Higher Dimensional Discrete Cheeger Inequalities

Higher Dimensional Discrete Cheeger Inequalities

Anna Gundert111Universität zu Köln, Weyertal 86-90, D-50923 Köln, Germany. anna.gundert@uni-koeln.de. Research supported by the Swiss National Science Foundation (SNF Projects 200021-125309 and 200020-138230).    May Szedlák222Institut für Theoretische Informatik, ETH Zürich, CH-8092 Zürich, Switzerland. may.szedlak@inf.ethz.ch.
Abstract

For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that , where is the second smallest eigenvalue of the Laplacian of a graph and is the Cheeger constant measuring the edge expansion of . We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs).

Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generalization of edge expansion to simplicial complexes is not straightforward. Recently, a topologically motivated notion analogous to edge expansion that is based on -cohomology was introduced by Gromov and independently by Linial, Meshulam and Wallach. It is known that for this generalization there is no direct higher dimensional analogue of the lower bound of the Cheeger inequality.

A different, combinatorially motivated generalization of the Cheeger constant, denoted by , was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed , where is the smallest non-trivial eigenvalue of the (-dimensional upper) Laplacian, for the case of -dimensional simplicial complexes with complete -skeleton.

Whether this inequality also holds for -dimensional complexes with non-complete -skeleton has been an open question. We give two proofs of the inequality for arbitrary complexes. The proofs differ strongly in the methods and structures employed, and each allows for a different kind of additional strengthening of the original result.

Introduction

Roughly speaking, a graph is an expander if it is sparse and at the same time well-connected. Such graphs have found various applications, in theoretical computer science as well as in pure mathematics. Expander graphs have, e.g., been used to construct certain classes of error correcting codes, in a proof of the PCP Theorem ([4], see also [22]), a deep result in computational complexity theory, and in the theory of metric embeddings. See, e.g., the surveys [10] and [14] for these and other applications.

In recent years, the combinatorial study of simplicial complexes - considering them as a higher-dimensional generalization of graphs - has attracted increasing attention and the profitability of the concept of expansion for graphs has inspired the search for a corresponding higher-dimensional notion, see, e.g., [9, 15, 21, 24].

The expansion of a graph can be measured by the Cheeger constant333Often the Cheeger constant is defined in a slightly different but closely related way, see Section 2.

Here is the set of edges with one endpoint in and the other in . A straightforward higher-dimensional analogue is the following Cheeger constant of a -dimensional simplicial complex with complete -skeleton, studied in [21]:

Here is the set of -dimensional faces of with exactly one vertex in each set . A different, more topologically motivated notion based on -cohomology was introduced by Gromov [8] and independently by Linial, Meshulam and Wallach [13, 20]. We will not work with this notion but we later consider an adapted version of inspired by it. See discussion of results and Section 2 for more details.

For graphs, this combinatorial notion of expansion is connected to the spectra of certain matrices associated with the graph: the adjacency matrix and the Laplacian. This connection between combinatorial and spectral expansion properties of a graph is established, e.g., by the discrete Cheeger inequality [1, 2, 5, 25]. For a graph with second smallest eigenvalue of the Laplacian (see Section 1) and maximum degree , it states that

A different approach to generalizing expansion is hence to consider higher-dimensional analogues of graph Laplacians. Higher-dimensional Laplacians were first introduced by Eckmann [6] in the 1940s and have since then been used in various contexts, see [12] for an example. We denote by the smallest non-trivial eigenvalue of this Laplacian. More precisely, is the smallest eigenvalue of the upper Laplacian on . (See Section 1 for further details.)

The Cheeger inequality for graphs has proven to be a useful tool. Computing the Cheeger constant is difficult, from the standpoint of complexity theory [19, 3] but often also for explicit examples. The lower bound – even though easy to prove – hence gives a helpful, polynomially computable, lower bound on the Cheeger constant. Many constructions of families of expander graphs (graphs on vertices with constant edge degree, where the Cheeger constant is bounded from below by a constant) use eigenvalues to establish a lower bound on the combinatorial expansion [7, 16, 17, 18, 23].

