# On the Stability of Low Pass Graph Filter with A Large Number of Edge
Rewires

###### Abstract

Recently, the stability of graph filters has been studied as one of the key
theoretical properties driving the highly successful graph convolutional
neural networks (GCNs). The stability of a graph filter characterizes the
effect of topology perturbation on the output of a graph filter, a fundamental
building block for GCNs. Many existing results have focused on the regime of
small perturbation with a small number of edge rewires. However, the number of
edge rewires can be large in many applications. To study the latter case, this
work departs from the previous analysis and proves a bound on the stability of
graph filter relying on the filter’s frequency response. Assuming the graph
filter is low pass, we show that the stability of the filter depends on
perturbation to the _community structure_. As an application, we show that for
stochastic block model graphs, the graph filter distance converges to zero
when the number of nodes approaches infinity. Numerical simulations validate
our findings.

Index Terms—  stability of graph filter, low pass graph filter, stochastic
block model

##  1 Introduction

Nowadays, developing models for graph structured data is popular across
disciplines [1, 2] with the increased availability of data in domains such as
social networks and biological networks. One of the main thrusts is to exploit
the irregular relationships between data samples through modeling the latter
using graphs that encode arbitrary pairwise relationships. Among others, graph
convolutional networks (GCNs) [3, 4, 5] are shown to be efficient in learning
graph structured data.

Despite the highly successful applications, there has been few theoretical
studies on _“why GCNs are powerful models for graph structured data?”_. As
GCNs are built on graph filters which replace the classical convolution
operation in convolutional neural networks, one possible approach is to
evaluate the efficacy of GCNs via analyzing the _stability of graph filter_
[6]. Formally, we consider an original graph and its perturbation such that we
measure the maximum change in the graph filter output in $\ell_{2}$ norm when
subjected to a unit norm excitation signal. The latter is also known as the
graph filter distance. The concept of stability is related to transferability
which studies the generalization property of a trained GCN when applied on
datasets with different distribution or embedded graph topology.

Among the existing attempts, [6] provided a comprehensive study on the
stability of graph filters utilizing an integral Lipschitz property, while [7,
8, 9, 10] have derived bounds surrounding the polynomial graph filter models
with normalized Laplacian matrices as graph shift operator (GSO); also see
[11, 12, 13]. As a common trail, many of the above works yielded an upper
bound on the graph filter distance as ${\mathcal{}O}(\|{\bf S}-\hat{\bf
S}\|_{2})$ which is proportional to the number of edge rewires, where ${\bf
S},\hat{\bf S}$ are respectively the original and perturbed graph
Laplacian/adjacency matrices. As such, they reach the conclusion that the
stability of graph filter holds in scenarios when only a few edges are rewired
in the event of graph perturbation. When the graph perturbation involves a
large number of edge rewiring, the above bounds may become uninformative. This
raises an interesting question: _what are the conditions for graph filter to
be stable when subjected to a large number of edge rewiring_?

This work provides an affirmative answer to the above question. Our intuition
is that as long as the community structure of a graph is stable upon graph
perturbation, e.g., only the edges within the same community are rewired, then
the graph filter shall be stable regardless of the number of edge rewires
since the perturbed graph has similar ‘shape’ as the original one. To confirm
this intuition, we depart from existing analysis which rely on properties of
polynomial graph filter. Instead, we treat the graph filter distance using
analysis in the graph frequency domain. Assuming that the graph filter is _low
pass_ [14] such that the filter suppresses high graph frequency components of
its input, our analysis relates the stability of graph filter to changes in
community structure of graph topology. Finally, we yield a graph distance
bound that can be _invariant_ to the number of edge rewires in the graph
perturbation. Our contributions are:

  * •

We derive the first _community structure_ dependent bound on graph filter
distance which accounts for the perturbation to the graph topology in terms of
community structure and the low pass filtering capability of graph filter, see
Theorem 1.

  * •

Our bound is specialized to the class of graph perturbation generated from a
homogeneous stochastic block model. When the graph filter is sufficiently low
pass, we show that the _graph filter distance approaches zero_ as the number
of nodes grows, see Corollaries 1 and 2. We emphasize that our result is
applicable to _unnormalized Laplacian_ as GSO, which is not covered by most of
the previous works on stability of graph filters.

Lastly, numerical simulations on synthetic and real graphs are provided to
verify our findings.

Related Works. The community structure of a graph emerges when the number of
nodes is large. As such, our study is closely related to works which study the
convergence properties of graph filters with very large graphs. For instance,
[15] studied the generalization of graph signal processing (GSP) [16] to
graphons, [17, 18] analyzed the stability of GCNs with an infinite number of
nodes/data. We remark that our Corollary 1 can be viewed as an extension of
[18] to the unnormalized Laplacian as GSO.

Notations. We use $||\cdot||_{2}$ to denote spectral norm for matrices and
Euclidean norm for vectors. For a vector (resp. matrix) ${\bf x}$ (resp. ${\bf
X}$), we use $[{\bf x}]_{i}$ (resp. $[{\bf X}]_{ij}$) to denote its $i$th
(resp. $(i,j)$th) entry.

##  2 Preliminaries

Graphs and Graph Signals. We consider undirected and connected graphs
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is set of $n$
nodes and $\mathcal{E}$ is the edge set. The graph ${\mathcal{}G}$ is endowed
with a graph shift operator (GSO), which is defined as a symmetric matrix
$\mathbf{S}\in\mathbb{R}^{n\times n}$ such that $[\mathbf{S}]_{ij}\neq 0$ if
and only if $i=j$ or $(i,j)\in\mathcal{E}$ [16]. We focus on two common GSOs:
unnormalized Laplacian, normalized Laplacian. Given the adjacency matrix
$\mathbf{A}\in\mathbb{R}^{n\times n}$ and the degree matrix $\mathbf{D}={\rm
Diag}(\mathbf{A}\mathbf{1})$, the unnormalized Laplacian is defined as
$\mathbf{L}_{U}=\mathbf{D}-\mathbf{A}$ and the normalized Laplacian is defined
as $\mathbf{L}_{\sf norm}=\mathbf{D}^{-1/2}\mathbf{L}_{U}\mathbf{D}^{-1/2}$.
The GSO ${\bf S}$ admits an eigendecomposition as
$\mathbf{S}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^{\top}$, where
$\mathbf{V}=(\mathbf{v}_{1},...,\mathbf{v}_{n})\in\mathbb{R}^{n\times n}$
collects the orthonormal eigenvectors and $\mathbf{\Lambda}={\rm
Diag}(\lambda_{1},...,\lambda_{n})$ is a diagonal matrix of the eigenvalues.
Throughout this paper, we order the eigenvalues such that
$0=\lambda_{1}<\lambda_{2}\leq\cdots\leq\lambda_{n}$.

The eigenvalues of ${\bf S}$ can be interpreted as the _graph frequencies_
with the smaller eigenvalues correspond to lower frequencies. This can be
understood with the graph total variation ${\bf x}^{\top}{\bf S}{\bf
x}/2=\sum_{(i,j)\in{\mathcal{}E}}(x_{i}-x_{j})^{2}$ for the graph signal on
${\mathcal{}G}$ represented by ${\bf x}\in\mathbb{R}^{n}$. As ${\bf
v}_{i}^{\top}{\bf S}{\bf v}_{i}=\lambda_{i}$, we observe that the eigenvector
${\bf v}_{i}$ has higher variations with respect to ${\mathcal{}G}$ as the
frequency index $i$ increases. We define the _graph Fourier transform_ (GFT)
of ${\bf x}$ as $\widehat{\bf x}={\bf V}^{\top}{\bf x}$ such that
$[\widehat{\bf x}]_{i}$ denotes the magnitude of the $i$th frequency
component.

Low Pass Graph Filters. A graph filter
$\mathcal{H}(\mathbf{S})\in\mathbb{R}^{n\times n}$ maps the input signal
$\mathbf{x}\in\mathbb{R}^{n}$ to the output signal
$\mathbf{y}=\mathcal{H}(\mathbf{S})\mathbf{x}\in\mathbb{R}^{n}$. Specifically,
we consider graph filter of the form:

$\textstyle\mathcal{H}(\mathbf{S})=\sum_{t=0}^{T-1}h_{t}\mathbf{S}^{t}.\vspace{-.1cm}$
(1)

Note that $T\in\mathbb{N}$ is the filter’s order. Based on the
eigendecomposition of GSO
$\mathbf{S}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^{\top}$, we define the
frequency responses as $h(\lambda):=\sum_{t=0}^{T-1}h_{t}\lambda^{t}$ and
subsequently the graph filter can be written as
$\mathcal{H}(\mathbf{S})=\mathbf{V}h(\mathbf{\Lambda})\mathbf{V}^{\top}$,
where $h(\mathbf{\Lambda})={\rm
Diag}\left(h(\lambda_{1}),...,h(\lambda_{n})\right)$. We note that the GFT of
${\bf y}$ can be written as $\widehat{\bf y}=h(\bm{\Lambda})\widehat{\bf x}$.

