Inverse Expander Mixing for Hypergraphs

We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.


Introduction
Beginning with the seminal work of Thomason [1] and Chung-Graham-Wilson [10] the theory of quasirandom graphs has served as an important tool in graph theory and the study of random structures. Their work showed that many graphs share a collection of common properties, including local properties like subgraph counts, and global properties like edge distribution and eigenvalues. The connection between these different concepts has proved to be an invaluable tool in graph theory and computer science over the past two decades. Even the recent theory of graph limits has borrowed many insights from quasirandom graphs [18]. A fundamental result in this theory is the quantitative relationship between spectral and expansion properties of a graph. Although early versions of this were proved by Tanner [21] and Alon [4], the formulation of the result below, which has been termed as the Expander Mixing Lemma for graphs, is perhaps the most popular one. For a graph G, let λ(G) be the second largest eigenvalue, in absolute value, of the adjacency matrix A(G). Theorem 1.1 (Expander Mixing Lemma for graphs, Alon and Chung [5]). If G is an r-regular graph then for any S, T ⊆ V (G), where E(S, T ) is the number of (s, t) ∈ S × T such that st ∈ E(G).
The converse to Theorem 1.1 is essentially a question about approximating certain quadratic forms on the sphere by points in a cube. This was achieved by Bilu and Linial [6] who proved the following converse to the Expander Mixing Lemma.

Simplicial Complex Setting
Our main result for simplicial complexes (Theorem 2.2) requires some notation and preliminaries which are discussed in the next section.

Notation for simplicial complexes
For a fuller discussion of notation and preliminaries, refer to [20]. Throughout, let X be a ddimensional simplicial complex with vertex set V of size n. Let X i denote the set of i-cells of X, where an i-cell consists of i + 1 vertices so that the simplex defined by these points has dimension i. The simplicial complex X is said to have a complete skeleton if X i = V i for each i = 0, . . . , d−1. For any subsets S 0 , . . . , S d ⊆ V , we write F (S 0 , . . . , S d ) for the number of ordered tuples (s 0 , . . . , s d ) ∈ S 0 × · · · × S d such that {s 0 , . . . , s d } ∈ X d .
If i > 0, an i-cell {σ 0 , . . . , σ i } has two orientations given by the orderings of its vertices up to an even permutation. Denote one orientation by σ = (σ 0 , . . . , σ i ) and the other orientation by σ. Let X i ± denote the set of all oriented i-cells. Sometimes, abusing this notation, we also use σ to refer to the unoriented i-cell which corresponds to the oriented i-cells σ and σ.
Definition 2.1. Let Ω i be the vector space of real-valued, skew-symmetric functions on X i ± , i.e., functions f : For example, we can think of Ω 0 as the set of vertex weightings, and Ω 1 as the set of flow functions.
Definition 2.2. Define an inner product on Ω i by noting that f (σ)g(σ) = f (σ)g(σ), and that we only take one of these terms in the sum.
With this inner product comes an associated norm f := f, f on Ω i . Recall that for an operator M : Ω i → Ω i (or on any normed vector space) we also have an operator norm, and let Z d−1 := ker ∂ d−1 .
The boundary of a weight function is the total weight of vertices, while the boundary of a flow is the function that assigns to each vertex the net flow at that vertex. Correspondingly, Z 0 is the set of vertex-weightings with weights summing to 0, while Z 1 is the set of conservative flows.
Let A := τ ∈X d−2 A τ be the adjacency operator, and let J := τ ∈X d−2 J τ . Denote by I the identity operator on Ω d−1 .
Remark 2.1. We could also view these definitions in a more linear-algebraic light. For each σ ∈ X d−1 we can choose a canonical "positive" orientation σ = (σ 0 , . . . , σ d−1 ). We identify Ω d−1 with the vector space R X d−1 + and operators such as the adjacency operator A above could then be considered as matrices indexed by the canonical orientations of i-cells. With these identifications the inner product and norms defined above correspond to their usual counterparts.
The matrix for the adjacency operator A τ would be the signed adjacency matrix of the graph induced on d-cells and d − 1-cells containing τ , with wτ ∼ vτ (i.e, a ±1 in the coordinate corresponding to the positive orientations of wτ and vτ ) if and only if wvτ ∈ X d . The signs are determined by the orientations of wτ and vτ relative to the canonical positive orientations of those cells (for instance, if the positive orientations are wτ and vτ then the entry is negative).
Similarly, the matrix for J τ would have a ±1 for any pair of (d−1)-cells (not necessarily distinct) both containing τ , with the signs determined in the same fashion.
More precisely, if σ, σ ′ ∈ X d−1 are positively-oriented cells that differ by exactly one vertex, let π σ,σ ′ be the unique permutation of {0, . . . , d − 1} so that σ π σ,σ ′ (i) = σ ′ i whenever σ ′ i ∈ σ. Then we can calculate the σ, σ ′ entry of A and J: Note that for ease of analysis positive orientations can be chosen for any particular τ to make all of the signs in A τ positive, but this cannot be maintained across all (d−2)-cells τ simultaneously and so the matrix for A must exhibit both signs, regardless of the choice of canonical orientations.
For graphs each cell has only one orientation, so we have X 0 = V and Ω 0 = R V and we can think of the usual adjacency matrix of a graph as an operator A : R V → R V . Indeed, for d = 1 the only (−1)-cell is the empty set, and so A = A ∅ is just the adjacency matrix of the graph, while J ∅ is the all-ones matrix. Note that for d = 1, ∆ + is the graph Laplacian.

