Linear embeddings of random complexes

Recall the multiparameter random simplicial complex model introduced by Costa and Farber [5]. Given $n\in\mathbb{N}$ and a sequence of probabilities $\textbf{p}=(p_{i})_{i\geq 0}$ we sample $X\sim X(n;\textbf{p})$ starting with the ground set $[n]$. We first include each element of the ground set as a vertex independently with probability $p_{0}$, from here we include each edge between exisiting vertices with probability $p_{1}$, and then each triangle whose boundary is included with probability $p_{2}$, and so on. This model interpolates between the clique complex model, introduced first by Kahle [11], which is the case where $p_{1}$ can vary but all other $p_{i}$’s are 1 and the Linial–Meshulam–Wallach $d$-dimensional model [12, 15] where $p_{d}$ can vary, $p_{i}=1$ for $i<d$ and $p_{i}=0$ for $i>d$ (note that the $d=1$ case for Linial–Meshulam–Wallach is the Erdős–Rényi random graph).

Here we establish thresholds for linear embeddings of random simplicial complexes in Euclidean space. Recall that a linear map from a simplicial complex $\Delta$ into Euclidean space $\mathbb{R}^{d}$ is a map $\phi:\Delta\rightarrow\mathbb{R}^{d}$ that is determined linearly by the image of vertices. That is, the image of any face $\sigma\in\Delta$ under $\phi$ is the convex hull in $\mathbb{R}^{d}$ of the image of the vertices of $\sigma$. We say that $\Delta$ embeds linearly in $\mathbb{R}^{d}$ if there is an injective linear map $\phi:\Delta\hookrightarrow\mathbb{R}^{d}$.

Perhaps the most well known situation concerning linear embeddings of simplicial complexes is graph planarity. A graph $G$ is planar if and only if it admits a linear embedding into $\mathbb{R}^{2}$. We could take this as a definition of planar graphs, but it turns out that when it comes to planar graphs it doesn’t matter if we characterize them by existence of linear embeddings or existence of topological embeddings. The result that a graph is planar if and only if it can be drawn in the plane with all edges as straight-line segments is known as Fáry’s theorem. The analogous statement for higher dimensions was first demonstrated to be false by Brehm [3] who gave a triangulation of the Möbius strip on nine vertices that does not linearly embed in $\mathbb{R}^{3}$. Results and discussion in [6] examine certain cases of equivalence or nonequivalence of topological and linear embeddings.

In between linear embeddings and topological embeddings are PL-embeddings (piecewise-linear embeddings). A simplicial complex is PL-embeddable in $\mathbb{R}^{d}$ if it admits a subdivision that is linearly embeddable in $\mathbb{R}^{d}$. It is known that PL-emdeddability is strictly weaker than linear embeddability in general. Brehm and Sarkaria [4] disproved a conjecture of Grünbaum that PL-embeddability of a $d$-complex in $2d$-dimensional space implies linear embeddability. A result of Matoušek, Tancer, and Wagner [14] establishing complexity results on PL-embeddings of simplicial complexes gives another proof that PL-embeddability does not imply linear embeddability by demonstrating that the two decision problems are not in the same complexity class. Here as a corollary to the main result we give a proof of this same nonequivalence via the probabilistic method by establishing threshold results for linear embeddability of random complexes and contrasting them with known threshold results for PL-embeddability, see the concluding remarks for a precise statement on this.

We consider multiparameter random complexes in a sparse regime where each $p_{i}=n^{-\alpha_{i}}$ for some fixed $\alpha_{i}\in[0,\infty]$ (with $\alpha_{i}=\infty$ corresponding to $p_{i}=0$). Furthermore, we assume that $p_{0}=1$. This is a convenience for the proofs, but by a straightforward but perhaps somewhat tedious application of large deviation inequalities on the number of vertices the natural extension of the main result also holds when $p_{0}$ is allowed to depend on $n$ provided that $p_{0}n\rightarrow\infty$ as $n\rightarrow\infty$.