This study is concerned with ${\mathcal{}H}(\cdot)$ that are _low pass graph
filters_ which retain the low frequency components of its input while
suppressing the high frequency components. We define this class of graph
filters of interest with respect to the frequency response function
$h(\lambda)$. Our definition is extended from [14] as follows:

######  Definition 1.

The graph filter $\mathcal{H}(\cdot)$ is said to be $\bar{\lambda}$-low pass
if

$\eta=\big{(}\min\limits_{\lambda\in\left[0,\bar{\lambda}\right]}\left|h(\lambda)\right|\big{)}^{-1}{\max\limits_{\lambda\in\left[\bar{\lambda},\infty\right)}\left|h(\lambda)\right|}<1,\vspace{-.2cm}$
(2)

where $\eta<1$ is known as the low pass ratio for the graph filter.

The above definition requires the graph filter ${\mathcal{}H}(\cdot)$ to have
a cutoff frequency at $\bar{\lambda}$ such that any components with graph
frequency above $\bar{\lambda}$ are attenuated by a factor of at least
$\eta<1$. Unlike [14], our definition is applicable regardless of the GSO
${\bf S}$ used as it only constraints on the form of the graph filter function
determined by the filter coefficients $\\{h_{t}\\}_{t=0}^{T-1}$. Lastly, the
cutoff frequency $\bar{\lambda}$ maybe different depending on the graph size
$n$ to achieve a reasonable low pass filtering performance.

Stability of Graph Filters. We consider a perturbation of ${\mathcal{}G}$ into
$\hat{\mathcal{}G}$, e.g., via edge rewiring, which results in the GSO
$\hat{\bf S}$. Our goal is to study the _stability of ${\mathcal{}H}({\bf
S})$_ via analyzing the amount of changes in the filter’s output under
perturbation. As inspired by [6], we measure the _stability of
${\mathcal{}H}({\bf S})$_ using the quantity:

$\displaystyle\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})=\sup_{{\bf
x}\in\mathbb{R}^{n},{\bf x}\neq{\bf
0}}\frac{\big{\|}\mathcal{H}(\mathbf{S})\mathbf{x}-\mathcal{H}(\mathbf{\hat{S}})\mathbf{x}\big{\|}_{2}}{||\mathbf{x}||_{2}}=\|\mathcal{H}(\mathbf{S})-\mathcal{H}(\hat{\bf
S})\|_{2}.$ (3)

We call the quantity $\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})$ the graph
filter distance between the filters $\mathcal{H}(\mathbf{S})$ and
$\mathcal{H}(\mathbf{\hat{S}})$. Notice that (3) yields an upper bound to the
operator distance modulo permutation measure $\|\cdot\|_{\mathcal{}P}$ defined
in [6] since we have ignored the ambiguity due to node permutations.

Stochastic Block Model. To derive stability properties of graph filters under
a large number of edge rewires, we concentrate on the stochastic block models
(SBMs) random graphs to develop insights. Here, ${\sf SBM}(n,k,{\bf B},{\bf
Z})$ [19] denotes a random graph model with $n$ nodes partitioned into $k$
blocks, the membership matrix $\mathbf{Z}\in\\{0,1\\}^{n\times k}$ such that
$[\mathbf{Z}]_{ij}=1$ if and only if node $i$ is in community $j$, and a
connectivity matrix $\mathbf{B}=[b_{ij}]_{1\leq i,j\leq k}\in[0,1]^{k\times
k}$, whose entries $b_{ij}$ being the probability of edges between nodes in
block $i$ and block $j$. Let ${\mathcal{}G}\sim{\sf SBM}(n,k,{\bf B},{\bf
Z})$, its empirical adjacency matrix $\mathbf{A}$ satisfies
$\mathbb{E}[\mathbf{A}]=\mathcal{A}=\mathbf{Z}\mathbf{B}\mathbf{Z}^{\top}$,
which is referred to as the population adjacency matrix. Lastly, we also
consider a special case of SBMs given by the planted partition model ${\sf
PPM}(n,k,a_{n},b_{n})$ which has $k$ equally sized blocks with $n/k$ nodes per
block. The PPM is a special case of an SBM where the connectivity matrix is
given by $\mathbf{B}=a_{n}\mathbf{I}+b_{n}\mathbf{1}\mathbf{1}^{T}$.

##  3 Main results

This section presents our main findings on the stability property of low pass
graph filters. We first present a bound on $\mathbb{D}_{\mathcal{}H}({\bf
S},\hat{\bf S})$ which explicitly considers the strength of low pass filtering
and the structural difference between ${\bf S},\hat{\bf S}$. Let us introduce
two assumptions for simplifying the constants in our bound:

######  H 1 ($\bar{\lambda}$).

There exists a constant $\mathbb{H}_{\max}$ such that

$\textstyle\sup_{\lambda\in[0,\bar{\lambda}]}|h(\lambda)|\leq\mathbb{H}_{\max}.\vspace{-.2cm}$
(4)

######  H 2 ($\bar{\lambda}$).

There exists a constant $L_{\mathbb{H}}$ such that

$|h(\lambda)-h(\lambda^{\prime})|\leq
L_{\mathbb{H}}\,|\lambda-\lambda^{\prime}|,~{}\forall~{}\lambda,\lambda^{\prime}\in[0,\bar{\lambda}].\vspace{-.1cm}$
(5)

H1 assumes that the frequency response function of the graph filter is bounded
in the interval $[0,\bar{\lambda}]$. Meanwhile, H2 imposes a Lipschitz
continuity condition on the frequency response function. Note that H2 is
weaker than a uniform Lipschitz filter condition as we only consider the graph
frequencies up to $\lambda\leq\bar{\lambda}$, and when $\bar{\lambda}$ is
small, it can imply the integral Lipschitz condition in [6], i.e.,
$\sup_{\lambda\geq 0}|\lambda h^{\prime}(\lambda)|\leq C$. An example
satisfying Definition 1, H1, H2 is the exponential graph filter
${\mathcal{}H}({\bf S})=e^{-\sigma{\bf S}}$, where $\sigma>0$.

The following theorem bounds the graph filter distance
$\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})$. For brevity, we denote
$\mathbf{\Lambda}_{k}$ and $\mathbf{V}_{k}$ (resp.
$\mathbf{\hat{\Lambda}}_{k}$ and $\mathbf{\hat{V}}_{k}$) as the matrices of
smallest-$k$ eigenvalues and eigenvectors of $\mathbf{S}$ (resp. $\hat{\bf
S}$):

######  Theorem 1.

Let ${\mathcal{}H}(\cdot)$ be a $\bar{\lambda}$-low pass filter with ratio
$\eta$. Suppose that H1($\bar{\lambda}$), H2($\bar{\lambda}$) hold, and the
graph frequencies of the GSOs ${\bf S},\hat{\bf S}$ satisfy
$\lambda_{k}\leq\bar{\lambda}\leq\lambda_{k+1}$ and
$\hat{\lambda}_{k}\leq\bar{\lambda}\leq\hat{\lambda}_{k+1}$ for some $1\leq
k\leq n-1$. Then, the graph filter distance satisfies:

$\displaystyle\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})$
$\displaystyle\leq 2\mathbb{H}_{\max}\,\eta$ (6)
$\displaystyle\quad+L_{\mathbb{H}}||\mathbf{\Lambda}_{k}-\mathbf{\hat{\Lambda}}_{k}||_{2}+2\mathbb{H}_{\max}||\mathbf{V}_{k}-\mathbf{\hat{V}}_{k}||_{2}.$

###### Proof.

Using the triangular inequality, we observe that

$\displaystyle\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})$
$\displaystyle\leq\|\mathbf{V}_{k}h(\mathbf{\Lambda}_{k})\mathbf{V}_{k}^{T}-\mathbf{\hat{V}}_{k}h(\mathbf{\hat{\Lambda}}_{k})\mathbf{\hat{V}}_{k}\|_{2}$
$\displaystyle\textstyle+\big{\|}\sum_{i=k+1}^{n}\\{h(\lambda_{i})\mathbf{v}_{i}\mathbf{v}_{i}^{\top}-h(\hat{\lambda}_{i})\mathbf{\hat{v}}_{i}\mathbf{\hat{v}}_{i}^{\top}\\}\big{\|}_{2}.$
(7)

By H1($\bar{\lambda}$) and the property of $\bar{\lambda}$-low pass graph
filters as specified in Definition 1, it is easy to derive that

$\textstyle\max_{i\in\\{k+1,...,n\\}}\max\\{|h(\lambda_{i})|,|h(\hat{\lambda}_{i})|\\}\leq\eta\,\mathbb{H}_{\max}.$
(8)

This bounds the second term in (7) as $2\eta\,\mathbb{H}_{\max}$ where we have
further used $\|\sum_{i=k+1}^{n}{\bf v}_{i}{\bf v}_{i}^{\top}\|_{2}\leq 1$. On
the other hand, by H1($\bar{\lambda}$), H2($\bar{\lambda}$), we can upper
bound the first term in (7) as

