#  Generations of random hypergraphs and random simplicial complexes by the
map algebra

Shiquan Ren

###### Abstract

We consider the random hypergraph on a finite vertex set by choosing each set
of vertices as an hyperedge independently at random. We express the
probability distributions of the (lower-)associated simplicial complex and the
(lower-)associated independence hypergraph of the random hypergraph in terms
of the probability distributions of certain random simplicial complex and
certain random independence hypergraph of Erdös-Rényi type. We construct a
graded structure of the map algebra explicitly and give algorithms to generate
random hypergraphs and random simplicial complexes.

2010 Mathematics Subject Classification. Primary 05C80, 05E45, Secondary
55U10, 68P05

Keywords and Phrases. hypergraph, simplicial complex, randomness, probability

##  1 Introduction

Let $V$ be a finite vertex set. Let $\Delta[V]$ be the collection of all the
nonempty subsets of $V$. Let $p:\Delta[V]\longrightarrow[0,1]$ be a function
on $\Delta[V]$ with values in the unit interval. A hypergraph on $V$ is a
subset of $\Delta[V]$. A hypergraph is an (abstract) simplicial complex if any
nonempty subset of any hyperedge is still a hyperedge in the hypergraph. For
any simplicial complex $\mathcal{K}$ on $V$, an external face of $\mathcal{K}$
is a nonempty subset $\sigma$ of $V$ such that $\sigma\notin\mathcal{K}$ and
$\tau\in\mathcal{K}$ for any proper subset $\tau$ of $\sigma$. Let
$E(\mathcal{K})$ be the set of all the external faces of $\mathcal{K}$. We
call a hypergraph an independence hypergraph if any superset, which is a
subset of $V$, of any hyperedge is still a hyperedge in the hypergraph. The
complement of a simplicial complex in $\Delta[V]$ is an independence
hypergraph and vice versa. For any independence hypergraph $\mathcal{L}$ on
$V$, a co-external face of $\mathcal{L}$ is a subset $\sigma$ of $V$ such that
$\sigma\notin\mathcal{L}$ and $\tau\in\mathcal{L}$ for any proper superset
$\tau$ of $\sigma$ such that $\tau$ a subset of $V$. Let
$\bar{E}(\mathcal{L})$ be the set of all the co-external faces of
$\mathcal{L}$.

Let $\mathcal{H}$ be a hypergraph on $V$. The associated simplicial complex
$\Delta\mathcal{H}$ of $\mathcal{H}$ is the smallest simplicial complex
containing $\mathcal{H}$ (cf. [7, 37, 9]). The lower-associated simplicial
complex $\delta\mathcal{H}$ of $\mathcal{H}$ is the largest simplicial complex
contained in $\mathcal{H}$ (cf. [39]). We define the associated independence
hypergraph $\bar{\Delta}\mathcal{H}$ of $\mathcal{H}$ as the smallest
independence hypergraph containing $\mathcal{H}$ and define the lower-
associated independence hypergraph $\bar{\delta}\mathcal{H}$ of $\mathcal{H}$
as the largest independence hypergraph contained in $\mathcal{H}$. Consider
the random hypergraph whose probability is given by

$\displaystyle\bar{\rm
P}_{p}(\mathcal{H})=\prod_{\sigma\in\mathcal{H}}p(\sigma)\prod_{\sigma\notin\mathcal{H}}\big{(}1-p(\sigma)\big{)},$

where $\mathcal{H}$ is any hypergraph on $V$, the random simplicial complex
whose probability is given by

$\displaystyle{\rm
P}_{p}(\mathcal{K})=\prod_{\sigma\in\mathcal{K}}p(\sigma)\prod_{\sigma\in
E(\mathcal{K})}\big{(}1-p(\sigma)\big{)},$

where $\mathcal{K}$ is any simplicial complex on $V$, and the random
independence hypergraph whose probability is given by

$\displaystyle{\rm
Q}_{p}(\mathcal{L})=\prod_{\sigma\in\mathcal{L}}p(\sigma)\prod_{\sigma\in\bar{E}(\mathcal{L})}\big{(}1-p(\sigma)\big{)},$

where $\mathcal{L}$ is any independence hypergraph on $V$. Note that $\bar{\rm
P}_{p}$, ${\rm P}_{p}$ and ${\rm Q}_{p}$ are well-defined probability
functions, i.e.

$\displaystyle\sum_{\mathcal{H}}\bar{\rm
P}_{p}(\mathcal{H})=\sum_{\mathcal{K}}{\rm
P}_{p}(\mathcal{K})=\sum_{\mathcal{L}}{\rm Q}_{p}(\mathcal{L})=1.$

The summations are over all the hypergraphs on $V$, all the simplicial
complexes on $V$ and all the independence hypergraphs on $V$ respectively.

######  Theorem 1.1.

Let $\mathcal{H}\sim\bar{\rm P}_{p}$ be a randomly generated hypergraph on
$V$. Then

  1. (1).

the complement of $\mathcal{H}$ is a randomly generated hypergraph
$\gamma\mathcal{H}\sim\bar{\rm P}_{1-p}$,

  2. (2).

the complement of the associated simplicial complex of $\mathcal{H}$ is a
randomly generated independence hypergraph $\gamma\Delta\mathcal{H}\sim{\rm
Q}_{1-p}$,

  3. (3).

the complement of the associated independence hypergraph of $\mathcal{H}$ is a
randomly generated simplicial complex $\gamma\bar{\Delta}\mathcal{H}\sim{\rm
P}_{1-p}$,

  4. (4).

the lower-associated simplicial complex of $\mathcal{H}$ is a randomly
generated simplicial complex $\delta\mathcal{H}\sim{\rm P}_{p}$,

  5. (5).

the lower-associated independence hypergraph of $\mathcal{H}$ is a randomly
generated independence hypergraph $\bar{\delta}\mathcal{H}\sim{\rm Q}_{p}$.

We will prove Theorem 1.1 in Section 3. In fact, Theorem 1.1 (1) follows from
Lemma 3.2, Theorem 1.1 (2), (3) follow from Corollary 3.6 (1), (2)
respectively, and Theorem 1.1 (4), (5) follow from Theorem 3.5 (3), (4)
respectively. As by-products, we have the following two corollaries.

######  Corollary 1.2.

Let $V^{\prime}$ and $V^{\prime\prime}$ be two disjoint vertex sets. Let
$p^{\prime}:\Delta[V^{\prime}]\longrightarrow[0,1]$ and let
$p^{\prime\prime}:\Delta[V^{\prime\prime}]\longrightarrow[0,1]$. Define
$p^{\prime}*p^{\prime\prime}:\Delta[V^{\prime}\sqcup
V^{\prime\prime}]\longrightarrow[0,1]$ by letting
$(p^{\prime}*p^{\prime\prime})(\sigma)=p^{\prime}(\sigma\cap
V^{\prime})p^{\prime\prime}(\sigma\cap V^{\prime\prime})$ for any
$\sigma\in\Delta[V^{\prime}\sqcup V^{\prime\prime}]$.

  1. (1).

Let $\mathcal{H}^{\prime}\sim\bar{\rm P}_{p^{\prime}}$ and
$\mathcal{H}^{\prime\prime}\sim\bar{\rm P}_{p^{\prime\prime}}$ be randomly
generated hypergraphs. Then their join is a randomly generated hypergraph
$\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}\sim\bar{\rm
P}_{p^{\prime}*p^{\prime\prime}}$;

  2. (2).

Let $\mathcal{K}^{\prime}\sim{\rm P}_{p^{\prime}}$ and
$\mathcal{K}^{\prime\prime}\sim{\rm P}_{p^{\prime\prime}}$ be randomly
generated simplicial complexes. Then their join is a randomly generated
simplicial complex $\mathcal{K}^{\prime}*\mathcal{K}^{\prime\prime}\sim{\rm
P}_{p^{\prime}*p^{\prime\prime}}$;

  3. (3).

Let $\mathcal{L}^{\prime}\sim{\rm Q}_{p^{\prime}}$ and
$\mathcal{L}^{\prime\prime}\sim{\rm Q}_{p^{\prime\prime}}$ be randomly
generated independence hypergraphs. Then their join is a randomly generated
independence hypergraph
$\mathcal{L}^{\prime}*\mathcal{L}^{\prime\prime}\sim{\rm
Q}_{p^{\prime}*p^{\prime\prime}}$.

Corollary 1.2 (1) follows from Lemma 3.4 and Corollary 1.2 (2), (3) follow
from Corollary 3.8 (1), (2) respectively.

######  Corollary 1.3.

Let $p^{\prime},p^{\prime\prime}:\Delta[V]\longrightarrow[0,1]$. Define
$p^{\prime}\cap p^{\prime\prime},p^{\prime}\cup
p^{\prime\prime}:\Delta[V]\longrightarrow[0,1]$ by letting $(p^{\prime}\cap
p^{\prime\prime})(\sigma)=p^{\prime}(\sigma)p^{\prime\prime}(\sigma)$ and
$(p^{\prime}\cup
p^{\prime\prime})(\sigma)=1-(1-p^{\prime}(\sigma))(1-p^{\prime\prime}(\sigma))$
for any $\sigma\in\Delta[V]$.

  1. (1).

Let $\mathcal{H}^{\prime}\sim\bar{\rm P}_{p^{\prime}}$ and let
$\mathcal{H}^{\prime\prime}\sim\bar{\rm P}_{p^{\prime\prime}}$. Then their
intersection is a randomly generated hypergraph
$\mathcal{H}^{\prime}\cap\mathcal{H}^{\prime\prime}\sim\bar{\rm
P}_{p^{\prime}\cap p^{\prime\prime}}$ and their union is a randomly generated
hypergraph $\mathcal{H}^{\prime}\cup\mathcal{H}^{\prime\prime}\sim\bar{\rm
P}_{p^{\prime}\cup p^{\prime\prime}}$;

  2. (2).

Let $\mathcal{K}^{\prime}\sim{\rm P}_{p^{\prime}}$ and let
$\mathcal{K}^{\prime\prime}\sim{\rm P}_{p^{\prime\prime}}$. Then their
intersection is a randomly generated simplicial complex
$\mathcal{K}^{\prime}\cap\mathcal{K}^{\prime\prime}\sim{\rm P}_{p^{\prime}\cap
p^{\prime\prime}}$;

  3. (3).

Let $\mathcal{L}^{\prime}\sim{\rm Q}_{p^{\prime}}$ and let
$\mathcal{L}^{\prime\prime}\sim{\rm Q}_{p^{\prime\prime}}$. Then their
intersection is a randomly generated independence hypergraph
$\mathcal{L}^{\prime}\cap\mathcal{L}^{\prime\prime}\sim{\rm Q}_{p^{\prime}\cap
p^{\prime\prime}}$.

Corollary 1.3 (1) follows from [38, Lemma 4.5] (Lemma 3.3) and Corollary 1.3
(2), (3) follow from Corollary 3.7 (1), (2) respectively.

The generation of random hypergraphs as well as random simplicial complexes is
useful in computer science (for example, [2]). The map algebra on the space of
random sub-hypergraphs of a fixed simplicial complex was initially studied by
C. Wu, J. Wu and the present author [38]. In Section 4 of this paper, we
consider the map algebra on the space of random sub-hypergraphs of the
complete complex $\Delta[V]$. By considering the compositions of $\Delta$,
$\bar{\Delta}$, $\delta$, $\bar{\delta}$, $\gamma$, the intersections, the
unions, the joins and the products, we construct a graded structure of the map
algebra in (4.1). We consider the action of the map algebra on the random
hypergraph $\mathcal{H}\sim\bar{\rm P}_{p}$, the random simplicial complex
$\mathcal{K}\sim{\rm P}_{p}$ and the random independence hypergraph
$\mathcal{L}\sim{\rm Q}_{p}$ in (4.2). Using the graded structure in (4.2), we
give some algorithms generating random hypergraphs, random simplicial
complexes and random independence hypergraphs. Each element in (4.2) will
randomly generate a hypergraph and each element of certain particular forms
will generate a random simplicial complex or a random independence hypergraph.

