Beyond the Expanders

Abstract

Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the vertex subsets. Implications between spectral and regularity properties are discussed.

1. Introduction

We want to go beyond the expander graphs that—for four decades—have played an important role in communication networks; for a summary, see for example, Chung [1] and Hoory et al. [2]. Roughly speaking, the expansion property means that each subset of the graph’s vertices has “many” neighbors (combinatorial view), and hence, information gets through such a graph very “quickly” (probabilistic view). We will not give exact definitions of an expander here as those contain many parameters which are not used later. We rather refer to the spectral and random walk characterization of such graphs, as discussed, among others by Alon [3] and Meilă and Shi [4].

The general framework of an edge-weighted graph will be used. Expanders have a spectral gap bounded away from zero, where—for a connected graph—this gap is defined as the minimum distance between the normalized Laplacian spectrum (apart from the trivial zero eigenvalue) and the endpoints of the [0,2] interval, the possible range of the spectrum. The larger is the spectral gap, the more our graph resembles a random graph and exhibits some quasirandom properties, for example, the edge densities within any subset and between any two subsets of its vertices do not differ too much of what is expected, see the Expander Mixing Lemma 2.2 of Section 2. Quasirandom properties and spectral gap of random graphs with given expected degrees are discussed in Chung and Graham [5] and Coja-Oghlan and Lanka [6].

However, the spectral gap appears not at the ends of the normalized Laplacian spectrum in case of generalized random or generalized quasirandom graphs that, in the presence of 𝑘≥2 underlying clusters, have 𝑘 eigenvalues (including the zero) separated from 1, while the bulk of the spectrum is located around 1, see for example, [7]. These structures are usual in social or biological networks having 𝑘 clusters of vertices (that belong to social groups or similarly functioning enzymes) such that the edge density within the clusters and between any pair of the clusters is homogeneous. Such a structure is theoretically guaranteed for any large graph by the Szemerédi Regularity Lemmas [8] with possibly large 𝑘, where 𝑘 does not depend on the number of vertices, it merely depends on the constant ruling the regularity of the cluster pairs.

Our conjecture is that 𝑘 so-called structural eigenvalues (separated from 1) in the normalized Laplacian spectrum are indications of such a structure, while the near 1 eigenvalues are responsible for the pairwise regularities. The clusters themselves can be recovered by applying the 𝑘-means algorithm for the vertex representatives obtained by the eigenvectors corresponding to the structural eigenvalues (apart from the zero). For the 𝑘=2 case, we will give an exact relation between the eigenvalue separation (of the nontrivial structural eigenvalue from the bulk of the spectrum) and the volume regularity of the cluster pair that is obtained by the 𝑘-means algorithm applied for the coordinates of the transformed eigenvector belonging to the nontrivial structural eigenvalue, see Theorem 3.1 of Section 3. To eliminate the trivial eigenvalue-eigenvector pair, we shall rather use the normalized modularity spectrum of [9] that plays an important role in finding the extrema of some penalized versions of the Newman-Girvan modularity introduced in [10].

2. Preliminaries and Statement of Purpose

Let 𝐺=(𝑉,𝐖) be a graph on 𝑛 vertices, where the 𝑛×𝑛 symmetric matrix 𝐖 has nonnegative real entries and zero diagonal. Here 𝑤_𝑖𝑗 is the similarity between vertices 𝑖 and 𝑗, where 0 similarity means no connection/edge at all. A simple graph is a special case of it with 0-1 weights. Without loss of generality,^𝑛𝑛_𝑖=1𝑗=1𝑤_𝑖𝑗=1 (2.1) will be supposed. Hence, 𝐖 is a joint distribution, with marginal entries 𝑑_𝑖=^𝑛_𝑗=1𝑤_𝑖𝑗,𝑖=1,…,𝑛, (2.2) which are the generalized vertex degrees collected in the main diagonal of the diagonal degree matrix 𝐃=diag(𝐝), 𝐝=(𝑑₁,…,𝑑_𝑛)^𝑇. In [11, 12], we investigated the spectral gap of the normalized Laplacian 𝐋_𝐷=𝐈−𝐃^−1/2𝐖𝐃^−1/2, where 𝐈 denotes the identity matrix of appropriate size.

