Random simplicial complexes
The study of random topological objects (such as random simplicial complexes and random manifolds) is motivated by potential applications to modelling of large complex systems in various engineering and computer science applications. Random topological objects are also of interest from pure mathematical point view since they can be used for constructing curious examples of topological objects with rare combinations of topological properties.
Several models of random manifolds and random simplicial complexes were suggested and studied recently, see  for a survey. One may mention random surfaces , random 3-manifodls , random configuration spaces of linkages . The present paper is was inspired by the model of random simplicial complexes developed by Linial, Meshulam and Wallach  , . In the first paper  the authors studied an analogue of the classical Erdös - Rényi  model of random graphs in the situation of 2-dimensional simplicial complexes. In the following paper  a more general model of -dimensional random simplicial complexes was studied. The random simplicial complexes of  and  have the complete -skeleton and their “randomness is concentrated in the top dimension”. More specifically, one starts with the full -skeleton of an -dimensional simplex and adds -faces at random, independently of each other, with probability .
A different model of random simplicial complexes was studied by M. Kahle ,  and by some other authors. These are clique complexes of random Erdös - Rényi graphs; here one takes a random graph in the Erdös - Rényi model and declares as a simplex every subset of vertices which form a clique, i.e. such that every two vertices of the subset are connected by an edge. Compared with the Linial - Meshulam model, the clique complex has “randomness” in dimension one but it influences the structure in all the higher dimensions.
In this paper we propose a more general and more flexible model of random simplicial complexes with randomness in all dimensions. We start with a set of vertices and retain each of them with probability ; on the next step we connect every pair of retained vertices by an edge with probability , and then fill in every triangle in the obtained random graph with probability , and so on. As the result we obtain a random simplicial complex depending on the set of probability parameters
Our multi-parameter random simplicial complex includes both Linial-Meshulam and random clique complexes as special case. Topological and geometric properties of this random simplicial complex depend on the whole set of parameters and their thresholds can be understood as convex subsets and not as single numbers as in all the previously studied models. We mainly focus on foundations and on containment properties of our multi-parameter random simplicial complexes. One may associate to any finite simplicial complex a reduced density domain (which is a convex domain) which fully controls information about the values of the multi-parameter for which the random complex contains as a simplicial subcomplex. We also analyse balanced simplicial complexes and give positive and negative examples. We apply these results to describe dimension of a random simplicial complex.
In a following paper we shall address other topological and geometric properties of random simplicial complexes depending on multiple parameters (such as their homology and the fundamental group).
The authors thank Thomas Kappeler for useful discussions.
2 The definition and basic properties.
2.1 The model
Let denote the simplex with the vertex set . We view as an abstract simplicial complex of dimension . Given a simplicial subcomplex , we denote by the number of -faces of (i.e. -dimensional simplexes of contained in ).
An external face of a subcomplex is a simplex such that but the boundary of is contained in , .
We shall denote by the number of -dimensional external faces of . Note that for , we have and for ,
Fix an integer and a sequence
of real numbers satisfying
We consider the probability space consisting of all subcomplexes
with where the probability function
is given by the formula
for We shall show below that is indeed a probability function, i.e.
see Corollary 2.3.
If for some then according to (1) we shall have unless , i.e. if contains no simplexes of dimension (in this case contains no simplexes of dimension ). Thus, if the probability measure is concentrated on the set of subcomplexes of of dimension .
In the special case when one of the probability parameters satisfies one has and from formula (1) we see unless , i.e. if the subcomplex has no external faces of dimension . In other words, we may say that if the measure is concentrated on the set of complexes satisfying , i.e. such that any boundary of the -simplex in is filled by an -simplex.
be two subcomplexes satisfying the following condition: the boundary of any external face of of dimension is contained in . Then
We act by induction on . For , are discrete sets of vertices and the condition of the Lemma is automatically satisfied (since the boundary of any 0-face is the empty set). A subcomplex satisfying is determined by a choice of vertices out of vertices. Hence using formula (1),
Now suppose that formula (3) holds for and consider the case of . Note the formula
where is the number of boundaries of -simplexes contained in . Note that the first two factors in (4) depend only on the skeleton .
