# On the spectrum of the Hodge Laplacian on sequences

Hannah Santa Cruz Baur
Pennsylvania State University
<EMAIL_ADDRESS>
&Vladimir Itskov
Pennsylvania State University
<EMAIL_ADDRESS>

###### Abstract

Hodge Laplacians have been previously proposed as a natural tool for
understanding higher-order interactions in networks and directed graphs. Here
we introduce a Hodge-theoretic approach to spectral theory and dimensionality
reduction for probability distributions on sequences and simplicial complexes.
We demonstrate that this Hodge theory has desirable properties with respect to
natural null-models, where the underlying vertices are independent.

We prove that for the case of independent vertices in simplicial complexes,
the appropriate Laplacians are multiples of the identity and thus have no
meaningful Fourier modes. For the null model of independent vertices in
sequences, we prove that the appropriate Hodge Laplacian has an integer
spectrum, and describe its eigenspaces. We also prove that the underlying cell
complex of sequences has trivial reduced homology. Our results establish a
foundation for developing Fourier analyses of probabilistic models, which are
common in theoretical neuroscience and machine-learning.

_K_ eywords Hodge Laplacian, dimensionality reduction, simplicial complexes,
sequences, neural codes.

##  1 Introduction

Graph Laplacians and diffusion maps [1] are a common tool in network analysis,
dimensionality reduction and deep learning networks [2, 3, 4]. In these
settings a pairwise similarity function on a finite set of vertices is
converted into a weighted graph; then a Laplacian is defined over the space of
vertices so that the corresponding heat equation describes the diffusion
process on the weighted graph [5]. The eigenfunctions of the Laplacian are
understood as an analog of the standard Fourier basis in a Euclidean domain,
and they are often used to embed the vertices of the graph into a low-
dimensional space. The resulting geometric representation accurately reflects
the similarity function, interpreted as a proxy for a distance [2, 3].

Pairwise correlations are often used for characterizing similarity on a vertex
set. However, in many application domains, such as theoretical neuroscience
and machine learning, higher-order correlations play an important role, and
essential features of the underlying data cannot be detected using only
second-order correlations. Hodge Laplacians for simplicial complexes, dating
back to [6], have been proposed as a proper replacement for the graph
Laplacians. Basic properties of the Hodge Laplacians with respect to the
structure of the simplicial complexes were investigated in the pioneering work
[7, 8], and later generalized to the context of cellular sheaves in [9, 10].
Similar to the graph Laplacians, the diffusion equation for the Hodge
Laplacians was reinterpreted in terms of random walks on simplicial complexes
[11, 12, 13]. Recently, there has been a resurgence of interest in Hodge-
theoretic tools for generalizing various deep learning networks [14, 15, 16,
17, 18, 19, 20, 21]111This list of papers is very incomplete..

While Hodge Laplacians are well-suited for a spectral theory on binary vectors
(or subsets), many scientific fields, such as genomics, theoretical
neuroscience and language models study data that come in the form of a
collection of sequences. This kind of data can be thought of as a
generalization of directed graphs, similar to how simplicial complexes and
hypergraphs generalize undirected graphs. The sequence data can be formalized
in terms of a cell complex $\mathcal{X}$, that comprises all possible
sequences, and a probability distribution $p:\mathcal{X}\to[0,1]$ that
describes the statistical properties of the data. Here, the role of
similarity, or correlations is played by the probability; an equivalent
formulation can be made in terms of the higher-order correlations.

Similar to the case of binary vectors, the interpretation and machine-learning
pipelines for sequential data require tools such as dimensionality reduction
and the notion of convolution. This necessitates a Laplacian formalism for
sequences. Importantly, such a Laplacian needs to have particularly simple
properties in the case of null models, where the statistical properties of
sequences can be explained by a simple probabilistic model with no other
underlying structure.

To this end we developed a Hodge Laplacian formalism associated with
probability distributions on sequence complexes. We accompany this
construction with an analogous one for the case of simplicial complexes. We
consider natural null models of independent vertices and prove that for these
models the Hodge Laplacians have particularly simple properties.

The paper is organized as follows. In section 2 we provide relevant background
on abstract cell complexes, then we introduce weight functions associated to
probability distributions and define the null models of independent vertices
(subsections 2.2, 2.3); we end the section with an exposition of classical
definitions and results in Hodge theory (section 2.5) in the context of
abstract cell complexes. In section 3 we present the main results, that
describe the Laplacian spectrum for independent vertices models, which we
prove in sections 4 and 5 for sequence and simplicial complexes respectively.

##  2 Hodge theory on abstract cell complexes

###  2.1 Abstract Cell Complexes

Sequence complexes are a special type of abstract cell complexes; thus start
with their formal definition.

######  Definition 2.1.

An _Abstract Cell Complex_ is a quadruple
$\mathcal{X}=(\mathcal{X},\leq,\dim,\kappa)$ that consists of a poset
$(\mathcal{X},\leq)$ along with two functions, the dimension function
$\operatorname{dim}:\mathcal{X}\rightarrow\mathbb{Z}$, and the incidence
function $\kappa:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{Z}$ that
satisfy the following conditions:

$\displaystyle 1.\hskip 2.84526pt\dim\text{ is a poset morphism;}$
$\displaystyle 2.\hskip 2.84526pt\text{For all
}\xi,\xi^{\prime}\in\mathcal{X},\hskip
28.45274pt\kappa\left(\xi,\xi^{\prime}\right)\neq
0\quad\implies\quad\xi^{\prime}\leq\xi,\text{ and
}\operatorname{dim}(\xi)=\operatorname{dim}\left(\xi^{\prime}\right)+1;$
$\displaystyle 3.\hskip 2.84526pt\text{For all
}\xi,\xi^{\prime\prime}\in\mathcal{X},\hskip
28.45274pt\sum_{\xi^{\prime}\in\mathcal{X}}\kappa\left(\xi,\xi^{\prime}\right)\kappa\left(\xi^{\prime},\xi^{\prime\prime}\right)=0$

Note that we will only consider abstract cell complexes where
$\mathcal{X}_{n}=\dim^{-1}(n)$ is a finite set for all $n\in\mathbb{Z}$. The
most popular example of abstract cell complexes are simplicial complexes.

######  Definition 2.2.

Let $V$ be a finite set of vertices, an _Abstract Simplicial Complex_ ,
$K\subseteq 2^{V}$, is a collection of subsets of $V$, that is closed under
inclusion, meaning that

$\forall\xi\in K,\quad\xi^{\prime}\subseteq\xi\implies\xi^{\prime}\in K.$

Here the poset structure is chosen to be the set inclusion $\subseteq$, and
the dimension function is $\dim(\xi)=|\xi|-1,$ where $|\xi|$ is the
cardinality of the set $\xi$. The incidence function is defined using a
particular choice of a total order $v_{1}<v_{2}<\dots<v_{m}$ on the vertex set
$V$. Given a set $\xi=\\{u_{0},u_{1},\dots,u_{n}\\}$, where $u_{j}<u_{j+1}$
for all $j=0,\dots n-1$, and a subset $\xi^{\prime}\subseteq\xi$ such that
$\dim(\xi^{\prime})=\dim(\xi)-1$,

$\kappa\left(\xi,\xi^{\prime}\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(-1)^{j},\quad\text{where
}\xi^{\prime}=\xi\setminus\\{u_{j}\\}.$

If $\dim(\xi^{\prime})\neq\dim(\xi)-1$, or $\xi^{\prime}\not\subseteq\xi$,
then $\kappa\left(\xi,\xi^{\prime}\right)=0$.

Another example of interest is a collection of sequences. Let $V$ be a finite
set of vertices, denote by $\mathcal{X}_{n}[V]$ the set of all length $(n+1)$
sequences with vertices in $V$. Here the same vertex can appear in the
sequence more than once, and sequences with different orderings are distinct.
We say that a sequence $\tau\in\mathcal{X}_{n-k}[V]$ is a subsequence of a
sequence $\sigma\in\mathcal{X}_{n}[V]$, with the notation $\tau\leq\sigma$, if
$\tau$ can be obtained by removing $k$ vertices (in any position) from the
sequence $\sigma$. For example, $(a,b,c)\leq(c,a,b,b,c,a)$. We also denote by
$\mathcal{X}_{-1}=\\{\varnothing\\}$ the set consisting of the empty sequence
in $V$.

######  Definition 2.3.

Let $V$ be a finite set of vertices. A _Sequence Complex_ is a collection of
sequences

$\mathcal{X}=\bigcup_{n\geq-1}\mathcal{X}_{n},\quad\text{ where
}\mathcal{X}_{n}\subseteq\mathcal{X}_{n}[V],$

which is closed under inclusion, i.e.
$\forall\sigma\in\mathcal{X},\,\,\tau\leq\sigma\implies\tau\in\mathcal{X}$.

This is a special case of an abstract cell complex, where the poset structure
is the subsequence relation $\leq$, and the dimension is defined as
$\dim(\sigma)=n=|\sigma|-1$ for $\sigma\in\mathcal{X}_{n}[V]$. The incidence
function $\kappa$ is defined for all $\sigma,\tau\in\mathcal{X},$ with
$\tau\leq\sigma$ and $\dim(\tau)=\dim(\sigma)-1$, as

$\kappa(\sigma,\tau)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i:{\sigma}^{\times_{i}}=\tau}(-1)^{i},$

where ${\sigma}^{\times_{i}}$ denotes the sequence, obtained by removing the
$i$’th vertex from sequence $\sigma$. If $\dim(\tau)\neq\dim(\sigma)-1$, or
$\tau\not\leq\sigma$ then $\kappa\left(\sigma,\tau\right)=0$.

In this paper, we will only consider $\mathcal{X}_{n}=\mathcal{X}_{n}[V]$ the
full collection of sequences over the vertex set $V$. We refer to the
corresponding sequence complex as the full sequence complex, even though all
the properties and results, except for Theorem 3.1, and its corollary hold in
the more general case.

###  2.2 Distributions on Simplicial and Sequence Complexes

A probability distribution on a sequence complex $\mathcal{X}$, is a function
$p:\mathcal{X}\to[0,1]$ that satisfies the condition

$\sum_{\sigma\in\mathcal{X}}p(\xi)=1.$

We introduce weights $w_{p}:\mathcal{X}\to[0,1]$, that are induced by the
distribution $p$, as the conditional probability given the sequence length:

$w_{p}(\sigma)=p(\sigma\bigm{|}|\sigma|=n+1)={p(\sigma)}\left(\sum_{|\tau|=|\sigma|}p(\tau)\right)^{-1},\qquad\sigma\in\mathcal{X}_{n}.$
(1)

An immediate consequence of this weight function construction is that all
sequences of same length have weights that sum to one.

$\sum_{\sigma\in\mathcal{X}_{n}}w_{p}(\sigma)=1,\hskip 5.69054pt\text{for all
}n\in\mathbb{Z}.$ (2)

In parallel to the distribution induced weight function on sequences, we
introduced two weight functions on simplicial complexes. A simplicial
distribution on an abstract simplicial complex $K\subseteq 2^{[m]}$, is a
probability distribution $p:K\to[0,1]$ that satisfies the condition

$\displaystyle\sum_{\xi\in K}\hskip 2.84526pt$ $\displaystyle p(\xi)=1$

We consider either the simplicial distribution directly as a weight function
on the simplicial complex, or the associated moment map as the weight
function. The moment map $m_{p}:K\to[0,1]$, associated to a simplicial
distribution, $p$ is defined for all $\xi\in K$ as:

$\displaystyle
m_{p}(\xi):=\mathbb{E}_{p}\Bigg{(}\prod_{i\in\xi}x_{i}\Bigg{)}=\sum_{\zeta\supseteq\xi}p(\zeta)$

Here we use the notation $x_{i}:2^{[m]}\to\\{0,1\\},x_{i}(\xi)=1\text{ iff
}i\in\xi$, to refer to the indicator function, viewed as a random variable on
$2^{[m]}$. The equivalence of the two definitions for the moment map is easy
to see,

$\mathbb{E}_{p}\Bigg{(}\prod_{i\in\xi}x_{i}\Bigg{)}=\sum_{\zeta\supseteq\xi}p(\zeta)\prod_{i\in\xi}x_{i}(\zeta)+\sum_{\zeta\not\supset\xi}p(\zeta)\prod_{i\in\xi}x_{i}(\zeta)=\sum_{\zeta\supseteq\xi}p(\zeta)\cdot
1+\sum_{\zeta\not\supseteq\xi}p(\zeta)\cdot
0=\sum_{\zeta\supseteq\xi}p(\zeta)$

Notice the moment map, as opposed to the simplicial distribution, takes value
one on the empty set, $m_{p}(\varnothing)=1$, and is monotone decreasing on
the faces of the simplicial complex,

$\zeta\subseteq\xi\implies m_{p}(\zeta)>m_{p}(\xi).$

###  2.3 The independence models for simplices and sequences

We define null models on sequence and simplicial complexes as weight functions
arising from an assumption that the weights assigned to the sequences (or
simplices) depend solely on the vertices involved, and not the precise
position of each vertex. We will refer to complexes with such weight functions
as following the Independent Vertices Model. Notice we would have one such
model for the sequence complex context, and another for the simplicial complex
context. We make this formal in the following two definitions.