Mixing Lemmas for Simplicial Complexes
The following is a recent mixing lemma due to Parzanchevski, Rosenthal & Tessler: Theorem 2.1 (Mixing Lemma for simplicial complexes, Parzanchevski et al. [20]). Let X be a ddimensional complex with a complete skeleton and fix α ∈ R. For any disjoint sets S 0 , . . . , S d ⊆ V , This is not quite the statement given in the original paper, which concludes with a slightly looser but more symmetric result. Note that if X is r-regular and α = r then ρ α is the second-largest eigenvalue of A = rI − ∆ + .
The first result we prove is an inverse of the mixing lemma for simplicial complexes.
Theorem 2.2 (Inverse Mixing Lemma for simplicial complexes). Let X be a d-dimensional, rregular simplicial complex with a complete skeleton, and suppose that for every collection of disjoint Again, when the complex is regular with a complete skeleton this quantity is the second-largest eigenvalue of A. Remark 2.2. It is possible to generalize this result by replacing the r in (2.2) with an arbitrary value α ∈ R, and by replacing A with αI − ∆ + , as in the statement of 2.1. Remark 2.3. Adding or removing d − 1-cells does not change the eigenvalues of A, but does change Z d−1 . The complete skeleton requirement is not necessary to make the statement true of the secondlargest eigenvalue of A, but is necessary to make λ 2 = A| Z d−1 . Observe that Z d−1 is basically the space of functions orthogonal to the all-ones function, but the eigenfunction of A corresponding to eigenvalue r has zeroes where there are isolated (i.e. missing) (d − 1)-cells. Remark 2.4. In our proof, we do not use the full strength of the hypothesis. We will always take S 2 , . . . , S d to be singletons.

