Definition 1. A hypergraph is a pair where is a finite set and is a nonempty collection of subsets of . is called -uniform if
. is called a graph if it is 2-uniform.
Our goal for this lecture is to explore the rudiments of the spectral theory of -uniform hypergraphs. The natural starting point, if only for inspiration, is the case of graphs. Letting be a graph, and identifying , we can begin by writing down the adjacency matrix of defined by
where . In this case are said to be adjacent.
The spectral theory of is rich. For example, it made an appearance in our 202C class for the special case where is the Cayley graph of a group. Our main focus however will be the matrix , defined by
where the degree is the number of edges containing . If is the diagonal matrix of degrees, then we may also write . This matrix is called the Laplacian or Kirchoff matrix of . The terminology arises from the fact that as an operator on , displays various properties canonically associated with the Laplacian on ; e.g., mean value property, maximum principle, etc.. A very interesting paper of Wardetsky, Mathur, Kälberer, and Grinspun explores these connections in detail, but they are outside of the scope of this lecture.
A few facts are worth collecting here- first, is a symmetric, positive semidefinite matrix in and hence admits nonnegative eigenvalues . always has zero as an eigenvalue, for example, via the vector . But the multiplicity of said eigenvalue could be higher, depending on the number of connected components (Exercise 1). However when we assume is connected (i.e. contains a single connected component), it follows that . In much of the literature, we are often interested in the interplay and connections between geometric and combinatorial properties of – things like its size, connectedness, density, or sparseness, for example; and the first eigenvalue (either of or one of its many variants). For example…
(a) Show that the geometric multiplicity of the null eigenvalue of is the number of connected components of .
(b) Show with equality if and only if is the complete graph. Hint: if is not complete, relate its Laplacian and that of its “complement” (suitably defined) to the Laplacian of the complete graph, and use eigenvalue interlacing.
For a more detailed introduction to spectral graph theory, see Fan Chung’s book[1, Ch. 1].
Our next goal is to try and take these notions and structures and construct extensions, or analogues, for the case where is a -uniform hypergraph. As we shall see, this is not so easy. Consider if you desire the following perspective from simplicial homology (those unfamiliar or otherwise uninterested, skip to Definition 2).
One approach would be to try and construct not as a specific matrix, but as a descendant of an invariant of the topology of the underlying graph; which we could then hope admits a generalization to the case where the graph is instead a hypergraph. Specifically, the idea is to look at the homology of and build a Laplacian from its boundary operator- as follows.
Let be a graph and enumerate . To each unordered pair pick an orientation or and let be the set of all such oriented edges. Let be the matrix whose rows are indexed by and whose columns are indexed by the ordered edges , defined by
It is left as an (unassigned) exercise to verify that ; and in fact that the choice of orientations/signs in the preceding setup is completely arbitrary. Thinking of as a homomorphism mapping elements of the module to the module by matrix multiplication, we recognize as the boundary operator that is related to the homology groups of as a simplicial complex (see [3, Section 2.1, page 105]) and thus we have constructed as an invariant of the topology of (up to choices in the orientations of the edges).
But as it turns out, the capricious choice of orientation we made earlier to define is what makes graphs special. As soon as we allow 3-edges, or 1-edges for that matter, it is not at all obvious how to “orient” them, or how to distinguish between orientations (compare to the natural symbolic relationship ). This, in fact, is a critically important distinction between graphs and hypergraphs. The answer is that we need more than linear operators, and so we look to tensors. The exposition that follows roughly follows two papers of Qi[4,5].
Definition 2. A tensor of order and dimension over a field is a multidimensional array
where each entry . We call an -tensor over .
We can think of an -tensor as a hypercubic array of size , times). All of the tensors presented here are understood to be over . The case recovers the usual square matrices. If then we define the product by
(The expression is conventional notation.) Also as a matter of notation, for and , let be given by . There are lots of analogous notions and properties for tensors as compared to usual matrices; for example, we have the following formulation of the spectrum of a tensor.
Definition 3. Let be an -tensor. Then we say is an eigenvalue of if there exists a nonzero vector , called an eigenvector of , which satisfies
Notice that no tensor for defines a true linear operator on , and hence we may not a priori speak of eigenspaces, kernels, etc. and their linear algebraic structure. (These notions do have meaning, but they require the tools of algebraic geometry to properly formulate, and are outside of our scope.) Nevertheless, we can construct tensors associated to -uniform hypergraphs that will give rise to various eigenvalues. And if so, can we find relationships between the geometry of a hypergraph and its eigenvalues?
Definition 4. Let be a -uniform hypergraph on vertices. The adjacency tensor of is the -tensor defined as follows. For each edge ,
and likewise for any rearrangment of the vertices in . The values of are defined to be zero otherwise. The degree tensor of is the -tensor defined by
where is the number of edges containing , and the values of are defined to be zero otherwise. The Laplacian tensor is the tensor defined by .