To state our main result we first introduce some notation. For $\operatorname{\overline{\alpha}}=(\alpha_{1},\alpha_{2},...)$ we denote $X(n;1,n^{-\alpha_{1}},n^{-\alpha_{2}}...)$ by $X(n;n^{-\operatorname{\overline{\alpha}}})$. For each $s\geq 1$ we let $v_{s}$ denote the vector of binomial coefficients ${v}_{s}=\left(\binom{s}{i+1}\right)_{i\geq 1}$. By linearity of expectation the expected number of $(s-1)$-simplices in $X(n,n^{-\operatorname{\overline{\alpha}}})$ is

Our main result establishes a threshold in $\operatorname{\overline{\alpha}}$ for embeddability in even dimensions.

Stated another way, our main result says if the $\operatorname{\overline{\alpha}}$ is selected so that the expected number of $d$-faces is at most $n^{1-\varepsilon}$ for some $\varepsilon>0$ then the random complex will embed in $\mathbb{R}^{2d}$ with high probability, while an average number of $d$-faces of order at least $n^{1+\varepsilon}$ implies that the random complex will not embed in $\mathbb{R}^{2d}$. Somewhat surprisingly then the threshold to embed a random complex in $\mathbb{R}^{2d}$ turns out to only depends on the $d$-skeleton of the random complex.

Recall that a $d$-complex $\Delta$ is alway embeddable in $\mathbb{R}^{2d+1}$. A generic set of points in $\mathbb{R}^{m}$ is one where for any $k$ there is no affine subspace of $\mathbb{R}^{m}$ containing more than $k+1$ of the points and for any two disjoint subsets $A$ and $B$ of the point set each of size at most $m$, the affine subspace of dimension $|A|-1$ spanned by the points in $A$ and the affine subspace of dimension $|B|-1$ spanned by the points in $B$ intersect as an affine subspace of dimension $|A|+|B|-2-m$, if this quantity is nonnegative and don’t intersect at all otherwise. If we place the points of a $d$-complex $\Delta$ in $\mathbb{R}^{2d+1}$ generically then it will be an embedding as two $d$-dimensional subspaces generically don’t intersect in $\mathbb{R}^{2d+1}$.

Observe that any set of points in $\mathbb{R}^{m}$ are moved to generic position by an arbitrarily small perturbation. The proof of the sparse side of Theorem 1 will give a generic embedding of our random complex in $\mathbb{R}^{2d}$, and we will rule out the existence of a generic embedding on the dense side of the claimed threshold. If there is no generic embedding of our finite simplicial complex there is no embedding at all as a small perturbation of an embedding will still be an embedding.

The proof for the sparse side will follow a collapsibility argument. We will prove a linear embedding analogue of a result of Horvatić from [10] about PL-embeddability. A face of a simplicial complex is said to be free if it is properly contained in only one other face. The removal of a free face and the face that properly contains it is called an elementary collapse, and a complex is $d$-collapsible if there is a sequence of elementary collapse on the complex that leave behind only faces of dimension at most $d-1$. Theorem 3.2 of [10] implies that a $d$-complex which is $d$-collapsible PL embeds in $\mathbb{R}^{2d}$. We’ll prove on the sparse side that if a stronger collapsibility property holds for a $d$-complex it will linearly embed in $\mathbb{R}^{2d}$ and then show that with high probability our random complex satisfies that strong collapsibility property.

For the dense side we make use of Radon’s Theorem. Recall that Radon’s Theorem states that for any set of $d+2$ points in $\mathbb{R}^{d}$ there is a partition into two sets $A$ and $B$ so that the convex hull of $A$ intersects the convex hull of $B$. For the dense side we start with a generic embedding of the points into $\mathbb{R}^{2d}$ and then generate the complex and count how many sets of $2d+2$ points, in expectation, have a Radon partition $(A,B)$ with the simplex on $A$ and the simplex on $B$ both included in the complex. We show that the expected number of such point sets is large and then prove a concentration of measure inequality that survives multiplication by the number of combinatorially distinct ways to place the $n$ points in $\mathbb{R}^{2d}$ (the number of order types).

