorithmsimplicitly optimize the LambdaCC objective for various λ. In sub-sequent experiments, we apply our methods to clique detection incollaboration and gene networks, and to social network analysis.

5.1 LambdaCC on Small NetworksIn our first experiment, we show that Lambda-Louvain is thebest general-purpose correlation clustering method for minimizingthe LambdaCC objective. We test this on four small networks:Karate [40], Les Mis [26], Polbooks [27], and Football [22]. Figure 3shows the performance of our algorithms, Pivot, and ICM, for arange of λ values. Pivot is the fast algorithm of Ailon et al. [1],which selects a uniform random node and clusters its neighborswith it. ICM is the energy-minimization heuristic algorithm ofBagon and Galun [3].

We find that threeLP gives much better than a 3-approximationin practice. Pivot is much faster, but performs poorly for λ closeto 0 or 1. ICM is also much quicker than solving the LP relaxation,but is still limited in scalability, as it is intended for correlationclustering problems where most edge weights are 0 (not the casefor LambdaCC). On the other hand, GrowCluster and Lambda-Louvain are scalable and give good approximations for all inputnetworks and values of λ.

5.2 Standard Clustering AlgorithmsMany existing clustering algorithms implicitly optimize differentparameter regimes of the LambdaCC objective. We show this byrunning several clustering algorithms on a 1000-node syntheticgraph generated from the BTER model [35]. We do the same on thelargest component (4158 nodes) of the ca-GrQc collaboration net-work from the arXiv e-print website. We then compute LambdaCCobjective scores for each algorithm for a range of λ values. Wefirst cluster each graph using Graclus [18] (forming two clusters),Infomap [9], and Louvain [7]. To form dense clusters, we also parti-tion the networks by recursively extracting the maximum clique(called RMC), and by recursively extracting the maximum quasi-clique (RMQC), i.e., the largest set of nodes with inner edge density

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Figure 3: We optimize the standard LambdaCC objectivewith five correlation clustering algorithms on four smallnetworks. The y-axis reports the ratio between each algo-rithm’s score and the lower bound on the optimal objectivedetermined by solving the LP relaxation. Lambda-Louvain(black) andGrowCluster (purple) performwell for all λ, inaddition to being the most scalable algorithms. In each plot,a dashed vertical line indicates the optimal scaled sparsestcut value, λ∗, for that network.

bounded below by some ρ < 1 (here we use ρ = 0.6). The last twoprocedures must solve an NP-hard objective at each step, but forreasonably sized graphs there is available clique and quasi-cliquedetection software [28, 34].

After each algorithm has produced a single clustering of theunsigned network, we evaluate how the LambdaCC objective scoreof that clustering changes as we vary λ. This allows us to observewhether an algorithm is effective in approximating the LambdaCCobjective for a certain range of λ values. For comparison we runLambda-Louvain for each different value of λ. Figure 4 reports theratio between the LambdaCC objective score of each clusteringand the LP-relaxation lower bound. These plots illustrate that ourframework and Lambda-Louvain effectively interpolate betweenseveral well-established strategies in graph partitioning, and canserve as a good proxy for any clustering task for which any one ofthese algorithms is known to be effective.

5.3 Cluster Deletion in Large CollaborationNetworks

The connection between LambdaCC and cluster deletion providesa new approach for enumerating groups in large networks. Here weevaluate GrowCliqe for cluster deletion and use it to cluster twolarge collaboration networks, one formed from a snapshot of theauthor-paper DBLP dataset in 2007, and the other generated using

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http://netwiki.amath.unc.edu/GenLouvain/

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Table 2: Cluster deletion scores for GrowCliqe (GC), Pro-jectCliqe (PC) and RMC on two collaboration networks.GrowCliqe is unaware of the underlying player-projectnetwork, and does not solve an NP-hard objective at eachiteration, yet returns very good results. Best score for eachdataset is emphasized.

Graph Nodes Edges GC PC RMC

Actors 341,185 10,643,420 8,085,286 8,086,715 8,087,241DBLP 526,303 1,616,814 945,489 946,295 944,087

actor-movie information from the NotreDame actors dataset [5].The original data in both cases is a bipartite network indicatingwhich players (i.e., authors or actors) have parts in different projects(papers or movies respectively). We transform each bipartite net-work into a graph in which nodes are players and edges representcollaboration on a project.

At each iteration GrowCliqe grows 500 (possibly overlapping)cliques from random seeds and selects the largest to be included inthe final output. We compare against RMC, an expensive methodwhich provably returns a 2-approximation to the optimal clusterdeletion objective [17]. We also design ProjectCliqe, a methodthat looks at the original bipartite network and recursively identifiesthe project associated with the largest number of players not yetassigned to a cluster. These players form a clique in the collaborationnetwork, so ProjectCliqe clusters them together, then repeatsthe procedure on remaining nodes.