###  1.1 Literature review on random simplicial complexes

Let $V$ be a vertex set of cardinality $N$. Denote $\Delta[V]$ as
$\Delta_{N}$. Let ${\rm sk}_{r}(\Delta_{N})$ be the $r$-skeleton of
$\Delta_{N}$, $0\leq r\leq N-1$. Consider the space $\Omega_{N}^{r}$
consisting of all sub-complexes $\mathcal{K}$ of ${\rm sk}_{r}(\Delta_{N})$.
Consider the probability function ${\rm
P}_{r,N,\mathbf{p}}:\Omega_{N}^{r}\longrightarrow\mathbb{R}$ given by

$\displaystyle{\rm
P}_{r,N,\mathbf{p}}(\mathcal{K})=\prod_{\sigma\in\mathcal{K}}p(\sigma)\prod_{\sigma\in
E(\mathcal{K})}\big{(}1-p(\sigma)\big{)},$

where $\mathbf{p}=(p_{0},p_{1},\ldots,p_{r})$, $0\leq
p_{0},p_{1},\ldots,p_{r}\leq 1$ are constants, and $p(\sigma)=p_{i}$ for any
$i$-simplex $\sigma\in{\rm sk}_{r}(\Delta_{N})$, $0\leq i\leq r$. This model
${\rm P}_{r,N,\mathbf{p}}$ was given by A. Costa and M. Farber in [15, 16,
19]. The connectivity, the fundamental group, the dimension, and the Betti
number of the random simplicial complex with the probability ${\rm
P}_{r,N,\mathbf{p}}$ have been studied by A. Costa and M. Farber in [15, 16,
17, 18, 19]. In particular, let $0\leq p\leq 1$ be a constant. The followings
are special cases of ${\rm P}_{r,N,\mathbf{p}}$:

(1). ${\rm P}_{1,N,\mathbf{p}}$ with $\mathbf{p}=(1,p)$ is the Erdös-Rényi
model $G(N,p)$. P. Erdös and A. Rényi [20] and E.N. Gilbert [23] constructed
the Erdös-Rényi model $G(N,p)$ by choosing each pair of vertices in $V$ as an
edge independently at random with probability $p$. Thresholds for the
connectivity of $G(N,p)$ were proved by P. Erdös and A. Rényi in [21].

(2). ${\rm P}_{2,N,\mathbf{p}}$ with $\mathbf{p}=(1,1,p)$ is the Linial-
Meshulam model $Y_{2}(N,p)$. N. Linial and R. Meshulam [33] constructed a
random $2$-complex $Y_{2}(N,p)$ by taking the complete graph on $V$ as the
$1$-skeleton and then choosing each $2$-simplex independently at random with
probability $p$. The fundamental group, the homology groups, and the
asphericity as well as the hyperbolicity of $Y_{2}(N,p)$ were respectively
studied in [5], [10, 11], and [12, 13].

(3). ${\rm P}_{d,N,\mathbf{p}}$ with $\mathbf{p}=(1,\ldots,1,p)$ is the
Meshulam-Wallach model $Y_{d}(N,p)$. R. Meshulam and N. Wallach [35]
constructed a random $d$-complex $Y_{d}(N,p)$ by taking the $(d-1)$-skeleton
of the complete complex on $V$ and then choosing each $d$-simplex
independently at random with probability $p$. The (co)homology groups, the
phase transition of the homology groups, the eigenvalues of the Laplacian, the
collapsibility and the topological minor were respectively studied in [4, 27,
31, 32], [34], [24], [3, 4] and [25].

(4). ${\rm P}_{N-1,N,\mathbf{p}}$ with $\mathbf{p}=(1,p,1,\ldots,1)$ is the
random flag complex (random clique complex) $X(N,p)$ of $G(N,p)$. This random
simplicial complex was studied by M. Kahle in [28, 30] and A. Costa, M. Farber
and D. Horak in [14]. Sharp vanishing thresholds for the cohomology of
$X(N,p)$ were proved by M. Kahle in [30].

(5). ${\rm P}_{2,N,\mathbf{p}}$ with $\mathbf{p}=(p_{0},p_{1},p_{2})$ was
considered by M. Farber and T. Nowik in [22]. The multi-parameter threshold
for the property that every $2$-dimensional simplicial complex admits a
topological embedding into the ${\rm P}_{2,N,\mathbf{p}}$-generated random
$2$-complex asymptotically almost surely was established.

##  2 Simplicial models for hypergraphs

###  2.1 Hypergraphs and simplicial complexes

Let $V$ be a finite set with a total order $\prec$. The elements of $V$ are
called vertices. Let $2^{V}$ be the power set of $V$. Let
$\Delta[V]=2^{V}\setminus\\{\emptyset\\}$. For any nonnegative integer $n$ and
any distinct vertices $v_{0},v_{1},\ldots,v_{n}\in V$ such that $v_{0}\prec
v_{1}\prec\cdots\prec v_{n}$, we call the set
$\sigma=\\{v_{0},v_{1},\ldots,v_{n}\\}$ an $n$-hyperedge on $V$. Let $\sigma$
be an $n$-hyperedge on $V$. The subset closure $\Delta\sigma$ of $\sigma$ is
an $n$-dimensional simplicial complex whose set of simplices consists of all
the non-empty subsets of $\sigma$. The superset closure $\bar{\Delta}\sigma$
of $\sigma$ is $(\Delta[V]\setminus\Delta\sigma)\cup\sigma$, in other words,
$\bar{\Delta}\sigma$ consists of all the supersets $\tau\supseteq\sigma$ such
that $\tau\subseteq V$. Note that $\Delta\sigma$ does not depend on the choice
of $V$ while $\bar{\Delta}\sigma$ depends on the choice of $V$.

######  Definition 2.1.

[6, 37, 9] A hypergraph $\mathcal{H}$ on $V$ is a collection of hyperedges on
$V$. In particular,

  1. (1).

$\mathcal{H}$ is an (abstract) simplicial complex if $\tau\in\mathcal{H}$ for
any $\sigma\in\mathcal{H}$ and any nonempty subset $\tau\subseteq\sigma$. A
hyperedge of a simplicial complex is called a simplex (cf. [26, 36]);

  2. (2).

$\mathcal{H}$ is called an independence hypergraph if $\tau\in\mathcal{H}$ for
any $\sigma\in\mathcal{H}$ and any superset $\tau\supseteq\sigma$ with
$\tau\subseteq V$.

Let $\mathcal{H}$ be a hypergraph on $V$.

######  Definition 2.2.

The complement of $\mathcal{H}$ is a hypergraph on $V$ given by

$\displaystyle\gamma\mathcal{H}=\\{\sigma\in\Delta[V]\mid\sigma\notin\mathcal{H}\\}.$

By Definition 2.2, $\gamma\mathcal{H}$ is a simplicial complex (resp. an
independence hypergraph) iff $\mathcal{H}$ is an independence hypergraph
(resp. a simplicial complex). Note that $\gamma^{2}={\rm id}$.

######  Definition 2.3.

The associated simplicial complex (cf. [7, 37, 9]) is

$\displaystyle\Delta\mathcal{H}=\bigcup_{\sigma\in\mathcal{H}}\Delta\sigma,$

which is the smallest simplicial complex containing $\mathcal{H}$. The lower-
associated simplicial complex is (cf. [39])

$\displaystyle\delta\mathcal{H}=\bigcup_{\Delta\sigma\subseteq\mathcal{H}}\\{\sigma\\},$

which is the largest simplicial complex contained in $\mathcal{H}$. The
associated independence hypergraph is

$\displaystyle\bar{\Delta}\mathcal{H}=\bigcup_{\sigma\in\mathcal{H}}\bar{\Delta}\sigma,$

which is the smallest independence hypergraph containing $\mathcal{H}$. The
lower-associated independence hypergraph is

$\displaystyle\bar{\delta}\mathcal{H}=\bigcup_{\bar{\Delta}\sigma\subseteq\mathcal{H}}\\{\sigma\\},$

which is the largest independence hypergraph contained in $\mathcal{H}$.

By Definition 2.3, we have a diagram

such that each arrow is a canonical inclusion of hypergraphs.

######  Lemma 2.1.

$\bar{\Delta}=\gamma\delta\gamma$ and $\bar{\delta}=\gamma\Delta\gamma$.

###### Proof.

For any hypergraph $\mathcal{H}$ on $V$, we have

$\displaystyle\gamma\bar{\Delta}\gamma\mathcal{H}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\sigma\notin\bar{\Delta}\gamma\mathcal{H}\\}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\sigma\not\supseteq\tau{\rm~{}for~{}any~{}}\tau\in\gamma\mathcal{H}\\}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\sigma\not\supseteq\tau{\rm~{}for~{}any~{}}\tau\notin\mathcal{H},{\rm~{}where~{}}\tau\in\Delta[V]\\}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\tau\in\mathcal{H}{\rm~{}for~{}any~{}}\tau\subseteq\sigma,{\rm~{}where~{}}\tau\in\Delta[V]\\}$
$\displaystyle=$ $\displaystyle\delta\mathcal{H}.$

Therefore, $\bar{\Delta}=\gamma\delta\gamma$. Moreover, For any hypergraph
$\mathcal{H}$ on $V$, we have

$\displaystyle\gamma\Delta\gamma\mathcal{H}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\sigma\notin\Delta\gamma\mathcal{H}\\}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\sigma\not\subseteq\tau{\rm~{}for~{}any~{}}\tau\in\gamma\mathcal{H}\\}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\sigma\not\subseteq\tau{\rm~{}for~{}any~{}}\tau\notin\mathcal{H},{\rm~{}where~{}}\tau\in\Delta[V]\\}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\Delta[V]\mid\tau\in\mathcal{H}{\rm~{}for~{}any~{}}\tau\supseteq\sigma,{\rm~{}where~{}}\tau\in\Delta[V]\\}$
$\displaystyle=$ $\displaystyle\bar{\delta}\mathcal{H}.$

Therefore, $\bar{\delta}=\gamma\Delta\gamma$. ∎

By [38, Subsection 2.1] and Lemma 2.1, the following relations among $\gamma$,
$\Delta$, $\delta$, $\bar{\Delta}$ and $\bar{\delta}$ hold: (1).
$\gamma^{2}={\rm id}$, (2). $\bar{\Delta}=\gamma\delta\gamma$ and
$\bar{\delta}=\gamma\Delta\gamma$, (3). $\Delta\delta=\delta$,
$\bar{\Delta}\bar{\delta}=\bar{\delta}$, $\delta\Delta=\Delta$ and
$\bar{\delta}\bar{\Delta}=\bar{\Delta}$, (4). $\Delta^{2}=\Delta$,
$\bar{\Delta}^{2}=\bar{\Delta}$, $\delta^{2}=\delta$ and
$\bar{\delta}^{2}=\bar{\delta}$, (5).
$(\delta\bar{\Delta})^{2}=(\delta\gamma\delta\gamma)^{2}=\delta\gamma\delta\gamma=\delta\bar{\Delta}$,
$(\bar{\Delta}\delta)^{2}=(\gamma\delta\gamma\delta)^{2}=\gamma\delta\gamma\delta=\bar{\Delta}\delta$,
$(\Delta\bar{\delta})^{2}=(\Delta\gamma\Delta\gamma)^{2}=\Delta\gamma\Delta\gamma=\Delta\bar{\delta}$
and
$(\bar{\delta}\Delta)^{2}=(\gamma\Delta\gamma\Delta)^{2}=\gamma\Delta\gamma\Delta=\bar{\delta}\Delta$.
Moreover, since the associated simplicial complex of any nonempty independence
hypergraph is $\Delta[V]$, we have

$\displaystyle\Delta\bar{\Delta}(\mathcal{H})$ $\displaystyle=$
$\displaystyle\begin{cases}\Delta[V],&\mathcal{H}\neq\emptyset,\\\
\emptyset,&\mathcal{H}=\emptyset,\end{cases}$
$\displaystyle\Delta\bar{\delta}(\mathcal{H})$ $\displaystyle=$
$\displaystyle\begin{cases}\Delta[V],&\\{V\\}\in\mathcal{H},\\\
\emptyset,&\\{V\\}\notin\mathcal{H}.\end{cases}$

Thus the following relations hold as well: (6).
$\Delta^{2}\bar{\Delta}=\bar{\Delta}\Delta\bar{\Delta}=\delta\Delta\bar{\Delta}=\bar{\delta}\Delta\bar{\Delta}=\Delta\bar{\Delta}$
and
$\Delta^{2}\bar{\delta}=\bar{\Delta}\Delta\bar{\delta}=\delta\Delta\bar{\delta}=\bar{\delta}\Delta\bar{\delta}=\Delta\bar{\delta}$.