We denote by the number of -simplexes of such that their boundary lies in . Clearly the number depends only on the skeleton . Our assumption that the boundary of any external -face of is contained in for implies that for any subcomplex one has
A complex is uniquely determined by its skeleton and by the set of its -faces. Note that, given the skeleton , the number is arbitrary satisfying
Thus using (4) we find that the probability
Here we used the equation (5). Next we may combine the obtained equality with the inductive hypothesis
to obtain (3). ∎
Note that the assumption that any external face of is an external face of is essential in Lemma 2.2; the lemma is false without this assumption.
Taking the special case , in (3) we obtain the following Corollary confirming the fact that is a probability function.
The probability of the empty subcomplex equals
If then Hence, if then , i.e. we may say that , a.a.s.
as . Thus, we see that for the empty subset appears with probability , a.s.s.
Since we intend to study non-empty large random simplicial complexes, we shall always assume that where tends to .
2.2 The number of vertices of
Consider a random simplicial complex , where is the probability multiparameter. Assume that where . Then the number of vertices of is approximately ; more precisely for any a.a.s. one has
2.3 Special cases
The models of random simplicial complexes which were studied previously contained randomness in a single dimension while our model allows various probabilistic regimes in different dimensions simultaneously. Thus we obtain more flexible constructions of random simplicial complexes.
The model we consider turns into some well known models in special cases:
When and we obtain the classical model of random graphs of Erdös and Rényi .
When and we obtain the Linial - Meshulam model of random 2-complexes .
When is arbitrary and fixed and we obtain the random simplicial complexes of Meshulam and Wallach .
For and one obtains the clique complexes of random graphs studied in .
2.4 Gibbs formalism
In this subsection we briefly describe a more general class of models of random simplicial complexes which includes the model of §2.1 as a special case.
On the set of all subcomplexes , one considers an energy function having the form
where and are real parameters, . The partition function
is a function of the parameters and and
is a probability measure on the set of all subcomplexes . Here the case is not excluded; then runs over all subcomplexes.
In the special case when the parameters , satisfy
we may define the probability parameters by
One can easily check that under the assumptions (10) the probability measure coincides with the measure given by (1). The relation (10) implies that the partition function equals one, according to Corollary 2.3.
3 The containment problem
Consider a random simplicial complex where . As in the classical containment problem for random graphs we ask under which conditions has a simplicial subcomplex isomorphic to a given -dimensional finite simplicial complex . The answer is slightly different from the random graph theory: we associate with a convex set
in the space of exponents of probability parameters which (as we show here) is fully responsible for the containment.
For simplicity we shall assume in this paper that the numbers are constant (do not depend on ). We shall use the following notation
Clearly, we must assume that
since for the complex is either empty or has one vertex, a.a.s. (see Example 2.4).
3.1 The density invariants
Let be a fixed finite simplicial -dimensional complex. As usual, denotes the number of -dimensional faces in . Define the following ratios (density invariants):
We do not exclude the case when ; then . The 0-th number is always one, . Compare , Definition 11.
then the probability that is embeddable into tends to zero as .
Let denote the set of vertices of . An embedding of into is determined by an embedding . For any such embedding define a random variable given by
Then is the random variable counting the number of isomorphic copies of in . One has
by Lemma 2.2. Thus we have
We see that (26) implies . Hence
also tends to zero. ∎
3.2 The density domains
Consider the Euclidean Space with coordinates .
For a finite simplicial complex of dimension , we denote by
the convex domain given by the following inequalities:
The domain is a simplex of dimension which has the origin as one of its vertices and the other vertices are of the form where with standing on the -th position, where .