Suppose that our graph is connected (𝐖 is irreducible). Let 0=𝜆₁<𝜆₂≤⋅⋅⋅≤𝜆_𝑛≤2 denote the eigenvalues of the symmetric normalized Laplacian 𝐋_𝐷 with corresponding unit-norm, pairwise orthogonal eigenvectors 𝐮₁,…,𝐮_𝑛. Namely, 𝐮₁=(𝑑₁,…,𝑑_𝑛)^𝑇=√𝐝. In the random walk setup, 𝐃⁻¹𝐖 is the transition matrix (its entry in the (𝑖,𝑗)th position is the conditional probability of moving from vertex 𝑖 to vertex 𝑗 in one step, given that we are in 𝑖) which is a stochastic matrix with eigenvalues 1−𝜆_𝑖 and corresponding eigenvectors 𝐃^−1/2𝐮_𝑖(𝑖=1,…,𝑛). “Good” expanders have a 𝜆₂ bounded away from zero, that also implies the separation of the isoperimetric numberℎ(𝐺)=min_{𝑈⊂𝑉∶Vol(𝑈)≤1/2}𝑒𝑈,𝑈Vol(𝑈), (2.3) where for 𝑋,𝑌⊂𝑉: 𝑒(𝑋,𝑌)=∑_𝑖∈𝑋∑_𝑗∈𝑌𝑤_𝑖𝑗 is the weighted cut between 𝑋 and 𝑌, while Vol(𝑈)=∑_𝑖∈𝑈𝑑_𝑖 is the volume of 𝑈⊂𝑉. In view of (2.1), Vol(𝑉)=1, this is why the minimum is taken on vertex sets having volume at most 1/2. In [12], we proved that12𝜆₂≤ℎ(𝐺)≤min1,2𝜆₂, (2.4) while in the 𝜆₂≤1 case the stronger upper estimation ℎ(𝐺)≤𝜆₂2−𝜆₂ (2.5) holds.

If a network does not have a “large” 𝜆₂ (compared to the natural lower bound), or equivalently—in view of the above inequalities—it has a relatively “small” isoperimetric number, then the 2-partition of the vertices giving the minimum in (2.3) indicates a bottleneck, or equivalently, a low conductivity edge-set between two disjoint vertex clusters such that the random walk gets through with small probability between them, but—as some equivalent notions will indicate—it is rapidly mixing within the clusters. To find the clusters, the coordinates of the transformed eigenvector 𝐃^−1/2𝐮₂ will be used. In [11], we proved that for the weighted 2-variance of this vector's coordinates𝑆²₂𝐃^−1/2𝐮₂≤𝜆₂𝜆₃ (2.6) holds. For a general 2≤𝑘≤𝑛, the notion of 𝑘-variance—in the Analysis of Variance sense—is the following. The weighted 𝑘-variance of the 𝑘-dimensional vertex representatives 𝐱₁,…,𝐱_𝑛 comprising the row vectors of the 𝑛×𝑘 matrix 𝐗 is defined as 𝑆²_𝑘(𝐗)=min_{𝑃𝑘∈𝒫𝑘}𝑆²_𝑘𝑃_𝑘,𝐗=min_{𝑃𝑘=𝑉1,…,𝑉𝑘}^𝑘_{𝑎=1𝑗∈𝑉𝑎}𝑑_𝑗‖𝐱_𝑗−𝐜_𝑎‖², (2.7) where 𝐜_𝑎=(1)/Vol(𝑉_𝑎)∑_{𝑗∈𝑉𝑎}𝑑_𝑗𝐱_𝑗 is the weighted center of cluster𝑉_𝑎 (𝑎=1,…,𝑘) and 𝒫_𝑘 denotes the set of 𝑘-partitions of the vertices. We remark that 𝑆²₂(𝐃^−1/2𝐮₁,𝐃^−1/2𝐮₂)=𝑆²₂(𝐃^−1/2𝐮₂), since 𝐃^−1/2𝐮₁=𝟏 is the all 1’s vector.