Parzanchevski, Rosenthal and Tessler [21] recently showed the following analogue of this lower bound of the Cheeger inequality for -dimensional simplicial complexes with complete -skeleton.

Theorem 1 (Parzanchevski et al. [21]).

Let be a -dimensional simplicial complex with complete -skeleton. Then .

Here, we present two ways to extend this result to -dimensional complexes with non-complete -skeleton, addressing an open question that was posed in [21]. Both proofs allow for an additional strengthening of the original result.

To make an extension to arbitrary complexes possible, it is necessary to adapt the definition of , as it is easily seen that as defined above is non-zero only for -dimensional with complete -skeleton. For any -dimensional complex , define its -dimensional completion as . If has a complete -skeleton, we get , the complete -dimensional complex on vertices. We then define, as suggested in [21],

where is defined as

Note that this is the set corresponding to in the completion – and hence the largest possible set of -simplices with one vertex in each in a simplicial complex with the -skeleton of . For a partition with define . Our first result is as follows:

Theorem 2.

Let be a -dimensional simplicial complex and let be a partition of that minimizes . Let

where for a -face . Then

In particular, if every -face is contained in at most -faces of , i.e.,

then and hence

The latter inequality is useful, since in general is hard to compute. Note that as well as depend on the minimizing partition , which is not necessarily unique. However, for better readability we omit the partition from the notation.

Observe that if is the unique block not containing a vertex of , then and this bound is tight for -complexes with complete -skeleton. So by definition and Theorem 2 implies the statement of Theorem 1 for arbitrary -dimensional simplicial complexes. While for with complete -skeleton, in extreme cases can be arbitrary small compared to .

Our second result strengthens Theorem 1 in a different way. It is possible to rephrase in terms of -coboundaries as follows (see Section 1 for the necessary definitions): For a partition let be the set of -dimensional faces of with exactly one vertex in each set , . Let be its characteristic function, interpreted as a -cochain. Then the support of the -coboundary in , the set of -simplices of with an odd number of -faces in , is exactly the set . The corresponding coboundary in has support . Thus,

where denotes the Hamming norm. In order to strengthen the bound on given by Theorem 1, we define

If , we again define . As an illustration think of the case of dimension . Then the cochains considered here are those whose support describes the edge set of a bipartite graph. We compare the number of triangles in with an odd number of egdes, i.e., exactly one edge, in the support of with the number of possible such triangles.

Note that here we consider partitions of into parts. For a partition into parts, clearly

Hence, as we minimize over a larger set of cochains, we have . See below for an example where . We show:

Theorem 3.

Let be a -dimensional simplicial complex. Then

Discussion of Results.

  1. The inspiration for the definition of is a different analogue of the Cheeger constant for graphs, introduced by Gromov and independently by Linial, Meshulam and Wallach. It is based on -cohomology and emerged in various contexts as a useful notion, see, e.g., [8, 13, 20]. For a -complex with complete -skeleton, this notion can be described444Usually, one considers a different notion. For -complexes with complete -skeleton, the two notions are closely related, see Section 2 for more details. by

    similar to the definitions of and , but without any restriction on the cochains considered. As this seems to be an important and useful concept, one might wish for an inequality as in Theorems 1, 2 and 3 also for this notion of expansion. It was, however, shown that such an inequality can not exist, see [9, 24].

    So far – except for a technical explanation, see Lemma 9 – we have no deeper insight into why the Cheeger inequality does hold when the definition of expansion is restricted to the special class of cochains appearing in the definition of .

  2. Theorem 2 and Theorem 3 can indeed give a stronger bound than Theorem 1, see Section 3 for examples that also show that it depends on the complex whether or presents the stronger upper bound on .

  3. Recall that the Cheeger inequality for graphs also gives an upper bound of in terms of . As does not imply , see [21], a higher-dimensional analogue of this upper bound of the form is hence not possible.

1 Preliminaries

Graph Laplacian.

Let be a finite simple undirected graph, . The Laplacian of is the -matrix

Here, is the adjacency matrix – given by if and only if – and is the diagonal matrix with entries , the degrees of the vertices.