$\displaystyle||\mathbf{V}_{k}h(\mathbf{\Lambda}_{k})\mathbf{V}_{k}^{\top}-\mathbf{\hat{V}}_{k}h(\mathbf{\hat{\Lambda}}_{k})\mathbf{\hat{V}}_{k}^{\top}||_{2}$
$\displaystyle\leq||h(\mathbf{\Lambda}_{k})-h({\bm{\hat{\Lambda}}_{k}})||_{2}+||\mathbf{V}_{k}h(\mathbf{\hat{\Lambda}}_{k})\mathbf{V}_{k}-\mathbf{\hat{V}}_{k}h(\mathbf{\hat{\Lambda}}_{k})\mathbf{\hat{V}}_{k}||_{2}$
$\displaystyle\leq
L_{\mathbb{H}}\|\bm{\Lambda}_{k}-\bm{\hat{\Lambda}}_{k}\|_{2}+2\mathbb{H}_{\max}\|{\bf
V}_{k}-\hat{\bf V}_{k}\|_{2}$ (9)

Combining (8), (9) yields the desired bound. ∎

The condition which requires the cutoff frequency $\bar{\lambda}$ to satisfy
$\lambda_{k}\leq\bar{\lambda}\leq\lambda_{k+1}$ and
$\hat{\lambda}_{k}\leq\bar{\lambda}\leq\hat{\lambda}_{k+1}$ impose a _spectral
gap_ requirement on the eigenvalues of ${\bf S},\hat{\bf S}$ where a common
$\bar{\lambda}$ separates the ‘low’ and ‘high’ frequencies of the two graphs.
This condition can be satisfied if the two graphs ${\bf S},\hat{\bf S}$ are
$k$-modular [20]. We remark that ${\mathcal{}H}({\bf S})$,
${\mathcal{}H}(\hat{\bf S})$ are also $k$-low pass graph filter [14] in this
case.

The graph filter distance bound (6) comprises of three terms. The first
depends on the low pass ratio $\eta$ which relies on the graph filter’s
frequency response. The last two terms capture the similarity in the graph
structure by comparing the pairs ${\bf V}_{k},\hat{\bf V}_{k}$ and
$\bm{\Lambda}_{k},\bm{\hat{\Lambda}}_{k}$. We expect the pairs $\|{\bf
V}_{k}-\hat{\bf V}_{k}\|_{2}$ and
$\|\bm{\Lambda}_{k}-\bm{\hat{\Lambda}}_{k}\|_{2}$ to be small when ${\bf
S},\hat{\bf S}$ are similar in terms of their community structures.

Theorem 1 implies that with a low pass filter ${\mathcal{}H}(\cdot)$
satisfying $\eta\ll 1$, the graph filter distance only depends on the
difference between ${\bf S},\hat{\bf S}$ in terms of their _community
structure_. The distance will be insensitive to the _number of edge rewiring_
unlike in [6, 9]. Our next endeavor is to showcase examples of graph
perturbation where the community structures are invariant, thereby applying
Theorem 1 concludes that the graph filter is stable.

Edge Rewiring Scheme. We focus on a simple case based on SBM by fixing the
number of blocks $k$, the membership matrix ${\bf Z}$ and connectivity matrix
${\bf B}$. In this example, both the original graph ${\mathcal{}G}$ and the
perturbed graph $\hat{\mathcal{}G}$ are generated from the SBM with the same
parameters such that ${\mathcal{}G},\hat{\mathcal{}G}\sim{\sf SBM}(n,k,{\bf
B},{\bf Z})$. The above setting covers an edge rewiring scheme as follows:

  1. 1.

Generate the original graph as ${\mathcal{}G}\sim{\sf SBM}(n,k,{\bf B},{\bf
Z})$.

  2. 2.

For each inter/intra-cluster block $(i,j)$, (i) delete a portion of $p_{\sf
re}\in[0,1]$ edges uniformly, and (ii) add edges to the node pairs without
edges with probability $[b_{ij}^{-1}-(1-p_{\sf re})]^{-1}p_{\sf re}$, selected
independently.

Notice that edge rewiring in the above only occurs for the edges within the
same inter/intra-cluster block. It can be shown that the resultant perturbed
graph satisfies $\hat{\mathcal{}G}\sim{\sf SBM}(n,k,{\bf B},{\bf Z})$.

###  3.1 Stability with Unnormalized Laplacian as GSO

Our plan is to study the graph filter distance bound through borrowing recent
consistency result of SBMs when $n\rightarrow\infty$. Notice that works on the
convergence of _unnormalized Laplacian_ are scarce. This may be due to the
fact that unnormalized Laplacian does not have good concentration due to the
involved diagonal degree matrix [21], [22]. Hence, we cannot use known
perturbation bounds such as Weyl’s inequality [23] and Davis-Kahan theorem
[24] to derive concentration of eigenvalues and eigenvectors. In the
following, we adopt the results from [21] and concentrate on a special case of
the SBM model given by a _sparse_ PPM with $k=2$ blocks such that
${\mathcal{}G}\sim{\sf PPM}(n,2,\alpha\log n/n,\beta\log n/n)$, where
$\alpha,\beta\in\mathbb{R}^{+}$.

We first observe the following high probability bounds on the bottom-$2$
eigenvectors and eigenvalues:

######  Corollary 1.

Let $\alpha,\beta\in\mathbb{R}^{+}$, consider PPM graphs as
${\mathcal{}G},\hat{\mathcal{}G}\sim{\sf PPM}(n,2,\alpha\log n/n,\beta\log
n/n)$. Denote their unnormalized Laplacian as ${\bf L}_{U}$, $\hat{\bf
L}_{U}$. Moreover, $\mathbf{\Lambda}_{2}$ and $\mathbf{V}_{2}$ (resp.
$\mathbf{\hat{\Lambda}}_{2}$ and $\mathbf{\hat{V}}_{2}$) are the matrices of
smallest-$2$ eigenvalues and eigenvectors of $\mathbf{L}_{U}$ (resp. $\hat{\bf
L}_{U}$). Suppose that $\sqrt{\alpha}-\sqrt{\beta}>\sqrt{2}$, then with
probability at least $1-o(1)$, it holds

$\|{\bf V}_{2}-\hat{\bf
V}_{2}\|_{2}=o(1),\quad\|\bm{\Lambda}_{2}-\bm{\hat{\Lambda}}_{2}\|_{2}={\mathcal{}O}(\log
n/n),\vspace{-.1cm}$ (10)

where $o(1)\rightarrow 0$ as $n\rightarrow\infty$.

The corollary is obtained by Theorem 8, Lemma 9 and Theorem 11 of [21] and the
randomness is due to the generation of PPM graphs. The condition
$\sqrt{\alpha}-\sqrt{\beta}>\sqrt{2}$ is known as the spectral gap criterion
which is necessary to distinguish the blocks in PPM [25, Theorem 13]. We shall
skip the proof in the interest of space.

To apply Theorem 1, we choose $k=2$ as the PPM contains 2 blocks which results
in a graph with $2$ densely connected clusters. We observe that for the
unnormalized Laplacian of sparse PPM graph, its non-zero eigenvalues grow with
$\Theta(\log n)$. Satisfying the spectral gap condition
$\lambda_{2}\leq\bar{\lambda}\leq\lambda_{3}$,
$\hat{\lambda}_{2}\leq\bar{\lambda}\leq\hat{\lambda}_{3}$ therefore requires
the cutoff frequency $\bar{\lambda}$ in Definition 1 to grow with $\log n$ as
well. To this end, if we consider the exponential graph filter111In fact, this
holds even for the filter ${\mathcal{}H}({\bf L}_{U})=e^{-\sigma{\bf
L}_{U}/\log n}$. ${\mathcal{}H}({\bf L}_{U})=e^{-\sigma{\bf L}_{U}}$, we
observe that the low pass ratio $\eta$, the constants in H1($\bar{\lambda}$),
H2($\bar{\lambda}$) are insensitive to the growth of
$\bar{\lambda}=\Theta(\log n)$. Under the above premises, we have

$\mathbb{D}_{\mathcal{}H}({\bf L}_{U},\hat{\bf L}_{U})\leq
2\eta\,\mathbb{H}_{\max}+\mathbb{H}_{\max}\,o(1)+L_{\mathbb{H}}\,{\mathcal{}O}(\log
n/n),\vspace{-.1cm}$ (11)

with high probability. As such, the graph filter distance between
${\mathcal{}H}({\bf L}_{U})$ and ${\mathcal{}H}(\hat{\bf L}_{U})$ is small
when $\eta\ll 1$ and $n\rightarrow\infty$.