The above results were generalized for minimizing the normalized 𝑘-way cut𝑓_𝑘𝑃_𝑘,𝐺=^𝑘−1𝑘_{𝑎=1𝑏=𝑎+1}1Vol𝑉_𝑎+1Vol𝑉_𝑏𝑒𝑉_𝑎,𝑉_𝑏=𝑘−^𝑘_𝑎=1𝑒𝑉_𝑎,𝑉_𝑎Vol𝑉_𝑎 (2.8) of the 𝑘-partition 𝑃_𝑘=(𝑉₁,…,𝑉_𝑘) over the set of all possible 𝑘-partitions. Hence, 𝑓_𝑘(𝐺)=min_{𝑃𝑘∈𝒫𝑘}𝑓_𝑘𝑃_𝑘,𝐺 (2.9) is the minimum normalized k-way cut of the underlying weighted graph 𝐺=(𝑉,𝐖). In fact, 𝑓₂(𝐺) is the symmetric version of the isoperimetric number and 𝑓₂(𝐺)≤2ℎ(𝐺). In [12], we proved that^𝑘_𝑖=1𝜆_𝑖≤𝑓_𝑘(𝐺)≤𝑐^2𝑘_𝑖=1𝜆_𝑖, (2.10) where the upper estimation is relevant only in the case when 𝑆²_𝑘(𝐃^−1/2𝐮₁,…,𝐃^−1/2𝐮_𝑘) is small enough and the constant 𝑐 depends on this minimum 𝑘-variance of the vertex representatives.

The normalized Newman-Girvan modularity is defined in [9] as the penalized version of the Newman-Girvan modularity [10] in the following way. The normalized 𝑘-way modularity of 𝑃_𝑘=(𝑉₁,…,𝑉_𝑘) is𝑄_𝑘𝑃_𝑘,𝐺=^𝑘_𝑎=11Vol𝑉_{𝑎𝑖,𝑗∈𝑉𝑎}𝑤_𝑖𝑗−𝑑_𝑖𝑑_𝑗=^𝑘_𝑎=11Vol𝑉_𝑎𝑒𝑉_𝑎,𝑉_𝑎−Vol²𝑉_𝑎=^𝑘_𝑎=1𝑒𝑉_𝑎,𝑉_𝑎Vol𝑉_𝑎−1=𝑘−1−𝑓_𝑘𝑃_𝑘, (2.11)𝑄_𝑘(𝐺)=max_{𝑃𝑘∈𝒫𝑘}𝑄_𝑘𝑃_𝑘,𝐺 (2.12) is the maximum normalized k-way Newman-Girvan modularity of the underlying weighted graph 𝐺=(𝑉,𝐖). For given 𝑘, maximizing this modularity is equivalent to minimizing the normalized cut and can be solved by the same spectral technique. In fact, it is more convenient to use the spectral decomposition of the normalized modularity matrix 𝐁_𝐷=𝐈−𝐋_𝐷−√𝐝√𝐝^𝑇 with eigenvalues 𝛽₁≥⋅⋅⋅≥𝛽_𝑛, that are the numbers 1−𝜆_𝑖 with eigenvectors 𝐮_𝑖 (𝑖=2,…,𝑛) and the zero with corresponding unit-norm eigenvector √𝐝. In [9, 12], we also show that a spectral gap between 𝜆_𝑘 and 𝜆_𝑘+1 is an indication of 𝑘 clusters with low intercluster connections; further, the intracluster connections (𝑤_𝑖𝑗) between vertices 𝑖 and 𝑗 of the same cluster are higher than expected under the hypothesis of independence (in view of which vertices 𝑖 and 𝑗 are connected with probability 𝑑_𝑖𝑑_𝑗). In the random walk framework, the random walk stays within the clusters with high probability.

Conversely, minimizing the above modularity will result in clusters with high inter- and low intra-cluster connections. In [9], we proved thatmin_{𝑃𝑘∈𝒫𝑘}𝑄_𝑘𝑃_𝑘,𝐺≥^𝑘_𝑖=1𝛽_{𝑛+1−𝑖}. (2.13) The existence of 𝑘 “large” (significantly larger than 1) eigenvalues in the normalized Laplacian spectrum, or equivalently, the existence of 𝑘 negative eigenvalues (separated from 0) in the normalized modularity spectrum is an indication of 𝑘 clusters with the above property. In the random walk setup, the walk stays within the clusters with low probability.

These two types of network structures are frequently called community or anticommunity structure. Some networks exhibit a more general, still regular behavior: the vertices can be classified into 𝑘 clusters such that the information-flow within them and between any pair of them is homogeneous. In terms of random walks, the walk stays within clusters or switches between clusters with probabilities characteristic for the cluster pair. That is, if the random walk moves from a vertex of cluster 𝑉_𝑎 to a vertex of cluster 𝑉_𝑏, then the probability of doing this does not depend on the actual vertices, it merely depends on their cluster memberships, 𝑎,𝑏=1,…,𝑘.

