spectrum, homology and random walks
Random walks on a graph reflect many of its topological and spectral properties, such as connectedness, bipartiteness and spectral gap magnitude. In the first part of this paper we define a stochastic process on simplicial complexes of arbitrary dimension, which reflects in an analogue way the existence of higher dimensional homology, and the magnitude of the high-dimensional spectral gap originating in the works of Eckmann and Garland.
The second part of the paper is devoted to infinite complexes. We present a generalization of Kesten’s result on the spectrum of regular trees, and of the connection between return probabilities and spectral radius. We study the analogue of the Alon-Boppana theorem on spectral gaps, and exhibit a counterexample for its high-dimensional counterpart. We show, however, that under some assumptions the theorem does hold - for example, if the codimension-one skeletons of the complexes in question form a family of expanders.
Our study suggests natural generalizations of many concepts from graph theory, such as amenability, recurrence/transience, and bipartiteness. We present some observations regarding these ideas, and several open questions.
- 1 Introduction
- 2 Finite complexes
- 3 Infinite complexes
- 4 Open Questions
There are well known connections between dynamical, topological and spectral properties of graphs: The random walk on a graph reflects both its topological and algebraic connectivity, which are reflected by the -homology and the spectral gap, respectively.
In this paper we present a stochastic process which takes place on simplicial complexes of arbitrary dimension and generalizes these connections. In particular, for a finite -dimensional complex, the asymptotic behavior of the process reflects the existence of a nontrivial -homology, and its rate of convergence is dictated by the -dimensional spectral gap111The high-dimensional spectral gap originates in the classic works of Eckmann [Eck44] and Garland [Gar73], and appears in Definition 5 here.. The study of the process on finite complexes occupies the first half of the paper. In the second half we turn to infinite complexes, studying the high-dimensional analogues of classic properties and theorems regarding infinite graphs. Both in the finite and the infinite cases, one encounters new phenomena along the familiar ones, which reveal that graphs present only a degenerated case of a broader theory.
In order to give a flavor of the results without plunging into the most general definitions, we present in §1.1, without proofs, the special case of regular triangle complexes. A summary of the paper and its main results both for finite and infinite complexes is presented in §1.2.
This manuscript is part of an ongoing research seeking to understand the notion of high-dimensional expanders. Namely, the analogue of expander graphs in the realm of simplicial complexes of general dimension. Here we discuss the dynamical aspect of expansion, i.e. asymptotic behavior of random walks, and its relation to spectral expansion and homology. In a previous paper we studied expansion from combinatorial and isoperimetric points of view [PRT12]. The study of high-dimensional expanders is currently a very active one, comprising the notions of geometric and topological expansion in [Gro10, FGL10, MW11], -coboundary expansion in [LM06, MW09, DK10, GW12, SKM12], and Ramanujan complexes in [CSŻ03, Li04, LSV05]. We refer the reader to [Lub13] for a survey of the current state of the field.
1.1 Example - regular triangle complexes
First, let us observe the -lazy random walk on a -regular graph : the walker starts at a vertex , and at each step remains in place with probability or moves to each of its neighbors with probability . Let denote the probability of finding the walker at the vertex at time . The following observations are classic:
If is finite, then exists, and it is constant if and only if is connected.
Furthermore, the rate of convergence is given by
where is the spectral gap of (the definition follows below).
When is infinite and connected, the spectral gap is related to the return probability of the walk by
We recall the basic definitions: the Laplacian of , which we denote by , is the operator which acts on by
(where denotes neighboring vertices). If is finite, then its spectral gap is defined as the minimal Laplacian eigenvalue on a function whose sum on vanishes. When is infinite, its spectral gap is defined to be (for more on this see §3.1).
Moving one dimension higher, let be a -regular triangle complex. This means that (i.e. consists of subsets of of size , the edges of ), (the triangles), every edge is contained in exactly triangles in , and for every triangle in , the edges forming its boundary, , and , are in .
For we denote the directed edge by , and the set of all directed edges by (so that ). For , denotes the edge with the same vertices and opposite direction, i.e. .
The following definition is the basis of the process which we shall study:
Two directed edges are called neighbors (indicated by ) if they have the same origin or the same terminus, and the triangle they form is in the complex. Namely, if and , then means that either and , or and .
We study the following lazy random walk on : The walk starts at some directed edge . At every step, the walker stays put with probability , or else move to a uniformly chosen neighbor. Figure 1.1 illustrates one step of the process, in two cases (the right one is non-regular, but the walk is defined in the same manner).