###  3.2 Stability with Normalized Laplacian as GSO

Similar to the previous subsection, our plan is to borrow the classical
consistency result on SBMs from [21] and show that the low pass graph filter
is stable as $n\rightarrow\infty$. Let $\alpha,\beta\in\mathbb{R}^{+}$, we
consider the $k$-blocks sparse PPM with ${\mathcal{}G}\sim{\sf
PPM}(n,k,\alpha\log n/n,\beta\log n/n)$, we observe the following high
probability bound on the difference between the original and perturbed
normalized Laplacian:

######  Corollary 2.

Let $\alpha,\beta\in\mathbb{R}^{+}$, consider PPM graphs as
${\mathcal{}G},\hat{\mathcal{}G}\sim{\sf PPM}(n,k,\alpha\log n/n,\beta\log
n/n)$. Denote their normalized Laplacian as ${\bf L}_{\sf norm}$, $\hat{\bf
L}_{\sf norm}$. With probability at least $1-o(1)$, it holds

$\displaystyle\|{\bf L}_{\sf norm}-\hat{\bf L}_{\sf
norm}\|_{2}={\mathcal{}O}\left(1/\sqrt{\log n}\right).$ (12)

The above corollary is obtained from extending [21, Theorem 4], which is based
on [26]. In particular, in the PPM graphs considered, we have
$\min_{ij}{\mathcal{}A}_{ij}\geq c_{0}\log n/n$ such that the minimum degrees
of ${\mathcal{}G},\hat{\mathcal{}G}$ are $\Omega(\log n)$. We skip the proof
in the interest of space. Notice that compared to the previous case with
unnormalized Laplacian matrices, we obtained a stronger concentration with
respect to the normalized Laplacian matrices themselves.

To apply Theorem 1, using the Weyl’s inequality and Davis-Kahan theorem one
can show that $\|{\bf V}_{k}-\hat{\bf V}_{k}\|_{2}={\mathcal{}O}(1/\sqrt{\log
n})$,
$\|\bm{\Lambda}_{k}-\bm{\hat{\Lambda}}_{k}\|_{2}={\mathcal{}O}(1/\sqrt{\log
n})$ for the bottom-$k$ eigenvectors and eigenvalues. Furthermore, if we
consider the special case of $k=2$ blocks PPM graphs, then $\lambda_{2}({\bf
L}_{\sf norm})$ will be approximately $2\beta/(\alpha+\beta)$ while
$\lambda_{3}({\bf L}_{\sf norm})$ is at least $1-{\mathcal{}O}(1/\sqrt{\log
n})$ [21, Theorem 10]. With the spectral gap condition
$\sqrt{\alpha}-\sqrt{\beta}>\sqrt{2}$ and let $\beta\leq 1$ for simplicity, as
long as the graph filter ${\mathcal{}H}(\cdot)$ satisfies Definition 1 with a
constant cutoff frequency $\bar{\lambda}\approx 1/2$, we obtain:

$\mathbb{D}_{\mathcal{}H}({\bf L}_{\sf norm},\hat{\bf L}_{\sf norm})\leq
2\eta\,\mathbb{H}_{\max}+(\mathbb{H}_{\max}+L_{\mathbb{H}})\,{\mathcal{}O}\left(1/\sqrt{\log
n}\right),\vspace{-.1cm}$ (13)

with high probability. In other words, the graph filter distance between
${\mathcal{}H}({\bf L}_{\sf norm})$, ${\mathcal{}H}(\hat{\bf L}_{\sf norm})$
is small when $\eta\ll 1$ and $n\rightarrow\infty$.

######  Remark 1.

One may apply [10, Proposition 1] to yield

$\textstyle\mathbb{D}_{\mathcal{}H}({\bf L}_{\sf norm},\hat{\bf L}_{\sf
norm})\leq\sum_{t=0}^{T-1}t\,2^{t-1}\,|h_{t}|\,\|{\bf L}_{\sf norm}-\hat{\bf
L}_{\sf norm}\|_{2},\vspace{-.1cm}$ (14)

where we have used $\|{\bf L}_{\sf norm}\|_{2}\leq 2$. Therefore, we observe
that when $T<\infty$, (14) also yields the stability property for the graph
filter as $n\rightarrow\infty$, without using the low pass condition on
${\mathcal{}H}(\cdot)$. In comparison, our result allows the filter order $T$
to be infinite. Note [10] also considered a special case with
${\mathcal{}H}({\bf L})=({\bf I}+\alpha{\bf L})^{-1}$.

##  4 Numerical experiments

Synthetic Experiment. We compare the graph filter distances
$\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})$ [cf. (3)] under different
configurations of graph filters and GSOs, as summarized in Table 1. Notice
that in addition to the _low pass_ graph filters ${\mathcal{}H}_{\sf LP}({\bf
S})$ which satisfy Definition 1, as a control experiment we also consider
_high pass_ graph filters ${\mathcal{}H}_{\sf HP}({\bf S})$. The latter aims
to illustrate if the low pass property is necessary for the stability of graph
filter in both situation.

Table 1: Graph filter settings for synthetic experiments | Unnormalized Laplacian | Normalized Laplacian
---|---|---
${\mathcal{}H}_{\sf LP}(\cdot)$ | $\exp(-(1/\log n)\mathbf{L}_{U})$ | $\exp(-\mathbf{L}_{\sf norm})$
${\mathcal{}H}_{\sf HP}(\cdot)$ | $\exp((1/\log n)\mathbf{L}_{U})$ | $\exp(\mathbf{L}_{\sf norm})$

200400600800100012001400160018002000$2$$4$$6$$\cdot 10^{5}$Number of nodes
$\displaystyle n$$\displaystyle\mathbb{D}_{\mathcal{}H}({\bf L}_{U},\hat{\bf
L}_{U})$200400600800100012001400160018002000$1$$2$$3$$4$$\cdot 10^{-2}$Number
of nodes $\displaystyle n$$p_{\sf re}$ = 0.1$p_{\sf re}$ = 0.5$p_{\sf re}$ =
0.9

(a) Unnormalized Laplacian ${\bf L}_{U}$ as GSO.

200400600800100012001400160018002000$0.2$$0.4$$0.6$$0.8$$1$Number of nodes
$\displaystyle n$$\displaystyle\mathbb{D}_{\mathcal{}H}({\bf L}_{\sf
norm},\hat{\bf L}_{\sf
norm})$200400600800100012001400160018002000$0$$0.05$$0.1$$0.15$Number of nodes
$\displaystyle n$

(b) Normalized Laplacian ${\bf L}_{\sf norm}$ as GSO.

Fig. 1: Comparing the average graph filter distance
$\mathbb{D}_{\mathcal{}H}({\bf S},\hat{\bf S})$ against the number of nodes
$n$. (Left) High pass filter ${\mathcal{}H}_{\sf HP}(\cdot)$ (Right) Low pass
filter ${\mathcal{}H}_{\sf LP}(\cdot)$. The vertical bars indicate $95\%$
confidence intervals of the graph filter distance.

To simulate the edge rewiring graph perturbations, we generate the original
graphs ${\mathcal{}G}$ from a $k=2$ blocks PPM with ${\mathcal{}G}\sim{\sf
PPM}(n,2,13\log n/n,2\log n/n)$ with $n\in[200,2000]$. An intra-block edge is
assigned with probability $15\log n/n$ while an inter-block edge is assigned
with probability $2\log n/n$. The perturbed graph is generated from
${\mathcal{}G}$ by the edge rewiring process in Section 1. We perform Monte-
Carlo simulations with $100$ trials to estimate
$\mathbb{E}[{\mathcal{}D}_{\mathcal{}H}({\bf S},\hat{\bf S})]$ under the
random graph model and its perturbations.

0.010.050.10.150.2$10^{0}$$10^{10}$$10^{20}$Rewiring ratio $\displaystyle
p_{\sf re}$$\displaystyle\mathbb{D}_{\mathcal{}H}({\bf L}_{U},\hat{\bf
L}_{U})$$\exp({\bf L}_{U}/\log n)$$\exp(-{\bf L}_{U}/\log n)$

0.010.050.10.150.2$10^{-1}$$10^{0}$$10^{1}$$10^{2}$Rewiring ratio
$\displaystyle p_{\sf re}$$\exp(0.1{\bf L}_{U}/\log n)$$\exp(-0.1{\bf
L}_{U}/\log n)$

Fig. 2: Comparing the averaged graph filter distance
$\mathbb{D}_{\mathcal{}H}({\bf L}_{U},\hat{\bf L}_{U})$ against rewiring ratio
$p_{\sf re}$ over $100$ trials for different graph filters on the email-Eu-
core dataset. Note that the red (resp. green) curves correspond to low pass
(resp. high pass) graph filters.

The simulation results can be found in Fig. 1. In the figure, we compare the
averaged graph filter distance for the four settings in Table 1. Particularly,
we consider three rewiring ratios $p_{\sf re}\in\\{0.1,0.5,0.9\\}$ and make
the following observations in order. In the case when unnormalized Laplacian
is taken as the GSO, we observe that the low pass graph filters stabilize with
${\mathcal{}D}_{{\mathcal{}H}_{\sf LP}}({\bf L}_{U},\hat{\bf
L}_{U})\rightarrow 0$ as $n\rightarrow\infty$; in contrast, the high pass
graph filters become unstable as the graph filter distance increases rapidly
with $n$, note that ${\mathcal{}D}_{{\mathcal{}H}_{\sf HP}}({\bf
L}_{U},\hat{\bf L}_{U})\approx 10^{5}$. In the case when normalized Laplacian
is taken as the GSO, both low pass and high pass graph filters stabilize as
$n\rightarrow\infty$. As observed, the convergence rate of graph filter
distance with low pass filters is slightly faster than the high pass filters.

