Graphs that are sparse but well connected, called expander graphs, have numerous applications in various areas of computer science (see for example [HLW06]). Recently, there has been much interest in generalizing the notion of expansion to higher-dimensional simplicial complexes, beginning with the work of Lineal and Meshulam [LM06], Meshulam and Wallach [MW09], and Gromov [Gro10]. While multiple notions of high-dimensional expansion have been introduced (see the survey by Lubotzky [Lub18]
), these notions agree in that random simplicial complexes are not good high-dimensional expanders. In contrast, ordinary random graphs are near-optimal expanders with high probability. Therefore constructions of high-dimensional expanders are of particular interest. Early constructions of high-dimensional expanders, namely Ramanujan complexes[LSV05b, LSV05a], were quite mathematically involved, whereas more simple and combinatorial constructions have been introduced recently.
In this paper, we introduce a modification of the high-dimensional expander construction of Liu, Mohanty, and Yang [LMY20], which is based on a sort of high-dimensional tensor product. We then show that this modification gives an exponential improvement in the dependence of the th order spectral gap on the dimension , and we discuss why our results suggest this modification is optimal among constructions with the same general product structure. These results also address a question posed by Liu et al. [LMY20] pertaining to the limitations of their product-based construction.
1.1 High-dimensional expanders
A high-dimensional expander is a simplicial complex with certain expansion properties. A simplicial complex is a hypergraph with downward-closed hyperedges, called faces. That is, a simplicial complex on vertices is a collection of faces , where for any face , all subsets of are also faces in . The dimension of a face is , and the dimension of is the maximum dimension of any face in . The 1-skeleton of a simplicial complex is the undirected -vertex graph whose edges are given by the 1-dimensional faces. We restict attention to pure simplicial complexes, meaning that every face is contained in a face of maximal dimension.
We consider the notion of high-dimensional expansion introduced by Kaufman and Mass [KM17], which requires rapid mixing for the high-order “up-down” or “down-up” random walks on simplicial complexes. The -dimensional up-down walk on a simplicial complex specifies transition probabilities for a walk that alternates between faces of dimension and . The 0-dimensional up-down walk is an ordinary lazy random walk on the 1-skeleton of the simplicial complex. Therefore 1-dimensional expanders are just ordinary (spectral) expander graphs. The spectral gaps of higher-order walks are difficult to bound directly, so they are instead typically bounded using a line of work [KM17, DK17, KO18, AL20], which has shown that a large spectral gap for high-order walks is implied by good local expansion, that is, good spectral expansion of a specific set of graphs that describe the local structure of the simplicial complex. Formal definitions of high-order walks and local expansion are provided in Section 2.
We are interested in constructions of infinite families of high-dimensional expanders with bounded degree and spectral gap for all dimensions. Specifically, for any fixed , a -dimensional expander family is a family of -dimensional simplicial complexes such that there is no finite upper bound on the number of vertices of elements , and the following two properties hold:
Bounded degree: There exists some such that for every , each vertex in belongs to at most faces.
Bounded spectral gap: There exists some such that for every and every , the -dimensional up-down walk on has spectral gap .
In general, the spectral gap of the -dimensional up down walk cannot be greater than (see for example Proposition 3.3 of [AL20]). Our goal is to prove good lower bounds for this spectral gap for specific constructions of -dimensional expander families. That is, we are interested in the optimal relationship between dimension and spectral gap, and only require an arbitrary upper bound on degree. This goal differs from the study of expander graphs, and specifically Ramanujan graphs, which focuses on the optimal relationship between degree and spectral gap.
While Kaufman and Mass [KM17] showed that Ramanujan complexes are high-dimensional expanders, multiple more elementary constructions have since been introduced [Con19, CTZ20, CLP20, LMY20, KO20]. However, only two of these constructions [LMY20, KO20] provide constant-degree high-dimensional expander families of all dimensions. The construction of Kaufman and Oppenheim [KO20] is based on coset geometries, and achieves near-optimal expansion in all dimensions. In contrast, the construction of Liu et al. [LMY20] is much more elementary, as it consists of a sort of high-dimensional tensor product between an expander graph and a constant-sized complete simplicial complex. However, this construction has suboptimal expansion in high dimensions. Specifically, Alev and Lau [AL20] showed that the -dimensional up-down walk on the -dimensional construction of Liu et al. [LMY20] has spectral gap at least , where is a constant depending on and on the expander graph used in the construction. Note that this bound has exponential dependence on , in contrast to the optimal linear dependence .