In this context, we examined the following generalized random graph model, that corresponds to the ideal case: given the number of clusters 𝑘, the vertices of the graph independently belong to the clusters; further, conditioned on the cluster memberships, vertices 𝑖∈𝑉_𝑎 and 𝑗∈𝑉_𝑏 are connected with probability 𝑝_𝑎𝑏, independently of each other, 1≤𝑎,𝑏≤𝑘. Applying the results [7] for the spectral characterization of some noisy random graphs, we are able to prove that the normalized modularity spectrum of a generalized random graph is the following: there exists a positive number 𝛿<1, independent of 𝑛, such that there are exactly 𝑘−1 so-called structural eigenvalues of 𝐁_𝐷 that are greater than 𝛿−𝑜(1), while all the others are o(1) in absolute value. It is equivalent that 𝐋_𝐷 has 𝑘 eigenvalues (including the zero) separated from 1.

The 𝑘=1 case corresponds to quasirandom graphs, and the above characterization corresponds to the eigenvalue separation of such graphs, discussed in [5]. The authors also prove some implications between the so-called quasirandom properties. For example, for dense graphs, “good” eigenvalue separation is equivalent to “low” discrepancy (of the induced subgraphs’ densities from the overall edge density).

For the 𝑘≥2 case, generalized quasirandom graphs were introduced by Lovász and Sós [13]. These graphs are deterministic counterparts of generalized random graphs with the same spectral properties. In fact, the authors define so-called generalized quasirandom graph sequences by means of graph convergence that also implies the convergence of spectra. Though, the spectrum itself does not carry enough information for the cluster structure of the graph, together with some classification properties of the structural eigenvectors it does. We want to prove some implication between the spectral gap and the volume-regularity of the cluster pairs, also using the structural eigenvectors.

The notion of volume regularity was introduced by Alon et al. [14]. We shall use a slightly modified version of this notion.

Definition 2.1. Let 𝐺=(𝑉,𝐖) be weighted graph with Vol(𝑉)=1. The disjoint vertex pair (𝐴,𝐵) is 𝛼-volume regular if for all 𝑋⊂𝐴, 𝑌⊂𝐵 we have 𝑒(𝑋,𝑌)−𝜌(𝐴,𝐵)Vol(𝑋)Vol(𝑌)≤𝛼Vol(𝐴)Vol(𝐵), (2.14) where 𝜌(𝐴,𝐵)=𝑒(𝐴,𝐵)/Vol(𝐴)Vol(𝐵) is the relative inter-cluster density of (𝐴,𝐵).

Our definition was inspired by the Expander Mixing Lemma stated for example, in [2] for regular graphs and in [1] for simple graphs in the context of quasirandom properties. Now we formulate it for edge-weighted graphs on a general degree sequence. We also include the proof as a preparation for the proof of Theorem 3.1 of Section 3.

Lemma 2.2 (Expander Mixing Lemma for Weighted Graphs). Let 𝐺=(𝑉,𝑊) be a weighted graph and suppose that Vol(𝑉)=1. Then for all 𝑋,𝑌⊂𝑉: |𝑒(𝑋,𝑌)−Vol(𝑋)Vol(𝑌)|≤‖𝐁_𝐷‖⋅Vol(𝑋)(1−Vol(𝑋))Vol(𝑌)(1−Vol(𝑌))≤‖𝐁_𝐷‖⋅Vol(𝑋)Vol(𝑌), (2.15) where ‖𝐁_𝐷‖ is the spectral norm of the normalized modularity matrix of 𝐺.