A brief remark- our choice to focus on -uniform hypergraphs is merely to simplify the exposition. There are well-developed versions of Definition 4 for general hypergraphs, but they are harder to write down and explore in this setting. With these fundamental notions established, the goal for the remainder will be to write down and explore a few basic propositions regarding the eigenvalues and eigenvectors of the tensors of a hypergraph .
Proposition 1. Let be a -uniform hypergraph, with , on vertices and its Laplacian tensor.
- Let denote the -th standard basis vector. Then is an eigenvector of with eigenvalue .
- The vector occurs as an eigenvector of with eigenvalue 0.
Proof. This is an exercise in calculations with tensors. For the first claim let be fixed, and notice that for any , and hence for each , it holds
Notice that if any of the ‘s are repeated, and that if and 0 otherwise. It follows that . For the second claim, again we have for any , and hence
where, with some abuse of notation in the second line, we emphasize that the sum contains all permutations of the terminal points in each edge . -QED.
Exercise 2. Let be a -uniform hypergraph and its adjacency tensor. Show that occurs as an eigenvector of with eigenvalue 0 for each . Sanity check: why does this not imply that ?
Our next proposition is a statement about the distribution of eigenvalues.
Proposition 2. Let be a -uniform hypergraph on vertices, and its Laplacian tensor. Let denote the maximum degree of . Then all of the eigenvalues of lie in the disk .
Proof. ([4, Thm. 6]) Let be an eigenvalue for , and let be a corresponding eigenvector. Assume we have
Then by definition, it holds , which in the -th component reads
Removing a single term from the right-hand side, we get
But from the definition of we recognize as and hence by choice of , we can estimate
where in the final equality we are using the definition of and are recalling the calculation in the proof of proposition 1. We have hence proved that all eigenvalues lie in a closed disk centered on of the same radius; but this depends on the unknown eigenvector , so by observing that for any ,
the claim follows. -QED
Exercise 3. Let be a -uniform hypergraph on vertices and its adjacency tensor. Show that all of the eigenvalues of lie in the disk .
For our final proposition, we strive for an analogue of Exercise 1(a)- that is, a relationship between the “size of the kernel” and the connected components of . However for -uniform hypergraphs, will not be a linear operator for any , and hence a kernel is not well-defined a priori. But as we saw in Proposition 1, the same vector shows up as an eigenvector for the null eigenvalue, and hence we are suspicious that more can be said.
A few small preliminaries are in order. A vector is called a binary vector if each of its components are either 0 or 1 for . The support of a vector is the set . If is a -tensor over , then an an eigenvector associated to eigenvalue is called a minimal binary eigenvector associated to if is a binary vector, and if there does not exist another binary eigenvector with eigenvalue for which is a proper subset of .
Proposition 3. Let be a -uniform hypergraph on vertices with Laplacian tensor . A binary vector is a minimal binary eigenvector associated to if and only if is the vertex set of a connected component of
Proof. We first observe using the proof of Proposition 1 that a nonzero vector is an eigenvector associated to if and only if, for each , it holds
(). Let be any minimal binary eigenvector associated to and let . It follows from equation (1) that for any it holds (otherwise, there will not be enough terms in the sum to achieve ). Applying this observation inductively to adjacent vertices in , we see that the vertices form either the vertex set of one component of , or the disjoint union of several. But the latter case is incompatible with minimality, since any binary vector supported on a single component will give rise to an eigenvector with eigenvalue (check!).
(). Now assume simply that is a binary vector with support equal to a single connected component of . A short calculation verifies that the components for satisfy equation (1), and hence that is a binary eigenvector for assosicated to eigenvalue . All that remains to be checked is that is minimal; to this end, suppose is another binary eigenvector, and that . Then there must exist some vertex for which there exists some edge and an index which does not belong to . But then,
and equation (1) cannot hold for , a contradiction. -QED
Corollary 1. A -uniform hypergraph is connected if and only if the vector is the unique minimal binary eigenvector with eigenvalue .
References and Further Reading.
- Fan Chung. Spectral Graph Theory. 1997.
- Cunxiang Duan, Ligong Wang, Xihe Li, et al. Some properties of the signless laplacian andnormalized laplacian tensors of general hypergraphs. Taiwanese Journal of Mathematics, 24(2). 2020.
- Allen Hatcher. Algebraic topology. 2005.
- Liqun Qi. Eigenvalues of a real supersymmetric tensor. Journal of Symbolic Computation, 40(6). 2005.
- Liqun Qi. H+-eigenvalues of laplacian and signless laplacian tensors. Communications in Mathematical Sciences, 12(6). 2013.
- Max Wardetzky, Saurabh Mathur, Felix Kälberer, and Eitan Grinspun. Discrete laplace operators: no free lunch. In Symposium on Geometry processing. 2007.