1. Threshold Functions in Random Simplicial Complexes
Amirim Project
by
Avichai Cohen
Supervised by
Prof. Nati Linial
October 2012
I want to thank Lior Aronshtam for her help during the work on
the project, Noga Rotman for her help with the design of this
document and nally professor Nati Linial for his supervision.
1

2. Contents
I Introduction: What is a Random Simplicial Complex? 3
1 Simplicial Complexes 3
2 Random Graphs 3
3 Random Simplicial Complexes 3
II Questions and Prior work 4
1 Threshold Functions in higher dimensions 4
2 Cycles of higher dimensions 4
2.1 Prior Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Koslov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 ALLM - upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Collapsability 5
3.1 Prior Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.1 Costa, Farber and Kappeler. . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.2 ALLM - lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Summary 7
III My Work 8
1 The General Program 8
2 Non-vanishing Homology Threshold with d=2 8
3 Connected Components Collapse with d=2 9
3.1 What is Connected Components Collapse? . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Results for d=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2

3. Part I
Introduction: What is a Random
Simplicial Complex?
1 Simplicial Complexes
Denition 1.1. Let V be a nite set of vertices (usually V = [n] = {1, 2, . . . , n} for some
n ∈ N). A family of subsets: F ⊆ 2V
is called a simplicial complex if it is closed down i.e.
A ∈ F, B ⊆ A → B ∈ F. We call the members of F simplexes or faces. The dimension
of a face A is dened as: dim (A) = |A| − 1, and dim (F) = max {dim (A) : A ∈ F}. A d-
dimensional face is called a d-face for short. We denote by fj the number of j-dimensional faces
in F. The k-dimensional skeleton of F is {A ∈ F : dim (A) ≤ k} and we say that F has the full
k-dimensional skeleton if ∀A ⊆ V , dim (A) ≤ k → A ∈ F (e.g. if V = [n] the full 1-dimensional
skeleton is the complete graph Kn).
Many properties of a simplicial complex can be expressed in terms of the inclusion matrix
of the (d − 1) vs. d faces. This is a fd−1 × fd matrix , denoted M such that: Mij = 1 i
the j-th d-face Aj in F and the i-th (d-1)-face Bi satises: Bi ⊂ Aj (the rows represent the
(d-1)-faces the columns represent the d-faces with a predened order upon them). The linear
operator corresponding to this matrix is called the (d-1)-st boundary operator over F2. It is
also interesting some times to deal with the boundary operator over other elds, but we will not
discuss this here.
2 Random Graphs
The concept of random graphs was introduced by P.Erd®s and A.Rényi in the 1960's. We will
mainly use one model which is denoted by G (n, p). As Alon and Spencer [1] state random
graph is a misnomer; G (n, p) is a probability space over graphs with vertex set [n]. Namely:
we have n ∈ N labelled vertices and Pr ({i, j} ∈ G) = Pr(edge) = p ∈ [0, 1] for 1 ≤ i j ≤ n for
each of the
n
2 possible edges and these events are mutually independent. In other words, for a
specic graph G0 with M edges: Pr ({G0}) = Pr (G = G0) = pM
(1 − p)(n
2)−M
. In addition, for
a family of graphs: A (e.g. all of the connected graphs) we denote Pr (G (n, p) ∈ A) as Pr (A)
in the probability space G (n, p).
3 Random Simplicial Complexes
As in the case of random graphs this is also a probability space. It was introduced several years
ago by Nati Linial and Roy Meshulam [2] as a higher dimensional analogue to G (n, p) random
graphs. Let n ∈ N and p ∈ [0, 1], Denote by Xd (n, p) the space of simplicial complexes of
dimension ≤ d that have a full (d-1)-dimensional skeleton. For each d-face A: Pr (A ∈ X) =
p and these events are again mutually independent. Another way to view it is through the
mentioned incidence matrix - here each column appear in the matrix with probability p and the
choice of columns is mutually independent. Note that: X1 (n, p) is identical with G (n, p).
3