Proof. It is clear from the graph interpretation of
On the other hand, if τ ⊂ σ then we can write σ = vτ for some v / ∈ τ . In this case This allows us to say that What remains is to bound (1). We use a lemma of Bilu & Linial: Lemma 2.3 (Bilu and Linial [6]). Let B be a symmetric, real-valued n × n matrix in which the diagonal entries are all 0. Suppose that the ℓ 1 -norm of every row of B is O(m), and also that for any vectors x, y ∈ {0, 1} n with disjoint support We will apply this lemma to As mentioned in Remark 2.1, we can interpret B as a matrix indexed by positive orientations of elements in X d−1 . Combining the calculations of A and J in that remark, we can calculate that for each σ, σ ′ ∈ X d−1 , Then we can see that B is symmetric (because π σ ′ ,σ = π −1 σ,σ ′ ), real-valued and its diagonal entries are 0. Since X is r-regular, the ℓ 1 -norm of each row σ in B is Indeed, there are (n − d) total sets η of size d + 1 containing σ, and r of those are d-cells; each such set η contains d other cells Note that each σ ∈ supp x is in the support of exactly d of the functions x τ . Observe that By definition, for any fixed τ = (τ 2 , . . . , τ d ) Using a similar decomposition for x τ , J τ y τ we obtain We would like to interpret the first half of the sum as a number of edges and the second half as the product of the sizes of some vertex sets, since for any disjoint sets S 0 , S 1 we have by assumption that However, this differs from what we have above by the signs from x and y. Instead we break the sum apart according to these values, and for each η ∈ {±1} we write These four sets are pairwise disjoint and where the inequality in (2) follows from Cauchy-Schwarz. Finally we can apply Lemma 2.3 with m = 2rd and β = 2ρd to get Combining the results for each τ using the triangle inequality gives As long as ∅ . . , S d to be singletons corresponding to a subset which is either a d-cell or not), so d = O(ρd) and we can replace the above bound by The same bound holds trivially for the empty complex (which has A = 0), while for the complete complex (with r = n − d) we can take S 2 , . . . , S d to be singletons and when 1 ≤ d < n, so this simpler bound holds in general.
Corollary 2.4. Let X be an r-regular, d-dimensional complex with a complete skeleton, and suppose that for every collection of disjoint sets S 0 , . . . , Then

Notation for hypergraph eigenvalues
The notion of eigenvalues for hypergraphs that we now describe was developed by Friedman & Wigderson in [12]. Further discussion can be found in [16].
As with the adjacency form we will suppress the subscript H when the hypergraph is clear from context. If V 1 , . . . , V k are pairwise disjoint, this is the number of edges that intersect each V i in exactly one vertex. Alternatively, if we take x i to be the indicator vector of V i then we could equivalently define e(V 1 , . . . , V k ) = A(x 1 , . . . , x k ).
Let J denote the k-linear form with J(e i 1 , e i 2 , . . . , e i k ) = 1 for all choices of standard basis vectors e i 1 , e i 2 , . . . , e i k . Let K = (V, V k ) denote the complete k-uniform hypergraph on vertex set V (with corresponding adjacency form A K which evaluates to 1 on any distinct standard basis vectors.) In the case where φ is symmetric, as shown in [12] we in fact have that Observe that both A and J are symmetric.
Recall that the first (largest) eigenvalue of a graph can be defined as the operator norm of its adjacency matrix, A G , and if the graph is r-regular then the second-largest eigenvalue is A G − r n J . This motivates a definition of the second eigenvalue for hypergraphs given by Friedman and Wigderson in [12]: if H is r-regular, they define the second eigenvalue to be For any H (not necessarily r-regular), the quantity in (3) is called the second eigenvalue of H with respect to r-regularity.
For technical reasons, we will take a slightly different definition for the second eigenvalue. We also define a parallel parameter measuring the combinatorial expansion of a hypergraph.
Definition 3.5. For a k-uniform hypergraph H, define for each α ≥ 0 where the maximum is taken over all tuples V 1 , . . . , V k of pairwise disjoint nonempty subsets of V .
Remark 3.1. Our aim is to bound the second eigenvalue in terms of ρ = ρ α (H). Unfortunately, we find that independently of ρ, λ 2 = Ω(rn k−2 ) with high probability for random hypergraphs with edge density r/n, making an inverse mixing lemma for λ 2 impossible. Our new definition λ 2,α allows us to avoid this problem. A further discussion of this problem is found in Section 3.3.
It is natural in our definition to choose α = |E(H)| / |E(K)|, i.e., the edge-density of H. However, to more closely parallel the Friedman-Wigderson definition one can choose α = r n . For now we will proceed without specifying a fixed value for α.
Remark 3.2. Even if one fixes α as suggested above to be the edge density of H, our definition does not quite agree with the usual definition of graph eigenvalues in the case of r-regular graphs (k = 2). In particular, where λ(G) = A − r n J we use λ 2,α (G) = A − r n−1 A K . However, it is easy to see that the two values never differ by more than r n−1 I − 1 n J = r n−1 ≤ 1. The following simple upper bound will come in handy in later analysis. Proof. Working directly from the definition, we have