We may restate Lemma 3.1 as follows:
If the vector of exponents does not belong to the closure
then the complex is not embeddable a.a.s. into a random simplicial complex , where , i.e.
Next we define the domain as the intersection
Here runs over all subcomplexes of . Clearly, is an -dimensional convex polytope.
then is embeddable into a random complex where , a.a.s.
As in the proof of Lemma 2.2, let denote the set of vertices of and let be an embedding. Any such embedding uniquely determines a simplicial embedding . Consider a random variable given by
Then counts the number of isomorphic copies of in and by Lemma 2.2. Hence,
We shall use the Chebyshev inequality
Given two simplicial embeddings , the product is a 0-1 random variable, it has value 1 on a subcomplex if and only if contains the union . Thus, by Lemma 2.2, we have
where denotes the intersection . If and are disjoint then equals ; thus in formula (16) we may assume that and are such that the intersection
Denote by by the subcomplex . For a fixed subcomplex the number of pairs of embeddings such that is bounded above by
where is the number of isomorphic copies of in .
Thus we obtain
On the other hand,
Here runs over all nonempty subcomplexes of . If then for any we have
Thus we see that the ratio tends to zero as . The result now follows from (15). ∎
We may summarise the obtained results as follows:
Let be a fixed finite simplicial complex of dimension .
If then a random simplicial complex contains as a simplicial subcomplex, a.a.s.
If then a random simplicial complex does not contain as a simplicial subcomplex, a.a.s.
3.3 The reduced density domain
Since the simplex contains the point as one of its vertices and is the cone with apex over the simplex
Hence the convex domain
is a cone with apex over the domain which is defined as the intersection
We call the reduced density domain associated to .
For a subset of vertices , denote by the simplicial complex induced on . Then
This follows from the observation that for a simplicial subcomplex one has
where is the set of vertices of .
3.4 The Invariance Principle
If and lie on a line passing trough then the probability spaces and have identical embedability properties with respect to any fixed finite simplicial complex , , i.e. embeds into a.a.s. if and only if is embeds into , a.a.s.
Thus, instead of a vector of exponents we may consider the vector
which has the first coordinate , i.e. in this case .
Conjecture: We conjecture that all geometric and topological properties of the the random complex remain invariant when the multi-exponent moves along any line passing through the point .
Let be a closed triangulated surface, . Then the number of edges and the number of faces are related by . We obtain that
i.e. the simplex has a fixed slope independent of the topology of the surface and of the details of a particular triangulation.
Let be the following 2-complex. Here is a triangulated real projective plane having a cycle of length 5 representing the non-contractible loop. is a triangulated disc with boundary of length 5 which is identified with . To compute we shall use the formulae
where and denote the numbers of edges and faces of and denotes
For an integer , let be a 2-complex constructed as follows. The vertex set is , the set of 1-simplexes is the set of all pairs where (i.e. the 1-skeleton of is a complete graph on vertices), and the set of 2-simplexes consists of triples where . To describe the reduced density domain we shall use the formula (19). Consider a subset If does not contain the vertex then the induced complex is the 2-skeleton of the -dimensional simplex where and we have
If contains the last vertex then
In the first case, and in the second case . We see that
(is achieved for ) and
(is achieved for ). The two lines given by the equations
intersect at the point
It is easy to check that this point satisfies the inequality
for arbitrary subset . This argument shows that in this example
and justifies our picture Figure 5.
4 Balanced simplicial complexes
We shall say that an -dimensional simplicial complex is balanced if
The following properties are equivalent:
a) is balanced;
b) for any subcomplex one has .
c) for any subcomplex and for any one has .
The proof is obvious.
Let be a 2-dimensional triangulated disc having vertices such that among them are internal. It is easy to check (using the Euler - Poincaré formula) that and . Hence,
Let us assume that ; then and . Suppose that there exists a proper subdisc containing all the internal vertices. Then
where and we see that