Table 2 shows that GrowCliqe outperforms ProjectCliqein both cases, and slightly outperforms RMC on the actor network.Our method is therefore competitive against two algorithms thatin some sense have an unfair advantage over it: ProjectCliqeemploys knowledge not available to GrowCliqe regarding theoriginal bipartite dataset, and RMC performs very well mainlybecause it solves an NP-hard problem at each step.

5.4 Clustering Yeast GenesThe study of cluster deletion and cluster editing (equivalent to ±1-correlation clustering, or to λ = 1/2) was originally motivated byapplications to clustering genes using expression patterns [6, 36].Standard LambdaCC is a natural framework for this, since it gener-alizes both objectives and interpolates between them as λ rangesfrom 1/2 to m/(m + 1). We cluster genes of the Saccharomycescerevisiae yeast organism using microarray expression data col-lected by Kemmeren et al. [25]. With the 200 expression valuesfrom the dataset, we compute correlation coefficients between allpairs of genes. We threshold these at 0.9 to obtain a small graphof 131 nodes corresponding to unique genes, which we cluster withtwoCD. For this cluster deletion experiment, our algorithm returnsthe optimal solution: solving the LP-relaxation returns a solutionthat is in fact integral. We validate each clique of size at least threereturned by twoCD against known gene-association data from theSaccharomyces Genome Database (SGD) and the String ConsortiumDatabase (see Table 3). With one exception, these cliques matchgroups of genes that are known to be strongly associated, accordingto at least one validation database. The exception is a cluster withfour genes (YHR093W, YIL171W, YDR490C, and YOR225W), threeof which, according to the SGD are not known to be associated withany Gene Ontology term. We conjecture that this may indicate arelationship between genes not previously known to be related.

Table 3: We list cliques of size ≥ 3 in the optimal clustering(found by twoCD) of a network of 131 yeast genes. We vali-date each cluster using the SGDGO slimmapper tool, whichidentifies any GO term (function, process, or component ofthe organism) for a given gene. We list one GO term sharedby all genes in the cluster, if one exists. The Term % columnreports the percentage of all genes in the organism associ-ated with this term. A low percentage indicates a cluster ofgenes that share a GO term that is not widely shared amongother genes. The final column shows theminimumString as-sociation score between every pair of genes in the cluster, anumber between 0 and 1000 (higher is better). Any non-zeroscore is a strong indication of gene association, as the ma-jority of String scores between genes of S. cerevisiae are zero.All clusters, except the third, either have a high minimumString score or are all associated with a specific GO term.

Clique # Size Shared GO term Term % String

1 6 nucleus 34.3 02 4 nucleus 34.3 2023 4 N/A - 04 4 vitamin metabolic process 0.7 9805 3 cytoplasm 67.0 9906 3 cytoplasm 67.0 9987 3 N/A - 9628 3 cytoplasm 67.0 9969 3 N/A - 97310 3 transposition 1.7 0

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5.5 Social Network Analysis with LambdaCCClustering a social network using a range of resolution parameterscan reveal valuable insights about how links are formed in thenetwork. Here we examine several graphs from the Facebook100dataset, each of which represents the induced subgraph of theFacebook network corresponding to a US university at some pointin 2005. The networks come with anonymized meta-data, reportingattributes such as major and graduation year for each node. Whilemeta-data attributes are not expected to correspond to ground-truthcommunities in the network [32], we do expect them to play a rolein how friendship links and communities are formed. In this exper-iment we illustrate strong correlations between the link structureof the networks and the dorm, graduation year, and student/facultystatus meta-data attributes. We also see how these correlations arerevealed, to different degrees, depending on our choice of λ.

Given a Facebook subgraph with n nodes, we cluster it withdegree-weighted Lambda-Louvain for a range of λ values between0.005/n and 0.25/n. In this clustering, we refer to two nodes inthe same cluster as an interior pair. We measure how well a meta-data attribute M correlates with the clustering by calculating theproportion of interior pairs that share the same value forM . Thisvalue, denoted by P (M ), can also be interpreted as the probabilityof selecting an interior pair uniformly at random and finding thatthey agree on attribute M . To determine whether the probabilityis meaningful, we compare it against a null probability P (M̃ ): theprobability that a random interior pair agree at a fake meta-dataattribute M̃ . We assign to each node a value for the fake attribute M̃by performing a random permutation on the vector storing val-ues for true attribute M . In this way, we can compare each trueattributeM against a fake attribute M̃ that has the same exact pro-portion of nodes with each attribute value, but does not impart anytrue information regarding each node.