######  Definition 2.4.

Given a vertex set $V$, and vector $(w_{v})_{v\in V}\in(0,1)^{|V|}$ which
satisfies:

$\displaystyle\sum_{v\in V}w_{v}$ $\displaystyle=1$ (3)

We define the associated _independent vertices model_ over a sequences complex
$\mathcal{X}$, with $\mathcal{X}_{0}=V$, as the weighted sequence complex
$(\mathcal{X},w)$ with weight function $w$, such that:

$\displaystyle w((v_{0},...,v_{n}))$
$\displaystyle=\prod_{i=0}^{n}w_{v_{i}},\hskip 14.22636pt$
$\displaystyle\text{for all }(v_{0},...,v_{n})\in\mathcal{X}$ $\displaystyle
w(\varnothing)$ $\displaystyle=1$

It is quick to check that this weight function satisfies the condition

$\sum_{\sigma\in\mathcal{X}_{n}}w(\sigma)=1\quad\text{ for all }n\geq-1.$

One can interpret this weight function as a probability distribution on
sequences of a fixed length $=n+1$, where the appearance of a vertex in each
sequence position is independent from the vertices in all the other positions,
and $w_{v}$ is the probability of the vertex $v$ appearing in each position.

In the case of simplicial complexes, we define the independent simplicial
distribution associated to a vector $(p_{1},...,p_{m})\in(0,1)^{m}$ as
follows. For each face $\xi\in 2^{[m]}$, interpreted as a binary vector on $m$
vertices, we define the probability $p(\xi)$, as the joint probability of each
of the vertices $i\in[m]$ taking the prescribed binary value in $\xi$
following a Bernoulli distribution given by $p_{i}$. In other words, the
probability is defined as:

$\displaystyle p(\xi)$
$\displaystyle=\prod_{i\in\xi}p_{i}\prod_{j\not\in\xi}(1-p_{j}),\hskip
14.22636pt$ $\displaystyle\text{ for all }\xi\in K.$

Here $p_{i}$ has the meaning of probability of the $i$-th vertex appearing in
a subset $\sigma$. Note that these probabilities do not sum to $1$. We define
the independent vertices model on simplicial complexes, inspired in the moment
map weight function associated to the independent simplicial distribution.

######  Observation 2.5.

The moment map $m_{p}$, on a simplicial complex $K\subseteq 2^{[m]}$ induced
by the independent simplicial distribution associated to
$(p_{1},...,p_{m})\in(0,1)^{m}$, is defined as:

$\displaystyle m_{p}(\xi)$ $\displaystyle=\prod_{i\in\xi}p_{i},\hskip
14.22636pt$ $\displaystyle\text{ for all }\xi\in K$ $\displaystyle
w(\varnothing)$ $\displaystyle=1$

We give the definition for the independent vertices model in the context of
simplicial complexes, for a general vector $(w_{1},...,w_{m})\in(0,1)^{m}$
determining the weight of each vertex. The definition is thus very reminiscent
of the one in the sequence complex context.

######  Definition 2.6.

Given a vertex set $[m]$, and vector $(w_{1},...,w_{m})\in(0,1)^{m}$ (we allow
in this context the possibility of $\sum_{i\in[m]}w_{i}\neq 1$), we define the
associated _independent vertices model_ over a simplicial complex $K\subseteq
2^{[m]}$, as the weighted simplicial complex $(K,w)$ with weight function $w$,
such that:

$\displaystyle w(\xi)$ $\displaystyle=\prod_{i\in\xi}^{n}w_{i},\hskip
14.22636pt$ $\displaystyle\text{for all }\xi\in K$ $\displaystyle
w(\varnothing)$ $\displaystyle=1$

In applications, we may interpret this weight function as either the moment
map associated to an independent simplicial distribution (as in observation
2.5), or alternatively as the weight function that normalizes the distribution
by the probability of the empty set, as in the following observation 2.7. We
will show, that in this null hypothesis setting, both weight functions yield
an equivalent result, which is what we would like to see.

######  Observation 2.7.

Given a simplicial complex $K$, and the independent simplicial distribution,
$p$, associated to the vector $(p_{1},...,p_{m})\in(0,1)^{m}$, consider the
weight function $p_{\varnothing}$ defined over each face as it’s probability
normalized by the probability of the empty set.

$\displaystyle p_{\varnothing}(\xi)=\dfrac{p(\xi)}{p(\varnothing)},\hskip
14.22636pt\text{ for all }\xi\in K$

In the case of the independent simplicial distribution, this yields

$\displaystyle
p_{\varnothing}(\xi)=\displaystyle\prod_{i\in\xi}p_{i}\cdot\displaystyle\prod_{i\not\in\xi}(1-p_{i})\cdot\Bigg{(}\displaystyle\prod_{i\in[m]}(1-p_{i})\Bigg{)}^{-1}=\displaystyle\prod_{i\in\xi}p_{i}(1-p_{i})^{-1}$

Consequently the weighted simplicial complex associated to this weight
function, $(K,p_{\varnothing})$, follows an independent vertices model
associated to the vector
$\Big{(}p_{1}(1-p_{1})^{-1},...,p_{m}(1-p_{m})^{-1}\Big{)}\in(0,1)^{m}$.

###  2.4 Homological algebra of Abstract Cell Complexes

Given an abstract cell complex $\mathcal{X}$, we denote the set of
$n$-dimensional elements as
$\mathcal{X}_{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\dim^{-1}(n)$, and
identify the space of $n$-dimensional cochains $C^{n}(\mathcal{X})$ with
$\mathbb{R}$-valued functions on $\mathcal{X}_{n}$:

$C^{n}(\mathcal{X})=\left\\{f:\mathcal{X}_{n}\to\mathbb{R}\right\\}.$

Note that if $\mathcal{X}_{n}$ is empty for some $n\in\mathbb{Z}$, then by
convention, $C^{n}(\mathcal{X})=\\{0\\}$. We will refer to the graded space of
cochains as,

$C^{*}(\mathcal{X})=\bigoplus_{n=-1}^{+\infty}C^{n}(\mathcal{X})$ (4)

Note that we restrict ourselves to working over the field of real numbers;
this is because we will make use of inner products on $C^{*}(\mathcal{X})$.

The coboundary operator $\delta_{n}:C^{n}(\mathcal{X})\rightarrow
C^{n+1}(\mathcal{X})$, is defined on $f\in C^{n}(\mathcal{X})$ via

$(\delta_{n}f)(\xi)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{\xi^{\prime}\in\mathcal{X}_{n}}\kappa\left(\xi,\xi^{\prime}\right)f\left(\xi^{\prime}\right).$

The coboundary operator satisfies $\delta_{n+1}\delta_{n}=0$ for all $n$; this
is because of the condition (3) in Definition 2.1. This property enables the
usual definition of cohomology
$H^{n}(\mathcal{X})=\frac{\ker\delta_{n}}{\operatorname{im}\delta_{n-1}}$ of
the chain complex

$\displaystyle\\{0\\}{\longrightarrow}\hskip
2.84526ptC^{-1}(\mathcal{X})\stackrel{{\scriptstyle\delta_{-1}}}{{\longrightarrow}}C^{0}(\mathcal{X})\stackrel{{\scriptstyle\delta_{0}}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle\delta_{n-1}}}{{\longrightarrow}}C^{n}(\mathcal{X})\stackrel{{\scriptstyle\delta_{n}}}{{\longrightarrow}}C^{n+1}(\mathcal{X}){\longrightarrow}\ldots.$

###  2.5 Hodge Theory on Abstract Cell Complexes

Here we expose the Hodge theory on abstract cell complexes. The definitions
and results we present here are classical results for simplicial complexes,
dating back to [6] and expanded in e.g. [7, 9]. We present them in the context
of abstract cell complexes, for sake of self-containment and notation
consistency. The results and proofs of this section are not new.

Let $\langle\cdot,\cdot\rangle$ be an inner product on $C^{*}(\mathcal{X})$,
such that the direct sum decomposition (4) is an orthogonal decomposition with
respect to the inner product $\langle\cdot,\cdot\rangle$. Denote by
$\delta_{n}^{*}$ the adjoint of $\delta_{n}$, with respect to the chosen inner
product $\langle\cdot,\cdot\rangle$. That is, for every $f,g\in
C^{*}(\mathcal{X})$,

$\langle\delta_{n}^{*}f,g\rangle=\langle f,\delta_{n}g\rangle.$

It is easy to see that $\delta_{n+1}\delta_{n}=0$ if and only if
$\delta^{*}_{n}\delta^{*}_{n+1}=0$. Therefore there are two versions of the
cohomology space, one for $\delta_{n}$ and the other for $\delta_{n}^{*}$, and
these two are isomorphic. We introduce the Laplacian operator, which is
closely related to the cohomology, $H^{n}(\mathcal{X})$, of the coboundary
operator ${\delta}_{n}$.

######  Definition 2.8.

The Laplacian of a triple
$(C^{*}(\mathcal{X}),\delta,\langle\cdot,\cdot\rangle)$ is an operator over
$C^{n}(\mathcal{X})\to C^{n}(\mathcal{X})$, for all $n\in\mathbb{Z}_{\geq-1}$.

$\displaystyle\text{defined as, }L_{n}$
$\displaystyle=L^{\mathrm{up}}_{n}+L^{\mathrm{down}}_{n}$
$\displaystyle\text{where, }L^{\mathrm{up}}_{n}$
$\displaystyle=\delta_{n}^{*}\delta_{n},\hskip
5.69054ptL^{\mathrm{down}}_{n}=\delta_{n-1}\delta_{n-1}^{*}.$

For the reader familiar with the combinatorial Laplacian, $L_{\text{comb}}$,
we note this operator corresponds to the $0$-dimensional up-Laplacian
associated to a weighted graph G,

$L_{\text{comb}}=L_{0}^{\text{up}}:C^{0}(G)\to C^{0}(G).$

We refer to Section 5 for a deeper discussion of this.

As a direct result of Definition 2.8 and standard Linear Algebra results, one
can easily prove the following properties.

######  Lemma 2.9.

The Laplacians $\ L^{\mathrm{up}}$, $\ L^{\mathrm{down}}$ and $\ L$ are self-
adjoint and positive semi-definite with respect to the inner product
$\langle\cdot,\cdot\rangle$. Moreover, the following identities hold for all
$n\in\mathbb{Z}_{\geq-1}$:

$\displaystyle\ker\ L_{n}=$ $\displaystyle\left(\ker\
L^{\mathrm{up}}_{n}\right)\cap\left(\ker\
L^{\mathrm{down}}_{n}\right)=(\ker\delta_{n-1}^{*})\cap(\ker\delta_{n}).$
$\displaystyle\operatorname{im}(\delta_{n}^{*})=$
$\displaystyle\operatorname{im}(\delta_{n}^{*}\delta_{n})\text{ and
}\operatorname{im}(\delta_{n+1})=\operatorname{im}(\delta_{n+1}\delta_{n+1}^{*}).$

As we mentioned earlier, the Laplacian is intimately related to the cohomology
of the coboundary operator ${\delta}_{n}$. This theorem dates back to 1944
[6], in the context of simplicial complexes, and is easily extended to the
case of abstract cell complexes with no changes to the proof.

######  Theorem 2.10.

The kernel of the Laplacian $L_{n}$ associated to an abstract cell complex
$\mathcal{X}$, referred to as the space of “harmonic vectors”, is isomorphic
to the cohomology of $\delta_{n}$, for all $n\in\mathbb{Z}_{\geq-1}$:

$\ker L_{n}\simeq H^{n}(\mathcal{X}).$

Theorem 2.10 is a specific case of a more detailed theorem below, we refer to
[7] for a proof. We use the following notations in what follows:

$B^{n}_{+}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\operatorname{im}\delta_{n}^{*},\quad
B^{n}_{-}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\operatorname{im}\delta_{n-1}.$

######  Theorem 2.11.

Given an abstract cell complex, $\mathcal{X}$, and a fixed
$n\in\mathbb{Z}_{\geq-1}$, the space $C^{n}(\mathcal{X})$ admits the following
$\langle\cdot,\cdot\rangle$-orthogonal decomposition,:

$C^{n}(\mathcal{X})=\ker L_{n}\oplus B^{n}_{-}\oplus B^{n}_{+}$

Theorem 2.11 Has many nice spectral interpretations, let us introduce some
notation so we may discuss them. Denote the multiset (set with multiplicity)
of strictly positive eigenvalues of $L_{n}$, $L^{\mathrm{up}}_{n}$ and
$L^{\mathrm{down}}_{n}$, by ${\text{Spec}}_{*}(L_{n})$,
${\text{Spec}}_{*}(L^{\mathrm{up}}_{n})$ and
${\text{Spec}}_{*}(L^{\mathrm{down}}_{n})$, respectively. For each
$\lambda\in{\text{Spec}}_{*}(L_{n})$, ${\text{Spec}}_{*}(L^{\mathrm{up}}_{n})$
and ${\text{Spec}}_{*}(L^{\mathrm{down}}_{n})$, denote the corresponding
eigenspace by $E^{n}_{\lambda}$, $E_{\lambda}^{n,+}$ and $E_{\lambda}^{n,-}$,
respectively.