As in the random walk on a graph, this process induces a sequence of distributions on ,
describing the probability of finding the walker at the directed edge at time (having started from ). However, studying the evolution of amounts to studying the traditional random walk on the graph with vertices and edges defined by . This will not take us very far, and in particular will not reveal the presence or absence of first homology in . Instead, we study the evolution of what we call the “expectation process” on :
i.e. the probability of finding the walker at time at , minus the probability of finding it at the opposite edge (for the reasons behind the name see Remark 4).
It is tempting to look at as is done in graphs, but a moment of reflection will show the reader that for any finite triangle complex, and any starting point . Namely, the probabilities of reaching and become arbitrarily close, for every . While this might cause initial worry, it turns out that the rate of decay of is always the same: for any finite triangle complex one has . It is therefore reasonable to turn our attention to the normalized expectation process,
and observe its limit,
For a finite triangle complex this limit always exists, and is nonzero. This is the object which reveals the first homology of the complex.
To see how, we need the following definition: We say that is exact if its sum along every closed path vanishes; namely, if
This is the one-dimensional analogue of constant functions (for reasons which will become clear in §2.2), and the following holds:
For a finite , is exact for every if and only if has a trivial first homology.
Furthermore, the rate of convergence is given by
where is the spectral gap of (see Definition 5).
If is infinite and every vertex in is of infinite degree, then its spectral gap (which is defined in §3.2) is revealed by the “return expectation”:
What if one is interested not only in the existence of a first homology, but also in its dimension? The answer is manifested in the walk as well. In graphs the number of connected components is given by the dimension of , and an analogue statement holds here (see Theorem 9).
If the non-lazy walk on a finite graph is observed, then apart from disconnectedness there is another obstruction for convergence to the uniform distribution: bipartiteness. We shall see that this is a special case of an obstruction in general dimension, which we call disorientability (see Definition 6). In our example we have avoided this problem by considering the lazy walk, both on graphs and on triangle complexes.
1.2 Summary of results
We give now a brief summary of the paper and its main results. The definitions of the terms which appear in this section are explained throughout the paper.
In §2.1 we define a -lazy random walk on the oriented -cells of a -dimensional complex , and associate with this walk the normalized expectation process . In §2.4 it is shown that the limit of this process always exists and captures various properties of , according to the amount of laziness (this is an abridged version of Theorem 9):
When , is exact for every starting point if and only if the -homology of is trivial. If furthermore then the rate of convergence is controlled by the spectral gap of :
Next, we move on to discuss infinite complexes, showing that they present new aspects which do not appear in infinite graphs. In §3.3 we define a family of simplicial complexes (which we call arboreal complexes) generalizing the notion of trees. In Theorem 3 we compute their spectra, generalizing Kesten’s classic result on the spectrum of regular trees [Kes59]. The spectra of the regular arboreal complexes of high dimension and low regularity exhibit a surprising new phenomenon - an isolated eigenvalue.
Sections 3.4 and 3.5 are devoted to study the behavior of the spectrum with respect to a limit in the space of complexes. In particular we are interested in the high-dimensional analogue of the Alon-Bopanna theorem, which states that if a sequence of graphs convergences to a graph , then . We first show that in general this need not hold in higher dimension (Theorem 10). This uses the isolated eigenvalue of the -regular arboreal complex of dimension two, which is shown in Figure 3.1, as well as a study of the spectrum of balls in this complex (shown in Figure 3.2).
Even though the Alon-Bopanna theorem does not hold in general in high dimension, we show that under a variety of conditions it does hold (Theorem 11):
If , and one of the following holds:
The spectral gap of is nonzero,
zero is a non-isolated point in the spectrum of , or
the -skeletons of the complexes form a family of -expanders,
In §3.7 we show that the connection between the spectrum of a graph, and the return probability of the random walk on it (see e.g. [Kes59, Lemma 2.2]), generalizes to higher dimensions (Proposition 14).
The final section on infinite complexes addresses the high-dimensional analogues of the concepts of amenability, recurrence and transience, proving some properties of these (Proposition 17), and raising many open questions.
We would like to thank Alex Lubotzky and Gil Kalai who prompted this research, and Jonathan Breuer for many helpful discussions. We are also grateful to Noam Berger, Emmanuel Farjoun, Nati Linial, Doron Puder and Andrzej Żuk for their insightful comments.