Proof. Let 𝑋⊂𝐴, 𝑌⊂𝐵, and 𝟏_𝑈∈ℝ^𝑛 denote the indicator vector of 𝑈⊂𝑉. Further, 𝐱∶=𝐃^1/2𝟏_𝑋 and 𝐲∶=𝐃^1/2𝟏_𝑌.
We use the spectral decomposition 𝐃^−1/2𝐖𝐃^−1/2=∑^𝑛_𝑖=1𝜌_𝑖𝐮_𝑖𝐮^𝑇_𝑖, where 𝜌_𝑖=1−𝜆_𝑖 (𝑖=2,…,𝑛) are eigenvalues of 𝐁_𝐷 and 𝜌₁=1 with corresponding unit-norm eigenvector 𝐮₁=√𝐝=𝐃^1/2𝟏. We remark that 𝐮₁ is also an eigenvector of 𝐁_𝐷 corresponding to the eigenvalue zero, hence ‖𝐁_𝐷‖=max_𝑖≥2|𝜌_𝑖|. Let 𝐱=∑^𝑛_𝑖=1𝑎_𝑖𝐮_𝑖 and 𝐲=∑^𝑛_𝑖=1𝑏_𝑖𝐮_𝑖 be the expansions of 𝐱 and 𝐲 in the orthonormal basis 𝐮₁,…,𝐮_𝑛 with coordinates 𝑎_𝑖=𝐱^𝑇𝐮_𝑖 and 𝑏_𝑖=𝐲^𝑇𝐮_𝑖, respectively. Observe that 𝑎₁=Vol(𝑋), 𝑏₁=Vol(𝑌) and ∑^𝑛_𝑖=1𝑎²_𝑖=‖𝐱‖²=Vol(𝑋), ∑^𝑛_𝑖=1𝑏²_𝑖=‖𝐲‖²=Vol(𝑌). Based on these,|𝑒(𝑋,𝑌)−Vol(𝑋)Vol(𝑌)|=^𝑛_𝑖=2𝜌_𝑖𝑎_𝑖𝑏_𝑖≤𝐁_𝐷⋅^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖≤𝐁_𝐷⋅^𝑛_𝑖=2𝑎²_𝑖^𝑛_𝑖=2𝑏²_𝑖≤𝐁_𝐷⋅(𝑋(𝑋(𝑌(𝑌≤𝐁Vol)(1−Vol))Vol)(1−Vol))_𝐷⋅Vol(𝑋)Vol(𝑌), (2.16) where we also used the triangle and the Cauchy-Schwarz inequalities.

We remark that the spectral gap of 𝐺 is 1−‖𝐁_𝐷‖; hence—in view of Lemma 2.2—the density between any two subsets of “good” expanders is near to what is expected. On the contrary, in the above definition of volume regularity, the 𝑋,𝑌 pairs are disjoint, and a “small” 𝛼 indicates that the (𝐴,𝐵) pair is like a bipartite expander, see for example, [1].

In the next section, we shall prove the following statement for the 𝑘=2 case: if one eigenvalue jumps out of the bulk of the normalized modularity spectrum, then clustering the coordinates of the corresponding transformed eigenvector into 2 parts (by minimizing the 2-variance of its coordinates) will result in an 𝛼-volume regular partition of the vertices, where 𝛼 depends on the spectral gap.

Our conjecture is that we may go further: if 𝑘−1 (so-called structural) eigenvalues jump out of the normalized modularity spectrum, then clustering the representatives of the vertices—obtained by the corresponding eigenvectors in the usual way—into 𝑘 clusters will result in 𝛼-volume regular pairs, where 𝛼 depends on the spectral gap (between the structural eigenvalues and the bulk of the spectrum) and the 𝑘-variance of the vertex representatives based on the eigenvectors corresponding to the structural eigenvalues.

3. Eigenvalue Separation and Volume Regularity (𝑘=2 Case)

Theorem 3.1. Let 𝐺=(𝑉,𝐖) is an edge-weighted graph on 𝑛 vertices, with generalized degrees 𝑑₁,…,𝑑_𝑛 and 𝐃=diag(𝑑₁,…,𝑑_𝑛). Suppose that Vol(𝑉)=1 and there are no dominant vertices: max_{1≤𝑖≤𝑛}𝑑_𝑖=𝑜(1) as 𝑛→∞. Let the eigenvalues of 𝐃^−1/2𝐖𝐃^−1/2, enumerated in decreasing absolute values, be 1=𝜌₁>𝜌₂=𝛿>𝜀=𝜌₃≥𝜌_𝑖,𝑖≥4. (3.1) The partition (𝐴,𝐵) of 𝑉 is defined in such a way that it minimizes the weighted 2-variance of the coordinates of 𝐃^−1/2𝐮, where 𝐮 is the unit-norm eigenvector belonging to 𝜌₂. Then the (𝐴,𝐵) pair is 𝒪((1−𝛿)/(1−𝜀)) -volume regular.

Remark 3.2. The statement of Theorem 3.1 has relevance only if 𝛿 is much larger than 𝜀. In this case, the spectral gap between the largest absolute value eigenvalue and the others in the normalized modularity spectrum indicates a regular 2-partition of the graph that can be constructed based on the eigenvector belonging to the structural eigenvalue.