4. Part II
Questions and Prior work
1 Threshold Functions in higher dimensions
One of the interesting phenomena in the theory of random graphs is the prevalence of threshold
functions and the occurrence of the phase transitions. First we dene a threshold function.
There are two types of threshold functions: coarse and sharp:
Denition 1.1. [1]
For a graph property P, a function: r : N → R will be called a coarse threshold function for
P if it satises:
1. If p = p (n) = o (r (n)) then Pr (G (n, p (n)) ∈ P) → 0 as n → ∞
2. If p = p (n) , r (n) = o (p (n)) then Pr (G (n, p (n)) ∈ P) → 1 as n → ∞
One can relate to this function as the point where a phase transition occurs.
We note that if r (n) is coarse threshold function of a property P then every other function
in Θ (r (n)) is also coarse threshold function (therefore one shall not write the coarse threshold
function etc.).
Denition 1.2. [3]
For a graph property P, a function: r : N → R will be called a sharp threshold function for
P if it satises:
1. If p = p (n) = (1 − o (1)) r (n) then Pr (G (n, p (n)) ∈ P) → 0 as n → ∞
2. If p = p (n) = (1 + o (1)) r (n) then Pr (G (n, p (n)) ∈ P) → 1 as n → ∞
A sharp threshold function will tell us where exactly that phase transition occurs.
One of the famous properties is: P = {G contain a cycle}. A coarse threshold function for
P is 1/n (as proved in the rst paper by Erd®s and Rényi). However, for each ε 0 if p = ε
n
then 1 Pr (G (n, p) ∈ P) 0 but if ε ≥1 then Pr (G (n, p) ∈ P) = 1 − o (1) and there is a
cycle almost surely.
2 Cycles of higher dimensions
Denition 2.1. For graphs, the right kernel of the incidence matrix is called the cycle space.
Thus the cycle space is trivial when there are no cycles in the graph i.e when it is a forest.
In the same manner, the existence of a cycle in a d-dimensional complex is the non-vanishing
right kernel of its (d-1)-st boundary operator or its incidence matrix. In topological terms this
is the non-vanishing of the d-th homology of the complex. We note that it may depend on the
underlying eld. When d = 1 the property does not depend on the underlying eld. However, in
higher dimensions this equivalence no longer holds and many questions remain in this context.
We will discuss the matter only with F2 as the underlying eld.
As a result of the above, another interpretation of P is that the incidence matrix has a
non-trivial right kernel. With this view of the matter we can now ask a similar question about
Xd (n, p) : what is the threshold function r = r (d, n) for the property:
Pd = {X ∈ Xd (n, p) : X s incidence matrix has a non trivial right kernel}
Another question is: given the coarse threshold function r = r (d, n), analogous to the
1-dimensional case, what is the constant cn s.t. for each ε cn if p = ε · r (d, n) then
Pr (Xd (n, p) ∈ Pd) = 1 − o (1) .
My goal in this paper is to review prior work regarding the threshold function for Pd and
the ways to get closer to the related constant and my contribution to this research activity.
4