Let $\mathcal{H}$ and $\mathcal{H}^{\prime}$ be two hypergraphs on $V$. We
have the following observations:

  1. (i).

$\gamma(\mathcal{H}\cap\mathcal{H}^{\prime})=\gamma\mathcal{H}\cup\gamma\mathcal{H}^{\prime}$,
$\gamma(\mathcal{H}\cup\mathcal{H}^{\prime})=\gamma\mathcal{H}\cap\gamma\mathcal{H}^{\prime}$,

  2. (ii).

$\Delta(\mathcal{H}\cap\mathcal{H}^{\prime})\subseteq\Delta\mathcal{H}\cap\Delta\mathcal{H}^{\prime}$,
$\Delta(\mathcal{H}\cup\mathcal{H}^{\prime})=\Delta\mathcal{H}\cup\Delta\mathcal{H}^{\prime}$,

  3. (iii).

$\delta(\mathcal{H}\cap\mathcal{H}^{\prime})=\delta\mathcal{H}\cap\delta\mathcal{H}^{\prime}$,
$\delta(\mathcal{H}\cup\mathcal{H}^{\prime})\supseteq\delta\mathcal{H}\cup\delta\mathcal{H}^{\prime}$,

  4. (iv).

$\bar{\Delta}(\mathcal{H}\cap\mathcal{H}^{\prime})\subseteq\bar{\Delta}\mathcal{H}\cap\bar{\Delta}\mathcal{H}^{\prime}$,
$\bar{\Delta}(\mathcal{H}\cup\mathcal{H}^{\prime})=\bar{\Delta}\mathcal{H}\cup\bar{\Delta}\mathcal{H}^{\prime}$,

  5. (v).

$\bar{\delta}(\mathcal{H}\cap\mathcal{H}^{\prime})=\bar{\delta}\mathcal{H}\cap\bar{\delta}\mathcal{H}^{\prime}$,
$\bar{\delta}(\mathcal{H}\cup\mathcal{H}^{\prime})\supseteq\bar{\delta}\mathcal{H}\cup\bar{\delta}\mathcal{H}^{\prime}$.

If both $\mathcal{H}$ and $\mathcal{H}^{\prime}$ are simplicial complexes
(resp. independence hypergraphs), then both
$\mathcal{H}\cap\mathcal{H}^{\prime}$ and
$\mathcal{H}\cup\mathcal{H}^{\prime}$ are simplicial complexes (resp.
independence hypergraphs).

Let $\sigma=\\{v_{0},v_{1},\ldots,v_{n}\\}$ be an $n$-hyperedge on $V$ and let
$\tau=\\{u_{0},u_{1},\ldots,u_{m}\\}$ be an $m$-hyperedge on $V^{\prime}$. The
box product $\sigma\square\tau$ is an $(mn+m+n+1)$-hyperedge

$\displaystyle\sigma\square\tau=\\{(v,u)\mid v\in\sigma,u\in\tau\\}$ (2.2)

on the Cartesian product $V\times V^{\prime}$. The right-hand side of (2.2) is
up to a rearrangement of the vertices with respect to the lexicographic order.
In addition, suppose $V\cap V^{\prime}=\emptyset$. The join $\sigma*\tau$ is
an $(n+m+1)$-hyperedge

$\displaystyle\sigma*\tau=\\{v_{0},v_{1},\ldots,v_{n},u_{0},u_{1},\ldots,u_{m}\\}$
(2.3)

on the disjoint union $V\sqcup V^{\prime}$, where the right-hand side of (2.3)
is with respect to the total order on $V\sqcup V^{\prime}$ given by $v\prec
v^{\prime}$ for any $v\in V$ and any $v^{\prime}\in V^{\prime}$. We observe

$\displaystyle\Delta(\sigma*\tau)$ $\displaystyle=$
$\displaystyle(\Delta\sigma)*(\Delta\tau),$
$\displaystyle\bar{\Delta}(\sigma*\tau)$ $\displaystyle=$
$\displaystyle(\bar{\Delta}\sigma)*(\bar{\Delta}\tau),$

where the associated independence hypergraphs of $\sigma*\tau$, $\sigma$ and
$\tau$ are taken with respect to $V\sqcup V^{\prime}$, $V$ and $V^{\prime}$
respectively.

######  Definition 2.4.

Let $V$ and $V^{\prime}$ be two finite sets. Let $\mathcal{H}$ be a hypergraph
on $V$ and let $\mathcal{H}^{\prime}$ be a hypergraph on $V^{\prime}$.

  1. (1).

We define the box product of $\mathcal{H}$ and $\mathcal{H}^{\prime}$ as

$\displaystyle\mathcal{H}\square\mathcal{H}^{\prime}=\\{\sigma\square\sigma^{\prime}\mid\sigma\in\mathcal{H},\sigma^{\prime}\in\mathcal{H}^{\prime}\\}.$

Then $\mathcal{H}\square\mathcal{H}^{\prime}$ is a hypergraph on $V\times
V^{\prime}$;

  2. (2).

Suppose in addition $V\cap V^{\prime}=\emptyset$. We define the join of
$\mathcal{H}$ and $\mathcal{H}^{\prime}$ as

$\displaystyle\mathcal{H}*\mathcal{H}^{\prime}=\\{\sigma*\sigma^{\prime}\mid\sigma\in\mathcal{H}{\rm~{}and~{}}\sigma^{\prime}\in\mathcal{H}^{\prime}\\}\cup\mathcal{H}\cup\mathcal{H}^{\prime}.$

Then $\mathcal{H}*\mathcal{H}^{\prime}$ is a hypergraph on $V\sqcup
V^{\prime}$.

By Definition 2.4, we observe the followings:

  1. (i)’.

$\Delta(\mathcal{H}*\mathcal{H}^{\prime})=\Delta\mathcal{H}*\Delta\mathcal{H}^{\prime}$,

  2. (ii)’.

$\delta(\mathcal{H}*\mathcal{H}^{\prime})=\delta\mathcal{H}*\delta\mathcal{H}^{\prime}$,

  3. (iii)’.

$\bar{\Delta}(\mathcal{H}*\mathcal{H}^{\prime})=\bar{\Delta}\mathcal{H}*\bar{\Delta}\mathcal{H}^{\prime}$,

  4. (iv)’.

$\bar{\delta}(\mathcal{H}*\mathcal{H}^{\prime})=\bar{\delta}\mathcal{H}*\bar{\delta}\mathcal{H}^{\prime}$,

  5. (v)’.

$\mathcal{H}_{1}*(\mathcal{H}_{2}\cup\mathcal{H}_{3})=(\mathcal{H}_{1}*\mathcal{H}_{2})\cup(\mathcal{H}_{1}*\mathcal{H}_{3})$,

  6. (vi)’.

$\mathcal{H}_{1}*(\mathcal{H}_{2}\cap\mathcal{H}_{3})=(\mathcal{H}_{1}*\mathcal{H}_{2})\cap(\mathcal{H}_{1}*\mathcal{H}_{3})$,

  7. (vii)’.

$\mathcal{H}_{1}\square(\mathcal{H}_{2}\cup\mathcal{H}_{3})=(\mathcal{H}_{1}\square\mathcal{H}_{2})\cup(\mathcal{H}_{1}\square\mathcal{H}_{3})$,

  8. (viii)’.

$\mathcal{H}_{1}\square(\mathcal{H}_{2}\cap\mathcal{H}_{3})=(\mathcal{H}_{1}\square\mathcal{H}_{2})\cap(\mathcal{H}_{1}\square\mathcal{H}_{3})$,

  9. (ix)’.

$\mathcal{H}_{1}\square(\mathcal{H}_{2}*\mathcal{H}_{3})=(\mathcal{H}_{1}\square\mathcal{H}_{2})*(\mathcal{H}_{1}\square\mathcal{H}_{3})$,

where $\bar{\Delta}$ and $\bar{\delta}$ on the left-hand sides are with
respect to $V\sqcup V^{\prime}$ while $\bar{\Delta}$ and $\bar{\delta}$ on the
right-hand sides are with respect to $V$ and $V^{\prime}$.

Let $\mathcal{K}$ be a simplicial complex on $V$ and let
$\mathcal{K}^{\prime}$ be a simplicial complex on $V^{\prime}$. Then
$\mathcal{K}*\mathcal{K}^{\prime}$ is a simplicial complex on $V\sqcup
V^{\prime}$ and both $\Delta(\mathcal{K}\square\mathcal{K}^{\prime})$ and
$\delta(\mathcal{K}\square\mathcal{K}^{\prime})$ are simplicial complexes on
$V\times V^{\prime}$. Let $\mathcal{L}$ be an independence hypergraph on $V$
and let $\mathcal{L}^{\prime}$ be an independence hypergraph on $V^{\prime}$.
Then $\mathcal{L}*\mathcal{L}^{\prime}$ is an independence hypergraph on
$V\sqcup V^{\prime}$ and both
$\bar{\Delta}(\mathcal{L}\square\mathcal{L}^{\prime})$ and
$\bar{\delta}(\mathcal{L}\square\mathcal{L}^{\prime})$ are independence
hypergraphs on $V\times V^{\prime}$.

######  Example 2.1.

Let $V=\\{v_{0},v_{1}\\}$ and let
$V^{\prime}=\\{v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\\}$.
Let $\mathcal{H}=\\{\\{v_{0}\\},\\{v_{0},v_{1}\\}\\}$ be a hypergraph on $V$
and let
$\mathcal{H}^{\prime}=\\{\\{v^{\prime}_{0},v^{\prime}_{1}\\},\\{v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\}\\}$
be a hypergraph on $V^{\prime}$. Let $v_{0}\prec v_{1}\prec
v^{\prime}_{0}\prec v^{\prime}_{1}\prec v^{\prime}_{2}\prec v^{\prime}_{3}$.
Then

$\displaystyle\mathcal{H}*\mathcal{H}^{\prime}$ $\displaystyle=$
$\displaystyle\\{\\{v_{0}\\},\\{v_{0},v_{1}\\},\\{v^{\prime}_{0},v^{\prime}_{1}\\},\\{v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\},\\{v_{0},v^{\prime}_{0},v^{\prime}_{1}\\},$
$\displaystyle\\{v_{0},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\},\\{v_{0},v_{1},v^{\prime}_{0},v^{\prime}_{1}\\},\\{v_{0},v_{1},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\}\\}$

is a hypergraph on $V\sqcup V^{\prime}$ and

$\displaystyle\mathcal{H}\square\mathcal{H}^{\prime}$ $\displaystyle=$
$\displaystyle\\{\\{(v_{0},v^{\prime}_{0}),(v_{0},v^{\prime}_{1})\\},\\{(v_{0},v^{\prime}_{0}),(v_{0},v^{\prime}_{1}),(v_{0},v^{\prime}_{2})\\},$
$\displaystyle\\{(v_{0},v^{\prime}_{0}),(v_{0},v^{\prime}_{1}),(v_{1},v^{\prime}_{0}),(v_{1},v^{\prime}_{1})\\},$
$\displaystyle\\{(v_{0},v^{\prime}_{0}),(v_{0},v^{\prime}_{1}),(v_{0},v^{\prime}_{2}),(v_{1},v^{\prime}_{0}),(v_{1},v^{\prime}_{1}),(v_{1},v^{\prime}_{2})\\}\\}$

is a hypergraph on $V\times V^{\prime}$. Moreover,

  1. (1).