Remark 3.3. The statement of the above theorem also has relevance in the dense case (supported by the condition that there are no dominant vertices). We remark that in the sparse case there is an exceptional set comprising low degree vertices. Authors of [14] prove equivalences of quasirandom properties for the core of the graph (the vertex set deprived of the exceptional set). Using the modularity spectrum of this core, the above theorem remains valid for it.

Proof of Theorem 3.1. We use the notations in Lemma 2.2’s proof. Let 𝑋⊂𝐴, 𝑌⊂𝐵. For short, 𝐱∶=𝐃^1/2𝟏_𝑋, 𝐲∶=𝐃^1/2𝟏_𝑌, 𝐚∶=𝐃^1/2𝟏_𝐴, and 𝐛∶=𝐃^1/2𝟏_𝐵. With the notations 𝜌=𝜌(𝐴,𝐵) and 𝐌=𝐖−𝜌𝐝𝐝^𝑇, 𝑒(𝑋,𝑌)−𝜌Vol(𝑋)Vol(𝑌)=𝟏^𝑇_𝑋𝐌𝟏_𝑌=𝐱^𝑇𝐃^−1/2𝐖𝐃^−1/2−𝜌√𝐝√𝐝^𝑇𝐲. (3.2) Using the spectral decomposition 𝐃^−1/2𝐖𝐃^−1/2=∑^𝑛_𝑖=1𝜌_𝑖𝐮_𝑖𝐮^𝑇_𝑖 and the fact that 𝐮₁=√𝐝=𝐃^1/2𝟏, we can write (3.2) as 1−𝜌𝑎₁𝑏₁+𝜌₂𝑎₂𝑏₂+^𝑛_𝑖=3𝜌_𝑖𝑎_𝑖𝑏_𝑖, (3.3) where 𝐱=∑^𝑛_𝑖=1𝑎_𝑖𝐮_𝑖 and 𝐲=∑^𝑛_𝑖=1𝑏_𝑖𝐮_𝑖 is the expansion of 𝐱 and 𝐲 in the orthonormal basis 𝐮₁,…,𝐮_𝑛 with coordinates 𝑎_𝑖=𝐱^𝑇𝐮_𝑖 and 𝑏_𝑖=𝐲^𝑇𝐮_𝑖, respectively.
By Lemma 2.2, for all 𝑋,𝑌⊂𝑉:𝜌(𝑋,𝑌)−1≤‖𝐁_𝐷‖⋅1−Vol(𝑋)(𝑋)⋅(𝑌)Vol1−VolVol(𝑌), (3.4) that is also applicable to the special two-partition 𝐴,𝐵⊂𝑉 satisfying the postulates. Hence, |1−𝜌| is governed by the spectral norm |𝜌₂| of the normalized modularity matrix. In [11], as a by-product of the proof of (2.6) it came out that to get the partition (𝐴,𝐵), the coordinates of 𝐃^−1/2𝐮₂ should be divided into two parts at the median, hence Vol(𝐴) and Vol(𝐵) are approximately equal to 1/2 (the approximation is good for large 𝑛 if the underlying graph does not have dominant vertex-weights). Further, the estimation 𝑆²₂𝐃^−1/2𝐮₂≤𝜆₂𝜆₃≤1−𝜌₂≤1−𝜀1−𝛿1−𝜀 (3.5) also follows. This applies to the 𝜌₂=1−𝜆₂ case. In the 𝜌₂=1−𝜆_𝑛 case, the optimum 𝐴,𝐵 is obtained by minimizing the 2-variance of the coordinates of the transformed eigenvector 𝐃^−1/2𝐮_𝑛 for which the following relation—like (2.6)—can be proved: 𝑆²₂𝐃^−1/2𝐮_𝑛≤2−𝜆_𝑛2−𝜆_𝑛−1≤𝜌₂+1≤1−𝜀1−𝛿1−𝜀, (3.6) and the optimum cut-point of the coordinates 𝐃^−1/2𝐮_𝑛 is also not far from the median. Summarizing, |1−𝜌|≤|𝜌₂|+𝜂, where 𝜂=𝑜(1).
Therefore, (3.3) can be estimated from above with𝜌₂⋅𝑎₁𝑏₁+𝑎₂𝑏₂+𝜂𝑎₁𝑏₁+𝜀⋅^𝑛_𝑖=3𝑎_𝑖𝑏_𝑖. (3.7) As for the second term, 𝜂𝑎₁𝑏₁=𝜂Vol(𝑋)Vol(𝑌).
Using the Cauchy-Schwarz inequality, the last term can be estimated from above with𝜀^𝑛_𝑖=3𝑎²_𝑖^𝑛_𝑖=3𝑏²_𝑖≤𝜀^𝑛_𝑖=2𝑎²_𝑖^𝑛_𝑖=2𝑏²_𝑖≤𝜀Vol(𝑋)(1−Vol(𝑋))Vol(𝑌)(1−Vol(𝑌))≤𝜀Vol(𝑋)Vol(𝑌)≤𝜀Vol(𝐴)Vol(𝐵), (3.8) since 𝑎₁=Vol(𝑋), 𝑏₁=Vol(𝑌) and ∑^𝑛_𝑖=1𝑎²_𝑖=‖𝐱‖²=Vol(𝑋), ∑^𝑛_𝑖=1𝑏²_𝑖=‖𝐲‖²=Vol(𝑌). The first term is reminiscent of an equation for the coordinates of orthogonal vectors. Therefore, we project the vectors 𝐮₁, 𝐮₂ onto the subspace 𝐹=Span{𝐚,𝐛}. In fact, 𝐮₁=𝐚+𝐛, therefore 𝐮₁∈𝐹. The vector 𝐮₂ can be decomposed as 𝐮₂=𝐮^𝑇₂𝐚(𝐴)𝐮Vol𝐚+^𝑇2𝐛Vol(𝐵)𝐛+𝐪, (3.9) where 𝐪 is the component orthogonal to 𝐹. For the squared distance ‖𝐪‖² between 𝐮₂ and 𝐹, in [11], we proved that it is equal to the weighted 2-variance 𝑆²₂(𝐃^−1/2𝐮₂) and in (3.5) we estimated it from above with (1−𝛿)/(1−𝜀). In the 𝜌₂=1−𝜆_𝑛 case similar estimation works using (3.6). First we estimate (𝐮^𝑇₁𝐚)(𝐮^𝑇₁𝐛)+(𝐮^𝑇₂𝐚)(𝐮^𝑇₂𝐛). The problem is that the pairwise orthogonal vectors 𝐮₁,𝐮₂ and 𝐚,𝐛 are not in the same subspace of ℝ^𝑛 as, in general, 𝐮₂∉𝐹. However, by an argument proved in [11], we can find orthogonal, unit-norm vectors 𝐮₁,𝐮₂∈𝐹 such that 𝐮₁−𝐮₁²+𝐮₂−𝐮₂²≤2𝑠², (3.10) where, in view of 𝐮₁∈𝐹, 𝐮₁=𝐮₁. Let 𝐫∶=𝐮₂−𝐮₂. Since 𝐮^𝑇₁𝐚,𝐮^𝑇₂𝐚 and 𝐮^𝑇₁𝐛,𝐮^𝑇₂𝐛 are