5. 2.1 Prior Works
2.1.1 Koslov
The rst result presented is of Koslov [4] who proved the following theorem:
Theorem 2.2. For any function ω : N → R satisfying: lim
n→∞
ω (n) = ∞ it holds that:
1. For p = p (n) = ω(n)
n : lim
n→∞
Pr (Xd (n, p) ∈ Pd) = 1
2. For p = p (n) = 1
n·ω(n) : lim
n→∞
Pr (Xd (n, p) ∈ Pd) = 0
The conclusion is that the coarse threshold function r (d, n) is
1
n however, it does not tell
much about the constant(s). We will try to bound that constant(s).
2.1.2 ALLM - upper bound
Another result is from a recent work by Nati Linial, Roy Meshulam, Tomasz Luczak and Lior
Aronshtam [7]. This result improves the upper bound as follows:
Observe the function: gd (x) = (d + 1) (x + 1) e−x
+x (1 − e−x
)
d+1
and let cd be the positive
root of gd (x) = d + 1.
Theorem 2.3. For a xed c cd : lim
n→∞
Pr Xd n, c
n ∈ Pd = 1
Remark: By calculation c2 ≈ 2.783 however, simulation shows that the actual threshold is
closer to 2.75.
3 Collapsability
For a graph T on n vertices with n − 1 edges the following three statements are equivalent:
1. T is connected.
2. T is acyclic.
3. If we repeatedly trim one leaf (meaning, removing an edge such that at least one of its
vertices is a leaf) we eventually (after n − 1 steps) eliminate all of the edges.
In higher dimension not all of the equivalences hold.
Denition 3.1. In a d-dimensional simlicial complex X = (V, E) a (d-1)-face e ∈ E is called
free if it is contained in exactly one d-face σ ∈ E. In that case we say that Xσ was gotten
from X by elementary collapse. If it is possible to eliminate, in this fashion - one by one, all the
d-faces in X we say that X d-collapsible.
We have dened the concepts of d-collapsability and acyclicity and now we shall prove the
following corollary:
Corollary 3.2. A d-collapsible d-dimensional simplicial complex is acyclic.
Proof. If e is a free (d-1)-face and σis a d-face and e ⊂ σ then σ is not part of any cycle (a
linear dependent set of columns in the (d-1)-boundry operator of the simlicial complex) then
eliminating e and σ from the simplicial comlex does not change the right kernel of the (d-
1)-boundry operator. Therefore if X is d-collapsible by the end of the elimination process
the incidence matrix has no columns and its kernel is trivial and the complex is acyclic by
denition.
We note that the reverse implication does not hold in dimension 1. To present a counter
example we rst dene Sum Complexes [5]:
5

6. Denition 3.3. For p prime number and a ∈ Fp {0, 1}, a Sum Complex is a 2-dimensional
simplicial complex above vertex set Fp containing the full 1-dimensional skeleton and a 2-face
{i, j, k} is in the complex i i + j + k ∈ {0, 1, a} .
Counter examples for the reverse implication i.e. acyclic complexes that are not d-collapsible
are sum complexes with parameters: p = 7 and a = 5 or p = 11 and a ∈ {3, 4, 7, 8, 9} and many
more.
Now we can ask: What is the threshold function r (n, d) for a complex X ∈ Xd (n, p) to be
d-collapsible and what is the critical probability for that phenomenon?
We note that as collapsability implies acyclicity, an answer to this question will give us a
lower bound to the critical probability of Pd.
3.1 Prior Works
3.1.1 Costa, Farber and Kappeler.
Theorem 3.4. For any function ω : N → R satisfying: lim
n→∞
ω (n) = ∞ X ∈ X2 n, 1
ω(n)n is
almost surely 2-collapsible. [6]
3.1.2 ALLM - lower bound
Denote Z the random variable that counts the number of boundaries of a (d+1)-simplex in
Xd (n, p) (a boundary of a (d+1)-simplex is a d+2 set from V for which all of the (d+1)-
dimensional faces are in X) when p = c
n . Then E [Z] = n
d+2
c
n
(d+2
d+1)
= n
d+2
c
n
d+2
=
(1 + o (1)) cd+2
(d+2)! and since Z is Poisson distributed the probability that a complex contains a
boundary of a (d+1)-simplex is positive but smaller than 1. It follows that we may as well
condition on there being no boundary of a (d+1)-simplex in X.
Denition 3.5. We denote
Fn,d = ∆(k−1)
n ⊆ X ⊆ ∆(k)
n : X does not contain the boundary of a (d+1)-simplex
for example all the simplexes in X2 (n, p) not containing a copy of a full
[4]
3 .
Now let's observe the function: u (γ, x) = −x + exp −γ (1 − x)
d
. For small positive γ the
only solution in [0, 1] of u (γ, x) = 0 is x = 1.
Denote: γd = inf {γ : u (γ, x) = 0has a root 0 x 1} .
Theorem 3.6. Let c γd be xed then in the probability space Xd n, c
n :
lim
n→∞
Pr (X is d-collapsible | X ∈ Fn,d) = 1
Lior Aronshtam calculated some of the γd-s. The most relevant for us here is γ2 2.455 for
which computer simulations suggests that it really is the actual threshold for 2-collapsibility for
random complexes.
6

7. 4 Summary
To sum up, we have two bounds on the threshold for the existence of a cycle in a complex in
Fn,d: if cd c then
Pr Xd n,
c
n
∈ Pd = 1 − o (1)
and if c γd then
Pr Xd n,
c
n
∈ Pd| X ∈ Fn,d = o (1)
In the next chapter I will present my attempts to improve these bounds using computer
programs.
7