In this paper, we present a modification of the high-dimensional expander family of Liu et al. [LMY20], for which we show that the spectral gap of the -dimensional up-down walk is at least . This quadratic dependence on provides an exponential improvement compared to the spectral gap bound of for the construction of Liu et al. [LMY20]. We attain this exponential improvement using the same product structure as Liu et al. [LMY20] while adjusting the weights of faces. Our modified construction also yields improved local expansion in high dimensions. For every , we show that the 1-skeleton of the link of any -dimensional face in our construction has spectral gap at least , an improvement over the analagous bound of for the construction of Liu et al. [LMY20].
The organization of the remainder of this paper is as follows. Section 2 presents preliminary notions, and Section 3 presents our main construction along with some basic properties. In Section 4, we compute the local expansion of the construction, from which a result of Alev and Lau [AL20] implies rapid mixing of high-order walks. Section 5 discusses potential generalizations and limitations.
2 Background and preliminaries
This section provides basic definitions pertaining to simplicial complexes and high-dimensional expanders, as well as relevant past results.
A simplicial complex on vertices is a subset such that if , then all subsets of also belong to . Let , and let the -skeleton of refer to the simplicial complex . The elements of will be referred to as -dimensional faces. The dimension of a simplicial complex is the maximum dimension of any of its faces, and if each face is contained in a face of maximal dimension, then the complex is pure. A balanced weight function on a pure simplicial complex is a function such that for every and every ,
The 1-skeleton of a simplicial complex with balanced weight function is the undirected weighted graph that has vertices , edges , and edge weights for . The weighted degree of a vertex in this graph is given by the weight .
This paper will restrict attention to pure weighted simplicial complexes with balanced weight functions. In this case, the faces and weight function may be defined only on faces of maximal dimension, then propagated downwards, and the following useful formula applies.
Lemma 2 ([Opp18]).
For every -dimensional simplicial complex , every , and every ,
Just as ordinary expander graph families are specified to have bounded degree, we are interested in families of simplicial complexes satisfying an analagous notion:
A family of simplicial complexes has bounded degree if there exists some constant such that for every and every vertex , there are at most faces in that contain .
The local properties of a simplicial complex are captured by the links of faces, defined below.
For a simplicial complex with weight function , the link of any is the simplicial complex defined by with weight function .
A common theme in the study of high-dimensional expanders is the “local-to-global” paradigm, which uses bounds on the expansion of the 1-skeletons of links to prove global expansion properties. To state such local-to-global results, it is first necessary to define graph expansion.
For a graph , the adjacency matrix is denoted , the diagonal degree matrix is denoted , and the random walk matrix is denoted
. The eigenvalues of the random walk matrix are denoted from greatest to least by. The expansion, or spectral gap, of is the quantity .
For any -vertex graph , all eigenvalues of the random walk matrix lie in , and , so because . Graphs with spectral gaps closer to are considered “better” expanders.
We now introduce a notion of expansion for simplicial complexes.
For , the -dimensional local expansion of a simplicial complex with weight function is the value
The local expansion of is the minimum of the -dimensional local expansion over all , while the global expansion equals the -dimensional local expansion.
That is, the -dimensional local expansion refers to the lowest expansion of the 1-skeleton of the link of any -dimensional face. Note that the terminology -dimensional local expansion is nonstandard, but will be useful here. The following result shows that good local expansion in higher dimensions implies good local expansion in lower dimensions.
Proposition 7 ([Opp18]).
Let be a simplicial complex in which all links of dimension are connected. Then for every ,
In particular, Proposition 7 implies that if , then .
The definition below presents the high-order random walks on simplicial complexes.
Fix a pure -dimensional simplicial complex . For , define the up-step random walk operator so that for ,
For , define the down-step random walk operator so that for ,
Define the up-down and down-up random walk operators by and respectively.
For context with this definition, consider a 1-dimensional simplicial complex , which may be viewed as a weighted, undirected graph. Then the up-step operator moves from a vertex to an adjacent edge with probability proportial to its weight, while the down-step operator moves from an edge to either of its vertices with probability . Thus is the ordinary graph lazy random walk operator, with stationary probability .