Hypergraph Mixing Lemmas
The following hypergraph mixing result is given in [12].
Before stating and proving a converse to Theorem 3.2 above, we mention the mixing result using our definition of the second eigenvalue λ 2,α , with respect to density α.

Theorem 3.3 (Mixing Lemma for hypergraphs). Let H be a k-uniform hypergraph. For any choice of subsets
Proof. Let V 1 , . . . , V k ⊂ V (H). If any V i is empty, it is clear that the inequality holds; we may assume that each V i is nonempty. For 1 ≤ i ≤ k let x i ∈ {0, 1} n be the indicator vector of V i . Then We now prove the main theorem of this section -a converse to the above Theorem 3.3:

Theorem 3.4 (Inverse Mixing Lemma for hypergraphs).
If H is a k-uniform hypergraph with maximum codegree r and ρ = ρ α (H) then Remark 3.3. We have left this result in what is perhaps not its simplest form, in order to show the difference between the cases k = 2 and k ≥ 3. In the case where k = 2 and α = Θ(r/n) the dependence on n disappears and this simplifies to the classic result λ 2,α = O(ρ(log(r/ρ) + 1)) for graphs. For larger (but still constant) uniformity, we can still simplify the result to λ 2,α = O(ρ (log k−1 ((r + αn)n/ρ) + 1)).
We prove the theorem through a series of lemmas. First we show that the partite expansion condition suffices to give expansion for any (not necessarily disjoint) sets of vertices. Throughout, b represents a constant independent of x (but which may depend on k, n, r, α, ρ or anything else).