######  Corollary 2.12.

Given an abstract cell complex $\mathcal{X}$, we may decompose the spectrum of
it’s associated Laplacian $L_{n}$, for any $n\in\mathbb{Z}_{\geq-1}$, as:

${\text{Spec}}_{*}(L_{n})={\text{Spec}}_{*}(L^{\text{up}}_{n})\sqcup{\text{Spec}}_{*}(L^{\text{down}}_{n})={\text{Spec}}_{*}(L^{\text{up}}_{n-1})\sqcup{\text{Spec}}_{*}(L^{\text{down}}_{n+1})$

Moreover, the eigenspaces associated to it’s eigenvalues may be described as:

$\displaystyle E_{\lambda}^{n,+}=E^{n}_{\lambda}\cap B^{n}_{+}\hskip
5.69054pt\text{ and }\hskip 5.69054ptE_{\mu}^{n,-}=E^{n}_{\mu}\cap
B^{n}_{-},\hskip 14.22636pt\text{for
}\lambda\in{\text{Spec}}_{*}(L^{\text{up}}_{n}),\mu\in{\text{Spec}}_{*}(L^{\text{down}}_{n})$

And isomorphically mapped onto the eigenspaces of the adjacent dimension
Laplacian by
$E_{\lambda}^{n,+}\underset{{\delta}^{*}}{\overset{{\delta}}{\rightleftarrows}}E_{\lambda}^{n+1,-}$.

The proof to the corollary is a straightforward linear algebra exercise.

##  3 The main results

Here we present theorems that completely describe the spectrum of the
Laplacians associated to weighted sequence and simplicial complexes under the
independent vertices model. We provide the proofs of these theorems in
Sections 4 and 5.

######  Theorem 3.1 (abridged).

Consider a weighted full sequence complex $(\mathcal{X},w)$, with the weights
of the independent vertices model, then for any $n\geq 0$, it’s $n$-Laplacian,
$L_{n}:C^{n}(\mathcal{X})\to C^{n}(\mathcal{X})$ has the following
eigenvalues:

$\displaystyle\lambda=\hskip 2.84526pt$ $\displaystyle 1,...,n+2$ with
multiplicities $\displaystyle\text{mult}(\lambda)=\hskip 2.84526pt$
$\displaystyle\binom{n+1}{\lambda-1}\cdot(|\mathcal{X}_{0}|-1)^{\lambda-1}.\hskip
142.26378pt$

We postpone the proof until Section 4, where we give a more complete theorem
statement, which includes explicit descriptions of the associated eigenspaces.
Let us describe here the eigenspaces for the smallest eigenvalues.

######  Example 3.2.

The Laplacians $L_{n}$, of dimension $n\geq 0$, associated to a weighted full
sequence complex $(\mathcal{X},w)$ with weights following the independent
vertices model, admit eigenvalues $\lambda=1\hskip 2.84526pt\&\hskip
2.84526pt2$. Furthermore, by fixing $a\in\mathcal{X}_{0}$, we may describe the
associated eigenspaces as:

$\displaystyle E(1,L_{n})$
$\displaystyle=\mathrm{Span}\left\\{\sum_{\sigma\in\mathcal{X}_{n}}e_{\sigma}\right\\},$
$\displaystyle E(2,L_{n})$
$\displaystyle=\mathrm{Span}\left\\{\sum_{\begin{subarray}{c}\sigma\in\mathcal{X}_{n}:\\\
\sigma_{i}=a\end{subarray}}w(x)\cdot
e_{\sigma}-\sum_{\begin{subarray}{c}\tau\in\mathcal{X}_{n}:\\\
\tau_{i}=x\end{subarray}}w(a)\cdot e_{\tau}\hskip 2.84526pt\Bigg{|}\hskip
5.69054pti\in[0:n],\hskip
2.84526ptx\in\mathcal{X}_{0}\setminus\\{a\\}\right\\}.$

This example follows immediately from the full version of Theorem 3.1 given in
Section 4.

A simple corollary of our theorem, is that the cohomology of the full sequence
complex is trivial.

######  Corollary 3.3.

Given the full sequence complex $\mathcal{X}$, the cohomology is trivial,
$H^{n}(\mathcal{X})=\\{0\\}$, for all $n\geq-1$.

###### Proof.

Fix $n\in\mathbb{Z}_{\geq-1}$, let $L_{n}$ be the Laplacian associated to the
$(\mathcal{X},w)$ with $w$ an independent vertices model weight function on
$\mathcal{X}$. From Theorem 2.10, we know $H^{n}(\mathcal{X})\simeq kerL_{n}$.
Applying Theorem 3.1, which tells us $L_{n}$ does not admit the $0$
eigenvalue, we conclude:

$H^{n}(\mathcal{X})\simeq kerL_{n}=\\{0\\}.$

∎

Here we should note that the homology of the complex of injective words was
computed previously in [22, 23], and is equal to the homology of a bouquet of
spheres. The complex of injective words is a special case of a sequence
complex (see Definition 2.3), where each vertex is allowed to appear only once
in a sequence. The full complex of sequence that we consider is the maximal
possible complex on a vertex set $V$, thus it is perhaps expected that it
should have a trivial homology/cohomology.

We contrast the Laplacian spectrum of the sequence complex under the
independent vertices model, with the Laplacian associated to a simplicial
complex under the same model. Interestingly, here the eigenfunctions reveal no
interesting structure. Although our weight functions differ sightly in the
different contexts, we cannot explain why the beautiful integer spectrum
appears in the sequence setting, but not the simplicial.

######  Theorem 3.4.

Let $(2^{[m]},w)$ be a weighted simplicial complex, with weight function
$w:2^{[m]}\to\mathbb{R}_{>0}$ such that $w(\varnothing)=1$. Then the following
two conditions are equivalent:

  1. (i)

$(2^{[m]},w)$ is an independent vertices model;

  2. (ii)

the associated Laplacians are multiples of identities:

$L_{n}=\alpha_{n}I_{C^{n}},\quad\forall n\geq-1,$ (5)

where $I_{C^{n}}$ denotes the identity on $C^{n}(2^{[m]})$, and
$\alpha_{n}\in\mathbb{R}$.

Moreover, if (5) holds, then $\alpha_{n}$ is constant with respect to $n$, and

$\alpha_{n}=\sum_{i=1}^{m}w_{i},$

where $(w_{1},\dots,w_{m})$ is the vector determining the independent vertices
model defining the weight function $w$.

We give the proof to Theorem 3.4 in Section 5. Note that this proof can be
seen as a natural consequence of the results in [7]. We can apply this theorem
to the aforementioned moment map and simplicial distribution weight functions
associated to the independent simplicial distribution.

######  Corollary 3.5.

Given the simplicial complex $2^{[m]}$, and the independent simplicial
distribution, $p$, associated to the vector $(p_{1},...,p_{m})\in(0,1)^{m}$,
the following two hold:

  1. (i)

The Laplacian associated to the moment map weighted simplicial complex
$(K,m_{p})$, is

$L_{n}=\left(\sum_{i=1}^{m}p_{i}\right)I_{C^{n}},\quad\forall n\geq-1$

  2. (ii)

The Laplacian associated to the distribution weighted simplicial complex
$(K,p)$, is

$L_{n}=\left(\sum_{i=1}^{m}\frac{p_{i}}{1-p_{i}}\right)I_{C^{n}},\quad\forall
n\geq-1.$

###### Proof.

Part (i) is immediate from Proposition 3.4 and Observation 2.5. To prove part
(ii), let $L_{n}^{p}$ be the Laplacian associated to the independent
simplicial distribution weight function. Define $p_{\varnothing}$ the weight
function which to each face assigns it’s probability normalized by that of the
empty set, as in Observation 2.7, and let $L_{n}^{p_{\varnothing}}$ be the
associated Laplacian. Lemma 5.2 from Section 5 tells us
$L_{n}^{p_{\varnothing}}=L_{n}^{p}$, hence we conclude the desired statement
by applying Proposition 3.4 and Observation ∎

##  4 Proof of Theorem 3.1.

In this section we provide explicit formulae for the Laplacian over weighted
sequence complexes, and prove Theorem 3.1 by computing the spectrum and the
associated eigenspaces in the case of the independent vertices model.

###  4.1 Computation of the Laplacian on Sequences

We begin by providing the formulae for the coboundary operator and its
conjugate on the sequence complex. We will consider a weighted sequence
complex $(\mathcal{X},w)$ with weight function
$w:\mathcal{X}\to\mathbb{R}_{>0}$, and vertices as $\mathcal{X}_{0}=V$. Let us
introduce a basis $\\{e_{\sigma}\\}_{\sigma\in\mathcal{X}}$ in
$C^{*}(\mathcal{X})$ defined as:

$e_{\sigma}(\tau)=\begin{cases}1,&\text{ if }\sigma=\tau\\\ 0&\text{ if
}\sigma\neq\tau.\end{cases}$

The coboundary operator can be written as

${\delta}_{n}e_{\tau}=\sum_{\sigma\in\mathcal{X}_{n+1}}\kappa({\sigma},{\tau})e_{\sigma}.$

We define an inner product on $C^{*}(\mathcal{X})$, associated to a positive
weight function, $w$, via its values on the basis elements $\\{e_{\sigma}\\}$:

$\langle
e_{\sigma},e_{\tau}\rangle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\begin{cases}w(\sigma),&\text{
if }\sigma=\tau\\\ 0&\text{ if }\sigma\neq\tau.\end{cases}$

The decomposition
$C^{*}(\mathcal{X})=\displaystyle\bigoplus_{n\geq-1}C^{n}(\mathcal{X})$ is an
orthogonal decomposition with respect to this inner product.

Given this inner product, consider the adjoint of ${\delta}_{n}$, the operator
${\delta}_{n}^{*}:C^{n+1}(\mathcal{X})\to C^{n}(\mathcal{X})$, where
$\langle{\delta}_{n}f,g\rangle=\langle f,{\delta}_{n}^{*}g\rangle$ for any
$f\in C^{n}(\mathcal{X})$ and $g\in C^{n+1}(\mathcal{X})$.

${\delta}_{n}^{*}e_{\sigma}=\sum_{\tau\in\mathcal{X}_{n}}\alpha_{\sigma}(\tau)e_{\tau}$

for some $\alpha:\mathcal{X}_{n+1}\times\mathcal{X}_{n}\to\mathbb{R}$.
Following this notation, for any $e_{\eta}\in C^{n}(\mathcal{X})$:

$\displaystyle\langle{\delta}_{n}^{*}e_{\sigma},e_{\eta}\rangle$
$\displaystyle=\sum_{\tau\in\mathcal{X}_{n}}\alpha_{\sigma}(\tau)\langle
e_{\tau},e_{\eta}\rangle=\alpha_{\sigma}(\eta)w(\eta)$

On the other hand, by definition of the adjoint, we may rewrite the inner
product as:

$\displaystyle\langle{\delta}_{n}^{*}e_{\sigma},e_{\eta}\rangle$
$\displaystyle=\langle
e_{\sigma},{\delta}_{n}e_{\eta}\rangle=\sum_{\omega\in\mathcal{X}_{n+1}}\kappa({\omega},{\eta})\langle
e_{\sigma},e_{\omega}\rangle=\kappa({\sigma},{\eta})w(\sigma)$

We conclude the adjoint of the coboundary operator ${\delta}_{n}$ for all
$n\in\mathbb{Z}_{\geq-1}$, can be written over any basis element,
$e_{\sigma}\in C^{n+1}(\mathcal{X})$, as:

${\delta}_{n}^{*}e_{\sigma}=\sum_{\tau\in\mathcal{X}_{n}}\frac{w(\sigma)}{w(\tau)}\kappa({\sigma},{\tau})e_{\tau}$

Our first step in the proof of Theorem 3.1 will be to give an explicit formula
for the Laplacian on a weighted sequence complex. We preface this with the
following notation.

######  Definition 4.1.

We introduce a relation on sequences, along with some operations.

  * (1)

We say $\sigma\bowtie\tau$ if $\sigma,\tau$ have a “swapped" vertex, that is:

$\sigma\bowtie\tau\iff\exists\text{ }i\text{ s.t. }\sigma_{j}=\tau_{j}\text{
}\forall j\neq i$

Notice in particular that we consider $\sigma\bowtie\sigma$.

  * (2)

We define an operation to “glue in" a vertex into a specific spot (thus
expanding the length of the sequence):

$\displaystyle\mathcal{X}_{n}\times[0:n]\times\mathcal{X}_{0}$
$\displaystyle\to\mathcal{X}_{n+1}$ $\displaystyle{\sigma}^{+_{i}:{a}}$
$\displaystyle\mapsto\tau\text{ }\text{ }\text{s.t.
}\tau_{j}=\begin{cases}\sigma_{j}\text{ , }\text{ }\text{ }\text{
}j=0,...,i-1\\\ a\text{ , }\text{ }\text{ }\text{ }\text{ }j=i\\\
\sigma_{j-1}\text{ , }j=i+1,...,n+1\\\ \end{cases}$

  * (3)

As well as an operation which “removes" the vertex from a specific spot (thus
shortening the length):

$\displaystyle\mathcal{X}_{n}\times[0:n]$ $\displaystyle\to\mathcal{X}_{n-1}$
$\displaystyle{\sigma}^{\times_{i}}$ $\displaystyle\mapsto\tau\text{ }\text{
}\text{s.t. }\tau_{j}=\begin{cases}\sigma_{j}\text{ , }\text{ }\text{ }\text{
}j=0,...,i-1\\\ \sigma_{j+1}\text{ , }j=i,...,n-1\\\ \end{cases}$

  * (4)

And finally, an operation that “swaps" in a vertex, into a specific spot (thus
eliminating the previous vertex in that spot, and preserving the length of the
sequence):

$\displaystyle\mathcal{X}_{n}\times[0:n]\times\mathcal{X}_{0}$
$\displaystyle\to\mathcal{X}_{n}$
$\displaystyle{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}$
$\displaystyle\mapsto\tau\text{ }\text{ }\text{ s.t. }\tau\bowtie\sigma\text{
and }\tau_{i}=a$

It will be of particular use, to understand how these functions interact with
each other. We summarize this in the following observation, which is
straightforward to verify.