Both for the proof of the sparse side and later for the proof of the concentration inequality on the dense side, we need the following lemma that gives basic facts about the $f$-vector of a random complex in the multiparameter model.

For the sparse side we will prove a certain collapsibility condition on our random complexes that implies the complex is embeddable. For $X$ a $d$-complex, we associate a $(d+1)$-uniform hypergraph $\mathcal{H}$ whose hyperedges are the $d$-faces of $X$. If $\mathcal{H}$ has no 2-core then $X$ will embed in $\mathbb{R}^{2d}$. By 2-core we mean a subhypergraph of $\mathcal{H}$ in which every vertex is contained in at least two hyperedges. If $\mathcal{H}$ has no 2-core then there is a sequence of vertex deletions obtained by removing a vertex of degree at most 1, along with the hyperedge that contains it, that removes all the vertices. We will say the pure part of $X$ has no 2-core to mean that the associated hypergraph has no 2-core. Recall that the pure part of a $d$-complex is the subcomplex generated by the downward closure of the $d$-faces. The next theorem shows that the absences of a 2-core in dimension $d$ implies embeddability in dimension $2d$.

By Theorem 3 we want to prove a collapsibility result about $X\sim X(n;n^{-\operatorname{\overline{\alpha}}})$. This is a different type of collapsibility than is studied in [13] and [16]. Here we are collapsing (vertex, $d$-simplex)-pairs and these other papers deal with collapsing ($(d-1)$-simplex, $d$-simplex)-pairs. The overall strategy though is the very similar: we prove that there are no large obstructions to collapsibility, and that local obstructions to collapsibility are too dense to appear in this sparse regime. We start with a definition.

We could also refer to weak connectivity as path connectivity but we use weak connectivity to contrast with strong connectivity from the other papers in the literature on collapsibility. Clearly a minimal 2-core of $X$ is weakly connected. We first rule out large weakly connected components as that will rule out large minimal 2-cores.

Next we show that small subcomplexes on at most $L$ vertices are too sparse to be 2-cores. To that end we prove the following lemma.

With the previous lemmas we are ready to prove the sparse side of Theorem 1

The proof for the dense side is based on Radon’s theorem. Recall that Radon’s theorem, originally proved in [17], states that any set of $d+2$ points in $\mathbb{R}^{d}$ can be partitioned into two sets whose convex hulls intersect and if the point set is generic the Radon partition is unique. Thus if we have a linear embedding of $X\sim X(n;n^{-\operatorname{\overline{\alpha}}})$ into $\mathbb{R}^{2d}$ then for any set of $2d+2$ vertices we get from Radon’s theorem a pair of simplices on those vertices that must be excluded from $X$. Given an embedding of $n$ points in $\mathbb{R}^{2d}$ and a simplicial complex $X$ on $[n]$ we say that $S\in\binom{[n]}{2d+2}$ is a Radon match if in the Radon partition on $S$ into two subsets $S_{1}$ and $S_{2}$, the simplex on $S_{1}$ and the simplex on $S_{2}$ are both present in $X$. If $X$ has a Radon match for every embedding of the vertices of $X$ into $\mathbb{R}^{2d}$ then $X$ is not linearly embeddable in $\mathbb{R}^{2d}$. Our strategy is to show that with high probability when $X\sim X(n;n^{-\operatorname{\overline{\alpha}}})$ on the dense side of the claimed phase transition every embedding of the vertices has at least one Radon match. We also will assume throughout that all embeddings have vertices placed generically in $\mathbb{R}^{2d}$ but it is clear that if there is a linear embedding of $X$ into $\mathbb{R}^{2d}$ then by a sufficiently small perturbation we have a generic embedding.