$\Delta\mathcal{H}=\\{\\{v_{0}\\},\\{v_{1}\\},\\{v_{0},v_{1}\\}\\}$,
$\delta\mathcal{H}=\\{\\{v_{0}\\}\\}$,
$\bar{\Delta}\mathcal{H}=\bar{\delta}\mathcal{H}=\mathcal{H}$,

  2. (2).

$\Delta\mathcal{H}^{\prime}=\\{\\{v^{\prime}_{0}\\},\\{v^{\prime}_{1}\\},\\{v^{\prime}_{2}\\},\\{v^{\prime}_{0},v^{\prime}_{1}\\},\\{v^{\prime}_{0},v^{\prime}_{2}\\},\\{v^{\prime}_{1},v^{\prime}_{2}\\},\\{v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\}\\}$,
$\delta\mathcal{H}^{\prime}=\emptyset$,
$\bar{\Delta}\mathcal{H}^{\prime}=\\{\\{v^{\prime}_{0},v^{\prime}_{1}\\},\\{v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\},\\{v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\\}\\}$,
$\bar{\delta}\mathcal{H}^{\prime}=\emptyset$,

  3. (3).

$\Delta\mathcal{H}*\Delta\mathcal{H}^{\prime}=\Delta\\{v_{0},v_{1},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\}$,
$\delta\mathcal{H}*\delta\mathcal{H}^{\prime}=\\{\\{v_{0}\\}\\}$,
$\bar{\Delta}\mathcal{H}*\bar{\Delta}\mathcal{H}^{\prime}=\\{\\{v_{0},v^{\prime}_{0},v^{\prime}_{1}\\},\\{v_{0},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\},\\{v_{0},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\\},\\{v_{0},v_{1},v^{\prime}_{0},v^{\prime}_{1}\\},\\\
\\{v_{0},v_{1},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2}\\},\\{v_{0},v_{1},v^{\prime}_{0},v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\\}\\}\cup\mathcal{H}\cup\mathcal{H}^{\prime}$,
$\bar{\delta}\mathcal{H}*\bar{\delta}\mathcal{H}^{\prime}=\mathcal{H}$.

###  2.2 Morphisms of hypergraphs and simplicial maps

Let $V$ and $V^{\prime}$ be two finite sets. Let $\mathcal{H}$ be a hypergraph
on $V$ and let $\mathcal{H}^{\prime}$ be a hypergraph on $V^{\prime}$. A
morphism $f:\mathcal{H}\longrightarrow\mathcal{H}^{\prime}$ of hypergraphs is
a map $f:V\longrightarrow V^{\prime}$ such that for any hyperedge
$\sigma=\\{v_{0},v_{1},\ldots,v_{n}\\}$ in $\mathcal{H}$, its image
$f(\sigma)$ is a hyperedge in $\mathcal{H}^{\prime}$ spanned by the (not
necessarily distinct) vertices $f(v_{0})$, $f(v_{1})$, $\ldots$, $f(v_{n})$.
Let $\mathcal{K}$ be a simplicial complex on $V$ and let
$\mathcal{K}^{\prime}$ be a simplicial complex on $V^{\prime}$. We call a
morphism $f:\mathcal{K}\longrightarrow\mathcal{K}^{\prime}$ a simplicial map.
Let $\mathcal{L}$ be an independence hypergraph on $V$ and let
$\mathcal{L}^{\prime}$ be an independence hypergraph on $V^{\prime}$. We call
a morphism $f:\mathcal{L}\longrightarrow\mathcal{L}^{\prime}$ a morphism of
independence hypergraphs.

Let $f:\mathcal{H}\longrightarrow\mathcal{H}^{\prime}$ be a morphism of
hypergraphs. Then $f$ canonically induces two simplicial maps

$\displaystyle\Delta f:$
$\displaystyle\Delta\mathcal{H}\longrightarrow\Delta\mathcal{H}^{\prime},$
$\displaystyle\delta f:$
$\displaystyle\delta\mathcal{H}\longrightarrow\delta\mathcal{H}^{\prime}$

and two morphisms of independence hypergraphs

$\displaystyle\bar{\Delta}f:$
$\displaystyle\bar{\Delta}\mathcal{H}\longrightarrow\bar{\Delta}\mathcal{H}^{\prime},$
$\displaystyle\bar{\delta}f:$
$\displaystyle\bar{\delta}\mathcal{H}\longrightarrow\bar{\delta}\mathcal{H}^{\prime}.$

We have the following naturalities:

(1). The canonical inclusions $i_{\Delta}$ and $i_{\delta}$ are natural. That
is, for any morphism $f:\mathcal{H}\longrightarrow\mathcal{H}^{\prime}$ we
have the commutative diagrams

(2). The join of hypergraphs is natural. That is, for any two morphisms
$f:\mathcal{H}_{1}\longrightarrow\mathcal{H}_{2}$ and
$f^{\prime}:\mathcal{H}^{\prime}_{1}\longrightarrow\mathcal{H}^{\prime}_{2}$,
we have a canonical induced morphism
$f*f^{\prime}:\mathcal{H}_{1}*\mathcal{H}^{\prime}_{1}\longrightarrow\mathcal{H}_{2}*\mathcal{H}^{\prime}_{2}$
such that $(f*f^{\prime})\mid_{\mathcal{H}_{1}}=f$,
$(f*f^{\prime})\mid_{\mathcal{H}^{\prime}_{1}}=f^{\prime}$ and for any
$\sigma\in\mathcal{H}_{1}$ and any
$\sigma^{\prime}\in\mathcal{H}^{\prime}_{1}$, the join
$\sigma*\sigma^{\prime}$ in $\mathcal{H}_{1}*\mathcal{H}^{\prime}_{1}$ is sent
to the hyperedge $f(\sigma)*f^{\prime}(\sigma^{\prime})$ in
$\mathcal{H}_{2}*\mathcal{H}^{\prime}_{2}$. The followings are direct

$\displaystyle\Delta(f*f^{\prime})=\Delta f*\Delta
f^{\prime},~{}~{}~{}~{}~{}~{}\delta(f*f^{\prime})=\delta f*\delta f^{\prime},$
$\displaystyle\bar{\Delta}(f*f^{\prime})=\bar{\Delta}f*\bar{\Delta}f^{\prime},~{}~{}~{}~{}~{}~{}\bar{\delta}(f*f^{\prime})=\bar{\delta}f*\bar{\delta}f^{\prime},$

where $\bar{\Delta}$ and $\bar{\delta}$ on the left-hand sides are with
respect to $V\sqcup V^{\prime}$ while $\bar{\Delta}$ and $\bar{\delta}$ on the
right-hand sides are with respect to $V$ and $V^{\prime}$.

(3). The box product of hypergraphs is natural. That is, for any two morphisms
$f:\mathcal{H}_{1}\longrightarrow\mathcal{H}_{2}$ and
$f^{\prime}:\mathcal{H}^{\prime}_{1}\longrightarrow\mathcal{H}^{\prime}_{2}$,
we have a canonical induced morphism $f\square
f^{\prime}:\mathcal{H}_{1}\square\mathcal{H}^{\prime}_{1}\longrightarrow\mathcal{H}_{2}\square\mathcal{H}^{\prime}_{2}$
sending $\sigma\square\sigma^{\prime}$ to $f(\sigma)\square
f^{\prime}(\sigma^{\prime})$.

##  3 Random hypergraphs and random simplicial complexes

###  3.1 General random hypergraphs and random simplicial complexes

Let ${\bf H}(V)$ be the category whose objects are hypergraphs on $V$ and
whose morphisms are morphisms of hypergraphs. Let ${\bf K}(V)$ be the category
whose objects are simplicial complexes on $V$ and whose morphisms are
simplicial maps. Let ${\bf L}(V)$ be the category whose objects are
independence hypergraphs on $V$ and whose morphisms are morphisms of
independence hypergraphs. Both ${\bf K}(V)$ and ${\bf L}(V)$ are full
subcategories of ${\bf H}(V)$.

A random hypergraph (resp. random simplicial complex and random independence
hypergraph) on $V$ is a probability function on ${\bf Obj}({\bf H}(V))$ (resp.
${\bf Obj}({\bf K}(V))$ and ${\bf Obj}({\bf L}(V))$). Let $D({\bf H}(V))$
(resp. $D({\bf K}(V))$ and $D({\bf L}(V))$) be the functional space of all the
probability functions on ${\bf Obj}({\bf H}(V))$ (resp. ${\bf Obj}({\bf
K}(V))$ and ${\bf Obj}({\bf L}(V))$). For any map

$\displaystyle f:$ $\displaystyle{\bf H}(V)\longrightarrow{\bf H}(V)$

there is an induced map

$\displaystyle Df:$ $\displaystyle D({\bf H}(V))\longrightarrow D({\bf H}(V))$

given by

$\displaystyle(Df)(\varphi)(\mathcal{H})=\sum_{f(\mathcal{H}^{\prime})=\mathcal{H}}\varphi(\mathcal{H}^{\prime}),$

where $\varphi\in D({\bf H}(V))$ and $\mathcal{H},\mathcal{H}^{\prime}\in{\bf
Obj}({\bf H}(V))$. Let $f$ be $\Delta,\delta,\bar{\Delta}$ and $\bar{\delta}$
respectively. We have the induced maps

$\displaystyle D\Delta,D\delta:$ $\displaystyle D({\bf H}(V))\longrightarrow
D({\bf K}(V)),$ $\displaystyle D\bar{\Delta},D\bar{\delta}:$ $\displaystyle
D({\bf H}(V))\longrightarrow D({\bf L}(V)).$

Moreover, for any map $\mu:{\bf Obj}({\bf H}(V))\times{\bf Obj}({\bf
H}(V))\longrightarrow{\bf Obj}({\bf H}(V))$ there is an induced map

$\displaystyle D\mu:$ $\displaystyle D({\bf H}(V))\times D({\bf
H}(V))\longrightarrow D({\bf H}(V))$

given by

$\displaystyle
D\mu(\varphi^{\prime},\varphi^{\prime\prime})(\mathcal{H})=\sum_{\mu(\mathcal{H}^{\prime},\mathcal{H}^{\prime\prime})=\mathcal{H}}\varphi^{\prime}(\mathcal{H}^{\prime})\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime}),$

where $\varphi^{\prime},\varphi^{\prime\prime}\in D({\bf H}(V))$ and
$\mathcal{H},\mathcal{H}^{\prime},\mathcal{H}^{\prime\prime}\in{\rm Obj}({\bf
H}(V))$. Let $\mu(\mathcal{H}^{\prime},\mathcal{H}^{\prime\prime})$ be
$\mathcal{H}^{\prime}\cap\mathcal{H}^{\prime\prime}$ and
$\mathcal{H}^{\prime}\cup\mathcal{H}^{\prime\prime}$ respectively. We obtain
the induced maps

$\displaystyle D\cap,D\cup:$ $\displaystyle D({\bf H}(V))\times D({\bf
H}(V))\longrightarrow D({\bf H}(V)).$

The restrictions of $D\cap$ and $D\cup$ give the maps

$\displaystyle D\cap,D\cup:$ $\displaystyle D({\bf K}(V))\times D({\bf
K}(V))\longrightarrow D({\bf K}(V)),$ $\displaystyle D\cap,D\cup:$
$\displaystyle D({\bf L}(V))\times D({\bf L}(V))\longrightarrow D({\bf
L}(V)).$

Furthermore, let $V^{\prime}$ and $V^{\prime\prime}$ be two finite sets. Then
the box product