The above results agree with Section 3 as the low pass graph filters stabilize
as $n\rightarrow\infty$. Moreover, our observations seem to suggest that
stability with large number of rewires does not hold for high pass filter with
${\bf L}_{U}$ as GSO. On the other hand, for high pass graph filters with
${\bf L}_{\sf norm}$, we suspect that the convergence of
${\mathcal{}D}_{\mathcal{}H_{\sf HP}}({\bf L}_{\sf norm},\hat{\bf L}_{\sf
norm})$ is due to the discussions in (14), where the facts $\|{\bf L}_{\sf
norm}\|_{2}\leq 2$ and $\|{\bf L}_{\sf norm}-\hat{\bf L}_{\sf
norm}\|_{2}\rightarrow 0$ suffice to ensure stability.

Real Data Experiment. Our last example considers the stability of graph
filters on real graph topology. We consider ${\mathcal{}G}$ as the email-Eu-
core network [27]. The latter is an email network containing $1,005$ nodes
with $25,571$ edges and $42$ communities representing the departmental
membership of researchers. To test the stability of a graph filter defined on
${\mathcal{}G}$, we perform a simplified edge rewiring process: for each inter
and intra-cluster block, (i) we delete a portion $p_{\sf re}\in[0,1]$ of edges
selected uniformly at random, then (ii) we add back the same number of edges
into the respective block, again selected uniformly at random.

Fig. 2 shows the graph filter distance against the rewiring ratio $p_{\sf re}$
for two pairs of low pass and high pass graph filters with unnormalized
Laplacians as GSO. In both cases, we observe that low pass filters are stable
over the considered range of rewiring ratios. Especially with a set of
‘weaker’ low/high pass filter [cf. right panel with ${\mathcal{}H}({\bf
L}_{U})=\exp(\pm 0.1{\bf L}_{U}/\log n)$], we observe that
$\mathbb{D}_{\mathcal{}H}({\bf L}_{U},\hat{\bf L}_{U})$ increases at slower
rate for the low pass filter.

Conclusions. We study the stability of low pass graph filters when subjected
to a large number of edge rewires. We propose a new stability bound given with
respect to the frequency response of filters. Our bound shows that the
stability property for low pass graph filters hinges on whether there are
changes to the community structure in the perturbation. Numerical experiments
validate our theories.

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#  What is A Wireless UAV?
A Design Blueprint for 6G Flying Wireless Nodes

John Buczek, Lorenzo Bertizzolo, Stefano Basagni, Tommaso Melodia  Institute
for the Wireless Internet of Things, Northeastern University, Boston, MA
02115, USA  Email: {buczek.j, bertizzolo.l, s.basagni,
<EMAIL_ADDRESS>
(2022)

###### Abstract.

Wireless Unmanned Aerial Vehicles (UAVs) were introduced in the world of $4$th
generation networks (4G) as cellular users, and have attracted the interest of
the wireless community ever since. In 5G, UAVs operate also as flying Base
Stations providing service to ground users. They can also implement
independent off-the-grid UAV networks. In 6G networks, wireless UAVs will
connect ground users to in-orbit wireless infrastructure. As the design and
prototyping of wireless UAVs are on the rise, the time is ripe for introducing
a more precise definition of what is a wireless UAV. In doing so, we revise
the major design challenges in the prototyping of wireless UAVs for future 6G
spectrum research. We then introduce a new wireless UAV prototype that
addresses these challenges. The design of our wireless UAV prototype will be
made public and freely available to other researchers.

Wireless Unmanned Aerial Vehicles, UAV networks, 6G.

This article is based upon material supported in part by the US National
Science Foundation under grant CNS #1923789.

††ccs: Networks Wireless access points, base stations and
infrastructure††journalyear: 2022††copyright: acmlicensed††conference: The
15th ACM Workshop on Wireless Network Testbeds, Experimental evaluation &
CHaracterization; January 31–February 4, 2022; New Orleans, LA,
USA††booktitle: The 15th ACM Workshop on Wireless Network Testbeds,
Experimental evaluation & CHaracterization (WiNTECH ’21), January 31–February
4, 2022, New Orleans, LA, USA††price: 15.00††doi:
10.1145/3477086.3480840††isbn: 978-1-4503-8703-3/22/01

##  1\. Introduction

Unmanned Aerial Vehicles (UAVs, also known as _drones_) have progressively
attracted the interest of the wireless community as a tool to deploy flexible
and on-demand network infrastructure. During the last decade, several bodies
and stakeholders have worked to regulate and facilitate the use of UAVs with
the goal of promoting their integration into current and next generation
networks. Among others, these include networking regulatory bodies (e.g., the
3GGP and the FCC), cellular wireless carriers (e.g., AT&T, China Mobile, and
Vodafone), aviation authorities (e.g., the FAA), and large corporations ready
to employ flying connected robots for their business operations (e.g., Amazon,
DHL and CNN) (bertizzolo2021streaming, ).

The main applications of wireless UAVs as networking nodes include the
following:

  * •

UAVs for networking and service provisioning: UAVs implement flying Base
Stations (BSs) that connect users on the ground to the Internet. UAVs here
extend missing or malfunctioning cellular infrastructure by routing the
traffic of unserved ground cellular users (User Equipments, or UEs) to an
Internet gateway, like the nearest cell tower. As such, the UAVs implement
both wireless access and backhaul infrastructure (sheshadri2020skyhaul, ).

  * •

UAVs as flying cellular users: UAVs are flying robots connected to the ground
cellular network. They can be controlled and commanded over the Internet, and
do not require a flight operator in the vicinity. Additionally, they can
upload drone-sourced sensor data and video streaming to a cloud server using
cellular connectivity (bertizzolo2021streaming, ).

  * •

UAV swarms for on-demand, private service networks: Multiple wireless UAVs
will form a flying mesh network to connect users on the ground with each
other. This way, UAVs implement a private service network that can be deployed
on-demand as an alternative to the unavailable or untrustworthy cellular
infrastructure (BertizzoloInfocom20SwarmControl, ; cheng2021learning, ).

  * •

UAVs for 6G: UAVs will implement key 6G network infrastructure operating in
the low atmosphere ($<1\mathrm{\>km}$) which will connect ground users with
the non-terrestrial 6G infrastructure deployed in-orbit ($20$ to
$35786\mathrm{\>km}$) (geraci2021will, ).

Due to the impracticality of employing power lines or fiber links on a flying
node for any of the above applications, wireless UAVs differ from most of the
network infrastructure: they are fully wireless—in control, radio access, and
backhaul. Additionally, UAVs are exclusively battery-powered. Thus, the design
of wireless UAV testing platforms for future wireless research requires a deep
understanding of interactions among the powering, wireless and motion
capabilities of UAVs. While recent years witnessed a surge in wireless UAVs-
related research, works that test their solutions on real UAV hardware are
limited to a handful. Authors mainly employed UAV hardware available on the
market and equipped it with compute and wireless capabilities to satisfy their
needs, leaving many designs and prototyping questions unanswered.

We believe the time is ripe to provide further insights on what a wireless UAV
is and to explain in detail what are the key design choices to be made toward
the prototyping of future wireless UAV testing platforms for 6G-related
research.

Our work provides the first formal definition of wireless UAV. We go over key
design principles for the design and development of future wireless UAV
testing platforms, and we introduce a new wireless UAV prototype that meets
the requirements of those principles. The wireless UAV that we present in this
article has been carefully designed to address the present and future
challenges for 6G spectrum research. Its design process, components and
assembling instructions are made public for the community to use.

##  2\. Wireless UAVs: A Formal Definition

Here and for the first time, we formally enunciate the design blueprint of a
Wireless UAV. While wireless UAVs can be employed for different wireless
applications, vary in size, or in manufacturer, they should include the
following.

  1. (1)

Frame: This is the physical structure of the robot, and concerns the main
plane, the motors, the electronic speed controllers (ESC), the propellers, the
batteries, and all the other structural components of a flying robot.

  2. (2)

Computing unit: Off-the shelf or added on, future wireless UAVs should be
equipped with a computing unit comprehensive of CPU, memory, RAM, disk, GPU,
hardware and software architecture to run an Operative System.

  3. (3)

Flight Control Unit (FCU): The FCU controls the mobility of the UAV by driving
its individual motors. The FCU hosts a flight controller firmware (FCF) that
regulates the electricity fed into the motors by the ESCs.

  4. (4)

Radio front-ends: To implement wireless communication, wireless UAVs feature
one or more radio front-ends on board. This can include Wi-Fi or Bluetooth
chip-sets, cellular modems or programmable Software-defined Radios (SDR).
These modules include the antennas and the on-board powering circuitry.

  5. (5)

Wireless stack: If the radio front-ends are in charge of modulating and
demodulating electromagnetic signals into information bits, upper-layers
operations are handled by the wireless stack. This can be partially
implemented in hardware or all in software. The wireless stack is in charge of
implementing Physical and MAC layer framing and functionalities, handling
traffic forwarding, operating session management and implement all the
protocol stack functionalities to supports the application layer. The wireless
stack is implemented on the on-board computing unit for its software
implementation side and on the radio front ends for its hardware
implementation side. When employing SDR as a radio front end, the
implementation if entirely in software.