2 Finite complexes
Throughout this section is a finite -dimensional simplicial complex on a finite vertex set . This means that is comprised of subsets of , called cells, and the subset of every cell is also a cell. A cell of size (where ) is said to be of dimension , and denotes the set of -cells - cells of dimension . The dimension of is the maximal dimension of a cell in it. The degree of a -cell , denoted , is the number of -cells containing it. We shall assume that is uniform, meaning that every cell is contained in some cell of dimension .
For , every -cell has two possible orientations, corresponding to the possible orderings of its vertices, up to an even permutation (-cells and the empty cell have only one orientation). We denote an oriented cell by square brackets, and a flip of orientation by an overbar. For example, one orientation of is , which is the same as and . The other orientation of is . We denote by the set of oriented -cells (so that for and for ), and we shall occasionally denote by a choice of orientation for , i.e. a subset of such that is the disjoint union of and .
The faces of a -cell are the -cells . An oriented -cell () induces an orientation on its faces as follows: the face is oriented as , where means taking the opposite orientation when is .
Finally, we define the space of -forms on : these are functions on which are antisymmetric w.r.t. a flip of orientation:
For there are no flips; is just the space of functions on the vertices, and can be identified in a natural way with . With every oriented -cell we associate the Dirac -form defined by
(for this is the standard Dirac function, and is the constant ).
2.1 The -walk and expectation process
Let be a uniform -dimensional complex and . The following process is the generalization of the edge walk from §1.1:
The -lazy -walk on a -complex is defined as follows:
Two oriented -cells are said to be neighbors (denoted ) if there exists an oriented -cell , such that both and are faces of with the orientations induced by it (see Figure 1.1).
The walk starts at an initial oriented -cell , and at each step the walker stays in place with probability , and with probability chooses uniformly one of its neighbors and moves to it.
Put differently, it is the Markov chain on with transition probabilities
(note that is contained in -cells, and thus has neighbors!)
We remark that neighboring cells can also be described in the following way: if and , then iff the unoriented cell is in , and the intersection inherits the same orientation from both and . For , this can be interpreted as follows: two edges are neighbors if they bound a triangle in the complex, and the vertex at which they intersect “inherits the same orientation from both of them”: it is either the origin of both and , or the terminus of both. Finally, for Definition 1 gives the standard neighboring relation and -lazy random walk on a graph.
We say that is -connected if the -walk on it is irreducible, i.e., for every pair of oriented -cells and there exist a chain . Moreover, having such a chain defines an equivalence relation on the -cells of , whose classes we call the -components of .
Due to the assumption of uniformity, it is enough to observe unoriented cells - is -connected iff for every there exists a chain of unoriented -cells with for all . This is also equivalent to the assertion that for any there is a chain of -cells with for all (this is sometimes referred to as a chamber complex). We note that it follows from uniformity that a -connected complex is connected as a topological space. The other direction does not hold: the complex is not -connected, even though it is connected (and uniform).
Observing the -walk on , we denote by the probability that the random walk starting at reaches at time . We then have:
For , the expectation process on starting at is the sequence of -forms defined by
For (i.e. graphs) we simply define 222The results in the paper hold for graphs as well, using this definition of , but they are all familiar theorems. In some cases the proofs are slightly different, and we will not trouble to handle this special case.
The normalized expectation process is defined to be
where is the laziness of the walk. In particular, for one has for all .
The reason for this particular normalization is that for a lazy enough process (in particular for ) one has (see (2.8)). Note that .
The name “expectation process” comes from the fact that for any -form
where, as implied by the notation, does not depend on the orientation of .
The evolution of the expectation process over time is given by , where is the transition operator acting on by
Note that the evolution of is given by the same , acting on all functions from to , and not only on forms.
It is sometimes useful to observe the Markov operator associated with this evolution, which is characterized by
and is given explicitly by
This is the transpose of , w.r.t. to a natural choice of basis for .
2.2 Simplicial complexes and Laplacians
For a cell and a vertex , we write if is a cell in . If is oriented, , and , then denotes the oriented cell .
For , the boundary operator is defined by
In particular is defined by
The sequence is the simplicial chain complex of , meaning that for all , giving rise to the -cycles , the -boundaries and the (real) -homology .
Given a weight function , become inner product spaces (for ) with
Note that the sum is over and not - this is well defined since the value of is independent of the orientation of .
Since is finite the spaces are finite dimensional, and there exist adjoint operators to the boundary operators . These are the co-boundary operators, which are denoted by , and are given by
We will stick with the notation until we get to infinite complexes, where sometimes is defined even though is not. The simplicial cochain complex of is , and , , are the cocycles, coboundaries and cohomology, respectively. Cocycles are also known as closed forms, and coboundaries as exact forms.