######  Observation 4.2.

Composing the glue and remove function at the appropriate index yield the swap
function. The order in which the functions are applied mean different indices
must be involved. Given a sequence complex $X$, for any $\sigma\in\mathcal{X}$
and $a\in\mathcal{X}_{0}$, $i=0,\dots,\mid\sigma\mid$ :

${\sigma}^{\mathrlap{\times}{+}_{i}:{a}}={({\sigma}^{\times_{i}})}^{+_{i}:{a}}={({\sigma}^{+_{i}:{a}})}^{\times_{i+1}}$

Although the glue and remove functions are not commutative, we may change the
order in which they are applied without altering the resulting sequence, by
modifying the involved indices. Specifically for any $\sigma\in\mathcal{X}$
and $a\in V$, $i=0,\dots,\dim(\sigma)$, $j=0,\dots,\dim(\sigma)+1$ :

$\displaystyle{({\sigma}^{+_{i}:{a}})}^{\times_{j}}$
$\displaystyle=\begin{cases}{({\sigma}^{\times_{j}})}^{+_{i-1}:{a}}\hskip
34.1433pt\text{ if }j<i\\\ \sigma\hskip 76.82243pt\text{ if }j=i\\\
{({\sigma}^{\times_{j-1}})}^{+_{i}:{a}}\hskip 32.72066pt\text{ if }j>i\\\
\end{cases}$

We can rewrite the coboundary operator and it’s adjoint in an explicit fashion
using these operations. We present these expressions in the next observation.

######  Observation 4.3.

Given a weighted sequence complex $(\mathcal{X},w)$, the coboundary operator
${\delta}_{n}$, and it’s adjoint ${\delta}_{n}^{*}$, for any $n\in\geq-1$, can
be written over any basis elements $e_{\sigma}\in C^{n}(\mathcal{X})$ and
$e_{\tau}\in C^{n+1}(\mathcal{X})$ as:

$\displaystyle{\delta}_{n}e_{\sigma}$
$\displaystyle=\sum_{\sigma^{\prime}\in\mathcal{X}_{n+1}}\kappa(\sigma^{\prime},\sigma)\cdot
e_{\sigma^{\prime}}=\sum_{i=0}^{n+1}\sum_{a\in\mathcal{X}_{0}}(-1)^{i}\cdot
e_{{\sigma}^{+_{i}:{a}}}$ $\displaystyle{\delta}_{n}^{*}e_{\tau}$
$\displaystyle=\sum_{\tau^{\prime}\in\mathcal{X}_{n}}\dfrac{w(\tau)}{w(\tau^{\prime})}\kappa(\tau,\tau^{\prime})\cdot
e_{\tau^{\prime}}=\sum_{i=0}^{n+1}\dfrac{w(\tau)}{w({\tau}^{\times_{i}})}(-1)^{i}\cdot
e_{{\tau}^{\times_{i}}}$

The observation stems immediately from the definition of incidence function in
Definition 2.3, and of the gluing and removing operations. We may apply the
observation to compute an explicit definition for the Laplacian operator over
a weighted sequence complex. We warn the reader this computation is quite
gruesome.

######  Proposition 4.4.

Given a weighted sequence complex $(\mathcal{X},w)$ , over vertices
$\mathcal{X}_{0}=V$, for all $n\in\mathbb{Z}_{\geq-1}$, it’s Laplacian of
dimension $n$ is given over basis element $e_{\sigma}\in C^{n}(\mathcal{X})$
by:

$\displaystyle L_{n}(e_{\sigma})=\hskip 14.22636pt\sum_{i=0}^{n+1}\sum_{a\in
V}\dfrac{w({\sigma}^{+_{i}:{a}})}{w(\sigma)}\cdot
e_{\sigma}+\sum_{i=0}^{n}\sum_{j=0}^{i}\sum_{a\in
V}(-1)^{i+j}\cdot\left[\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}-\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}\right]\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$
$\displaystyle-\sum_{i=0}^{n}\sum_{a\in
V}\left[\dfrac{w({\sigma}^{+_{i}:{a}})}{w({\sigma}^{\mathrlap{\times}{+}_{i}:{a}})}+\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w({\sigma}^{\mathrlap{\times}{+}_{i}:{a}})}-\dfrac{w(\sigma)}{w({\sigma}^{\times_{i}})}\right]\cdot
e_{{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}}+\sum_{i=0}^{n}\sum_{j=i+1}^{n}\sum_{a\in
V}(-1)^{i+j}\cdot\left[\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}-\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}\right]\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$

###### Proof.

We compute $L_{n}^{down}$ and $L_{n}^{up}$ separately. Consider a cochain
$e_{\sigma}\in C^{n}(\mathcal{X})$, for $L_{n}^{down}$ we simply plug in the
definitions of ${\delta}_{n-1}$ and ${\delta}_{n-1}^{*}$, and then split the
sum, so that we may substitute
${({\sigma}^{\times_{i}})}^{+_{i}:{a}}={\sigma}^{\mathrlap{\times}{+}_{i}:{a}}$
for one of the terms.

$\displaystyle
L_{n}^{down}e_{\sigma}={\delta}_{n-1}{\delta}_{n-1}^{*}e_{\sigma}={\delta}_{n-1}\left(\sum_{j=0}^{n}(-1)^{j}\cdot\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}\cdot
e_{{\sigma}^{\times_{j}}}\right)=\sum_{i=0}^{n}\sum_{j=0}^{n}\sum_{a\in
V}(-1)^{i+j}\cdot\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{i-1}\sum_{a\in
V}(-1)^{i+j}\cdot\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}+\sum_{i=0}^{n}\sum_{a\in
V}(-1)^{2i}\cdot\dfrac{w(\sigma)}{w({\sigma}^{\times_{i}})}\cdot
e_{{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}}+\sum_{i=0}^{n}\sum_{j=i+1}^{n}\sum_{a\in
V}(-1)^{i+j}\cdot\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$

For $L_{n}^{up}$ we also begin by plugging in the definitions of
${\delta}_{n}$ and ${\delta}_{n}^{*}$, and then splitting the sum so that we
may substitute ${({\sigma}^{+_{i}:{a}})}^{\times_{j}}=\sigma$ for one of the
terms. Moreover, we also swap ${({\sigma}^{+_{i}:{a}})}^{\times_{j}}$ for
${({\sigma}^{\times_{j}})}^{+_{i-1}:{a}}$, when $j<i$; and swap
${({\sigma}^{+_{i}:{a}})}^{\times_{j}}$ for
${({\sigma}^{\times_{j-1}})}^{+_{i}:{a}}$, when $j>i$, as per observation 4.2.

$\displaystyle
L_{n}^{up}e_{\sigma}={\delta}_{n}^{*}{\delta}_{n}e_{\sigma}={\delta}_{n}^{*}\left(\sum_{i=0}^{n+1}\sum_{a\in
V}(-1)^{i}\cdot
e_{{\sigma}^{+_{i}:{a}}}\right)=\sum_{i=0}^{n+1}\sum_{j=0}^{n+1}\sum_{a\in
V}(-1)^{i+j}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{+_{i}:{a}})}^{\times_{j}}\right)}\cdot
e_{{\left({\sigma}^{+_{i}:{a}}\right)}^{\times_{j}}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\sum_{j=0}^{i-1}\sum_{a\in
V}(-1)^{i+j}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i-1}:{a}}\right)}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i-1}:{a}}}+\sum_{i=0}^{n+1}\sum_{a\in
V}(-1)^{2i}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w(\sigma)}\cdot e_{\sigma}$
$\displaystyle\hskip 170.71652pt+\sum_{i=0}^{n+1}\sum_{j=i+1}^{n+1}\sum_{a\in
V}(-1)^{i+j}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{\times_{j-1}})}^{+_{i}:{a}}\right)}\cdot
e_{{\left({\sigma}^{\times_{j-1}}\right)}^{+_{i}:{a}}}$

We follow by shifting the first sum over $i$ and the third sum over $j$. We
then proceeded by eliminating the vacuous terms of the sums, and splitting the
first and third sum so as to substitute in
${({\sigma}^{\times_{i}})}^{+_{i}:{a}}={\sigma}^{\mathrlap{\times}{+}_{i}:{a}}$.

$\displaystyle L_{n}^{up}e_{\sigma}$
$\displaystyle=\sum_{i=-1}^{n}\sum_{j=0}^{i}\sum_{a\in
V}(-1)^{i+j+1}\cdot\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}+\sum_{i=0}^{n+1}\sum_{a\in
V}\dfrac{w({\sigma}^{+_{i}:{a}})}{w(\sigma)}\cdot e_{\sigma}$
$\displaystyle\hskip 5.69054pt+\sum_{i=0}^{n+1}\sum_{j=i}^{n}\sum_{a\in
V}(-1)^{i+j+1}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$
$\displaystyle=\sum_{i=0}^{n}\sum_{j=0}^{i-1}\sum_{a\in
V}(-1)^{i+j+1}\cdot\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}+\sum_{i=0}^{n}\sum_{a\in
V}(-1)^{2i+1}\cdot\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w({\sigma}^{\mathrlap{\times}{+}_{i}:{a}})}\cdot
e_{{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}}$ $\displaystyle\hskip
5.69054pt+\sum_{i=0}^{n+1}\sum_{a\in
V}\dfrac{w({\sigma}^{+_{i}:{a}})}{w(\sigma)}\cdot
e_{\sigma}+\sum_{i=0}^{n}\sum_{a\in
V}(-1)^{2i+1}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w({\sigma}^{\mathrlap{\times}{+}_{i}:{a}})}\cdot
e_{{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}}$ $\displaystyle\hskip
5.69054pt+\sum_{i=0}^{n}\sum_{j=i+1}^{n}\sum_{a\in
V}(-1)^{i+j+1}\cdot\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$

Adding our computations for $L_{n}^{down}$ and $L_{n}^{up}$, we conclude the
theorem statement. ∎

######  Corollary 4.5.

Given a weighted full sequence complex $(\mathcal{X},w)$, over vertices
$\mathcal{X}_{0}=V$, following the independent vertices model (Definition
2.4), the Laplacian of dimension $n$, for all $n\in\mathbb{Z}_{\geq-1}$, can
be simply described over basis element $e_{\sigma}\in C^{n}(\mathcal{X})$ as:

$L_{n}(e_{\sigma})=\left[(n+2)-\sum_{i=0}^{n}w(\sigma_{i})\right]\cdot
e_{\sigma}-\sum_{\begin{subarray}{c}\tau:\tau\bowtie\sigma,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot e_{\tau}$

###### Proof.

Let us simplify the terms that appear in proposition 4.4. Fix
$\sigma\in\mathcal{X}_{n}$ for some $n\in\mathbb{Z}_{\geq-1}$, and
$i\in[0:n]$, $a\in V$, by applying the independence model hypothesis we can
write out:

$\displaystyle\dfrac{w({\sigma}^{+_{i}:{a}})}{w(\sigma)}=\dfrac{\Big{(}\prod_{j=0}^{i-1}w(\sigma_{j})\Big{)}\cdot
w(a)\cdot\Big{(}\prod_{j=i}^{n}w(\sigma_{j})\Big{)}}{\prod_{j=0}^{n}w(\sigma_{j})}=w(a)$

Similarly, we may write out and cancel terms to get:

$\displaystyle w(\sigma_{i})$
$\displaystyle=\dfrac{w(\sigma)}{w({\sigma}^{\times_{i}})}=\dfrac{w({\sigma}^{+_{i}:{a}})}{w({\sigma}^{\mathrlap{\times}{+}_{i}:{a}})}=\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w({\sigma}^{\mathrlap{\times}{+}_{i}:{a}})}$
$\displaystyle w(\sigma_{j})$
$\displaystyle=\dfrac{w(\sigma)}{w({\sigma}^{\times_{j}})}=\dfrac{w({\sigma}^{+_{i+1}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}=\dfrac{w({\sigma}^{+_{i}:{a}})}{w\left({({\sigma}^{\times_{j}})}^{+_{i}:{a}}\right)}$

Substituting these equalities into the expression from Proposition 4.4,
simplifies it to:

$\displaystyle L_{n}(e_{\sigma})=$ $\displaystyle\sum_{i=0}^{n+1}\sum_{a\in
V}w(a)\cdot e_{\sigma}-\sum_{i=0}^{n}\sum_{a\in
V}\big{[}w(\sigma_{i})+0\big{]}\cdot
e_{{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}}+\sum_{i=0}^{n}\sum_{j=0}^{i}\sum_{a\in
V}0\cdot
e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}+\sum_{i=0}^{n}\sum_{j=i+1}^{n}\sum_{a\in
V}0\cdot e_{{\left({\sigma}^{\times_{j}}\right)}^{+_{i}:{a}}}$
$\displaystyle=$ $\displaystyle(n+2)\cdot e_{\sigma}-\sum_{i=0}^{n}\sum_{a\in
V}w(\sigma_{i})\cdot
e_{{\sigma}^{\mathrlap{\times}{+}_{i}:{a}}}=\left[(n+2)-\sum_{i=0}^{n}w(\sigma_{i})\right]\cdot
e_{\sigma}-\sum_{\begin{subarray}{c}\tau:\tau\bowtie\sigma,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot e_{\tau}$

Where the second equality follows from $\sum_{v\in V}w(v)=1$. This concludes
the corollary statement. ∎

###  4.2 Eigenspace for Laplacian on Sequences under Independent Vertices
Model

For the study of the independent vertices model, it is useful to introduce a
tensor product on the space of cochains, the definition follows:

######  Definition 4.6.

For cochains $u\in C^{k}(\mathcal{X})$ and $v\in C^{l}(\mathcal{X})$, the
_tensor product_ $u\otimes v\in C^{k+l+1}(\mathcal{X})$ is the cochain whose
value on a sequence $\sigma\in\mathcal{X}_{k+l+1}$ is given by the formula:

$(u\otimes v)(\sigma)=u(\sigma|_{[0:k]})\cdot v(\sigma|_{[k+1:k+l+1]})$

Here $\sigma|_{[i:j]}$ refers to the restriction of the sequence $\sigma$ to
the vertices in it’s $i$ through $j$ spots (preserving the ordering), yielding
a subsequence of length $j-i+1$.