$\displaystyle\square:$ $\displaystyle{\bf Obj}({\bf H}(V^{\prime}))\times{\bf
Obj}({\bf H}(V^{\prime\prime}))\longrightarrow{\bf Obj}({\bf
H}(V^{\prime}\times V^{\prime\prime}))$

sending $(\mathcal{H}^{\prime},\mathcal{H}^{\prime\prime})$ to
$\mathcal{H}^{\prime}\square\mathcal{H}^{\prime\prime}$ induces a map

$\displaystyle D\square:$ $\displaystyle D({\bf H}(V^{\prime}))\times D({\bf
H}(V^{\prime\prime}))\longrightarrow D({\bf H}(V^{\prime}\times
V^{\prime\prime}))$

given by

$\displaystyle(D\square)(\varphi^{\prime},\varphi^{\prime\prime})(\mathcal{H})=\sum_{\mathcal{H}^{\prime}\square\mathcal{H}^{\prime\prime}=\mathcal{H}}\varphi^{\prime}(\mathcal{H}^{\prime})\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime}).$
(3.1)

Here in (3.1), we take $\varphi^{\prime}\in D({\bf H}(V^{\prime}))$,
$\varphi^{\prime\prime}\in D({\bf H}(V^{\prime\prime}))$,
$\mathcal{H}^{\prime}\in{\bf Obj}({\bf H}(V^{\prime}))$,
$\mathcal{H}^{\prime\prime}\in{\bf Obj}({\bf H}(V^{\prime\prime}))$ and
$\mathcal{H}\in{\bf Obj}({\bf H}(V^{\prime}\times V^{\prime\prime}))$. Suppose
in addition $V^{\prime}\cap V^{\prime\prime}=\emptyset$. Then the join

$\displaystyle\ast:$ $\displaystyle{\bf Obj}({\bf H}(V^{\prime}))\times{\bf
Obj}({\bf H}(V^{\prime\prime}))\longrightarrow{\bf Obj}({\bf
H}(V^{\prime}\sqcup V^{\prime\prime}))$

sending $(\mathcal{H}^{\prime},\mathcal{H}^{\prime\prime})$ to
$\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}$ induces a map

$\displaystyle D\ast:$ $\displaystyle D({\bf H}(V^{\prime}))\times D({\bf
H}(V^{\prime\prime}))\longrightarrow D({\bf H}(V^{\prime}\sqcup
V^{\prime\prime}))$

given by

$\displaystyle(D\ast)(\varphi^{\prime},\varphi^{\prime\prime})(\mathcal{H})=\sum_{\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}=\mathcal{H}}\varphi^{\prime}(\mathcal{H}^{\prime})\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime}).$
(3.2)

Here in (3.2), we take $\varphi^{\prime}\in D({\bf H}(V^{\prime}))$,
$\varphi^{\prime\prime}\in D({\bf H}(V^{\prime\prime}))$,
$\mathcal{H}^{\prime}\in{\bf Obj}({\bf H}(V^{\prime}))$,
$\mathcal{H}^{\prime\prime}\in{\bf Obj}({\bf H}(V^{\prime\prime}))$ and
$\mathcal{H}\in{\bf Obj}({\bf H}(V^{\prime}\sqcup V^{\prime\prime}))$.

######  Lemma 3.1.

  1. (1).

$(D\cup)(D\Delta,D\Delta)=(D\Delta)(D\cup)$ and
$(D\cup)(D\bar{\Delta},D\bar{\Delta})=(D\bar{\Delta})(D\cup)$,

  2. (2).

$(D\cap)(D\delta,D\delta)=(D\delta)(D\cap)$ and
$(D\cap)(D\bar{\delta},D\bar{\delta})=(D\bar{\delta})(D\cap)$;

  3. (3).

$(D\ast)(D\Delta,D\Delta)=(D\Delta)(D\ast)$,
$(D\ast)(D\delta,D\delta)=(D\delta)(D\ast)$,
$(D\ast)(D\bar{\Delta},D\bar{\Delta})=(D\bar{\Delta})(D\ast)$ and
$(D\ast)(D\bar{\delta},D\bar{\delta})=(D\bar{\delta})(D\ast)$.

###### Proof.

For any $\varphi^{\prime},\varphi^{\prime\prime}\in D({\bf H}(V))$ and any
$\mathcal{K}\in{\bf K}(V)$, with the help of Subsection 2.1 (ii),

$\displaystyle(D\cup)(D\Delta,D\Delta)(\varphi^{\prime},\varphi^{\prime\prime})(\mathcal{K})$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{K}^{\prime}\cup\mathcal{K}^{\prime\prime}=\mathcal{K}}\Big{(}(D\Delta)(\varphi^{\prime})(\mathcal{K}^{\prime})\Big{)}\Big{(}(D\Delta)(\varphi^{\prime\prime})(\mathcal{K}^{\prime\prime})\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{K}^{\prime}\cup\mathcal{K}^{\prime\prime}=\mathcal{K}}\Big{(}\sum_{\Delta\mathcal{H}^{\prime}=\mathcal{K}^{\prime}}\varphi^{\prime}(\mathcal{H}^{\prime})\Big{)}\Big{(}\sum_{\Delta\mathcal{H}^{\prime\prime}=\mathcal{K}^{\prime\prime}}\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime})\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{\Delta\mathcal{H}^{\prime}\cup\Delta\mathcal{H}^{\prime\prime}=\mathcal{K}}\varphi^{\prime}(\mathcal{H}^{\prime})\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime})$
$\displaystyle=$
$\displaystyle\sum_{\Delta(\mathcal{H}^{\prime}\cup\mathcal{H}^{\prime\prime})=\mathcal{K}}\varphi^{\prime}(\mathcal{H}^{\prime})\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime})$
$\displaystyle=$
$\displaystyle\sum_{\Delta\mathcal{H}=\mathcal{K}}\sum_{\mathcal{H}^{\prime}\cup\mathcal{H}^{\prime\prime}=\mathcal{H}}\varphi^{\prime}(\mathcal{H}^{\prime})\varphi^{\prime\prime}(\mathcal{H}^{\prime\prime})$
$\displaystyle=$
$\displaystyle(D\Delta)(D\cup)(\varphi^{\prime},\varphi^{\prime\prime})(\mathcal{K}).$

We obtain $(D\cup)(D\Delta,D\Delta)=(D\Delta)(D\cup)$. Other identities can be
proved analogously. The second identity in (1) is proved with the help of
Subsection 2.1 (iv). The two identities in (2) are respectively proved with
the help of Subsection 2.1 (iii) and (v). The identities in (3) are proved
with the help of Subsection 2.1 (i)’ - (iv)’. ∎

###  3.2 Random hypergraphs and random simplicial complexes of Erdös-Rényi
type

Let $p:\Delta[V]\longrightarrow[0,1]$. Let $\mathcal{K}$ be a simplicial
complex on $V$. An external face of $\mathcal{K}$ is a hyperedge
$\sigma\in\Delta[V]$ such that $\sigma\notin\mathcal{K}$ and
$\tau\in\mathcal{K}$ for any nonempty proper subset $\tau\subsetneq\sigma$.
Let $E(\mathcal{K})$ be the collection of all the external faces of
$\mathcal{K}$. Let $\mathcal{L}$ be an independence hypergraph on $V$. A co-
external face of $\mathcal{L}$ is a hyperedge $\sigma\in\Delta[V]$ such that
$\sigma\notin\mathcal{L}$ and $\tau\in\mathcal{L}$ for any proper superset
$\tau\supsetneq\sigma$, where $\tau\in\Delta[V]$. Let $\bar{E}(\mathcal{L})$
be the collection of all the co-external faces of $\mathcal{L}$.

(1). Consider the Erdös-Rényi-type model ${\rm\bar{P}}_{p}$ of random
hypergraphs given by

$\displaystyle\bar{\rm
P}_{p}(\mathcal{H})=\prod_{\sigma\in\mathcal{H}}p(\sigma)\prod_{\sigma\notin\mathcal{H}}\big{(}1-p(\sigma)\big{)}$

for any $\mathcal{H}\in{\bf Obj}({\bf H}(V))$. We choose each element
$\sigma\in\Delta[V]$ to be a hyperedge of $\mathcal{H}$ independently at
random with probability $p(\sigma)$. This randomly generated hypergraph
$\mathcal{H}$ satisfies the probability distribution $\bar{\rm P}_{p}$,
written $\mathcal{H}\sim\bar{\rm P}_{p}$.

(2). Consider the Erdös-Rényi-type model ${\rm P}_{p}$ of random simplicial
complexes given by

$\displaystyle{\rm
P}_{p}(\mathcal{K})=\prod_{\sigma\in\mathcal{K}}p(\sigma)\prod_{\sigma\in
E(\mathcal{K})}\big{(}1-p(\sigma)\big{)},$

where $\mathcal{K}\in{\rm Obj}({\bf K}(V))$. We generate the $0$-skeleton of
$\mathcal{K}$ by choosing each $0$-hyperedge $\\{v\\}\in\Delta[V]$
independently at random with probability $p(\\{v\\})$. For any nonnegative
integer $k$, once the $k$-skeleton of $\mathcal{K}$ is randomly generated, we
generate the $(k+1)$-skeleton of $\mathcal{K}$ by choosing each external
$(k+1)$-face $\sigma$ of the $k$-skeleton of $\mathcal{K}$ independently at
random with probability $p(\sigma)$. By an induction on $k$, the final
randomly generated simplicial complex $\mathcal{K}$ satisfies the probability
distribution ${\rm P}_{p}$, written $\mathcal{K}\sim{\rm P}_{p}$.

(3). Consider the Erdös-Rényi-type model ${\rm Q}_{p}$ of random independence
hypergraphs given by

$\displaystyle{\rm
Q}_{p}(\mathcal{L})=\prod_{\sigma\in\mathcal{L}}p(\sigma)\prod_{\sigma\in\bar{E}(\mathcal{L})}\big{(}1-p(\sigma)\big{)},$

where $\mathcal{L}\in{\rm Obj}({\bf L}(V))$. We choose the $(|V|-1)$-hyperedge
$V\in\Delta[V]$ at random with probability $p(V)$ to be a hyperedge of
$\mathcal{L}$. Once all the $i$-hyperedges, $i=k+1,k+2,\ldots,|V|-1$, of
$\mathcal{L}$ are randomly generated, we generate the $k$-hyperedges of
$\mathcal{L}$ by choosing each co-external face $\sigma$ of the collection of
all the $(k+1)$-hyperedges of $\mathcal{L}$ independently at random with
probability $p(\sigma)$. By an induction on $k$ in reverse, The final randomly
generated independence hypergraph $\mathcal{L}$ satisfies the probability
distribution ${\rm Q}_{p}$, written $\mathcal{L}\sim{\rm Q}_{p}$.

######  Lemma 3.2.

(cf. [38, Lemma 4.4]). $(D\gamma)(\bar{\rm P}_{p})=\bar{\rm P}_{1-p}$.

######  Lemma 3.3.

(cf. [38, Lemma 4.5]). Let
$p^{\prime},p^{\prime\prime}:\Delta[V]\longrightarrow[0,1]$. Write
$(p^{\prime}\cap
p^{\prime\prime})(\sigma)=p^{\prime}(\sigma)p^{\prime\prime}(\sigma)$ and
$(p^{\prime}\cup
p^{\prime\prime})(\sigma)=1-(1-p^{\prime}(\sigma))(1-p^{\prime\prime}(\sigma))$
for any $\sigma\in\Delta[V]$. Then $(D\cap)(\bar{\rm P}_{p^{\prime}},\bar{\rm
P}_{p^{\prime\prime}})=\bar{\rm P}_{p^{\prime}\cap p^{\prime\prime}}$ and
$(D\cup)(\bar{\rm P}_{p^{\prime}},\bar{\rm P}_{p^{\prime\prime}})=\bar{\rm
P}_{p^{\prime}\cup p^{\prime\prime}}$. In other words, if
$\mathcal{H}^{\prime}\sim\bar{\rm P}_{p^{\prime}}$ and
$\mathcal{H}^{\prime\prime}\sim\bar{\rm P}_{p^{\prime\prime}}$, then
$\mathcal{H}^{\prime}\cap\mathcal{H}^{\prime\prime}\sim\bar{\rm
P}_{p^{\prime}\cap p^{\prime\prime}}$ and
$\mathcal{H}^{\prime}\cup\mathcal{H}^{\prime\prime}\sim\bar{\rm
P}_{p^{\prime}\cup p^{\prime\prime}}$.