  6. (6)

Control APIs: Future wireless UAVs will have to expose control of their
wireless and motion functionalities in (near) real time. Accordingly, the
radio front-end(s), the wireless stack, and the FCU must expose control APIs.
Through the control APIs, a control program running on the on-board computer,
or communicating with it, can change the wireless UAV operational parameters
for a specific control objective in (near) real time.

According to this definition of a wireless UAV we classify the most important
related work on the design and prototyping of wireless UAV testing platforms.
Table 1 reports wireless UAV prototypes indicating frame, FCU, computing unit
and radio front-end.

Paper | UAV model | FCU | Computing unit | Radio front-end
---|---|---|---|---
(BertizzoloInfocom20SwarmControl, ) | Intel Aero | Pixhawk for Intel Aero / PX4 | Intel Aero Board | USRP B205-mini-i
(bertizzolo2021streaming, ) | DJI M100 | Pixhawk / PX4 | Intel NUC | USRP B210
(polese2020experimental, ; BertizzoloMmnets19, ; sanchez2020robust, ; sanchez2020millimeter, ) | DJI M600 Pro | DJI FCU | Intel NUC | Facebook Terragraph
(BertizzoloHotmobile20, ) | Intel Aero | Pixhawk for Intel Aero / PX4 | Intel Aero Board | ZTE LTE USB dongle
(sheshadri2020skyhaul, ) | DJI M600 | DJI FCU |  Unknown (Intel Core i7-6600U CPU) | MikroTik WAP 60G
(moradi2018skycore, ) | DJI M600 | DJI FCU |  Unknown (4-core, 1.9 GHz CPU, 8 GB RAM) | S60 nanoLTE
(ferranti2020skycell, ) | DJI M600 | DJI FCU | Intel NUC | USRP B210
(amorim2017radio, ) | DJI M600 | DJI FCU | — | R&S TSMA
(chakraborty2018skyran, ) | DJI M600 | DJI FCU |  Unknown (Intel Core i7) + Unknown (J1900 CPU) | USRP B210
(amorim2018measured, ) | DJI M600 | DJI FCU | — | R&S QualiPoc Smartphone
(d2019quality, ) | DJI M100 | Pixhawk / PX4 | Intel NUC | USRP B200mini-i
(ferranti2019hiro, ) | Intel Aero | Pixhawk for Intel Aero / PX4 | Intel Aero Board | NXP i.MX7D
(laclau2021signal, ) | Crazyflie 2.1 | Crazyflie Bolt | Crazyflie Board | Crazyradio
(mishra2020survey, ) | DJI Phantom 3 | DJI FCU | Raspberry Pi 3B | — / VNF
(khawaja2017uav, ) | DJI S-1000 | DJI FCU | — | PEM009-KIT & USRP X310
(sundqvist2015cellular, ) | Custom | Pixhawk / PX4 | Raspberry Pi 3B | LTE USB dongle
(muzaffar2020first, ) | Asctec Pelican | Asctec Atomboard | Asctec Board | Wistron NeWeb mobile platform
(burke2019safe, ) | Custom | Omnibus F4 Pro | Raspberry Pi Zero | Verizon USB dongle 730L
(petrolo2018astro, ) | Custom | Pixhawk / PX4 | Zynq 7030 | Iris-030 SDR
(semkin2021lightweight, ) | DJI M210 | DJI FCU | Intel compute stick | Spectrum Master MS2760A-0100
(mohanti2019airbeam, ) | DJI M100 | DJI FCU | ARMv8 NVIDIA TX2 | USRP B210
This work | Custom | Pixhawk / Ardupilot | Intel NUC | USRP B210
Table 1. Literature review of wireless UAV models, their FCU, computing, and
radio front-end components.

##  3\. A Wireless UAV Prototype for Future 6G Wireless Research

###  3.1. Key design principles

Given the definition of wireless UAV, in this section we outline key design
principles for prototyping future wireless UAVs for 6G spectrum research.

  1. i)

Software-defined motion control: The FCU must expose a wide range of motion
control APIs. These APIs should be made accessible via a standardized
interface (e.g., UART or USB), and should be controllable by software running
on the on-board computer. The flight control firmware should be open-source to
allows for modification of flight control’s primitives, if needed, and
guarantee operators’ privacy. Additionally, it should expose on-board sensors’
readings to the on-board computing unit in real time.

  2. ii)

Software-defined RF front-end: Future 6G non-terrestrial nodes will be
extremely constrained regarding the payload size and weight. Additionally,
hardware replacements and new hardware rollout will be reduced to the minimum.
Future non-terrestrial networks are thus being designed with programmable
hardware in mind, and Software-defined Radios (SDR) are the designated radio
boards to implement wireless communications. Different from hardware
implementations ‘baked’ into the chipsets, SDR allow for full protocol
implementation programmability, in software. Additionally, SDR allow for fine-
tuning of a wide range of wireless parameters in real time. As wireless
protocol will have to be re-designed and adjusted to the specific non-
terrestrial deployment scenario, employing SDR is paramount for the success of
future 6G networks.

  3. iii)

Multi-connectivity: Alongside in-orbit satellites and cell towers on the
ground, future wireless UAVs will be integral in multi-layered hierarchical 6G
networks (wang2020potential, ). To relay traffic from the ground to the orbit
infrastructure and viceversa, the availability of multi-connectivity on board
will be paramount. Multi-connectivity can be implemented, for example, by
multiple TX and RX chains on the same radio front end, or by multiple radio
front-ends on board. Multiple TX and RX chains can additionally be employed to
implement robust MIMO communications, which will be needed to cover the long
distances of aerial connectivity.

  4. iv)

Software-programmable wireless stack: Similar to the previous point, the
protocol implementations of the upper layers of the wireless stack must be re-
programmable. MAC, Networking, Transport, and Session-layer functionalities
must allow for swift software updates, new protocols roll out, and tuning of
their parameters in real time, in software.

  5. v)

Control plane on board: A key aspect of future 6G networks is the presence of
intelligence at the edge of the network. In a nutshell, this means that the
control and optimization of the operations of wireless nodes—formerly
delegated to a central controller in the cloud—will be carried out at the very
edge nodes of the network: the network nodes. In SDN terms, this concept
translates into having a control plane on board of UAVs. The control plane
consists in a set of algorithms that control and optimize the data handling
operations of the wireless stack and the motion operations of the flight
controller.

  6. vi)

On-board computing unit: Intelligence at the edge of the network goes along
with edge computing. Accordingly, the UAVs’ computing unit must feature high-
bandwidth hardware and software necessary to support a wide range of control
capabilities. The computing unit must be powerful enough to support the
previous points: drive the FCU and the radio front end(s), implement fully
programmable wireless protocol stacks, and execute intelligent control logic
on board. Additionally, it should feature a powerful enough GPU to support the
latest cutting edge AI algorithms.

  7. vii)

Mobile powering: Even though tethered options have been proposed in
literature, a fully scalable wireless UAV fabric should operate with on-board-
powered hardware only. This should include powering of the compute unit and
the radio front end(s) without posing excessive constraint on the flight time.

  8. viii)

Electromagnetic noise: The proximity of radio front ends, computing unit,
batteries, power amplifiers, and sensors, might result in undesirable
operational points with effects from noisy wireless communications to impaired
sensor readings. Consequences can be as harsh as self-jamming to unstable
flight operations. In designing a wireless UAV, it is important to keep in
mind the electromagnetic effects that might impact the different hardware
components on board.

An illustration of the design blueprint for future UAVs is given in Figure 1.

Figure 1. Design blueprint for 6G wireless UAVs.

###  3.2. UAV design fundamentals

To facilitate the design process of wireless UAVs, in here we introduce some
notation. Let $P_{T}$ be the total average power consumption of a wireless UAV
during a flight. $P_{T}$ is expressed as:

(1) $\displaystyle P_{T}=P_{F}+P_{C}+P_{R}+P_{L},$

where $P_{F}$ is the average required flight power (the electrical power
required to keep the UAV air-born), $P_{C}$ is the average required
computation power (the power required from the UAV’s on-board computers and
micro-controllers), $P_{R}$ is the average required radio power (power
required to function the radio front end), and $P_{L}$ is the average lost
power. Accordingly, the total flight time of a UAV $t_{f}$ can be found as a
function of the total energy capacity on-board the UAV $E_{T}$ with the
equation:

(2) $\displaystyle
t_{f}=\frac{E_{T}}{P_{T}}=\frac{E_{T}}{P_{F}+P_{C}+P_{R}+P_{L}}.$

Let’s define the All Up Weight (AUW) of the UAV as the total weight of the UAV
at the time of flight. This includes the weight of the frame, battery,
computing unit, radios, and any attached payloads to the airframe (expressed
in [$\mathrm{kg}$]). For a rotor-craft UAV (a heavier-than-air aircraft that
flies thanks to the lift generated by one or more rotors(FAA-14-1, )), the AUW
is a good approximation of the average thrust $\bar{T}$ (expressed in
Kilogram-force [$\mathrm{kgf}$]) needed from the motors to keep the UAV up in
the air