Note that this tensor product is reminiscent of, but different from the cup
product222 The tensor product $\otimes:C^{k}(\mathcal{X})\times
C^{l}(\mathcal{X})\to C^{k+l+1}(\mathcal{X})$ and cup product
$\smile:C^{k}(\mathcal{X})\times C^{l}(\mathcal{X})\to C^{k+l}(\mathcal{X})$
differ in the dimension of the resulting cochain. Recall that $u\smile v\in
C^{{k+l}}(\mathcal{X})$ takes value on a sequence $\tau\in\mathcal{X}_{{k+l}}$
as follows:  $(u\smile v)(\tau)=u(\tau|_{[0:k]})\cdot v(\tau|_{[{k}:k+l]})$ .
Using the tensor product on the space of cochains, we introduce a new basis
for $C^{n}(\mathcal{X})$.

######  Definition 4.7.

Fix $a\in\mathcal{X}_{0}$ and $w:\mathcal{X}\to(0,1]$ a weight function. We
define the associated map

$f_{0}=f_{0}^{a,w}:\mathcal{X}_{0}\to C^{0}(\mathcal{X})$

associating to all $x\in\mathcal{X}_{0}$, a $0$-cochain defined over each
$y\in\mathcal{X}_{0}$ as:

$\displaystyle f_{0}(x):\mathcal{X}_{0}$ $\displaystyle\to\mathbb{R}$
$\displaystyle y$ $\displaystyle\mapsto\big{(}f_{0}(x)\big{)}(y)$
$\displaystyle=\mathds{1}_{a}(x)+\Big{(}1-\mathds{1}_{a}(x)\Big{)}\cdot\Big{(}w(x)\mathds{1}_{a}(y)-w(a)\mathds{1}_{x}(y)\Big{)}$
$\displaystyle=\begin{cases}1\hskip 32.72066pt\text{ if }x=a\\\ w(x)\hskip
17.07164pt\text{ if }y=a\neq x\\\ -w(a)\hskip 9.95845pt\text{ if }y=x\neq a\\\
0\hskip 34.1433pt\text{ if not}\end{cases}$

We further define the map

$f=f^{a,w}:\mathcal{X}\to C^{*}(\mathcal{X})$

which, for all $n\in\mathbb{Z}_{\geq-1}$, maps any $n$-dimensional sequence,
$\eta\in\mathcal{X}_{n}$, to an $n$-cochain:

$f(\eta)=f_{0}(\eta_{1})\otimes\dots\otimes f_{0}(\eta_{n})\text{ }\in
C^{n}(\mathcal{X})$

Here we denote by $\eta_{i}\in\mathcal{X}_{0}$ the $i$’th vertex of the
sequence $\eta$.

We give an example of a particular cochain $f(\eta)\in C^{6}(\mathcal{X})$
applied to two different sequences $\sigma,\tau\in\mathcal{X}_{6}$ in Figure
1.

$\eta$ $\sigma$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$ $a$ $w(\eta_{1})$
$\eta_{2}$ $\eta_{2}$ $-w(a)$ $\eta_{3}$ $\eta_{3}$ $-w(a)$ $\eta_{4}$
$\eta_{4}$ $-w(a)$ $a$ $a$ $1$ $a$ $\sigma_{6}$ $1$ $\shortparallel$
$\left(\vbox{\hrule
height=77.28752pt,depth=77.28752pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=77.28752pt,depth=77.28752pt,width=0.0pt}\right)$$f$:,

(

$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$

)

$w(a)^{4}\cdot w(\eta_{1})$

$\eta$ $\tau$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$ $\eta_{1}$ $-w(a)$
$\eta_{2}$ $\tau_{2}$ $0$ $\eta_{3}$ $a$ $w(\eta_{2})$ $\eta_{4}$ $\eta_{4}$
$-w(a)$ $a$ $\tau_{5}$ $1$ $a$ $\tau_{6}$ $1$ $\shortparallel$
$\left(\vbox{\hrule
height=77.28752pt,depth=77.28752pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=77.28752pt,depth=77.28752pt,width=0.0pt}\right)$$f$:

(

$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$

)

$0$

Figure 1: Example cochain $f(\eta)\in C^{6}(\mathcal{X})$,
$\eta=(\eta_{0},\eta_{1},\eta_{2},\eta_{3},\eta_{4},a,a)\in\mathcal{X}_{6}$
applied to two different sequences
$\sigma=(\eta_{0},a,\eta_{2},\eta_{3},\eta_{4},a,\sigma_{6})\in\mathcal{X}_{6}$,
and
$\tau=(\eta_{0},\eta_{1},\tau_{2},a,\eta_{4},\tau_{5},\tau_{6})\in\mathcal{X}_{6}$,
with $\tau_{2}\neq\eta_{2}$ (in red). We observe
$\left(f(\eta)\right)(\sigma)\neq 0$ whereas $\left(f(\eta)\right)(\tau)=0$.

Of interest to us, is that the cochains
$\\{f(\eta)\\}_{\eta\in\mathcal{X}_{n}}$ form a basis for
$C^{n}(\mathcal{X})$, the proof is quite simple, we give it in the following
lemma.

######  Lemma 4.8.

Given a sequence complex $\mathcal{X}$, and any $n\in\mathbb{Z}_{\geq 0}$, the
cochains $\\{f(\eta)\\}_{\eta\in\mathcal{X}_{n}}$ form a basis for
$C^{n}(\mathcal{X})$.

###### Proof.

We will show
$\mathrm{Span}\big{\\{}f(\eta)\mid\eta\in\mathcal{X}_{n}\big{\\}}\ni
e_{\sigma}$, for all $\sigma\in\mathcal{X}_{n}$, $n\in\mathbb{Z}_{\geq 0}$. We
do the proof by induction over $n$.

Consider n=0, then $\mathcal{X}_{0}=V=\\{a,b,c,...\\}$. By definition we have:

$\displaystyle f({(a)})$ $\displaystyle=\big{(}f_{0}(a)\big{)}(a)\cdot
e_{(a)}+\big{(}f_{0}(a)\big{)}(b)\cdot e_{(b)}+\big{(}f_{0}(a)\big{)}(c)\cdot
e_{(c)}+\dots=e_{(a)}+e_{(b)}+e_{(c)}+\dots$ $\displaystyle f({(b)})$
$\displaystyle=\big{(}f_{0}(b)\big{)}(a)\cdot
e_{(a)}+\big{(}f_{0}(b)\big{)}(b)\cdot e_{(b)}+\big{(}f_{0}(b)\big{)}(c)\cdot
e_{(c)}+\dots=w(b)\cdot e_{(a)}-w(a)\cdot e_{(b)}$ $\displaystyle f({(c)})$
$\displaystyle=\big{(}f_{0}(c)\big{)}(a)\cdot
e_{(a)}+\big{(}f_{0}(c)\big{)}(b)\cdot e_{(b)}+\big{(}f_{0}(c)\big{)}(c)\cdot
e_{(c)}+\dots=w(c)\cdot e_{(a)}-w(a)\cdot e_{(c)}$ $\displaystyle\vdots$

As we have $w(v)\neq 0$ for all $v\in V$ , we may take the following linear
combination of cochains:

$\displaystyle f({(a)})+\sum_{x\neq a}w(a)^{-1}\cdot f({(x)})=\sum_{x\in
A}e_{(x)}+\sum_{x\neq a}\Big{(}\dfrac{w(x)}{w(a)}\cdot
e_{(a)}-e_{(x)}\Big{)}=\Bigg{(}1+\frac{1-w(a)}{w(a)}\Bigg{)}\cdot
e_{(a)}=w(a)^{-1}\cdot e_{(a)}$

Notice that in the second equality, we made use of $\sum_{x\in V}w(x)=1$.
Proving
$e_{(a)}\in\mathrm{Span}\big{\\{}f(\eta)\mid\eta\in\mathcal{X}_{n}\big{\\}}$.

On the other hand, by definition, we may write $f({(y)})$ for any $y\in V$,
$y\neq a$ as:

$f({(y)})=w(y)\cdot e_{(a)}-w(a)\cdot e_{(y)}$

Solving for $e_{(y)}$ and plugging in our identity for $e_{(a)}$, we get

$\displaystyle e_{(y)}=-w(a)^{-1}\cdot\big{(}f({(y)})-w(y)\cdot
e_{(a)}\big{)}=-w(a)^{-1}\cdot f({(y)})+w(y)\cdot\Bigg{(}f({(a)})+\sum_{x\neq
a}w(a)^{-1}\cdot f({(x)})\Bigg{)}$

Proving for all $y\in X_{0}$,
$e_{(y)}\in\mathrm{Span}\big{\\{}f(\eta)\mid\eta\in\mathcal{X}_{0}\big{\\}}$,
concluding the base induction case.

Consider $\eta\in\mathcal{X}_{n}$ and denote $\eta_{0}\in\mathcal{X}_{0}$ it’s
first vertex and $\eta\mid_{[1,n]}\in\mathcal{X}_{n-1}$ it’s restriction to
it’s other $n$ vertices. By definition, $f(\eta)=f({\eta_{0}})\otimes
f({\eta\mid_{[1,n]}})$, allowing us to immediately apply our induction
hypothesis:

$\displaystyle\mathrm{Span}\\{f(\eta)\mid\eta\in\mathcal{X}_{n}+1\\}$
$\displaystyle=\mathrm{Span}\\{f({(x)})\mid
x\in\mathcal{X}_{0}\\}\otimes\mathrm{Span}\\{f(\tau)\mid\tau\in\mathcal{X}_{n}\\}$
$\displaystyle=\mathrm{Span}\\{e_{(x)}\mid
x\in\mathcal{X}_{0}\\}\otimes\mathrm{Span}\\{e_{\tau}\mid\tau\in\mathcal{X}_{n}\\}=\mathrm{Span}\\{e_{\eta}\mid\eta\in\mathcal{X}_{n+1}\\}$

proving the lemma. ∎

Having proven the cochains $\\{f(\eta):\eta\in\mathcal{X}_{n}\\}$ form a basis
for $C^{n}(\mathcal{X})$, we can finally detail the proof for Theorem 3.1,
presented in abridged form in the previous section. We give here the full
version of the theorem, which exposes the eigenspaces associated to each
eigenvalue.

######  Theorem 3.1.

The Laplacian of a weighted full sequence complex $(\mathcal{X},w)$, over
vertices $\mathcal{X}_{0}=V$, with weights following the independent vertices
model (Definition 2.4):

$\displaystyle L_{n}:\hskip 2.84526pt$ $\displaystyle C^{n}(\mathcal{X})\to
C^{n}(X)\hskip 14.22636pt\text{for }n\in\mathbb{Z}_{\geq 0}$ admits
eigenvalues $\displaystyle\lambda=\hskip 2.84526pt$ $\displaystyle 1,...,n+2$
with multiplicity: $\displaystyle\text{mult}(\lambda)=\hskip 2.84526pt$
$\displaystyle\binom{n+1}{\lambda-1}\cdot(|V|-1)^{\lambda-1}\hskip
142.26378pt$

By fixing one of the vertices, $a\in V$ , we can succinctly describe the
eigenspaces, associated over the usual basis $\\{e_{\sigma}\\}$ , to each of
our eigenvalues:

$\displaystyle E(\lambda,L_{n})$
$\displaystyle=\mathrm{Span}\big{\\{}f(\eta)\mid\eta\in\mathcal{X}_{n}:\\#\\{a\in\eta\\}=n+2-\lambda\big{\\}}\hskip
14.22636pt\text{, }\lambda=1,...,n+2$

Where $f(\eta)$ is as in Definition 4.7, and $\\#\\{a\in\eta\\}$ denotes the
number of vertex “$a$" in the sequence $\eta$.

###### Proof.

Let us show the cochains $f(\eta)$ are eigenvectors for the Laplacian
associated to a independent vertices model sequence complex under the usual
representation. From corollary 4.5, we know the Laplacian $L_{n}$, for all
$n\in\mathbb{Z}\geq-1$, has representation, over the usual basis
$\\{e_{\sigma}\\}$, given by:

$\displaystyle(L_{n})_{\tau\sigma}=\begin{cases}(n+2)-\sum_{i=0}^{n}w(\sigma_{i})\hskip
8.53581pt&\text{if }\tau=\sigma\\\ w(\sigma_{j})&\text{if
}\tau\bowtie\sigma\text{ , }\tau_{j}\neq\sigma_{j}\\\ 0&\text{if
}\tau\not\bowtie\sigma\end{cases}$

for any $\sigma,\tau\in\mathcal{X}_{n}$. We recall the use here of the
“swapped" sequences notation $\tau\bowtie\sigma$ from Definition4.1.

