Estimating minimum multiway cuts via spectral relaxation

Clusters (in other words, modules or communities) of graphs are typical (usually, loosely connected) subsets of vertices that can be identified, for example, with social groups or interacting enzymes in social or metabolic networks, respectively; they form special partition classes of the vertices. To measure the performance of a clustering, different kinds of multiway cuts are introduced and estimated by means of Laplacian spectra. The key motif of these estimations is that minima and maxima of the quadratic placement problems of Section 1.1 are attained on some appropriate eigenspaces of the Laplacian, while optimal multiway cuts are special values of the same quadratic objective function realized by step-vectors.

Hence, the optimization problem, formulated in terms of the Laplacian eigenvectors, is the continuous relaxation of the underlying maximum or minimum multiway cut problem.

For a fixed integer1≤k≤n, letPk= (V1, . . . , Vk)be aproper k-partition of the vertices, where the disjoint, non-empty vertex subsets V1, . . . , Vk will be referred to as clusters or modules. Let Pk denote the set of all k-partitions. Optimization over Pk is usually NP-complete, except some special classes of graphs.

Definition 2 The weighted cut between the non-empty vertex-subsetsU, T ⊂V of the edge-weighted graph G= (V,W) is

w(U, T) =X

i∈U

X

j∈T

wij.

The minimumk-way cut ofGis

mincutk(G) = min

Pk∈Pk

k−1

X

a=1

Xk b=a+1

w(Va, Vb). (1.4)

For a simple graphG, Fiedler [Fid73] called the quantitymincut2(G)the edge-connectivity ofG, because it is equal to the minimum number of edges that should be removed to make G disconnected. He used the notation e(G) for the edge-connectivity of the simple graph G, andv(G)for its vertex-connectivity (minimum how many vertices should be removed to make Gdisconnected). In his breakthrough papers [Fid72, Fid73], Fiedler proved that for any graphGonnvertices, that differs from the complete graphKn, the relation

λ1≤v(G)≤e(G) (1.5)

holds. In [Fid73], he also provided two lower estimates for λ1 bye(G):

λ1≥2e(G)(1−cosπ

n) (1.6)

and

λ1≥C1e(G)−C2dmax, (1.7)

where C1= 2(cos^π_n −cos^2π_n),C2= 2 cos^π_n(1−cos^π_n), anddmax= maxidi is the maximum vertex-degree. Compared to (1.5), this estimation makes sense in then≥3case. The bound of (1.7) is tighter than that of (1.6) if and only ife(G)≥¹₂dmax. The two estimates are equal and sharp for the path graphPn withe(G) = 1andλ1= 2(1−cos^π_n). The path graph can be split into two clusters by removing any of its edges, however, we would not state that it has two underlying clusters. The forthcoming ratio cut ofPn is minimized by removing the middle edge (for evenn) or one of the middle edges (for odd n), thus, it provides balanced clusters.

Because of this two-sided relation betweenλ1 ande(G), the smallest positive Laplacian eigenvalue of a connected graph is able to detect the strength of its connectivity; therefore, Fiedler calledλ1 thealgebraic connectivity ofG. This relation betweenλ1(G)ande(G)was also discovered by A. J. Hoffman [Hof70, Hof69], at the same time.

The proof of Fiedler gives us the following hint how to find the optimal 2-partition: the eigenvectoru₁should be close to a step-vector over an appropriate 2-partition of the vertices.

Note that because of its orthogonality to the vector 1, the vectoru₁ contains both positive and negative coordinates, and Juhász and Mályusz [Juh-Mály] separated the two clusters according to the signs. In the sequel, we will use thek-means algorithm for this purpose, in a more general setup. The vectoru₁is frequently calledFiedler-vector.

Even in the simplestk= 2case, the solution of the minimum cut problem is frequently at-tained by an uneven 2-partition, for example, if there is an almost isolated vertex (connected to few other vertices), it may form a cluster itself. To prevent this situation and rather find real-life loosely connected clusters, we require some balancing for the cluster sizes. For this purpose, in [Bol91, Bol-Tus94] (even in the preprint version) we defined a type of a weighted cut that, in addition, penalizes partitions with very unequal cluster sizes. This cut was later called ratio cut, see, e.g., [Hag-Kah].

Definition 3 LetG= (V,W)be an edge-weighted graph andPk = (V1, . . . , Vk)ak-partition of its vertices. The k-way ratio cut of Gcorresponding to thek-partition Pk is

g(Pk, G) = and the minimum k-way ratio cut ofGis

gk(G) = min

Pk∈Pk

g(Pk, G).

Assume that G is connected. Let 0 = λ0 < λ1 ≤ · · · ≤ λn−1 denote the eigenvalues of its Laplacian matrix L with corresponding unit-norm, pairwise orthogonal eigenvectors u₀,u₁, . . . ,u_n−1. Namely,u₀=√¹n1.

Theorem 2 ([Bol-Tus94]) For the minimumk-way ratio cut of the connected edge-wighted graph G= (V,W) the lower estimate

To illustrate the spectral relaxation technique, we describe the short proof here.

Proof. Thek-partitionPk is uniquely determined by then×k balanced partition matrix Zk = (z1, . . . ,z_k), where the a-th balanced k-partition vector z_a = (z1a, . . . , zna)^T is the

The matrix Zk is trivially suborthogonal, and the set of balanced k-partition matrices is denoted byZk^B. With the special representation in which the representatives˜r₁, . . . ,˜r_n∈R^k If we minimize it over balanced k-partition matrices Zk ∈ Zk^B, the so obtained minimum cannot go below the overall minimum Pk−1

i=0 λi. This finishes the proof.

Note that equality can be attained only in the k = 1 trivial case, otherwise the eigen-vectors u_i (i = 1, . . . , k−1) cannot be partition vectors, since their coordinates sum to 0 because of the orthogonality to the u₀ vector.

In the case of k = 2, in view of Theorem 2, g2(G)is bounded from below by λ1, akin to the edge-connectivity of [Fid73]. The proof also suggests that the quality of the above estimation depends on, how close the k bottom eigenvectors ofL are to partition vectors.

The measure of the closeness of the involved subspaces is thek-variance of thek-dimensional vertex representativesr₁, . . . ,r_n defined as

wherec_a =_|V¹_a|P

j∈Var_j is the center of clusterVa(a= 1, . . . , k). The minimum is obtained by the k-means algorithm. More precisely, we will apply the k-means algorithm for the optimal representatives, and if there is a large gap between λk−1 and λk, we may expect that the optimum, given by thek-means algorithm is not far from that of the minimumk-way ratio cut. These issues are investigated in [Bol13] thoroughly, together with hypergraph cuts, here we do not discuss the details. We will rather give similar estimates for the normalized cut with the normalized Laplacian eigenvalues in the next section.