The Laplacian is a symmetric positive semi-definite matrix and hence has real non-negative eigenvalues. As , the smallest eigenvalue is always , and we denote by the second smallest eigenvalue of . A graph is connected if and only if is non-zero (see, e.g., [10]).

Simplicial Complexes.

Let be a finite set. A (finite abstract) simplicial complex (or complex) with vertex set is a collection of subsets of that is closed under taking subsets, i.e., implies .

An element is called a simplex or face of , the dimension of is . A simplex with is also called an -simplex. The dimension of the complex is . A simplicial complex of dimension is called a -dimensional simplicial complex or a -complex. The one-element sets , , are the vertices of . We identify the singleton with its unique element . For an -simplex the degree of is defined as . The set of all -simplices of is denoted by , the collection of all simplices of dimension at most , the -skeleton of , by . The complete -complex has vertex set and for all .

Cohomology.

Let be a -dimensional simplicial complex with vertex set and assume that we have a fixed linear ordering on . We consider the faces of with the orientation given by the order of their vertices.

Formally, consider an -simplex where . For an -simplex the oriented incidence number is defined as if , for some and zero otherwise, i.e., if . In particular for and the unique face we have .

Let be an Abelian group (we will be concerned with the cases and ). The group of -dimensional cochains on (with coefficients in ) is

i.e., the group of maps from the set of -simplices to . For or we conveniently define . Note that since the empty set is the unique element of we have . The characteristic functions of faces form a basis of , they are called the elementary cochains.

The coboundary operator is the linear function given by

for an -simplex, and . We let otherwise. Define the group of -dimensional cocycles and the group of -dimensional coboundaries. A straightforward calculation shows that , i.e., . Hence, we can define the (reduced) -th cohomology group with coefficients in as

Real Coefficients and Higher-Dimensional Laplacians.

We endow with the inner product

for and denote the dual operator of by , i.e., for and we have

The map is also called the boundary operator and the groups and are the group of -dimensional cycles and the group of -dimensional boundaries, respectively.

Setting , one gets a Hodge decomposition of the vector space into pairwise orthogonal subspaces

(1)

in particular, (see [6, 11]).

The higher-dimensional analogue of the graph Laplacian is based on these notions. From now on, write for , for and for . The upper, lower and full Laplacian in dimension are defined as

respectively. We solely focus on the case .

Analogously to the case of graphs () we can express as a matrix: With respect to the orthogonal basis of elementary cochains it corresponds to the matrix

Here we let denote the diagonal matrix with entry for and define the adjacency matrix by

where and we write if and share a common -face and . Note that for a graph , we get for all non-zero entries of . This shows that and that agrees with the Laplacian . For , non-zero entries of do not necessarily have the same sign.

Note that (as well as and ) is a self-adjoint and positive semidefinite linear operator. It is furthermore not hard to see that . As , this implies that is zero on . Hence, non-zero eigenvalues can only occur in and we define the spectral gap of as {linenomath}

where the equality holds because we have by the Hodge decomposition (1). We remark that even though the spaces and depend on the choice of orientations for the faces of , the spectrum of and the value of do not.

Note that is also the minimal eigenvalue of the full Laplacian on , since . We have , i.e., there exist more zero eigenvalues than the ones corresponding to functions in , if and only if .

For a graph the space is the space of constant functions, spanned by the all-ones vector , so this definition of the spectral gap coincides with as defined previously.

2 Different Notions of Higher-Dimensional Expansion

The notions of expansion considered here are based on the Cheeger constant for graphs defined as Often the Cheeger constant is defined differently by

Since , these two concepts are closely related. Both are also called (edge) expansion ratio.

An analogue of the latter graph parameter for higher dimensional simplicial complexes was introduced by Gromov and independently by Linial, Meshulam and Wallach. For a -dimensional complex define

where , see, e.g., [8, 13, 20]. Here, denotes the -coboundary operator of the complex . For with , define . Note that if and only if the complex has a non-trivial -cocycle, i.e., iff .

Our definition of is inspired by the following parameter, generalizing the definition of for graphs

Complexes with Complete -Skeleton.

For -complexes with a complete -skeleton the two parameters and have a close relation, analogous to the situation for graphs. This close relation follows from a basic observation concerning the expansion of the complete complex that was made independently in different contexts, see, e.g., [8, 20]:

Proposition 4 (E.g., [20, Proposition 2.1]).