(3) $\displaystyle AUW\equiv\bar{T}.$

From the average thrust, the average per-motor thrust ($\bar{T}_{m}$) can be
found from the equation:

(4) $\displaystyle\bar{T}_{m}=\bar{T}/n,$

where $n$ is the number of motors on the UAV (e.g., $n=4$ for quadcopters,
$n=6$ for hexacopter). Conversely, the total average thrust can be found from
the inverse equation:

(5) $\displaystyle\bar{T}=\bar{T}_{m}*n.$

Finally, the design of a wireless UAV must ensure that the payload can
successfully be lifted with the available motors’ thrust. A condition to
ensure safe flying operations of the UAV is that the thrust generated by the
motors at $100\%$ throttle must be greater than twice the AUW
(biczyski2020multirotor, ). To guarantee that a $2:1$ thrust-to-weight ratio,
the design must ensure that the AUW is always below the thrust output of the
motors at $60\%$ throttle or at an ESC’s Pulse Width Modulation (PWM) of 1600,
denoted by $T_{m}^{1600}$:

(6) $\displaystyle max(AUW)\leq n*T_{m}^{1600}.$

The thrust generated by an individual motor (and its mounted propeller) is a
function of the electrical power supplied to the motor itself. The thrust
force versus power supply analysis can be performed via Computational Fluid
Dynamics (CFD) on the propeller’s airflow moved by the motor’s torque at
different speeds, using the motor performance data provided by the
manufacturer, or acquired via the use of thrust stands such as (rcbench_web,
). From the motor’s performance data, it is possible to determine the discrete
values of the motor thrust $T_{m}(P_{m})$ as a function of electrical power.
Conversely, the discrete values of the motor’s electrical power,
$P_{m}(T_{m})$, can be determined as a function of thrust. Finally, the total
average flight power $P_{F}$ can be determined by adding the individual
motor’s powers. Substituting in Eqs. 4 and 3 we obtain:

(7) $\displaystyle P_{F}\approx n*P_{m}(T_{m})=n*P_{m}(T/n)=n*P_{m}(AUW/n).$

From this and Eq. 2, the total flight time $t_{f}$ can be approximated as
(without loss of generality we assume $P_{L}$ negligible):

(8) $\displaystyle t_{f}\approx\frac{E_{T}}{n*P_{m}(AUW/n)+P_{C}+P_{R}}.$

For what concerns the onboard energy, the most common power source used on
modern UAVs are Lithium Polymer (LiPo) or Lithium Ion (LIon) batteries. LiPo
and LIon batteries cannot be discharged for their entire energy capacity due
to their internal chemistry. Typically, only approximately $80\%-90\%$ of a
LiPo battery capacity can be used without damaging the cells. Thus, only
approximately $80\%$ of the battery capacity on-board a UAV is usable. The
equation for the flight time of a multi-rotor UAV can then be further
expressed as:

(9) $\displaystyle
t_{f}\approx\frac{0.8*Batt_{Wh}}{n*P_{m}(AUW/n)+P_{C}+P_{R}},$

where $Batt_{Wh}$ is the battery energy capacity in Watts hour.

We will use the relationship between the UAV flight time, the battery
capacity, and power consumption expressed in Eq. 9 in the design process of a
new wireless UAV prototype in the next section.

###  3.3. UAV prototype

The design process of prototyping a wireless UAV is iterative. As seen in the
previous section, the choice of motors, propellers, and battery affect the AUW
of the UAV, the thrust-current efficiency, and the total energy capacity on
board. Indeed, increasing the battery size will increase the overall capacity
on-board the UAV, but also will increase the AUW, causing the hovering
throttle value to increase. In addition, unless the motors, propellers, and
batteries are being directly manufactured, commercial equipment must be used,
which limits the range of hardware to a few off-the-shelf models only.

Figure 2. Wireless UAV design process.

We use the iterative design process reported in Figure 2 as a guideline to
design a new wireless UAV prototype. In there, we approximate the motor’s PWM,
the UAV’s flight time, and AUW as described in Section 3.2.

  1. (I)

Choose Radio, Computing Unit, and minimum Flight Time: We have highlighted the
importance of programmable hardware such as SDRs in future wireless UAVs for
6G spectrum research. UAVs equipped with SDRs could implement different
solutions, from aerial BSs(BertizzoloMmnets19, ), to aerial
UEs(bertizzolo2021streaming, ; BertizzoloHotmobile20, ), to ad hoc UAV-to-UAV
communications (BertizzoloInfocom20SwarmControl, ). To accomplish this goal,
we chose NI’s USRP B210 as a radio front end. USRP B210 ($0.35\>\mathrm{kg}$)
is a fully-programmable lightweight software radio module featuring
$70\>\mathrm{MHz}$ – $6\>\mathrm{GHz}$ carrier frequency range,
$56\>\mathrm{MHz}$ of real-time bandwidth, 2 TX and 2 RX chains with MIMO
capabilities.
To operate USRP B210 and the wide range of supported software-based wireless
applications (e.g., Open Air Interface, srsLTE, GNU Radio), we select the
Intel NUC NUC7i7DN ($0.47\>\mathrm{kg}$) (nuc, ). The Intel NUC is a
commercial Mini PC, whose compact dimensions and good computational
capabilities (Intel Core i7 CPU with $32\>\mathrm{GB}$ RAM) make it
particularly suitable even to be carried on board of an UAV. Finally, we
specify a desired flight time of $45$ minutes.

  2. (II)

Determine Required $P_{C}$ and $P_{R}$: From the products’ data sheets, the
maximum power requirements for the computer ($P_{C}$) and radio ($P_{R})$ are
65W and 18W respectively (nuc, ; b200, ).

  3. (III)

Choose Flight Controller, UAV frame, Motors, ESCs and Propellers: The choice
of the UAV frame, UAV’s motors and propellers is subject to the availability
of off-the-shelf products. We chose a combination that can fit the chosen
computing and radio hardware onboard yet containing the UAV’s form factor and
cost. This choice can be iterated if necessary, as we will see later. The
motor must be chosen first. From the motor manufacturer’s specification or
suggestion, we can choose the battery voltage, propeller size, and ESC
voltage. For our design, we first considered the T-Motor MN501-S KV240 motor
($0.17\>\mathrm{kg}$ each) with T-Motor P20*6.5 Propellers
($0.044\>\mathrm{kg}$ each) operating with an 8 cell ($33.6\>\mathrm{V}$) LiPo
using TMotor Flame $60\>\mathrm{A}$ ESCs ($0.074\>\mathrm{kg}$ each)
(tmotor_mn501, ; tmotor_g20x6.5, ; tmotor_flame, ). As a UAV frame, we chose
the IFlight IXC15 frame ($0.643\>\mathrm{kg}$) as it easily accommodate the
motors and propellers, yet leaving enough room to mount radio and computing
unit on board. We finally select a small form-factor Pixhawk Mini (45.2g) as
FCU (holy_pix_mini, ).

Figure 3. T-Motor MN501-S power and thrust analysis.

  4. (IV)

Determine the $P_{m}$ and $T_{m}$ Relationship: To determine the non-linear
relationship between the motors’ input power and the motors’ generated thrust,
we preformed a static motor thrust analysis using an RCBenchmark Series 1580
Thrust Stand (rcbench_web, ). Specifically, we measure the motors’ input power
and the motors’ output thrust as a function of PWM. Characterizing this
relationship is necessary to calculate the battery capacity and weight, as we
will see later. We perform a series of discrete measurements and report the
interpolated power and thrust results in Figure 3.

  5. (V)

Choose the Battery: From Eq. 9 we can express the required battery capacity as
a function of the motor input power as:

$\displaystyle Batt_{Wh}\geq$
$\displaystyle\frac{\left(n*P_{m}(PWM)+P_{C}+P_{R}\right)*t_{f}}{0.8}$ (10)
$\displaystyle\geq$
$\displaystyle\frac{[4*P_{m}(PWM)+65\>\mathrm{W}+18\>\mathrm{W}]}{0.8}*\frac{45\>min}{60\>min}.$

Similarly, from Eqs. 3 and 5, we can express the required battery weight as a
function of the thrust generated by the motors as:

$\displaystyle Batt_{kg}\leq$
$\displaystyle\bar{T}-\text{AUW}_{\text{other}}=n*T_{m}(PWM)-\text{AUW}_{\text{other}}$
$\displaystyle\leq$ $\displaystyle n*T_{m}(PWM)-[4*(170g+44g+74g)+643g$
$\displaystyle+45.2g+470g+350g]$ (11) $\displaystyle\leq$ $\displaystyle
4*T_{m}(PWM)-2.66\>\mathrm{kg}.$

By employing the relationship between motors’ power and motors’ generated
thrust derived in (IV), we can calculate $Batt_{Wh}$ and $Batt_{kg}$ both as
function of PWM and plot them against each other. We report this relationship
in Figure 4.

Figure 4. Battery capacity vs. battery weight.