######  Lemma 3.4.

Let $V^{\prime}$ and $V^{\prime\prime}$ be two disjoint vertex sets. Let
$p^{\prime}:\Delta[V^{\prime}]\longrightarrow[0,1]$ and
$p^{\prime\prime}:\Delta[V^{\prime\prime}]\longrightarrow[0,1]$. Write
$(p^{\prime}*p^{\prime\prime})(\sigma)=p^{\prime}(\sigma\cap
V^{\prime})p^{\prime\prime}(\sigma\cap V^{\prime\prime})$ for any
$\sigma\in\Delta[V^{\prime}\sqcup V^{\prime\prime}]$. Then $(D\ast)(\bar{\rm
P}_{p^{\prime}},\bar{\rm P}_{p^{\prime\prime}})=\bar{\rm
P}_{p^{\prime}*p^{\prime\prime}}$. In other words, if
$\mathcal{H}^{\prime}\sim\bar{\rm P}_{p^{\prime}}$ and
$\mathcal{H}^{\prime\prime}\sim\bar{\rm P}_{p^{\prime\prime}}$, then
$\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}\sim\bar{\rm
P}_{p^{\prime}*p^{\prime\prime}}$.

###### Proof.

Consider the following two independent trials: (1). generate
$\mathcal{H}^{\prime}$, (2). generate $\mathcal{H}^{\prime\prime}$. Then for
any $\sigma\in\Delta[V^{\prime}\sqcup V^{\prime\prime}]$, both of the
followings hold:

  1. (1).

$\sigma\in\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}$ iff both
$(\sigma\cap V^{\prime})\in\mathcal{H}^{\prime}$ in trial (1) and $(\sigma\cap
V^{\prime\prime})\in\mathcal{H}^{\prime\prime}$ in trial (2). Thus the event
$\sigma\in\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}$ is the product
event of the two independent events $(\sigma\cap
V^{\prime})\in\mathcal{H}^{\prime}$ and $(\sigma\cap
V^{\prime\prime})\in\mathcal{H}^{\prime\prime}$. This event has the
probability $p^{\prime}(\sigma\cap V^{\prime})p^{\prime\prime}(\sigma\cap
V^{\prime\prime})$;

  2. (2).

$\sigma\notin\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}$ iff either
$(\sigma\cap V^{\prime})\notin\mathcal{H}^{\prime}$ in trial (1) or
$(\sigma\cap V^{\prime\prime})\notin\mathcal{H}^{\prime\prime}$ in trial (2).
Thus the event $\sigma\notin\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}$
has the probability $1-p^{\prime}(\sigma\cap
V^{\prime})p^{\prime\prime}(\sigma\cap V^{\prime\prime})$.

Therefore, $\mathcal{H}^{\prime}*\mathcal{H}^{\prime\prime}$ is the randomly
generated hypergraph on $V^{\prime}\sqcup V^{\prime\prime}$ satisfying the
probability distribution $\bar{\rm P}_{p^{\prime}*p^{\prime\prime}}$. ∎

######  Theorem 3.5.

For any $\mathcal{H}\in{\rm Obj}({\bf{H}}(V))$, any $\mathcal{K}\in{\rm
Obj}({\bf{K}}(V))$ and any $\mathcal{L}\in{\rm Obj}({\bf{L}}(V))$,

  1. (1).

$\big{(}(D\Delta)(\bar{\rm P}_{p})\big{)}(\mathcal{K})={\rm
Q}_{1-p}(\gamma\mathcal{K})$,

  2. (2).

$\big{(}(D\bar{\Delta})(\bar{\rm P}_{p})\big{)}(\mathcal{L})={\rm
P}_{1-p}(\gamma\mathcal{L})$,

  3. (3).

$\big{(}(D\delta)(\bar{\rm P}_{p})\big{)}(\mathcal{K})={\rm
P}_{p}(\mathcal{K})$,

  4. (4).

$\big{(}(D\bar{\delta})(\bar{\rm P}_{p})\big{)}(\mathcal{L})={\rm
Q}_{p}(\mathcal{L})$.

We will prove Theorem 3.5 in the next subsection. The following corollary is a
restatement of Theorem 3.5 (1) and (2).

######  Corollary 3.6.

For any $\mathcal{H}\in{\rm Obj}({\bf{H}}(V))$, any $\mathcal{K}\in{\rm
Obj}({\bf{K}}(V))$ and any $\mathcal{L}\in{\rm Obj}({\bf{L}}(V))$,

  1. (1).

$\big{(}(D\gamma)(D\Delta)(\bar{\rm P}_{p})\big{)}(\mathcal{L})={\rm
Q}_{1-p}(\mathcal{L})$,

  2. (2).

$\big{(}(D\gamma)(D\bar{\Delta})(\bar{\rm P}_{p})\big{)}(\mathcal{K})={\rm
P}_{1-p}(\mathcal{K})$.

The following two corollaries follow from Theorem 3.5.

######  Corollary 3.7.

Let $p^{\prime},p^{\prime\prime}:\Delta[V]\longrightarrow[0,1]$. Then

  1. (1).

$(D\cap)({\rm P}_{p^{\prime}},{\rm P}_{p^{\prime\prime}})={\rm
P}_{p^{\prime}\cap p^{\prime\prime}}$,

  2. (2).

$(D\cap)({\rm Q}_{p^{\prime}},{\rm Q}_{p^{\prime\prime}})={\rm
Q}_{p^{\prime}\cap p^{\prime\prime}}$.

###### Proof.

By Lemma 3.1, Lemma 3.3 and Theorem 3.5,

$\displaystyle(D\cap)({\rm P}_{p^{\prime}},{\rm P}_{p^{\prime\prime}})$
$\displaystyle=$ $\displaystyle(D\cap)(D\delta,D\delta)(\bar{\rm
P}_{p^{\prime}},\bar{\rm P}_{p^{\prime\prime}})$ $\displaystyle=$
$\displaystyle(D\delta)(D\cap)(\bar{\rm P}_{p^{\prime}},\bar{\rm
P}_{p^{\prime\prime}})$ $\displaystyle=$ $\displaystyle(D\delta)(\bar{\rm
P}_{p^{\prime}\cap p^{\prime\prime}})$ $\displaystyle=$ $\displaystyle{\rm
P}_{p^{\prime}\cap p^{\prime\prime}}$

and

$\displaystyle(D\cap)({\rm Q}_{p^{\prime}},{\rm Q}_{p^{\prime\prime}})$
$\displaystyle=$ $\displaystyle(D\cap)(D\bar{\delta},D\bar{\delta})(\bar{\rm
P}_{p^{\prime}},\bar{\rm P}_{p^{\prime\prime}})$ $\displaystyle=$
$\displaystyle(D\bar{\delta})(D\cap)(\bar{\rm P}_{p^{\prime}},\bar{\rm
P}_{p^{\prime\prime}})$ $\displaystyle=$
$\displaystyle(D\bar{\delta})(\bar{\rm P}_{p^{\prime}\cap p^{\prime\prime}})$
$\displaystyle=$ $\displaystyle{\rm Q}_{p^{\prime}\cap p^{\prime\prime}}.$

∎

######  Corollary 3.8.

Let $V^{\prime}$ and $V^{\prime\prime}$ be two disjoint vertex sets. Let
$p^{\prime}:\Delta[V^{\prime}]\longrightarrow[0,1]$ and
$p^{\prime\prime}:\Delta[V^{\prime\prime}]\longrightarrow[0,1]$. Then

  1. (1).

$(D\ast)({\rm P}_{p^{\prime}},{\rm P}_{p^{\prime\prime}})={\rm
P}_{p^{\prime}*p^{\prime\prime}}$,

  2. (2).

$(D\ast)({\rm Q}_{p^{\prime}},{\rm Q}_{p^{\prime\prime}})={\rm
Q}_{p^{\prime}*p^{\prime\prime}}$.

###### Proof.

By Lemma 3.1, Lemma 3.4 and Theorem 3.5,

$\displaystyle(D*)({\rm P}_{p^{\prime}},{\rm P}_{p^{\prime\prime}})$
$\displaystyle=$ $\displaystyle(D*)(D\delta,D\delta)(\bar{\rm
P}_{p^{\prime}},\bar{\rm P}_{p^{\prime\prime}})$ $\displaystyle=$
$\displaystyle(D\delta)(D*)(\bar{\rm P}_{p^{\prime}},\bar{\rm
P}_{p^{\prime\prime}})$ $\displaystyle=$ $\displaystyle(D\delta)(\bar{\rm
P}_{p^{\prime}*p^{\prime\prime}})$ $\displaystyle=$ $\displaystyle{\rm
P}_{p^{\prime}*p^{\prime\prime}}$

and

$\displaystyle(D*)({\rm Q}_{p^{\prime}},{\rm Q}_{p^{\prime\prime}})$
$\displaystyle=$ $\displaystyle(D*)(D\bar{\delta},D\bar{\delta})(\bar{\rm
P}_{p^{\prime}},\bar{\rm P}_{p^{\prime\prime}})$ $\displaystyle=$
$\displaystyle(D\bar{\delta})(D*)(\bar{\rm P}_{p^{\prime}},\bar{\rm
P}_{p^{\prime\prime}})$ $\displaystyle=$
$\displaystyle(D\bar{\delta})(\bar{\rm P}_{p^{\prime}*p^{\prime\prime}})$
$\displaystyle=$ $\displaystyle{\rm Q}_{p^{\prime}*p^{\prime\prime}}.$

∎

###  3.3 Proof of Theorem 3.5

Consider a map $p:\Delta[V]\longrightarrow[0,1]$. For any hypergraph
$\mathcal{H}$ on $V$ and any hyperedge $\sigma\in\mathcal{H}$, we call
$\sigma$ a maximal hyperedge of $\mathcal{H}$ if there does not exist any
$\tau\in\mathcal{H}$ such that $\sigma\subsetneq\tau$ and call $\sigma$ a
minimal hyperedge of $\mathcal{H}$ if there does not exist any
$\tau\in\mathcal{H}$ such that $\sigma\supsetneq\tau$. Let $\max(\mathcal{H})$
be the collection of all the maximal hyperedges of $\mathcal{H}$ and let
$\min(\mathcal{H})$ be the collection of all the minimal hyperedges of
$\mathcal{H}$.

######  Lemma 3.9.

(cf. [38, Theorem 1.5 (2)]). For any simplicial complex $\mathcal{K}$ on $V$,

$\displaystyle\big{(}(D\Delta)(\bar{\rm
P}_{p})\big{)}(\mathcal{K})=\prod_{\tau\in\max(\mathcal{K})}p(\tau)\prod_{\tau\in\Delta[V]\atop\tau\notin\mathcal{K}}\big{(}1-p(\tau)\big{)}.$
(3.3)

The proof of Lemma 3.9 is in [38, Lemma 4.2].