If has a complete -skeleton, . As trivially for any , we have the following relation between the above notions of expansion:

General Complexes.

For general -complexes with arbitrary -skeleton, this close relation does not necessarily exist. While the trivial lower bound continues to hold, the existence of an upper bound is not guaranteed. As an example consider the complex defined in Section 3 (see Figure 2). Since , we have , whereas , because .

If , i.e., , we get analogously to the case above

Thus, the existence (and the quality) of the upper bound depends on the expansion properties of the completion .

One might wonder if Theorem 3 can be improved to a bound of by

This, however, is impossible as is shown by the counterexamples in [24], originally constructed to show that is not an upper bound for . For these complexes, , since is attained by a cochain of this special form.

3 Examples

The following examples show that Theorem 2 and Theorem 3 can indeed give a stronger bound than Theorem 1. The first example shows how the three values, , and can all be different and nonzero. In the second example we have (and therefore ) but nonzero. The converse is not possible since implies , but we give an instance where is constant, while is linear.

Consider the complex given in Figure 1, a triangulation of the real projective plane. It has vertices, a complete -skeleton and all triangles that are visible in the figure. Let be the edge set depicted by bold lines and let be its characteristic function, interpreted as a -cochain. Then and and hence . By computing explicitly one can see that . We will show that . Note that, since has a complete -skeleton, we furthermore have .

Consider a -coloring of the vertices of . In the case where there exists a color class of size one, the five neighbors of (which belong to the other two colors classes) span at least two -colored triangles with . In the case where all color classes have size two, one can show in a similar fashion that every such -coloring (one has to distinguish between two cases) has exactly four -colored triangles. Therefore .

Figure 1: Real Projective Plane

In the second example we give a construction of a simplicial complex where , but . Note that here is bounded by a constant, while in general a nonzero bound for can be of order .

Let be the following -complex with complete -skeleton on the vertex set (): We set and as the set of triangles that contain zero or two edges from . Then by construction and hence . Formally the set of triangles is given by

Denote and . We determine by the following case distinction, which covers all possible cases up to permutation. Note that , since has a complete -skeleton.

  1. and contain at most as many elements of as of i.e.  and . Then for all , , , and therefore

  2. but . Since it follows that . Hence

  3. All elements of are in , i.e., . By checking all possible partitions, one can find that the minimum is obtained by , in which case .

In our last example consider the -dimensional complex depicted in Figure 2. It has vertex set and edge set . The set of triangles is . Depending on the parity of , this describes either a Möbius strip or a cylinder. Observe that and hence . For the partition , , , we obtain the smallest possible value of . Therefore gives a constant bound whereas yields a linear bound. Note that , as is non-trivial.

Figure 2: Möbius Strip or Cylinder

4 The Result for Complexes with Complete Skeleton

In the following part we describe the basic ideas of the proof of Theorem 1 from [21]. By the variational characterization of eigenvalues we know that

(2)

The key idea is to find a function that satisfies

In order to define a function satisfying this equation, we fix a partition of which realizes the minimum in . We call the ’s blocks of the partition or shortly just blocks. Since the value of does not depend on the chosen orientation, we are free to choose an orientation depending on this partition. For reasons of simplicity we choose a linear ordering on such that for all , , we have .

Let , with . Then is defined as

(3)

The following two statements describing essential properties of give the proof of Theorem 1.

Lemma 5.

[21] Let be a -dimensional simplicial complex with complete -skeleton and let be defined as above. Then and

Lemma 6.

[21] Let be any -dimensional simplicial complex and let be defined as above. Then

For the first lemma, which can be proven by a straightforward calculation, there is no trivial generalization for arbitrary simplicial complexes. The latter lemma does not require any assumptions on the -skeleton and we will be able to use it for our purposes. For completeness we here give the proof of Lemma 6.

Proof of Lemma 6.

Let with . By definition of the coboundary operator it is enough to prove that

First suppose that . If has three vertices in the same block or four vertices in two blocks, then every term in

has two vertices in the same block and hence the sum vanishes. If we assume that with