In Figure 5 we plot results for each of the three attributesM ∈{dorm, year, s/f (student/faculty)} on four Facebook networks, as λis varied. In all cases, we see significant differences between P (M )and P (M̃ ). In general, P (year ) and P (s/f ) reach a peak at smallvalues of λ when clusters are large, whereas P (dorm) is highestwhen λ is large and clusters are small. This indicates that the firsttwo attributes are more highly correlated with large sparse commu-nities in the network, whereas sharing a dorm is more correlatedwith smaller, denser communities. Caltech, a small residential uni-versity, is an exception to these trends and exhibits a much strongercorrelation with the dorm attribute, even for very small λ.

6 DISCUSSIONWe have introduced a new clustering framework that unifies severalother commonly-used objectives and offers many attractive theo-retical properties. We prove that our objective function interpolatesbetween the sparsest cut objective and the cluster deletion problem,as we vary a single input parameter, λ. We give a 3-approximationalgorithm for our objective when λ ≥ 1/2, and a related methodwhich improves the best approximation factor for cluster deletionfrom 3 to 2. We also give scalable procedures for greedily improvingour objective, which are successful in a wide variety of clustering ap-plications. These methods are easily modified to add must-clusterand cannot-cluster constraints, which makes them amenable to

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Figure 5: On four university Facebook graphs, we illustratethat the dorm (red), graduation year (green), and student/-faculty (S/F) status (blue) meta-data attributes all correlatehighly with the clustering found by Lambda-Louvain foreach λ. Above the x-axis we show the number of clustersformed,which increaseswith λ. They-axis reports the proba-bility that two nodes sharing a cluster also share an attributevalue. Each attribute curve is compared against a null prob-ability, shown as a dashed line of the same color. The largegaps between each attribute curve and its null probabilityindicate that the link structure of each network is highlycorrelated with these attributes. In general, probabilities foryear and s/f status are highest for small λ, whereas dormhas ahigher correlation with smaller, denser communities in thenetwork. Caltech is an exception to the general trend; seethe main text for discussion.

many applications. In future work, we will continue exploring ap-proximations when λ < 1/2.

ACKNOWLEDGEMENTSThis work was supported by several funding agencies: Nate Veldtand David Gleich are supported by NSF award IIS-154648, DavidGleich is additionally supported by NSF awards CCF-1149756 andCCF-093937 as well as the DARPA Simplex program and the SloanFoundation. Anthony Wirth is supported by the Australian Re-search Council. We thank Flavio Chierichetti for several helpfulconversations and also thank the anonymous reviewers for severalhelpful suggestions for improving our work, in particular for men-tioning connections to the work of van Zuylen andWilliamson [38],which led to significantly improved approximation results.

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https://doi.org/10.1126/science.286.5439.509https://doi.org/10.1126/science.286.5439.509http://stacks.iop.org/1742-5468/2008/i=10/a=P10008http://stacks.iop.org/1742-5468/2008/i=10/a=P10008https://doi.org/10.1016/j.ipl.2011.05.003https://doi.org/10.1007/978-3-642-11269-0_8https://doi.org/10.1007/978-3-540-39658-1_21https://doi.org/10.1007/3-540-44849-7_17https://doi.org/10.1007/s00453-004-1090-5http://networkdata.ics.uci.edu/data.php?d=polbookshttp://networkdata.ics.uci.edu/data.php?d=polbooks"https://doi.org/10.1007/978-3-540-87481-2_3""https://doi.org/10.1007/978-3-540-87481-2_3"https://doi.org/10.1126/sciadv.1602548http://arxiv.org/abs/http://advances.sciencemag.org/content/3/5/e1602548.full.pdfhttps://doi.org/10.1137/14100018Xhttps://doi.org/10.1137/14100018Xhttps://doi.org/10.1103/PhysRevE.85.056109https://doi.org/10.1287/moor.1090.0385http://arxiv.org/abs/https://doi.org/10.1287/moor.1090.0385https://arxiv.org/abs/1712.05825

Abstract1 Introduction2 Background and Related Work2.1 Correlation Clustering2.2 Modularity and the Hamiltonian2.3 Sparsest Cut and Normalized Cut2.4 Cluster Deletion

3 Theoretical Results3.1 Connection to Modularity3.2 Standard LambdaCC3.3 Connection to Sparsest Cut3.4 Connection to Cluster Deletion3.5 Equivalences and Approximations

4 Algorithms4.1 3-Approximation for LambdaCC4.2 2-Approximation for Cluster Deletion4.3 Scalable Heuristic Algorithms

5 Experiments5.1 LambdaCC on Small Networks5.2 Standard Clustering Algorithms5.3 Cluster Deletion in Large Collaboration Networks5.4 Clustering Yeast Genes5.5 Social Network Analysis with LambdaCC

6 DiscussionReferences