To satisfy Eqs. 10 and 11, any battery above the curve in Figure 4 is a good
choice. For our design, we chose to use 4 8S 6Ah LiPo batteries
($0.549\>\mathrm{kg}$ each) (8s_bat, ) reported as a dot in Figure 4. From the
power and thrust characterization data shown in Figure 3, we report the
required Battery Capacity as a function of Battery weight in Figure 4.

  6. (VI)

Calculate AUW and $T_{m}$: Following the battery selection, all the wireless
UAV components have been selected. Thus, the theoretical AUW and $T_{m}$ can
be calculated as:

$\displaystyle AUW$
$\displaystyle\approx(170\>\mathrm{kg}+0.044\>\mathrm{kg}+0.074\>\mathrm{kg})*4+0.643\>\mathrm{kg}$
$\displaystyle+0.045\>\mathrm{kg}+2.199\>\mathrm{kg}+0.47\>\mathrm{kg}+0.350\>\mathrm{kg}=4.856\>\mathrm{kg}$
$\displaystyle T_{m}$ $\displaystyle\approx AUW/4=1.214\>\mathrm{kgf}.$

  7. (VII)

Check Maximum PWM: From the motor analysis shown in Figure 3 we find that the
individual motors’ PWM required to lift the AUW is
$\text{PWM}(T_{m}=1.214\>\mathrm{kgf})\approx 1260\>\mathrm{\mu s}$. Since the
found PWM value is less than $1600\mu s$, the design process outlined in Fig.2
is satisfied and our prototype is completed.

Last, we select Ardupilot as flight controller firmware (FCF), we equip the
wireless UAV with 3D-printed landing gear and antenna mounts, use 4 VERT 2450
antennas for over-the-air communication and shield the wireless UAV components
with copper foil to limit the electromagnetic noise that could impair RF and
flight operations. The prototyped wireless UAV, termed Monarch, is shown in
Figure 5. Its design, landing gear CAD models, mounting instructions, and
components publicly available (wines-repo, ).

##  4\. Experimental Performance Analysis

We benchmarked the flight time performance of Monarch as well as its wireless
capabilities and electromagnetic shielding though a series of real-world
flight experiments.

Figure 5. The Monarch wireless UAV.

From the thrust and power analysis in Figure 3, we calculated the flight time
prediction for different AUWs. We report this analysis in Figure 6(a). To
support our calculations, we experimentally verified the UAV’s flight time
relative to different payload weights through real-world flight experiments.
To do this, we manually added variable mass payload to the UAV and
autonomously flight the UAV in a set square pattern until the battery voltage
under load went below $3.6V$ per LiPo cell. The field testing results are
consistent with our analysis and are also reported in Figure 6(a). We recorded
a flight time of 44.43 minutes for AUW of $5.08\>\mathrm{kg}$ with an onboard
NUC and B210. Compared to off-the-shelf UAV models equipped with wireless
modules, Monarch provides superior flight time. Our constructed design
performance are consistent with prediction, thus we validate our design and
our design methodology.

(a) Power and flight time vs. AUW.  (b) Flight autonomy benchmarking.

Figure 6. Monarch flight autonomy and power performance. Figure 7. GPS
Horizontal Dilution of Precision (HDOP) for different wireless applications.

We underline the importance of prototyping wireless UAVs adopting the
presented bottom-up approach by benchmarking Monarch’s flight autonomy with
relevant work in literature. We compare Monarch with similar off-the-shelf UAV
models equipped with wireless modules in Fig. 6(b). When compared to
(BertizzoloInfocom20SwarmControl, ), (bertizzolo2021streaming, ), and
(ferranti2020skycell, ), Monarch reports flight autonomy gains of $4.7$x
$2.9$x, and $1.4$x, respectively.

Last, we performed an experimental validation of the electromagnetic noise
figure on board. To do so, we used the GPS Horizontal Dilution of Precision
(HDOP) as an indicator of the electromagnetic noise figure to which the on-
board sensors are subject to. Specifically, we have measured the HDOP over
3-minute long flight experiments with different wireless applications running
onboard.  (i) NUC off, Radio off: all electronics on board is powered off.
This case represents ideal conditions.  (ii) NUC on, Radio off: The computing
unit (NUC) is turned on and runs the OS. No electromagnetic isolation is
applied here.  (iii) NUC on, Radio on (Gnuradio OFDM TX): The computing unit
(NUC) and the SDR (USRP B210) are on and run an OFDM transmitter at
$2.4\>\mathrm{GHz}$ with $1\>\mathrm{MHz}$ of bandwidth. No electromagnetic
isolation is applied here.  (iv) NUC on, Radio on (Gnuradio OFDM TX) with
shielding: The computing unit (NUC) and the SDR (USRP B210) are on and run an
OFDM transmitter at $2.4\>\mathrm{GHz}$ with $1\>\mathrm{MHz}$ of bandwidth.
We do apply electromagnetic isolation here.  The results are reported in
Figure 7. Without isolation, the computing unit and radio application rise the
average HDOP from $0.8\>\mathrm{m}$ to $1.04\>\mathrm{m}$ and
$1.07\>\mathrm{m}$, respectively, with outliers as high as $3\>\mathrm{m}$. On
the other hand, the applied copper shielding provides electromagnetic
isolation with respect to the on-board computing and wireless application and
features an even lower mean HDOP than when all electronics are powered off
($0.77\>\mathrm{m}$).

##  5\. Open Challenges

As we are still in the early stages of aerial wireless communications and
wireless UAVs development, we conclude this article with a few major
challenges to keep in mind for the design and development of future wireless
UAVs.

  * •

Small form-factor UAVs and long-range wireless communications: Long-range
communications with in-orbit 6G satellites or users on the ground might
require significant transmission power and large antenna gains. The latter can
be achieved with large antenna modules or high order antenna arrays. Large
antenna modules (e.g., log-periodic, yagi, or horn antennas) are bulky and
might take significant room on board. Despite these modules are passive and do
not require significant power, they are generally heavy and can reduce the
flight time when carried onboard. On the other hand, high-order antenna arrays
can be used with analog and digital beamforming techniques to ‘convey’ the
signal energy toward a single direction and extend the communication range.
However, these modules require a clear mounting surface and consume a
significant amount of power. Additionally, both solutions have a limited
coverage angle, which requires intelligent steering mechanisms or employing
multiple modules to extend coverage. Functional antenna design, such as curved
and frame-embedded antennas, similar to the ones in modern smartphones and
tablets, can help equip wireless UAVs with high-gain antenna modules and
address some of these issues.

  * •

Programmable radio front-end and software stack powering: As hardware
programmability is a must-have feature for future non-terrestrial networks,
efficient powering of high-performance programmable hardware is still an open
question. High-bandwidth and high-power programmable radio front ends are
still unsuitable to be mounted on board of a UAV. Similarly, high-performance
computing units necessary to process the baseband processing at ultra-wide
bands are large-form-factor server-like computing units, unfit to be carried
on board. Hardware and software advances in power-efficient protocol suites
implementations as well as power-efficient computing units and low-power SDR
boards will ease the portability of programmable radio front ends on small
UAVs.

  * •

UAV frame blockage and shadowing: One key design choice in the prototyping of
a wireless UAV is the placement of RF antennas on board. To maintain a small-
form factor and stable flight operations, these are generally placed above or
below the main frame. In close proximity with the antennas, the UAV frame can
‘shield’ the signal coming in and out the RF antennas and weaken the UAV
communication capabilities. This effect is known as ‘frame shadowing’ and
depends on the frame’s material, size, and the wireless transmission
direction. This problem is exacerbated when implementing wireless
communications with UAVs hovering at different altitudes. Employing multiple
antennas together with intelligent antenna selection or antenna steering
mechanisms can help alleviate this problem.

  * •

In-motion communications: The popularity of wireless UAVs can be found in
their ability to combine 3D mobility with wireless communication. While many
works have taken advantage of the combination of these two aspects, less
effort has been put in studying the problematic interaction of aerial mobility
and wireless communications. Involuntary micro-mobility, involuntary
fluctuation, and communications at different altitudes, can degrade or even
disrupt a wireless link. Even more problematic, the very principle at the base
of UAV’s mobility—tilting—can inadvertently change the antenna polarization,
significantly change the boresight of a directional antenna module, or impair
the effectiveness of digital and analog beamforming techniques. Thus,
developing intelligent in-flight transmission mechanism is fundamental to
support in-motion wireless communications for UAVs. Several approaches are
possible. These can be hardware implementations (mobile antennas, antenna
gimbals, phased array compensations, antenna selection) or software
implementations (e.g., motion-adaptive beamforming). Both cases can greatly
benefit from accessing the FCU sensors’ readings such as the accelerometer,
the Inertial Measurement Units (IMU), and the Global Positioning System (GPS).

##  6\. Conclusions

This article introduces a formal definition of a wireless UAV. We revise key
design choices for future 6G wireless UAVs and present Monarch, a new wireless
UAV prototype for 6G spectrum research. The Monarch design is publicly
available (wines-repo, ). Our work concludes indicating a few open challenges
in the domain of UAV-based wireless communications.

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