8. Part III
My Work
1 The General Program
Obviously, due to computational limitations, one cannot go over all of the probability space
Xd (n, p) for a certain n,d and p. Therefore the approach I took was to start from the full (d-
1)-dimensional skeleton and sequentially add random d-faces until the occurrence of a relevant
change.
Each run of that program employed a dierent random seed (in order to get independence).
The run produces a series of d-faces that in some point get the property we are looking for. The
main outcome of the program is the number of steps until the occurrence of that critical d-face.
After running the program and getting enough results we can calculate an estimate of the
threshold constant using the following calculation:
We denote by Z the random variable that counts the d-faces in a complex X ∈ Xd n, c
n .
The expectation Z is:
E [Z] =
c
n
n
d
By the law of large numbers the average of enough results will give us an estimation of that
expectation so if we denote the average by x we will get the threshold constant c by: c = nx
(n
d)
.
2 Non-vanishing Homology Threshold with d=2
The rst program I wrote was intended to nd a more accurate threshold constant for Pd.
I start with some graphic results showing there is indeed a threshold for Pd:
Figure 1: Number of complexes for which the rst cycle appears at step x
(The graph was produced from 929 runs for n = 100)
The constant c obtained from these results is around: 2.72 which is close to c2 we presented
above.
8

9. 3 Connected Components Collapse with d=2
3.1 What is Connected Components Collapse?
We presented above cd and γd and we saw that there is a gap between them in which the constant
we are looking for resides. We wanted to try and get a better lower bound with the following
method.
We can associate with a d-dimensional complex a graph whose vertices are the d-faces. There
is an edge between two d-faces if they intersect dimension in a (d-1)-face.
Here instead of applying the whole Gauss elimination we applied repeatedly the following
steps: 1. Elementary collapse. 2. Conaider a row i with exactly two 1 entries, say aij = aik = 1
and all other ait = 0. Then we remove column j and k and introduce the column that is their
sum. That process stops in one of two conditions: 1. All of the columns collapsed. 2. Each line
in the matrix has at least three 1-s in it.
In terms of the mentioned graph we reduce each connected component to a single column
by summing up all of its columns.
We call this process Connected Components Collapse. We will say that a complex is CCC
(Connected components collapsible) if that process stopped for reason 1.
We note that any d-collapsible complex is also CCC whereas if it contains a cycle it is
not CCC. Therefore the threshold constant of CCC will give us a new lower bound for the
Non-vanishing Homology.
3.2 Results for d=2
As we tried to nd a lower bound, for each series of faces we compared the number of faces that
after adding them the simplex was not collapsible anymore and the number of faces that after
adding them the simplex is not CCC anymore. Here is a chart presenting our results:
Table 1:
The 2-collapse average column shows where the regular 2-collapse stopped and the CCC
average shows where the regular 2-collapse stopped and in the right three columns we compared
the constants derived from those averages.
As we can see, the constants are very close to each other and getting closer as n grows.
Therefore the conclusion is that this method does not yield a new bound.
9

10. References
[1] N. Alon and J. Spencer, The Probabilistic Method, 2nd Edition. New York: Wiley-
Interscience, 2000
[2] N. Linial and R. Meshulam, Homological Connectivity of Random 2-dimensional Complexes,
Combinatorica, 26(2006) 475487.
[3] E. Friedgut, Sharp thresholds of graph properties, and the k-sat problem, J. Amer. Math.
Soc. 12 (1999), 1017-1054
[4] D. Kozlov, The threshold function for vanishing of the top homology group of random d-
complexes, Proc. Amer. Math. Soc. 138(2010) 45174527
[5] N. Linial, R. Meshulam and M. Rosenthal, Sum Complexes - a new family of hypertrees,
Discrete and Computational Gepmetry, 2010(44) 622-636
[6] A. Costa, M. Farber and T. Kappeler, Topology of random 2-complexes arXiv:1006.4229
[7] L. Aronshtam, N. Linial, T. Luczak and R. Meshulam, arXiv:1010.1400
10