Hence to show $v\in C^{n}(\mathcal{X})$ is an eigenvector of $L_{n}$
associated to $\lambda$, we need to prove:

$\displaystyle\lambda\cdot
v(\tau)=\left[(n+2)-\sum_{i=0}^{n}w(\tau_{i})\right]\cdot
v(\tau)-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot v({\sigma})\hskip
8.53581pt\text{ , }\forall\tau\in\mathcal{X}_{n}$ (6)

We will separate the proof into two parts. First we will deal with the
cochains of the form $f\big{(}{(a,\dots,a)}\big{)}\in C^{n}(\mathcal{X})$, for
some $n\in\mathbb{Z}_{\geq 0}$. We illustrate such a cochain in Figure 2.

$\eta$ $\sigma$ $a$ $\sigma_{0}$ $1$ $a$ $\sigma_{1}$ $1$ $a$ $\sigma_{2}$ $1$
$a$ $\sigma_{3}$ $1$ $a$ $\sigma_{4}$ $1$ $\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$1$ Figure 2: Example of a cochain associated to a sequence of all $a$’s.
Figured here $\eta=(a,a,a,a,a)\in\mathcal{X}_{4}$ is applied to a sequence
$\sigma=(\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4})\in\mathcal{X}_{4}$
and yields $\big{(}f(\eta)\big{)}(\sigma)=1$.

After showing the cochain $f({(a,\dots,a)})\in C^{n}(\mathcal{X})$ is an
eigenvector, we will show all other cochains $f(\eta)\in C^{n}(\mathcal{X})$,
$\eta\neq(a,\dots,a)$, are eigenvectors for the Laplacian $L_{n}$ as well. In
this general step, we distinguish the action of the cochain $f(\eta)$ on
$\tau\in\mathcal{X}_{n}$ such that $\big{(}f(\eta)\big{)}(\tau)=0$ and on
$\sigma\in\mathcal{X}_{n}$ such that $\big{(}f(\eta)\big{)}(\sigma)\neq 0$. In
both cases, we explore what the action of $f(\eta)$ would be on any
$\tau^{\prime}\bowtie\tau$ and $\sigma^{\prime}\bowtie\sigma$. We illustrate
some examples for the first case in Figure 3 and for the second case in Figure
4.

$\eta$ $\tau$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$ $a$ $w(\eta_{1})$
$\eta_{2}$ $\tau_{2}$ $0$ $a$ $\tau_{3}$ $1$ $a$ $\tau_{4}$ $1$
$\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$0$

$\eta$ $\tau^{\prime}$ $\eta_{0}$ $a$ $w(\eta_{0})$ $\eta_{1}$ $a$
$w(\eta_{1})$ $\eta_{2}$ $\tau_{2}$ $0$ $a$ $\tau_{3}$ $1$ $a$ $\tau_{4}$ $1$
$\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:,

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$0$

$\eta$ $\tau^{\prime\prime}$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$ $a$
$w(\eta_{1})$ $\eta_{2}$ $a$ $w(\eta_{2})$ $a$ $\tau_{3}$ $1$ $a$ $\tau_{4}$
$1$ $\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:,

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$-w(a)\cdot w(\eta_{1})\cdot w(\eta_{2})$

$\eta$ $\tau^{\prime\prime\prime}$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$
$a$ $w(\eta_{1})$ $\eta_{2}$ $\eta_{2}$ $-w(a)$ $a$ $\tau_{3}$ $1$ $a$
$\tau_{4}$ $1$ $\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$w(a)^{2}\cdot w(\eta_{1})$

Figure 3: Example of a cochain $f(\eta)$ applied to a sequence $\tau$ such
that $\left(f(\eta)\right)(\tau)=0$, and of the same cochain $f(\eta)$ applied
to sequences $\tau^{\prime},\tau^{\prime\prime}$ and
$\tau^{\prime\prime\prime}$ that have a single swapped vertex (highlighted in
yellow) with $\tau$. Figured here,
$\eta=(\eta_{0},\eta_{1},\eta_{2},a,a)\in\mathcal{X}_{4}$ and
$\tau=(\eta_{0},\eta_{1},\tau_{2},\tau_{3},\tau_{4})\in\mathcal{X}_{4}$ with
$\tau_{2}\neq\eta_{2}$ (in red), such that $\big{(}f(\eta)\big{)}(\tau)=0$. In
the second row we display the cochain $f(\eta)$ applied to
$\tau^{\prime},\tau^{\prime\prime},\tau^{\prime\prime\prime}\bowtie\tau$:
First, $\tau^{\prime}$ such that $\tau^{\prime}_{2}=\tau_{2}\neq\eta_{2}$
(still in red), in this case it would also hold that
$\big{(}f(\eta)\big{)}(\tau^{\prime})=0$. Then
$\tau^{\prime\prime}\&\tau^{\prime\prime\prime}$ such that
$\tau_{2}^{\prime\prime},\tau_{2}^{\prime\prime\prime}\neq\tau_{2}$ and
specifically $\tau_{2}^{\prime\prime}=a$ and
$\tau_{2}^{\prime\prime\prime}=\eta_{2}$, in these cases we would have
$\big{(}f(\eta)\big{)}(\tau^{\prime\prime}),\big{(}f(\eta)\big{)}(\tau^{\prime\prime\prime})\neq
0$.

$\eta$ $\sigma$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$ $a$ $w(\eta_{1})$
$\eta_{2}$ $\eta_{2}$ $-w(a)$ $a$ $\sigma_{3}$ $1$ $a$ $\sigma_{4}$ $1$
$\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$w(a)^{2}\cdot w(\eta_{1})$

$\eta$ $\sigma^{\prime}$ $\eta_{0}$ $\sigma^{\prime}_{0}$ $0$ $\eta_{1}$ $a$
$w(\eta_{1})$ $\eta_{2}$ $\eta_{2}$ $-w(a)$ $a$ $\sigma_{3}$ $1$ $a$
$\sigma_{4}$ $1$ $\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:,

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$0$

$\eta$ $\sigma^{\prime\prime}$ $\eta_{0}$ $a$ $w(\eta_{0})$ $\eta_{1}$ $a$
$w(\eta_{1})$ $\eta_{2}$ $\eta_{2}$ $-w(a)$ $a$ $\sigma_{3}$ $1$ $a$
$\sigma_{4}$ $1$ $\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:,

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$-w(a)\cdot w(\eta_{0})\cdot w(\eta_{1})$

$\eta$ $\sigma^{\prime\prime\prime}$ $\eta_{0}$ $\eta_{0}$ $-w(a)$ $\eta_{1}$
$\eta_{1}$ $-w(a)$ $\eta_{2}$ $\eta_{2}$ $-w(a)$ $a$ $\sigma_{3}$ $1$ $a$
$\sigma_{4}$ $1$ $\shortparallel$ $\left(\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=53.80931pt,depth=53.80931pt,width=0.0pt}\right)$$f$:

(

$\cdot$$\cdot$$\cdot$$\cdot$

)

$-w(a)^{3}$

Figure 4: Example of a cochain $f(\eta)$ applied to a sequence $\tau$ such
that $\big{(}f(\eta)\big{)}(\sigma)\neq 0$, and of the same cochain $f(\eta)$
applied to sequences $\sigma^{\prime},\sigma^{\prime\prime}$ and
$\sigma^{\prime\prime\prime}$ that have a single swapped vertex (highlighted
in yellow) with $\sigma$. Figured here
$\eta=(\eta_{0},\eta_{1},\eta_{2},a,a)\in\mathcal{X}_{4}$ and
$\sigma=(\eta_{0},\eta_{1},\eta_{2},\sigma_{3},\sigma_{4})\in\mathcal{X}_{4}$
such that $\big{(}f(\eta)\big{)}(\sigma)=w(a)^{2}\cdot w(\eta_{1})$. In the
second row, we display the cochain $f(\eta)$ applied to
$\sigma^{\prime},\sigma^{\prime\prime},\sigma^{\prime\prime\prime}\bowtie\sigma$:
First, $\sigma^{\prime}$ such that $\sigma^{\prime}_{0}\neq\eta_{0},a$ (now in
red), in this case it would hold that
$\big{(}f(\eta)\big{)}(\sigma^{\prime})=0$. Then
$\sigma^{\prime\prime}\&\sigma^{\prime\prime\prime}$ such that
$\sigma_{0}\neq\sigma_{0}^{\prime\prime}=a$ and
$\sigma_{1}\neq\sigma_{1}^{\prime\prime\prime}=\eta_{1}$, in these cases we
would have
$\big{(}f(\eta)\big{)}(\sigma^{\prime\prime}),\big{(}f(\eta)\big{)}(\sigma^{\prime\prime\prime})\neq
0,\big{(}f(\eta)\big{)}(\sigma)$.

As stated, we will begin by showing for all $n\in\mathbb{Z}_{\geq 0}$ the
cochain, $f({(a,\dots,a)})\in C^{n}(\mathcal{X})$, is an eigenvector.
Specifically, that it is associated to eigenvalue $\lambda=1$. By definition
$f_{0}(a)(y)=1$ for all $y\in V$, hence
$f({(a,\dots,a)})=f_{0}(a)\otimes\dots\otimes
f_{0}(a)=\sum_{\sigma\in\mathcal{X}_{n}}e_{\sigma}\in C^{n}(\mathcal{X})$ is
the constant one cochain, we will refer to it as
$\mathds{1}:=f({(a,\dots,a)})$. For any $\tau\in\mathcal{X}_{n}$, we may take
$v=\mathds{1}$ on the right hand side of (6), and get:

$\displaystyle\left[(n+2)-\sum_{i=0}^{n}w(\tau_{i})\right]\cdot\mathds{1}(\tau)-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\mathds{1}({\sigma})=(n+2)-\sum_{i=0}^{n}w(\tau_{i})-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})$ $\displaystyle=$
$\displaystyle(n+2)-\sum_{i=0}^{n}w(\tau_{i})-\sum_{j=0}^{n}\sum_{u\neq\tau_{j}}w(u)=(n+2)-\sum_{i=0}^{n}\sum_{u\in
V}w(u)=(n+2)-(n+1)=1$

Where the second to last equality was due to $\sum_{u\in V}w(u)=1$. We
conclude that ,

$\displaystyle
1\cdot\mathds{1}(\tau)=1=\left[(n+2)-\sum_{i=0}^{n}w(\tau_{i})\right]\cdot\mathds{1}(\tau)-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\mathds{1}({\sigma})\hskip
8.53581pt\text{ , }\forall\tau$

Proving $f({(a,\dots,a)})=\mathds{1}$ is an eigenvector corresponding to
$\lambda=1$.

We will now show that for any $k\in[0:n]$, and $\eta\in\mathcal{X}_{n}$ such
that $\\#\\{a\in\eta\\}=n-k$, the cochain $f(\eta)$ as defined in Definition
4.7, is an eigenvector associated to $\lambda=k+2$.

Fix $k\in\\{0,...,n\\}$ and $\eta\in\mathcal{X}_{n}$ such that
$\\#\\{a\in\eta\\}=n-k$. Denote by $I_{a}\subseteq[0:n]$ the set of indices
such that $\eta_{i}=a$ for all $i\in I_{a}$.As we mentioned earlier, we first
deal with sequences $\tau\in\mathcal{X}_{n}$ such that
$\big{(}f(\eta)\big{)}(\tau)=0$. Consider any sequence $\tau$ that admits
$i\not\in I_{a}$, such that the vertex $\tau_{i}\neq a,\eta_{i}$, by
definition, it holds that

$\big{(}f_{0}(\eta_{i})\big{)}(\tau_{i})=0\qquad\Longrightarrow\qquad\big{(}f(\eta)\big{)}(\tau)=0$

Fix such a $\tau$ and let $i^{*}\in[0:n]\setminus I_{a}$ be such that
$\tau_{i^{*}}\neq a,\eta_{i^{*}}$, again we look at the right hand side of
equation (6):

As $\big{(}f(\eta)\big{)}(\tau)=0$, we immediately have:

$\left[(n+2)-\sum_{i=0}^{n}w(\tau_{i})\right]\cdot\big{(}f(\eta)\big{)}(\tau)-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\big{(}f(\eta)\big{)}(\sigma)=-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\big{(}f(\eta)\big{)}(\sigma)$
(7)

Now, for $\sigma\bowtie\tau$ such that $\sigma_{j}\neq\tau_{j}$ and $j\neq
i^{*}$, we would still have $\big{(}f(\eta)\big{)}(\sigma)=0$. Moreover, if
$j=i^{*}$ but $\sigma_{i^{*}}\neq a,\eta_{i^{*}}$, we would also still have
that $\big{(}f(\eta)\big{)}(\sigma)=0$.
Hence the sum (7) reduces to:

$\displaystyle\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\big{(}f(\eta)\big{)}(\sigma)$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\sigma_{i^{*}}=a\end{subarray}}w(\sigma_{i^{*}})\cdot\big{(}f(\eta)\big{)}(\sigma)+\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\sigma_{i^{*}}={\eta_{i^{*}}}\end{subarray}}w(\sigma_{i^{*}})\cdot\big{(}f(\eta)\big{)}(\sigma)$
(8)

Furthermore, as there exists only one $\sigma$ such that $\tau\bowtie\sigma$
and $\sigma_{i^{*}}=a$, and one $\sigma$ such that $\tau\bowtie\sigma$ and
$\sigma_{i^{*}}=\eta_{i^{*}}$, we can reduce the sum (8) to the sum of exactly
two terms:

$\displaystyle\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\sigma_{i^{*}}=a\end{subarray}}w(\sigma_{i^{*}})\cdot\big{(}f(\eta)\big{)}(\sigma)+\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\sigma_{i^{*}}={\eta_{i^{*}}}\end{subarray}}w(\sigma_{i^{*}})\cdot\big{(}f(\eta)\big{)}(\sigma)$
$\displaystyle=$ $\displaystyle
w(a)\cdot\Biggl{(}(w(\eta_{i^{*}}))\cdot\prod_{\begin{subarray}{c}s\in[0:n],\\\
s\neq
i^{*}\end{subarray}}\big{(}f_{0}({\eta_{s}})\big{)}(\tau_{s})\Biggr{)}+w(\eta_{i^{*}})\cdot\Biggl{(}(-w(a))\cdot\prod_{\begin{subarray}{c}s\in[0:n],\\\
s\neq i^{*}\end{subarray}}\big{(}f_{0}({\eta_{s}})\big{)}(\tau_{s})\Biggr{)}$
$\displaystyle=$ $\displaystyle w(a)\cdot
w(\eta_{i})\cdot\prod_{\begin{subarray}{c}s\in[0:n],\\\ s\neq
i^{*}\end{subarray}}\big{(}f_{0}({\eta_{s}})\big{)}(\tau_{s})\cdot[1-1]=0$

We conclude that, for $\tau$ that admit an $i\in[0:n]\setminus I_{a}$ such
that the vertex $\tau_{i}\neq a,\eta_{i}$:

$\lambda\cdot\big{(}f(\eta)\big{)}(\tau)=0=\left[(n+2)-\sum_{i=0}^{n}w(\tau_{i})\right]\cdot\big{(}f(\eta)\big{)}(\tau)-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\big{(}f(\eta)\big{)}(\sigma)$

Proving the cochain $f(\eta)$ satisfies the system of equations corresponding
to the eigenvalue $\lambda$, on sequences $\tau$ with $\tau_{i}\neq
a,\eta_{i}$ for $i\in[0:n]\setminus I_{a}$. We will now show the system is
also satisfied for any other $\tau\in\mathcal{X}_{n}$. In this case for all
$i\in[0:n]\setminus I_{a}$ the vertices $\tau_{i}$ are forced to be either
$a,\eta_{i}$. Hence we deal as promised with the case of
$\tau\in\mathcal{X}_{n}$, such that $\big{(}f(\eta)\big{)}(\tau)=0$.

We use the swap operation from Definition 4.1 to rewrite the sum:

$\displaystyle\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\big{(}f(\eta)\big{)}(\sigma)=\sum_{j=0}^{n}\sum_{y\neq\tau_{j}}w(y)\cdot\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})$
(9)

We have assumed, $\eta_{j}=a$ for all $j\in I_{a}$. As such, for all $y\in V$:

$\displaystyle\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})=\big{(}f(\eta)\big{)}(\tau)\hskip
14.22636pt\forall j\in I_{a}$

Meaning we can reduce part of the sum in (9), when restricted to $j\in I_{a}$,
to:

$\displaystyle\sum_{j\in
I_{a}}\sum_{y\neq\tau_{j}}w(y)\cdot\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})=\sum_{j\in
I_{a}}\sum_{y\neq\tau_{j}}w(y)\cdot\big{(}f(\eta)\big{)}(\tau)=\sum_{j\in
I_{a}}\left(1-w(\tau_{j})\right)\cdot\big{(}f(\eta)\big{)}(\tau)$ (10)

Notice in the last line we used the fact that $\sum_{y\in V}w(y)=1$.

On the other hand, for $j\in[0:n]\setminus I_{a}$ we have:

$\displaystyle\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})$
$\displaystyle=\big{(}f_{0}(\eta_{j})\big{)}(y)\cdot\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})$

In particular, as we have assumed $\eta_{j}\neq a$, for $j\not\in I_{a}$, this
means

$\displaystyle\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})$
$\displaystyle=\begin{cases}\big{(}f_{0}(\eta_{j})\big{)}(y)\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})\neq 0\hskip 8.53581pt\text{if
}y=\eta_{j},a\\\ 0\hskip 142.26378pt\text{ if }y\neq\eta_{j},a\end{cases}$

We recall that for $j\in[0:n]\setminus I_{a}$ the vertices $\tau_{j}$ are
forced to be either $a,\eta_{j}$. Consider the case $\tau_{j}=a$, then we can
reduce part of the sum in (9) to:

$\displaystyle\sum_{y\neq\tau_{j}}w(y)\cdot\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})=w(\eta_{j})\cdot\big{(}f_{0}(\eta_{j})\big{)}(\eta_{j})\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})$ $\displaystyle=$ $\displaystyle
w(\eta_{j})\cdot(-w(a))\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})=\big{(}f_{0}(\eta_{j})\big{)}(\tau_{j})\cdot(-w(a))\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})=-w(\tau_{j})\cdot\big{(}f(\eta)\big{)}(\tau)$

Similarly, consider the case where $\tau_{j}=\eta_{j}$, again we can reduce
the sum to:

$\displaystyle\sum_{y\neq\tau_{j}}w(y)\cdot\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})=w(a)\cdot\big{(}f_{0}(\eta_{j})\big{)}(a)\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})$ $\displaystyle=$ $\displaystyle
w(a)\cdot w(\eta_{j})\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})=-\big{(}f_{0}(\eta_{j})\big{)}(\tau_{j})\cdot
w(\eta_{j})\cdot\displaystyle\prod_{s\neq
j}\big{(}f_{0}(\eta_{s})\big{)}(\tau_{s})=-w(\tau_{j})\cdot\big{(}f(\eta)\big{)}(\tau)$

Putting both these cases together we can rewrite the sum in (9) restricted to
$[0:n]\setminus I_{a}$, as:

$\displaystyle\sum_{j\not\in
I_{a}}\sum_{y\neq\tau_{j}}w(y)\cdot\big{(}f(\eta)\big{)}({{\tau}^{\mathrlap{\times}{+}_{j}:{y}}})$
$\displaystyle=\sum_{j\not\in
I_{a}}-w(\tau_{j})\cdot\big{(}f(\eta)\big{)}(\tau)$ (11)

Putting the restricted sums (10) and (11) together, and plugging into the
right hand side of (6), we see that for $\tau$ for which all vertices
$\tau_{i}$, for $i\not\in I_{a}$, are forced to be either $a,\eta_{i}$, we
have:

$\displaystyle\left[(n+2)-\sum_{j=0}^{n}w(\tau_{j})\right]\cdot\big{(}f(\eta)\big{)}(\tau)-\sum_{\begin{subarray}{c}\sigma:\sigma\bowtie\tau,\\\
\tau_{j}\neq\sigma_{j}\end{subarray}}w(\sigma_{j})\cdot\big{(}f(\eta)\big{)}({\sigma})$
$\displaystyle=$
$\displaystyle\left[(n+2)-\sum_{j=0}^{n}w(\tau_{j})\right]\cdot\big{(}f(\eta)\big{)}(\tau)+\sum_{j\not\in
I_{a}}w(\tau_{j})\cdot\big{(}f(\eta)\big{)}(\tau)-\sum_{j\in
I_{a}}\left(1-w(\tau_{j})\right)\cdot\big{(}f(\eta)\big{)}(\tau)=(k+2)\cdot\big{(}f(\eta)\big{)}(\tau)$

Notice in the last equality we used $|I_{a}|=n-k$, as per our assumption. With
this we conclude $f(\eta)$ satisfies the system of equations corresponding to
the eigenvalue $\lambda=k+2$ for all sequences $\tau\in\mathcal{X}_{n}$.
Proving $f(\eta)$ is an eigenvector for $\lambda=k+2$ for $\eta$ such that
$\\#\\{a\in\eta\\}=n-k$.

As we know the cochains $f(\eta)$ form a basis for $C^{n}$, they must span the
eigenspaces of the Laplacian, $L_{n}$. Hence the multiplicity of each
eigenvalue $\lambda=1,...,n+2$ is easily computed to be,

$\displaystyle\text{mult}(\lambda)$
$\displaystyle=\\#\Big{\\{}\eta\in\mathcal{X}_{n}:\\#\\{a\in\eta\\}=n+2-\lambda\Big{\\}}=\binom{n+1}{\lambda-1}\cdot(|V|-1)^{\lambda-1}$

With this, we finish the proof of the main theorem. ∎

##  5 Proof of Theorem 3.4 for Simplicial Complexes

In this section we prove the analogous results to the previous section’s, in
the setting of simplicial complexes. As we mentioned earlier, the results in
this section, if at all new, are simple consequences of classical results for
the Hodge Laplacian. We present here results, old and new, for completion, and
to provide a parallel to the sequence complex setting.

###  5.1 Laplacian over Simplicial Complexes

We will consider a weighted simplicial complex $K\subseteq 2^{[m]}$ with
weight function $w:K\to(0,+\infty)$ and associated inner product
$\langle\cdot,\cdot\rangle$ on $C^{*}(K)$, defined via its values on the usual
basis$\\{e_{\xi}\\}$ as:

$\langle
e_{\xi},e_{\xi^{\prime}}\rangle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\begin{cases}w(\xi),&\text{
if }\xi=\xi^{\prime}\\\ 0&\text{ if }\xi\neq\xi^{\prime}\end{cases}$

In this setting the coboundary ${\delta}$ and coboundary adjoint
${\delta}^{*}$ operators are defined for any face
$\xi=\\{i_{0},...,i_{n+1}\\}\in K_{n+1}$, with vertices labeled such that
$i_{0}<\dots<i_{n+1}$, as:

$\displaystyle{\delta}_{n}e_{\xi}$
$\displaystyle=\sum_{\begin{subarray}{c}i\in[m]\setminus\xi,\\\
\\{i\\}\cup\xi\in K\end{subarray}}\kappa({\\{i\\}\cup\xi},{\xi})\cdot
e_{\\{i\\}\cup\xi}$ $\displaystyle{\delta}_{n}^{*}e_{\xi}$
$\displaystyle=\sum_{\xi^{\prime}\in
K_{n}}\frac{w(\xi)}{w(\xi^{\prime})}\kappa({\xi},{\xi^{\prime}})e_{\xi^{\prime}}=\sum_{j=0}^{n+1}\frac{w(\xi)}{w(\xi\setminus\\{i_{j}\\})}(-1)^{j}e_{\xi\setminus\\{i_{j}\\}}$

Where the incidence function $\kappa$ is defined as following Definition 2.2.
In this setting, the Laplacian is easily computed as stated in the following
lemma.

######  Lemma 5.1.

Let $2^{[m]}$ be a weighted full simplex, with weight function
$w:2^{[m]}\to(0,1]$. For all $n\in\mathbb{Z}_{\geq-1}$ and faces
$\xi=\\{i_{0},...,i_{n}\\}\in K_{n}$, with vertices labeled such that
$i_{0}<\dots<i_{n}$, the Laplacian operator on the basis elements $e_{\xi}\in
C^{n}(2^{[m]})$, is given by:

$\displaystyle L_{n}(e_{\xi})=$
$\displaystyle\left(\sum_{i\in[m]\setminus\xi}\dfrac{w(\\{i\\}\cup\xi)}{w(\xi)}+\sum_{k=0}^{n}\dfrac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}\right)\cdot
e_{\xi}$
$\displaystyle+\sum_{i\in[m]\setminus\xi}\sum_{k=0}^{n}(-1)^{k}\cdot\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot\Bigg{(}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}-\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{i_{k}\\})}\Bigg{)}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}$

###### Proof.

We compute $L_{n}^{down}$ and $L_{n}^{up}$ separately, by plugging in our
computations for ${\delta}$ and ${\delta}^{*}$ each time. Consider a face
$\xi=\\{i_{0},...,i_{n}\\}\in K_{n}$, with vertices labeled such that
$i_{0}<\dots<i_{n}$:

$\displaystyle L_{n}^{up}e_{\xi}=$
$\displaystyle{\delta}^{*}{\delta}_{n}e_{\xi}={\delta}^{*}\left(\sum_{i\in[m]\setminus\xi}\kappa({\\{i\\}\cup\xi},{\xi})\cdot
e_{\\{i\\}\cup\xi}\right)$ $\displaystyle=$
$\displaystyle\sum_{i\in[m]\setminus\xi}\kappa({\\{i\\}\cup\xi},{\xi})\cdot\Bigg{[}\sum_{j\in\\{i\\}\cup\xi}\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{j\\})}\kappa(\\{i\\}\cup\xi,\\{i\\}\cup\xi\setminus\\{j\\})\cdot
e_{\\{i\\}\cup\xi\setminus\\{j\\}}\Bigg{]}$ $\displaystyle=$
$\displaystyle\sum_{i\in[m]\setminus\xi}\kappa({\\{i\\}\cup\xi},{\xi})\cdot\Bigg{[}\frac{w(\\{i\\}\cup\xi)}{w(\xi)}\kappa({\\{i\\}\cup\xi},{\xi})\cdot
e_{\xi}$ $\displaystyle+\sum_{\begin{subarray}{c}i_{k}\in\xi,\\\
i_{k}<i\end{subarray}}\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{i_{k}\\})}(-1)^{k}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}+\sum_{\begin{subarray}{c}i_{k}\in\xi,\\\
i_{k}>i\end{subarray}}\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{i_{k}\\})}(-1)^{k+1}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}\Bigg{]}$ (12)

Here we made use of the simple observation:

$\displaystyle\kappa(\\{i\\}\cup\xi,\\{i\\}\cup\xi\setminus\\{i_{k}\\})=\begin{cases}(-1)^{k}\hskip
14.22636pt&\text{if }i_{k}<i\\\ (-1)^{k+1}\hskip 14.22636pt&\text{if
}i_{k}>i\end{cases}$