The following weight functions will be used throughout this paper333Another natural weight function is the constant one. The obtained Laplacians are more convenient for isoperimetric analysis. For more details see [PRT12].:
Notice that for
Due to our choice of weights, the inner product and coboundary operators are given by
Finally, the upper, lower, and full Laplacians are defined by:
An explicit calculation gives
More generally, one can define the -th upper, lower and full Laplacians , and . Apart from , these will only make a brief appearance in §3.5. The kernel of is the space of harmonic -forms, denoted by .
The spaces defined so far are related by
The isomorphism between harmonic functions, homology and cohomology, which is sometimes called the discrete Hodge theorem, was first observed in [Eck44]. In a similar manner, there is a “discrete Hodge decomposition”
and all the Laplacians decompose with respect to it. All of these claims follow by linear algebra, using the fact that is finite-dimensional (see [PRT12, §2] for the details). For infinite complexes the situation is more involved, and is addressed in §3.2.
2.3 The upper Laplacian spectrum
In this section we study the spectrum of the upper Laplacian of a finite complex . First note that as , its spectrum is non-negative. Furthermore, zero is obtained precisely on closed forms, i.e. . The space of closed forms always contains the exact forms, , which are considered the trivial zeros in the spectrum of . The existence of nontrivial zeros in the spectrum of , i.e. closed forms which are not exact, indicates the existence of a nontrivial homology. This leads to the following definition:
The spectral gap of a finite -dimensional complex , denoted , is
The essential gap of , denoted , is
(the transition from to follows from the fact that vanishes on .)
Since vanishes precisely when the -homology of is nontrivial, it should be thought of as giving a quantitative measure for the “triviality of the homology”. For example, in graphs, having far away from zero is a measure of high-connectedness, or “high triviality of the -homology”.
In contrast, never vanishes, as . If the -homology is nontrivial then , so that is only of additional interest when the homology vanishes. In a disconnected graph controls the mixing rate as does for a connected graph, and we will see that the same happens in higher dimension (see (2.10)).
Until now we have studied one extremity of . The other side relates to the following definition:
A disorientation of a -complex is a choice of orientation of its -cells, so that whenever intersect in a -cell they induce the same orientation on it. If has a disorientation it is said to be disorientable.
A disorientable -complex is precisely a bipartite graph, and thus disorientability should be thought of as a high-dimensional analogue of bipartiteness. Another natural analogue is “-partiteness”: having some partition of so that every -cell contains one vertex from each . A -partite complex is easily seen to be disorientable, but the opposite does not necessarily hold for .
Notice the similarity to the notion of orientability: a -complex is orientable if there is a choice of orientations of its -cells, so that cells intersecting in a codimension one cell induce opposite orientations on it. However, orientability implies that -cells have degrees at most two, where as disorientability impose no such restrictions. Note that a complex can certainly be both orientable and disorientable (e.g. Figure 2.1(a)).
Let be a finite complex of dimension .
is the disjoint union of where are the -components of .
The spectrum of is contained in .
Zero is achieved on the closed -forms, .
If is -connected, then is in the spectrum iff is disorientable, and is achieved on the boundaries of disorientations (see (2.7)).
(1) follows from the observation that decomposes w.r.t. the decomposition . We have already seen (3), and the fact that the spectrum is nonnegative. Now, assume that . Choose which maximize . By (2.4),
(since ), hence and (2) is obtained.
Next, assume that is -connected and that is a disorientation. Define
and . For any , there exists some vertex with (since is uniform). Furthermore, by the assumption on , if and for vertices then if and only if , and thus
where is any -cell containing . If and are neighboring -faces in , then by definition, for some , is a face of and is a face of , so that
and consequently for any
so that is a -eigenform with eigenvalue .
In the other direction, assume that is -connected and that for some . Fix some which maximize , normalize so that , and define
We have by assumption, and we proceed to show that is a disorientation with the corresponding form as in (2.7). By the definition of
so that for every . Continuing in this manner, -connectedness implies that on all . Using again the definition of , for any in
Since the r.h.s is an average over terms whose absolute value is that of the l.h.s this gives whenever , hence
is always of absolute value one. Furthermore, if intersect in a face and induce opposite orientations on it, then and for some vertices , hence
which concludes the proof. ∎
2.4 Walk and spectrum
Observe the -lazy -walk on starting at . Then
The transition operator is given by