Lemma 3.5. Let H be a k-uniform hypergraph on n vertices with adjacency form A, and suppose that
x i for every choice of pairwise orthogonal vectors x 1 , . . . , x k ∈ {0, 1} n . Then for every choice of (not necessarily orthogonal) vectors x 1 , . . . , x k ∈ {0, 1} n .
Proof. Let V 1 , . . . , V k ⊆ [n] be any sets of vertices. Consider an ordered partition P = P 1 ∪ · · · ∪ P k of [n] into k nonempty parts. Then as every ordered edge (v 1 , . . . , v k ) shows up in the sum once for each partition P with v j ∈ P j for every j, and there are (n − k) k such partitions (the remaining n − k elements can be partitioned in any way among the k sets). Similarly, replacing H with the complete hypergraph gives For a fixed partition the subsets P i ∩ V i are disjoint, so by hypothesis we have Then Here k!S(n, k) ≤ k n is the number of ordered partitions of [n] into k nonempty sets (the number of terms in the sum over all choices of P), and the inequality in (4) follows by concavity of square root.
The main part of the work goes towards proving a hypergraph version of Lemma 2.3. We go through several steps to show that if the expansion bound holds for {0, 1} vectors then a somewhat relaxed bound holds for all real vectors. Lemma 3.6. Suppose B is a k-linear form such that Proof. Let x 1 , . . . , x k ∈ {0, ±1} n , and decompose Proof. Let x ∈ 0, ±2 −ℓ : ℓ ∈ N and write x = i∈N 2 −i x i with x i ∈ {0, ±1} n (the x i have pairwise disjoint support and are hence orthogonal). Define s i = supp x i = x i 2 so that Note that all sums have only finitely many nonzero terms. We are interested in bounding We split this sum into two parts, bounding separately the sums over the index sets for some γ ≥ 0 to be determined later. For the sum over i ∈ P we have where the final step uses the AM-GM inequality. Each ℓ ∈ N appears at most k(2γ) k−1 times in elements of P (as each time ℓ appears in some position the remaining k − 1 terms must all be between ℓ − γ and ℓ + γ), so where we have used that i a k/2 i ≤ ( i a i ) k/2 for nonnegative a i and k ≥ 2. Now we focus on bounding the sum over i ∈ Q. For each i ∈ Q we move min i to i 1 and max i to i k . Such a reordered index vector corresponds to at most k 2 original vectors, so we have For fixed i 1 , . . . , i k−1 , |B(e ℓ 1 , . . . , e ℓ k )| where the last inequality is due to hypothesis (5). Plugging this into the bound (6) above gives using the fact that x 1 ≤ √ n x (by Cauchy-Schwarz). Putting everything together, we have Finally, set γ = lg(amn (k−2)/2 /b) (which is non-negative by the restriction on a) to get |B(x, . . . , x)| / x k ≤ b (lg k−1 (a 2 m 2 n k−2 /b 2 ) + k 2 /a) as desired.
Proof. Let x ∈ R n be a vector which maximizes |B(x, . . . , x)| / x k = B . Without loss of generality, scale x so that |x i | ≤ 1/2 for all i ∈ [n]. Choose a random vector z ∈ 0, ±2 −ℓ : ℓ ∈ N n by picking each coordinate z i independently as follows: If x i = 0 then z i = 0. Otherwise, write |x i | = 2 ℓ i (1 + ε i ) for some integer ℓ i and some value of ε i ∈ [0, 1). Let z i = sign(x i )2 ℓ i with probability 1 − ε i and z i = sign(x i )2 ℓ i +1 with probability ε i .
Thus there is a vector z for which |B(z, . . . , z)| ≥ |B(x, . . . , x)|. Observe that by construction z ≤ 2 x , so Finally, we put all of these lemmas together to prove the theorem.

Comparison with the Friedman-Wigderson definition of λ 2
In this section we prove an inverse mixing lemma for the Freidman-Wigderson definition of the second eigenvalue. We will see that this result, while tight, is not as useful as theorem 3.4, and we briefly discuss the reason for this. First, we define a useful k-linear form and evaluate its norm.  Proof. First of all note that On the other hand, for any x ∈ R n |D(x, . . . , as desired. Theorem 3.10. Let H be a k-uniform hypergraph with maximum degree r, and suppose that for every choice of disjoint sets V 1 , . . . , V k ⊂ V (H), Then Proof. Set α = r n , we have that r = Θ(αn). Observe that if V 1 , . . . , V k are disjoint, then

By Theorem 3.4,
and hence by Proposition 3.9 A similar calculation to the one above also gives a lower bound of If the first term dominates in (7) then the asymptotics of λ 2 are independent of ρ and so there is no interesting inverse mixing for this definition of the second eigenvalue. We now show that this is in fact typically the case by examining ρ α for random hypergraphs.
To get some idea about the typical magnitude of ρ α , we analyze the Erdős-Renyi random hypergraph G(n, α, k), in which each of the n k k-tuples is taken as a hyperedge independently with probability α. Proof. For fixed disjoint sets of vertices V 1 , . . . , V k , note that e(V 1 , . . . , V k ) is a sum of k i=1 |V i | independent Bernoulli random variables each with mean α.
Assume that α = r n is a positive constant independent of n, in other words, that r = Θ(n).
For G that satisfies the bound in Proposition 3.13, the second term of (7) is Θ( √ n log k−1 (n)), which is dominated by the first term if k ≥ 4. So, almost every hypergraph with k ≥ 4 will not have an interesting inverse mixing lemma for the Friedman-Wigderson definition of the second eigenvalue. This is proven by combining Proposition 3.13 with the bounds on λ 2,α found in Theorem 3.3 and Theorem 3.4.