The nonzero elements of the spectra of and of are identical. Therefore when studying the expansion of these operators, we restrict attention without loss of generality to .
The spectral gap gives a measure of high-dimensional expansion. Local expansion, defined above, provides another notion of high-dimensional expansion. The following result, which follows the “local-to-global” paradigm, shows that these two notions are closely related.
Theorem 9 ([Al20]).
Let be an -dimensional simplicial complex, and let refer to the up-down walk operator on . Then for every ,
Thus if a simplicial complex has good local expansion, then its high-order walks have large spectral gaps. We apply this result to show our main high-dimensional expansion bound.
The bound in Theorem 9 is nearly tight for good local expanders:
Proposition 10 ([Al20]).
Let be an -dimensional simplicial complex with at least vertices, and let refer to the up-down walk operator on . Then for every ,
The main purpose of this paper is to present a construction of high-dimensional expander families, defined below.
A family of -dimensional simplicial complexes is an -dimensional expander family if the following conditions hold:
For every , there exists some with .
has bounded degree.
For every , there exists some such that for every , the -dimensional up-down walk operator on satisfies .
We are interested in constructing high-dimensional expander families of all dimensions with the up-down walk spectral gaps close to the upper bound in Proposition 10. By Theorem 9, item 3 in Definition 11 is implied by a uniform lower bound on the local expansion of all for every . We use this fact in the analysis of our construction.
The following definition introduces the simplicial complex with weight function considered in this paper. The construction takes as input an -vertex graph , which will typically be chosen from a family of expanders, as well as a dimension and a parameter , the latter of which will typically be taken as . The parameters and will typically be treated as fixed values, while and vary, so that the construction provides a family of -dimensional simplicial complexes.
Let be any connected undirected graph on vertices with no self-loops. For positive integers and , define the -dimensional simplicial complex with vertex set such that
Define a weight function on so that if is such that and for some edge , then let
In words, the -dimensional faces of are those sets of the form such that all are contained in a single edge of , and such that all are distinct. If the above definition is modified so that the condition is replaced with and so that for all -dimensional faces , then the resulting simplicial complex is exactly the construction of Liu et al. [LMY20]. The difference between the weight functions of and lead to the key insights of this paper. Also note that here may be a weighted graph, whereas Liu et al. [LMY20] only considered complexes derived from unweighted graphs.
If is unweighted so that for all , then every has . Thus may be viewed as an unweighted simplicial complex, meaning that all -dimensional faces have the same weight, but where multiple copies of faces are permitted.
The simplicial complex may be viewed as a sort of high-dimensional tensor product of the graph and the -vertex, -dimensional complete complex . In particular, the faces in are exactly those -element subsets of for which the projection onto the first component gives an edge in , and the projection onto the second component gives a face in . For comparison, an analagous property holds for the ordinary graph tensor product , in which the edges are given by those pairs of elements of whose projection onto either component gives an edge in the respective graph or .
Note that as with , the simplicial complex has bounded degree with respect to if has bounded degree. In particular, fix some values and , and let be chosen from a family of bounded degree graphs. For every and , by definition
It follows by the definition of that both the maximum weight and the maximum number of faces containing any face in are bounded by a constant. Furthermore, the number of faces in grows as .
An additional consequence of Equation (1) is that while the cardinality of the set on the left hand side, which equals the sum of the weights of the faces in this set under the weight function of , has an exponential dependence on , the sum of the weights of the faces in this set under the weight function of is equal to , which has only a quadratic dependence on . This observation provides some initial intuition for the exponential speedup of high-order walks on compared to .
From here on, the weight function and the operators will always refer to , unless explicitly stated otherwise.
3.1 Main result
Our main result, shown in Section 4, is stated below.
Theorem 13 (Restatement of Corollary 19).
Let be the up-down walk operator for the simplicial complex of Definition 12. If , , and , then for every ,
In contrast to this quadratic dependence on , Alev and Lau [AL20] showed that the spectral gap of the -dimensional up-down walk on is at least , where depends on the structure of . The discussion in Section 3.3 below provides intuition for why this exponential dependence in arises for , and how the adjusted weights in yield the improved quadratic dependence.