######  Lemma 3.10.

For any independence hypergraph $\mathcal{L}$ on $V$,

$\displaystyle\big{(}(D\bar{\Delta})(\bar{\rm
P}_{p})\big{)}(\mathcal{L})=\prod_{\tau\in\min(\mathcal{L})}p(\tau)\prod_{\tau\in\Delta[V]\atop\tau\notin\mathcal{L}}\big{(}1-p(\tau)\big{)}.$
(3.4)

###### Proof.

Let $\mathcal{L}$ be an independence hypergraph on $V$. Let
$S=\\{\sigma_{1},\sigma_{2},\ldots,\sigma_{s}\\}$ be any set (allowed to be
the emptyset) of distinct hyperedges in $\mathcal{L}$ such that for each
$\sigma_{i}$, $i=1,2,\ldots,s$, there exists $\tau\in\min(\mathcal{L})$ such
that $\sigma_{i}\supsetneq\tau$. Suppose $S$ runs over all such sets of
hyperedges in $\mathcal{L}$. Then $\mathcal{H}=\min(\mathcal{L})\sqcup S$ runs
over all the hypergraphs on $V$ such that
$\bar{\Delta}\mathcal{H}=\mathcal{L}$. Consequently,

$\displaystyle\big{(}(D\bar{\Delta})(\bar{\rm P}_{p})\big{)}(\mathcal{L})$
$\displaystyle=$
$\displaystyle\sum_{\bar{\Delta}\mathcal{H}=\mathcal{L}}~{}\bar{\rm
P}_{p}(\mathcal{H})$ (3.5) $\displaystyle=$
$\displaystyle\sum_{\bar{\Delta}\mathcal{H}=\mathcal{L}}~{}\prod_{\sigma\in\mathcal{H}}p(\sigma)\prod_{\sigma\notin\mathcal{H}}\big{(}1-p(\sigma)\big{)}$
$\displaystyle=$ $\displaystyle\sum_{\mathcal{H}=\min(\mathcal{L})\sqcup
S}~{}\prod_{\sigma\in\mathcal{H}}p(\sigma)\prod_{\sigma\in\Delta[V]\atop\sigma\notin\mathcal{H}}\big{(}1-p(\sigma)\big{)}$
$\displaystyle=$
$\displaystyle\prod_{\sigma\in\min(\mathcal{L})}p(\sigma)\prod_{\sigma\in\Delta[V]\atop\sigma\notin\mathcal{L}}\big{(}1-p(\sigma)\big{)}$
$\displaystyle\Big{(}\sum_{S\subseteq\mathcal{L}\setminus\min(\mathcal{L})}~{}\prod_{\sigma\in
S}p(\sigma)\prod_{\sigma\in\mathcal{L}\setminus\min(\mathcal{L})\atop\sigma\notin
S}\big{(}1-p(\sigma)\big{)}\Big{)}$ $\displaystyle=$
$\displaystyle\prod_{\sigma\in\min(\mathcal{L})}p(\sigma)\prod_{\sigma\in\Delta[V]\atop\sigma\notin\mathcal{L}}\big{(}1-p(\sigma)\big{)}$
$\displaystyle\prod_{\sigma\in\mathcal{L}\setminus\min(\mathcal{L})}\Big{(}p(\sigma)+\big{(}1-p(\sigma)\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\prod_{\tau\in\min(\mathcal{L})}p(\tau)\prod_{\tau\in\Delta[V]\atop\tau\notin\mathcal{L}}\big{(}1-p(\tau)\big{)}.$

We obtain (3.4). ∎

######  Lemma 3.11.

For any simplicial complex $\mathcal{K}$ on $V$,

$\displaystyle\big{(}(D\delta)(\bar{\rm
P}_{p})\big{)}(\mathcal{K})=\prod_{\tau\in\min(\gamma\mathcal{K})}\big{(}1-p(\tau)\big{)}\prod_{\tau\in\mathcal{K}}p(\tau).$

###### Proof.

By a direct calculation,

$\displaystyle\big{(}(D\delta)(\bar{\rm P}_{p})\big{)}(\mathcal{K})$
$\displaystyle=$
$\displaystyle\big{(}(D\gamma)\circ(D\bar{\Delta})\circ(D\gamma)(\bar{\rm
P}_{p})\big{)}(\mathcal{K})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\mathcal{L}=\mathcal{K}}(D\bar{\Delta}\circ
D\gamma)(\bar{\rm P}_{p})(\mathcal{L})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\mathcal{L}=\mathcal{K}}~{}\sum_{\bar{\Delta}\mathcal{H}=\mathcal{L}}(D\gamma)(\bar{\rm
P}_{p})(\mathcal{H})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\mathcal{L}=\mathcal{K}}~{}\sum_{\bar{\Delta}\mathcal{H}=\mathcal{L}}~{}\sum_{\gamma\mathcal{H}^{\prime}=\mathcal{H}}\bar{\rm
P}_{p}(\mathcal{H}^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{\bar{\Delta}\mathcal{H}=\gamma\mathcal{K}}\bar{\rm
P}_{p}(\gamma\mathcal{H})$ $\displaystyle=$
$\displaystyle\prod_{\tau\in\min(\gamma\mathcal{K})}\big{(}1-p(\tau)\big{)}\prod_{\tau\in\mathcal{K}}p(\tau).$

The last equality follows by a analogous calculation of (3.5). ∎

######  Lemma 3.12.

For any independence hypergraph $\mathcal{L}$ on $V$,

$\displaystyle\big{(}(D\bar{\delta})(\bar{\rm
P}_{p})\big{)}(\mathcal{L})=\prod_{\tau\in\max(\gamma\mathcal{L})}\big{(}1-p(\tau)\big{)}\prod_{\tau\in\mathcal{L}}p(\tau).$

###### Proof.

By a direct calculation,

$\displaystyle\big{(}(D\bar{\delta})(\bar{\rm P}_{p})\big{)}(\mathcal{L})$
$\displaystyle=$
$\displaystyle\big{(}(D\gamma)\circ(D\Delta)\circ(D\gamma)(\bar{\rm
P}_{p})\big{)}(\mathcal{L})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\mathcal{K}=\mathcal{L}}(D\Delta\circ
D\gamma)(\bar{\rm P}_{p})(\mathcal{K})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\mathcal{K}=\mathcal{L}}~{}\sum_{\Delta\mathcal{H}=\mathcal{K}}(D\gamma)(\bar{\rm
P}_{p})(\mathcal{H})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\mathcal{K}=\mathcal{L}}~{}\sum_{\Delta\mathcal{H}=\mathcal{K}}~{}\sum_{\gamma\mathcal{H}^{\prime}=\mathcal{H}}\bar{\rm
P}_{p}(\mathcal{H}^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{\Delta\mathcal{H}=\gamma\mathcal{L}}\bar{\rm
P}_{p}(\gamma\mathcal{H})$ $\displaystyle=$
$\displaystyle\prod_{\tau\in\max(\gamma\mathcal{L})}\big{(}1-p(\tau)\big{)}\prod_{\tau\in\mathcal{L}}p(\tau).$

The last equality follows by a analogous calculation of [38, Lemma 4.2]. ∎

###### Proof of Theorem 3.5.

For any simplicial complex $\mathcal{K}$ on $V$,

$\displaystyle E(\mathcal{K})=\min(\gamma\mathcal{K}).$

For any independence hypergraph $\mathcal{L}$ on $V$,

$\displaystyle\bar{E}(\mathcal{L})=\max(\gamma\mathcal{L}).$

Therefore, Theorem 3.5 follows from Lemma 3.9 \- Lemma 3.12. ∎

##  4 Generations of random hypergraphs and random simplicial complexes by the
map algebra

###  4.1 A graded construction of the map algebra

Let ${\bf H}$ be the category such that each object is a pair
$(V,\mathcal{H})$, where $V$ is a finite set and $\mathcal{H}$ is a hypergraph
on $V$, and each morphism from $(V,\mathcal{H})$ to
$(V^{\prime},\mathcal{H}^{\prime})$ is a morphism $f:V\longrightarrow
V^{\prime}$ of hypergraphs from $\mathcal{H}$ to $\mathcal{H}^{\prime}$. Let
$G^{1}$ be the semi-group generated by $\gamma$, $\delta$ and $\Delta$, where
the unit element is the identity map and the multiplication is the composition
of maps. Each element of $G^{1}$ is a map from ${\bf Obj}({\bf H})$ to itself.

######  Proposition 4.1.

Each of the triples $\\{\gamma,\delta,\Delta\\}$,
$\\{\gamma,\bar{\delta},\bar{\Delta}\\}$, $\\{\gamma,\Delta,\bar{\Delta}\\}$
and $\\{\gamma,\delta,\bar{\delta}\\}$ is a set of multiplicative generators
of $G^{1}$.

###### Proof.

By Lemma 2.1, each of the triples $\\{\gamma,\Delta,\delta\\}$,
$\\{\gamma,\bar{\Delta},\bar{\delta}\\}$, $\\{\gamma,\Delta,\bar{\Delta}\\}$
and $\\{\gamma,\delta,\bar{\delta}\\}$ could multiplicatively generate
$\gamma$, $\Delta$, $\delta$, $\bar{\Delta}$ and $\bar{\delta}$. Therefore,
each of the triple could multiplicatively generate $G^{1}$. ∎

For any positive integer $k$, Let $G^{k}$ be the collection of all the words

$\displaystyle(\cdots(w_{1}\bullet w_{2})\bullet\cdots\bullet w_{k}),$

where $w_{1},w_{2},\ldots,w_{k}\in G$ and $\bullet=\cap,\cup,*$ or $\square$,
with the binary operation $\bullet$ for $k-1$ times and $k-2$ brackets giving
the order of operations. For any $w\in G^{k}$, we call an element
$(\cdots,(V_{1},\mathcal{H}_{1}),(V_{2},\mathcal{H}_{2}),\cdots)$ in ${\bf
Obj}({\bf H})^{\times k}$ $w$-admissible if it is sent to an element in ${\bf
Obj}({\bf H})$ by $w$, i.e. all the binary operations
$w_{1}(\mathcal{H}_{1})\bullet w_{2}(\mathcal{H}_{2})$ in $w$ are well-
defined. For example, any element in ${\bf Obj}({\bf H})^{\times 2}$ is
$\square$-admissible, a $*$-admissible element in ${\bf Obj}({\bf H})^{\times
2}$ is of the form $((V_{1},\mathcal{H}_{1}),(V_{2},\mathcal{H}_{2}))$ where
$V_{1}\cap V_{2}=\emptyset$, and a $\cap$-admissible element as well as a
$\cup$-admissible element in ${\bf Obj}({\bf H})^{\times 2}$ is of the form
$((V,\mathcal{H}_{1}),(V,\mathcal{H}_{2}))$. Note that $w$ is a map from the
set of all $w$-admissible elements, which is a subset of ${\bf Obj}({\bf
H})^{\times k}$, to ${\bf Obj}({\bf H})$. We define the map algebra to be the
union

$\displaystyle G=\bigcup_{k\geq 1}G^{k}.$ (4.1)

Let $T$ be a binary tree. Let $x_{0}$ be the root of $T$. We label $x_{0}$
with $\cap$, $\cup$, $*$ or $\square$ if $\deg x_{0}=2$ and label $x_{0}$ with
$\Delta$, $\delta$ or $\gamma$ if $\deg x_{0}=0,1$. For any vertex $x$ of $T$
with $x\neq x_{0}$, we label $x$ with $\cap$, $\cup$, $*$ or $\square$ if
$\deg x=3$ and label $x$ with $\Delta$, $\delta$ or $\gamma$ if $\deg x=1,2$.
Let $V_{T}$ be the vertex set of $T$. Consider a map

$\displaystyle\alpha:$ $\displaystyle
V_{T}\longrightarrow\\{\cap,\cup,*,\square,\Delta,\delta,\gamma\\}$

sending each vertex of $T$ to its label satisfying the above labeling rule.
Let $k(T)=1$ if $T$ is the empty binary tree or $T$ has a single vertex
$x_{0}$ and let $k(T)$ be the number of vertices $x\neq x_{0}$ such that $\deg
x=1$ if $T$ has at least two vertices. Then each pair $(T,\alpha)$ represents
an element in $G^{k(T)}$. For each nonnegative integer $k$, each element in
$G^{k}$ has such a representation $(T,\alpha)$, while two pairs $(T,\alpha)$
and $(T^{\prime},\alpha^{\prime})$ may represent one element in $G^{k}$. The
pair $(\emptyset,\alpha)$, where $\emptyset$ is the empty binary tree,
represents ${\rm id}\in G^{1}$. A pair $(T,\alpha)$, where $T$ is a binary
tree such that $\deg x_{0}=1$ and $\deg x=1,2$ for any $x\neq x_{0}$,
represents an element in $G^{1}$.