Turning the strategy into a proof requires three ingredients. First we need some control on which type of splits of our $(2d+2)$-sets we see. If, for example, they were all split as $2d+1$ vertices in convex position surrounding a single vertex in the interior, we would not expect to see any Radon partitions just beyond the phase transition. Second, for our purposes any two embeddings of the $n$ vertices in $\mathbb{R}^{d}$ can be regarded as equivalent if they induce the same Radon partition on all $2(d+2)$-subsets. Therefore we need an upper bound on how many nonequivalent embeddings there are to consider. Third we need an upper bound on the probability that a particular embedding has no Radon match and this upper bound has to go to zero faster than the number of nonequivalent embeddings is going to infinity so that we can take a union bound.

For the first point we use the following stronger version of the classical van Kampen–Flores Theorem.

As a corollary to this we have the following regarding an embedding of $n$ points in $\mathbb{R}^{d}$

We now turn our attention to bounding the number of embeddings of the vertices we have to consider. We consider two general position embeddings $f,g:[n]\rightarrow\mathbb{R}^{d}$ to be equivalent if for every $S\in\binom{[n]}{d}$ the Radon partition of $S$ under the image of $f$ is the same as the Radon partition of $S$ under the image of $g$.

For this we are really counting labeled, simple order types of a configuration of $n$ points in $\mathbb{R}^{d}$. Recall that the order type of a labeled set of points $p_{1},...,p_{n}$ in $\mathbb{R}^{d}$ is the $\binom{n}{d+1}$-tuple of signs of determinants of $(d+1)\times(d+1)$ submatrices of

That is the order type is

By construction then for any set of $d+2$ points in general position the order type of those $d+2$ points determines the Radon partition. Indeed for $p_{1}$, …, $p_{d+1}$, $p_{d+2}$ we have $p_{1}$, …, $p_{d+1}$ are vertices of a simplex and for any $i$, the order type (either $-1$ or $1$) of $p_{1},..,p_{i-1},p_{i+1},..,p_{d+1},p_{d+2}$ determines on which side of the hyperplane through $p_{1},..,p_{i-1},p_{i+1},..,p_{d+1}$, $p_{d+2}$ sits. Thus the order type of $p_{1},...,p_{d+1},p_{d+2}$ determines which region of the hyperplane arrangement on $p_{1},...,p_{d+1}$ contains $p_{d+2}$. This region uniquely determines the Radon partition on $p_{1},..,p_{d+2}$. For more background on order types see the survey of Goodman and Pollack [9]. By simple order types we mean order types where none of the determinant signs are zero, these are all the order types that have to be considered for generic point sets.

The following enumeration result of Goodman and Pollack on order types will be sufficient for our proof. There is also a stronger result for all labeled order types, not just simple ones, due to [1]

We note that the number of order types is a technically a refinement of the number of embeddings which induce the same Radon partitions on every $(d+2)$-subset of $n$ points. If we have only $d+2$ points in general position we have a distinct Radon partition, $(A,B)$, but we can change the order type by permuting the points relabeling the points in $A$ and the points in $B$. However the direction that we need is that the order type determines the Radon partition discussed above. So we have the following lemma as an immediate corollary to Theorem 11.

For the last piece of the proof for the dense side of the phase transition we bound the probability that for any fixed embedding of the vertices of $X$ we have no Radon matches.

For the proof we will make use of the following form of Janson’s equality from Theorem 23.13 of [7]:

In our case $N$ is all the faces of the simplex on $n$ vertices other than the vertices and $R$ is the random complex in $X(n;n^{-\operatorname{\overline{\alpha}}})$. We might hesitate for a moment regarding the independence assumption as the faces are not included independently in $X$. However we can instead associate to $\sigma$ in the simplex on $n$ vertices $q_{\sigma}=n^{-\alpha_{|\sigma|-1}}$, in this way each face is set to be either on or off independently and then the complex $X$ consists of those faces that are switched on and have all their subfaces switched on as well. Each $D_{i}$ will be a Radon match which corresponds to some collection of faces all switched on independently.