coordinates of the orthogonal vectors 𝐚,𝐛 in the basis 𝐮₁,𝐮₂, 𝐮^𝑇₁𝐚𝐮^𝑇₁𝐛+𝐮^𝑇₂𝐚𝐮^𝑇₂𝐛=0, (3.11) and because of 𝐮^𝑇₂𝐚+𝐮^𝑇₂𝐛=𝐮^𝑇₂𝐮₁=0, 𝐮^𝑇₂𝐚=−𝐮^𝑇₂𝐛=Vol(𝐴)Vol(𝐵)=∶𝑐. (3.12) Therefore, 𝐮^𝑇₁𝐚𝐮^𝑇₁𝐛+𝐮^𝑇₂𝐚𝐮^𝑇₂𝐛=Vol(𝐴)Vol(𝐵)+𝐮₂+𝐫^𝑇𝐚,𝐮₂+𝐫^𝑇𝐛=Vol(𝐴)Vol(𝐵)+𝑐+𝐫^𝑇𝐚⋅−𝑐+𝐫^𝑇𝐛=𝑐−𝐫^𝑇𝐚+𝐫^𝑇𝐛+𝐫^𝑇𝐚𝐫^𝑇𝐛≤|𝑐|‖𝐫‖²‖𝐛−𝐚‖²+‖𝐫‖²‖𝐚‖²‖𝐫‖²‖𝐛‖²≤Vol(𝐴)Vol(𝐵)‖𝐫‖+‖𝐫‖²≤Vol(𝐴)Vol(𝐵)√2𝑠+2𝑠², (3.13) using (3.10) and the fact that ‖𝐛−𝐚‖²=1.
Now we estimate 𝑎₁𝑏₁+𝑎₂𝑏₂=(𝐮^𝑇₁𝐱)(𝐮^𝑇₁𝐲)+(𝐮^𝑇₂𝐱)(𝐮^𝑇₂𝐲). Going back to (3.9), we have𝐮^𝑇₂𝐮𝐱=^𝑇2𝐚(𝐴)𝐚Vol^𝑇𝐮𝐱+^𝑇2𝐛(𝐵)𝐛Vol^𝑇𝐱+𝐪^𝑇(𝑋)𝐱=Vol(𝐴)𝐮Vol^𝑇2𝐚+𝐪^𝑇𝐱, (3.14) and similarly, 𝐮^𝑇₂𝐮𝐲=^𝑇2𝐚(𝐴)𝐚Vol^𝑇𝐮𝐲+^𝑇2𝐛(𝐵)𝐛Vol^𝑇𝐲+𝐪^𝑇(𝑌)𝐲=Vol(𝐵)𝐮Vol^𝑇2𝐛+𝐪^𝑇𝐲, (3.15) that in view of ‖𝐪‖²=𝑠² yields 𝐮^𝑇₁𝐱𝐮^𝑇₁𝐲+𝐮^𝑇₂𝐱𝐮^𝑇₂𝐲=(𝑋)(𝑌)+(𝑋)VolVolVol(𝐴)𝐮Vol^𝑇2𝐚+𝐪^𝑇𝐱Vol(𝑌)(𝐵)𝐮Vol^𝑇2𝐛+𝐪^𝑇𝐲≤(𝑋)(𝑌)+(𝑋)VolVolVol(𝐴)𝐮Vol^𝑇2𝐚Vol(𝑌)(𝐵)𝐮Vol^𝑇2𝐛+𝐪^𝑇𝐱Vol(𝑌)(𝐵)𝐮Vol^𝑇2𝐛+𝐪^𝑇𝐲Vol(𝑋)(𝐴)𝐮Vol^𝑇2𝐚+𝐪^𝑇𝐱𝐪^𝑇𝐲≤Vol(𝑋)(𝐴)(𝑌)VolVol(𝐵)(𝐴)(𝐵)+𝐮VolVolVol^𝑇₂𝐚𝐮^𝑇₂𝐛+‖𝐪𝐱‖(𝑌)‖‖Vol(𝐵)𝐮Vol^𝟐𝐛‖+‖𝐪𝐲‖(𝑋)‖‖‖Vol(𝐴)𝐮Vol^𝟐𝐚‖+‖𝐪‖₂‖‖𝐱𝐲‖≤‖‖Vol(𝐴)Vol(𝐵)√2𝑠+2𝑠²+‖𝐪‖(𝑋)(𝑌)VolVolVol(𝐵)(𝐵)+‖𝐪‖Vol(𝑌)(𝑋)VolVolVol(𝐴)Vol(𝐴)+‖𝐪‖²Vol(𝑋)(𝑌)=VolVol(𝐴)Vol(𝐵)√2𝑠+2𝑠²+‖𝐪‖Vol(𝑋)(𝑌)×VolVol(𝑌)(𝐵)+VolVol(𝑋)(𝐴)+‖𝐪‖≤VolVol(𝐴)Vol(𝐵)√2𝑠+2𝑠²(=+𝑠2+𝑠)Vol(𝐴)Vol(𝐵)√2+2𝑠+3𝑠²≤Vol(𝐴)Vol(𝐵)√2+5𝑠≤Vol(𝐴)Vol(𝐵)√2+51−𝛿.1−𝜀 (3.16) Summarizing, the second and third terms in (3.7) are estimated from above with 𝜂Vol(𝑋)Vol(𝑌)≤𝜂Vol(𝐴)Vol(𝐵) and 𝜀Vol(𝐴)Vol(𝐵), respectively. Because of 𝜂=𝑜(1) and 𝜀<𝛿, by an easy calculation it follows that they are less than (1−𝛿)/(1−𝜀) for 𝑛 “large” enough. Therefore, 𝛼=√2+7 will do.

Acknowledgments

The author wishes to thank Vera T. Sós, László Lovász, and Miklós Simonovits for their useful advices. Research was supported by the Hungarian National Research Grants OTKA 76481 and OTKA-NKTH 77778.