Theorem 3.1 implies that for any fixed with , if is chosen from a family of bounded degree expanders with spectral gap , the resulting simplicial complexes form a family of -dimensional expanders with -dimensional up-down walk spectral gap at least . For comparison, an optimal -dimensional expander family, as is given by Ramanujan complexes, achieves a spectral gap of for the -dimensional up-down walk.
3.2 Decomposition into permutation-invariant subsets
This section formalizes the intuitive notion that the construction of treats elements of interchangeably, and introduces some notation to reflect this symmetry. For any , the set of -dimensional faces may be decomposed as follows. For any and , let
be the set of all -dimensional faces in that contain vertices in and vertices in . To remove redundancy when , let
Then by construction,
This decomposition simply partitions into faces that differ only by permutations of :
For every , there is a group action of on the set of faces of given by
The orbits of this action are exactly the sets . The action preserves weights, that is, .
The construction of directly implies that for all , if then , so the group action on is well defined. Similarly, for all , the definition of directly implies that . For any , if is any permutation such that and , then . Thus is the orbit of under the group action. For any , to verify that , note that by definition is constant over all values of for any given and . Thus for all , the characterization of the orbits above implies that . This equality then extends to of any dimension by Lemma 2. ∎
Loosely speaking, Lemma 14 says that elements of may be treated interchangably, which in particular permits the following definition.
For all , , and , choose any and define . This definition does not depend on the choice of by Lemma 14. To avoid redundancy, also define .
Note that is defined by letting . A basic property of these weights is that for any , the ratio is independent of the choice of edge , as is shown below.
For all , , and ,
For any , any -dimensional face must satisfy for some . Therefore by Lemma 2,
where the final equality holds because there are exactly elements such that , and for each one by definition. ∎
3.3 Relative weights of overlapping faces
The proposition below determines the relative weights of faces of that intersect at all but one of their vertices. This result is used in Section 4 to determine the local expansion of .
For all , , and , it holds that
Both statements are shown using induction. For the first equality, the base case follows by the definition of , so that
For the inductive step, assume for some that it holds for all that For any and any , by definition any -dimensional face is obtained from by adding either a vertex in or in . Therefore
Applying the exact same reasoning to a face gives that
completing the inductive step; note that the third equality above holds by the inductive hypothesis.
To show the second equality in the proposition statement, first note that the base case holds immediately as because the definition of the complex does not include, or equivalently assigns zero weight, to faces in . For the inductive step, assume that for some it holds that For any , by definition any -dimensional face is obtained from by adding either a vertex in or in for some . Therefore
Applying (2) with gives that
completing the inductive step; note that the third equality above holds by the inductive hypothesis, and because for all as was shown above. ∎
Proposition 17 provides the key insight for understanding why the spectral gap of the up-down walk on has a quadratic dependence in , whereas that of has an exponential dependence. For some , , , consider an element . Let
be the random variable for the face obtained by taking one step in the up-down walk starting at. Then for some . Let denote the subset of obtained by projecting the elements of to their second components. If , then the up step must add some vertex in and the down step must remove some vertex in , while if then the up step must add some vertex in and the down step must remove some vertex in . Thus by the definition of the up- and down-step transition probabilities,
) are heavily skewed to pushin the direction of . It is this property that results in an exponential dependence on in the th order up-down walk on ; the up-down walk becomes “trapped” in the set of faces contained in , with the transition probabilities pushing away from the “exit routes” and .
To understand why the weight function on resolves this issue, observe that for , Proposition 17 implies that both probabilities in (3) equal . Therefore the events and are equally likely. Thus the up-down walk moves across the sets for as a lazy random walk on an unweighted, undirected, -vertex path. The mixing time for such a walk grows quadratically in , thereby providing intuition for the quadratic dependence in for the mixing time of .
The intuition described above can be formalized to bound the mixing time of the high-order walks on . However, the following section takes a different approach by computing the local expansion of , which leads to tighter bounds on .
4 Local and global expansion
This section analyzes the local and global expansion of , which is then used to bound the mixing time of the up-down random walk using Theorem 9.
If , , and , then for every ,
Note that the local expansion for does not depend on . This property stems from the fact that the structure of any given link in depends only on the local structure of , that is, on the weights of edges adjacent to a single vertex.
For comparison, the construction of Liu et al. [LMY20] has local expansion in each dimension , and has global expansion approaching as