On the other hand, notice:

$\displaystyle\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})=\begin{cases}-\kappa(\\{i\\}\cup\xi,\xi)\hskip
14.22636pt&\text{if }i_{k}<i\\\ \kappa(\\{i\\}\cup\xi,\xi)\hskip
14.22636pt&\text{if }i_{k}>i\end{cases}$

We can use this, together with
$\Big{(}\kappa(\\{i\\}\cup\xi,\xi\Big{)}^{2}=1$, to simplify the expression
from (5.1) :

$\displaystyle
L_{n}^{up}e_{\xi}=\sum_{i\in[m]\setminus\xi}\frac{w(\\{i\\}\cup\xi)}{w(\xi)}\cdot
e_{\xi}+\sum_{i\in[m]\setminus\xi}\sum_{k=0}^{n}\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{i_{k}\\})}\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot(-1)^{k+1}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}$

Now let us look at $L_{n}^{down}$,

$\displaystyle
L_{n}^{down}e_{\xi}={\delta}{\delta}_{n-1}^{*}e_{\xi}={\delta}\left(\sum_{k=0}^{n}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}(-1)^{k}\cdot
e_{\xi\setminus\\{i_{k}\\}}\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}(-1)^{k}\Bigg{[}\kappa(\xi,\xi\setminus\\{i_{k}\\})\cdot
e_{\xi}+\sum_{i\in[m]\setminus\xi}\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}\Bigg{]}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}(-1)^{2k}\cdot
e_{\xi}+\sum_{i\in[m]\setminus\xi}\sum_{k=0}^{n}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}(-1)^{k}\cdot\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}$

Putting $L_{n}^{down}$ and $L_{n}^{up}$ together, we get

$\displaystyle L_{n}(e_{\xi})=$ $\displaystyle
L^{\text{down}}_{n}(e_{\xi})+L^{\text{up}}_{n}(e_{\xi})=\left(\sum_{i\in[m]\setminus\xi}\dfrac{w(\\{i\\}\cup\xi)}{w(\xi)}+\sum_{k=0}^{n}\dfrac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}\right)\cdot
e_{\xi}$
$\displaystyle+\sum_{i\in[m]\setminus\xi}\sum_{k=0}^{n}(-1)^{k}\cdot\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot\Bigg{(}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}-\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{i_{k}\\})}\Bigg{)}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}$

Proving the expression in the proposition statement. ∎

An immediate consequence of this lemma, tells us that scaling the weight
function, by the same factor over all dimensions, has no impact on the
associated Laplacian.

######  Lemma 5.2.

Let $2^{[m]}$ be a weighted full simplex, with weight function
$w:2^{[m]}\to(0,1]$. Consider a different weight function
$w^{\prime}:2^{[m]}\to(0,1]$, defined as $w^{\prime}(\xi)=\alpha\cdot w(\xi)$
for all $\xi\in 2^{[m]}$, for some $\alpha\in\mathbb{R}$. The respective
associated Laplacians $L^{w}_{n}$ and $L^{w^{\prime}}_{n}$ satisfy

$L^{w}_{n}=L^{w^{\prime}}_{n}\quad\text{ for all }n\geq-1.$

It is also sometimes useful to think of the matrix representation of $L_{n}$
directly in terms of the matrix representation of ${\delta}$. This
particularly comes in useful when relating to the combinatorial Laplacian. Let
$D_{n}$ be the matrix representation of ${\delta}_{n}$ and $W_{n}$ the
diagonal matrix representing the inner product on $C^{n}(K)$. Then we may
express the $L_{n}^{\text{up}},L_{n}^{\text{down}}$ operators as:

$\displaystyle L_{n}^{\text{up}}$
$\displaystyle=W_{n}^{-1}D_{n}^{T}W_{n+1}D_{n}$ $\displaystyle
L_{n}^{\text{down}}$ $\displaystyle=D_{n-1}W_{n-1}^{-1}D_{n-1}^{T}W_{n}$

Consider $K=G$ to be a weighted graph. Denote by $\mathbb{A}$ the diagonal
matrix of vertex weights, following our definition $\mathbb{A}=W_{0}$. On the
other hand, let $\mathbb{W}$ be the weighted adjacency matrix:
$\mathbb{W}_{ij}=w(\\{i,j\\})$, and $\mathbb{D}$ be the degree matrix given
by, $\mathbb{D}_{ii}=\sum_{j\neq i}\mathbb{W}_{ij}$ . It is easy to show that,

$\mathbb{D}-\mathbb{W}=D_{0}^{T}W_{1}D_{0}.$

Which implies

$L_{0}^{\text{up}}=W_{0}^{-1}D_{0}^{T}W_{1}D_{0}=\mathbb{A}^{-1}[\mathbb{D}-\mathbb{W}]=L_{\text{comb}}$

As we had referenced previously in Section 2.5.

###  5.2 Laplacian on Simplicial Complexes for the Independent Vertices Model

Here we now show the Laplacian for the independent vertices model is “boring";
we recall the theorem we stated in section 3:

######  Theorem 3.4.

Let $(2^{[m]},w)$ be a weighted simplicial complex, with weight function
$w:2^{[m]}\to\mathbb{R}_{>0}$ such that $w(\varnothing)=1$. Then the following
two conditions are equivalent:

  1. (i)

$(2^{[m]},w)$ is an independent vertices model;

  2. (ii)

the associated Laplacians are multiples of identities:

$L_{n}=\alpha_{n}\cdot I_{C^{n}},\quad\forall n\geq-1,$ (13)

where $I_{C^{n}}$ denotes the identity on $C^{n}(2^{[m]})$, and
$\alpha_{n}\in\mathbb{R}$.

Moreover, if (13) holds, then $\alpha_{n}$ is constant with respect to $n$,
and

$\alpha_{n}=\sum_{i=1}^{m}w_{i},$

where $(w_{1},\dots,w_{m})$ is the vector determining the independent vertices
model associated to the weight function $w$.

###### Proof.

We start by proving the reverse direction: we assume our simplicial complex
follows the independent weights model, and wish to prove the associated
Laplacian $L_{n}=\alpha_{n}\cdot I_{C^{n}(2^{[m]})}$. Notice that for
$n=-1,m-1$, we have $|(2^{[m]})_{n}|=1$, making the statement trivial. We now
focus on the cases for $n=0,...,m-1$: Fix $e_{\xi}\in C^{n}(2^{[m]})$, we call
upon Lemma 5.1:

$\displaystyle
L_{n}(e_{\xi})=\sum_{i\in[m]\setminus\xi}\sum_{k=0}^{n}(-1)^{k}\cdot\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot\Bigg{(}\frac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}-\frac{w(\\{i\\}\cup\xi)}{w((\\{i\\}\cup\xi)\setminus\\{i_{k}\\})}\Bigg{)}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}$ $\displaystyle+$
$\displaystyle\left(\sum_{i\in[m]\setminus\xi}\dfrac{w(\\{i\\}\cup\xi)}{w(\xi)}+\sum_{k=0}^{n}\dfrac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}\right)\cdot
e_{\xi}\hskip 5.69054pt=\hskip
5.69054pt\left(\sum_{i\in[m]\setminus\xi}w(i)+\sum_{k=0}^{n}w(i_{k})\right)\cdot
e_{\xi}$ $\displaystyle+$
$\displaystyle\sum_{i\in[m]\setminus\xi}\sum_{k=0}^{n}(-1)^{k}\cdot\kappa(\\{i\\}\cup\xi\setminus\\{i_{k}\\},\xi\setminus\\{i_{k}\\})\cdot\Big{(}w(i_{k})-w(i_{k})\Big{)}\cdot
e_{\\{i\\}\cup\xi\setminus\\{i_{k}\\}}\hskip 5.69054pt=\hskip
5.69054pt\left(\sum_{i\in[m]}w(i)\right)\cdot e_{\xi}+0$

Proving $L_{n}=\alpha_{n}\cdot I_{C^{n}(2^{[m]})}$ for
$\alpha_{n}=\sum_{i\in[m]}w(i)$ for all $n=0,...,m-2$. In the cases of
$n=-1,m-1$, the computation of $L_{n}$ easily shows that $\alpha_{n}$ must
also be $\alpha_{n}=\sum_{i\in[m]}w(i)$.

We now proceed to prove the forward direction, we assume for all
$n=-1,\dots,m$, it holds that $L_{n}=\alpha_{n}\cdot I_{C^{n}(2^{[m]})}$.
Putting this assumption together with Lemma 5.1 tells us that for all
$n\in[-1:m]$, face $\xi\in\left(2^{[m]}\right)_{n}$, and vertices
$i\in[m]\setminus\xi,i_{k}\in\xi$ it holds that

$\dfrac{w(\\{i\\}\cup\xi)}{w(\\{i\\}\cup\xi\setminus\\{i_{k}\\})}=\dfrac{w(\xi)}{w(\xi\setminus\\{i_{k}\\})}$

We will use this consequence of our hypothesis to prove the weights follow the
independent vertices model by applying induction over $n$. The statement is
assumed in the proposition statement for $n=-1$ and vacuous for $n=0$, let us
set $n=1$. Consider $\\{i,j\\}\in\left(2^{[m]}\right)_{1}$, by hypothesis:

$\displaystyle\dfrac{w(\\{i\\}\cup\\{j\\})}{w(\\{i\\}\cup\\{j\\}\setminus\\{j\\})}=\dfrac{w(\\{j\\})}{w(\\{j\\}\setminus\\{j\\})}\hskip
5.69054pt\iff\hskip
5.69054pt\dfrac{w(\\{i,j\\})}{w(\\{i\\})}=\dfrac{w(\\{j\\})}{w(\varnothing)}$

As the proposition statement assumes $w(\varnothing)=1$, we conclude
$w(\\{i,j\\})=w(\\{i\\})w(\\{j\\})$, proving the base case. Now assume the
result holds for faces of dimension $n$, we will show it holds for faces of
dimension $n+1$.
Consider $\\{i_{0},\dots,i_{n+1}\\}\in\left(2^{[m]}\right)_{n+1}$, by
hypothesis:

$\displaystyle\dfrac{w(\\{i_{0},\dots,i_{n}\\}\cup\\{i_{n+1}\\})}{w(\\{i_{0},\dots,i_{n}\\}\cup\\{i_{n+1}\\}\setminus\\{i_{0}\\})}=\dfrac{w(\\{i_{0},\dots,i_{n}\\})}{w(\\{i_{0},\dots,i_{n}\\}\setminus\\{i_{0}\\})}\hskip
5.69054pt\iff\hskip
5.69054pt\dfrac{w(\\{i_{0},\dots,i_{n+1}\\})}{w(\\{i_{1},\dots,i_{n+1}\\})}=\dfrac{w(\\{i_{0},\dots,i_{n}\\})}{w(\\{i_{1},\dots,i_{n})}$

Applying the induction hypothesis, we get

$\displaystyle
w(\\{i_{0},\dots,i_{n+1}\\})=\dfrac{\prod_{k=1}^{n+1}w(i_{k})\prod_{k=0}^{n}w(i_{k})}{\prod_{k=1}^{n}w(i_{k})}=\prod_{k=0}^{n+1}w(i_{k})$

proving the induction step, and the proof for the theorem. ∎

## Conclusion

We introduced a general framework for weighted sequence complexes, where the
weights are derived from probability distributions on sequences, and extended
the Hodge Laplacian formalism to this setting. An important requirement for
such a Laplacian is that it has a “simple” spectrum in the null models with
independent vertices.

We completely characterized the spectrum of the associated Laplacian for the
independent vertices model, both in the sequence and simplicial complex
setting. For simplicial complexes we found (Corollary 3.5) that the
appropriate Laplacians are multiples of the identity. This happens with both
natural choices of the weights: (i) if the weights are the probabilities and
(ii) if the weights are the moments, and confirms a “common sense” intuition
that there should not be any preferred eigenbasis in the independent model, as
the data does not have any “interesting” structure beyond the independence of
vertices.

We found, that unlike in the simplicial complex case, the Laplacian of the
independent vertices model for the sequence complex has a stereotypical
structure (Theorem 3.1). Specifically, it has distinct integer eigenvalues,
with multiplicities that scale exponentially in eigenvalues. Here the weight
function on the sequence complex was chosen to be the conditional probability
per equation (1). This choice is natural in this context, as the length of the
sequence is another random variable that does not depend on the assumption
that vertices appear in each sequence position independently. The eigenbasis
for this independent vertices model provides the analogue of the Fourier
transform in the context of the Laplacians in a Euclidean domain.

There are many open questions that stem from our results. For example, both of
our independent vertices models can be thought of as maximal entropy
distributions that are constrained by the first moments. What is special about
the spectrum of common maximal entropy distributions (such as the Ising
model), constrained by the higher moments? What is the interpretation of the
diffusion equation for the Hodge Laplacian on the weighted sequence complex?
What can be said about the probability distributions on sequence complexes
that are Markovian? Can one utilize the Hodge Laplacian on sequences for
construction of the language models, similar to how the Hodge Laplacian was
used in simplicial neural networks [14, 15, 16, 17, 18, 19, 20, 21]? We
believe that our results establish a foundation for dimensionality reduction
and Fourier analyses of probabilistic models, that are common in theoretical
neuroscience and machine-learning.

Acknowledgments: This work was supported by the NSF Next Generation Networks
for Neuroscience Program (award 2014217).

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