Let ${\bf DH}$ be the category such that each object is a triple
$(V,\mathcal{H},\varphi)$, where $V$ is a finite set, $\mathcal{H}$ is a
hypergraph on $V$ and $\varphi\in D({\bf H}(V))$ is a probability distribution
on ${\bf Obj}({\bf H}(V))$, and each morphism from $(V,\mathcal{H},\varphi)$
to $(V^{\prime},\mathcal{H}^{\prime},\varphi^{\prime})$ is a morphism
$f:V\longrightarrow V^{\prime}$ of hypergraphs from $\mathcal{H}$ to
$\mathcal{H}^{\prime}$ satisfying $(Df)(\varphi)=\varphi^{\prime}$. Let $S$ be
the subset of ${\bf Obj}({\bf DH})$ consisting of all the elements
$(V,\mathcal{H},\bar{\rm P}_{p})\in{\bf Obj}({\bf DH})$, where
$p:\Delta[V]\longrightarrow[0,1]$. For any $w_{1},w_{2}\in G^{1}$, define
$D(w_{1}w_{2})=(Dw_{1})(Dw_{2})$. For each $w\in G^{1}$, define

$\displaystyle w(V,\mathcal{H},\varphi)=(w(V,\mathcal{H}),(Dw)(\varphi)).$

Then $w$ is a map from ${\bf Obj}({\bf DH})$ to itself. Let $G^{1}(S)$ be the
union of $w(S)$ for all $w\in G_{1}$, where $w(S)$ is the image of $S$ under
$w$.

######  Proposition 4.2.

For any $p:\Delta[V]\longrightarrow[0,1]$, ${\rm P}_{p}$ and and ${\rm Q}_{p}$
are in $G^{1}(S)$.

###### Proof.

By Theorem 1.1 (4) or Theorem 3.5 (3), ${\rm P}_{p}\in G^{1}(S)$. By Theorem
1.1 (5) or Theorem 3.5 (4), ${\rm Q}_{p}\in G^{1}(S)$. ∎

Define $D\square$ to be the map from ${\bf Obj}({\bf DH})\times{\bf Obj}({\bf
DH})$ to ${\bf Obj}({\bf DH})$ by

$\displaystyle(D\square)((V,\mathcal{H},\varphi),(V^{\prime},\mathcal{H}^{\prime},\varphi^{\prime}))=(V\times
V^{\prime},\mathcal{H}\square\mathcal{H}^{\prime},(D\square)(\varphi,\varphi^{\prime})).$

Define $D*$ to be the map from the collection of $*$-admissible elements in
${\bf Obj}({\bf DH})\times{\bf Obj}({\bf DH})$ to ${\bf Obj}({\bf DH})$ by

$\displaystyle(D*)((V,\mathcal{H},\varphi),(V^{\prime},\mathcal{H}^{\prime},\varphi^{\prime}))=(V\sqcup
V^{\prime},\mathcal{H}*\mathcal{H}^{\prime},(D*)(\varphi,\varphi^{\prime})),$

where $V\cap V^{\prime}=\emptyset$. Define $D\cap$ and $D\cup$ to be maps from
the collection of $\cap$-admissible (equivalently, $\cup$-admissible) elements
in ${\bf Obj}({\bf DH})\times{\bf Obj}({\bf DH})$ to ${\bf Obj}({\bf DH})$ by

$\displaystyle(D\cap)((V,\mathcal{H},\varphi),(V,\mathcal{H}^{\prime},\varphi^{\prime}))$
$\displaystyle=$
$\displaystyle(V,\mathcal{H}\cap\mathcal{H}^{\prime},(D\cap)(\varphi,\varphi^{\prime})),$
$\displaystyle(D\cup)((V,\mathcal{H},\varphi),(V,\mathcal{H}^{\prime},\varphi^{\prime}))$
$\displaystyle=$
$\displaystyle(V,\mathcal{H}\cup\mathcal{H}^{\prime},(D\cup)(\varphi,\varphi^{\prime})).$

Define $D(w_{1}\bullet w_{2})=(D\bullet)(Dw_{1},Dw_{2})$ for
$\bullet=\cap,\cup,*$ or $\square$. Then each element in $G^{k}$ is a map from
an admissible subset of ${\bf Obj}({\bf DH})^{\times k}$ to ${\bf Obj}({\bf
DH})$. Let

$\displaystyle G^{k}(S^{\times k})$ $\displaystyle=$
$\displaystyle\\{w((V_{1},\mathcal{H}_{1},\bar{\rm
P}_{*}),\cdots,(V_{k},\mathcal{H}_{k},\bar{\rm P}_{*}))\mid w\in
G^{k}{\rm~{}and~{}}$ $\displaystyle((V_{1},\mathcal{H}_{1},\bar{\rm
P}_{*}),\cdots,(V_{k},\mathcal{H}_{k},\bar{\rm P}_{*}))\in S^{\times
k}~{}{\rm~{}is~{}}w{\rm\text{-}admissible}\\}.$

Take the union

$\displaystyle G(S)=\bigcup_{k\geq 1}G^{k}(S^{\times k}).$ (4.2)

Each element in $G(S)$ is of the form $(V,\mathcal{H},\varphi)$, where $V$ is
a finite vertex set, $\mathcal{H}$ is a hypergraph on $V$ and $\varphi$ is a
probability function on ${\bf Obj}({\bf H}(V))$. The probability function
$\varphi$ is given by the action of certain $Dw$, where $w\in G$, on multiple
probability functions of the form $\bar{\rm P}_{*}$, ${\rm P}_{*}$ and ${\rm
Q}_{*}$.

###  4.2 Algorithms for the generations of random hypergraphs and random
simplicial complexes

###### Algorithm generating random hypergraphs.

Step 1. Choose a positive integer $k$. Choose $k$ finite sets $V_{1}$,
$V_{2}$, $\ldots$, $V_{k}$ as the vertex sets without mutual intersections.

Step 2. For each $i=1,2,\ldots,k$, choose a positive integer $n_{i}$. Use the
Erdös-Rényi-type model ${\rm\bar{P}}_{p}$ to give $n_{i}$ randomly generated
hypergraphs $\mathcal{H}_{1,i}$, $\mathcal{H}_{2,i}$, $\ldots$,
$\mathcal{H}_{n_{i},i}$ on $V_{i}$.

Step 3. For each $i=1,2,\ldots,k$ and each $j=1,2,\ldots,n_{i}$, choose an
element $w_{j,i}\in G^{1}$. Apply $w_{j,i}$ to $\mathcal{H}_{j,i}$ and give a
randomly generated hypergraph $w_{j,i}(\mathcal{H}_{j,i})$ on $V_{i}$.

Step 4. For each $i=1,2,\ldots,k$, (4.a). Choose two randomly generated
hypergraphs from $w_{j,i}(\mathcal{H}_{j,i})$, $j=1$, $2$, $\ldots$, $n_{i}$,
and apply $\cap$ or $\cup$ to these two randomly generated hypergraphs. (4.b).
Choose a randomly generated hypergraph from the remaining $n_{i}-2$ randomly
generated hypergraphs $w_{j,i}(\mathcal{H}_{j,i})$, $j=1$, $2$, $\ldots$,
$n_{i}$, and apply $\cap$ or $\cup$ to this chosen randomly generated
hypergraph and the randomly generated hypergraph given in (4.a). (4.c). Repeat
(4.b) for $n_{i}-2$ times. Denote this final randomly generated hypergraph on
$V_{i}$ as $\mathcal{H}_{i}$.

Step 5. (5.a). Choose two randomly generated hypergraphs from
$\mathcal{H}_{i}$, $i=1$, $2$, $\ldots$, $k$, and apply $\square$ or $*$ to
these two randomly generated hypergraphs. (5.b). Choose a randomly generated
hypergraph from the remaining $k-2$ randomly generated hypergraphs
$\mathcal{H}_{i}$, $i=1$, $2$, $\ldots$, $k$, and apply $\square$ or $*$ to
this chosen randomly generated hypergraph and the randomly generated
hypergraph given in (5.a). (5.c). Repeat (5.b) for $k-2$ times. This gives a
randomly generated hypergraph $\mathcal{H}$.

Step 6. Choose $w\in G^{1}$. Apply $w$ to $\mathcal{H}$. The final randomly
generated hypergraph is $w(\mathcal{H})$. ∎

Some algorithms that generate random simplicial complexes and random
independence hypergraphs follow immediately.

###### Algorithm generating random simplicial complexes.

Step 1. Same as Step 1 in the algorithm generating random hypergraphs.

Step 2. For each $i=1,2,\ldots,k$, choose a positive integer $n_{i}$. Use the
algorithm generating random hypergraphs to give $n_{i}$ randomly generated
hypergraphs $\mathcal{H}_{1,i}$, $\mathcal{H}_{2,i}$, $\ldots$,
$\mathcal{H}_{n_{i},i}$ on $V_{i}$.

Step 3. For each $i=1,2,\ldots,k$ and each $j=1,2,\ldots,n_{i}$, apply
$\Delta$ or $\delta$ to $\mathcal{H}_{j,i}$. This gives a randomly generated
simplicial complex $\mathcal{K}_{j,i}$ on $V_{i}$.

Step 4. For each $i=1,2,\ldots,k$, similar to Step 4 of the previous
algorithm, apply the binary operations $\cap$ and $\cup$ to
$\mathcal{K}_{1,i}$, $\mathcal{K}_{2,i}$, $\ldots$, $\mathcal{K}_{n_{i},i}$.
After $n_{i}-1$ times of the binary operations, we obtain a randomly generated
simplicial complex $\mathcal{K}_{i}$ on $V_{i}$.

Step 5. Similar to Step 5 of the previous algorithm, apply the binary
operations $*$, $\Delta\square$ and $\delta\square$ to $\mathcal{K}_{1}$,
$\mathcal{K}_{2}$, $\ldots$, $\mathcal{K}_{k}$. After $k-1$ times of the
binary operations, we obtain a randomly generated simplicial complex
$\mathcal{K}$. ∎

###### Algorithm generating random independence hypergraphs.

Steps 1-2. Same as Steps 1-2 in the algorithm generating random simplicial
complexes.

Step 3. For each $i=1,2,\ldots,k$ and each $j=1,2,\ldots,n_{i}$, apply
$\bar{\Delta}$ or $\bar{\delta}$ to $\mathcal{H}_{j,i}$ with respect to
$V_{i}$. This gives a randomly generated independence hypergraph
$\mathcal{L}_{j,i}$ on $V_{i}$.

Step 4. For each $i=1,2,\ldots,k$, similar to Step 4 of the previous
algorithms, apply the binary operations $\cap$ and $\cup$ to
$\mathcal{L}_{1,i}$, $\mathcal{L}_{2,i}$, $\ldots$, $\mathcal{L}_{n_{i},i}$.
After $n_{i}-1$ times of the binary operations, we obtain a randomly generated
independence hypergraph $\mathcal{L}_{i}$ on $V_{i}$.

Step 5. Similar to Step 5 of the previous algorithms, apply the binary
operations $*$, $\bar{\Delta}\square$ and $\bar{\delta}\square$ to
$\mathcal{L}_{1}$, $\mathcal{L}_{2}$, $\ldots$, $\mathcal{L}_{k}$. After $k-1$
times of the binary operations, we obtain a randomly generated independence
hypergraph $\mathcal{L}$. ∎

Acknowledgement. The present author would like to express his deep gratitude
to the editor and the referee for their kind helps.

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Shiquan Ren

Address: School of Mathematics and Statistics, Henan University, Kaifeng
475004, China.

e-mail<EMAIL_ADDRESS>