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$\displaystyle=-\frac{1}{N^{2}}\sum_{j=1}^{N}\psi_{j}^{2}(z)\mathbbm{E}\text{Tr}D^{-1}(z)T_{j}D^{-1}\Sigma_{a}\mathbbm{E}\text{Tr}R^{-1}MR^{-1}T_{j}+a_{2}(z),$

with $\mathbbm{E}|a_{2}(z)|\leq O(N^{1/2})$.

##  Appendix D Proof of Lemma 2.3

Similar to our proof of Lemma 4.5, Let $\mathcal{C}$ be a contour containing
the interval (7), with endpoints at $(\pm r,\pm v_{0})$. We split
$\mathcal{C}$ into the union of $\mathcal{C}_{u}$,
$\overline{\mathcal{C}_{u}}$ and $\mathcal{C}_{j}$, where
$\mathcal{C}_{u}=\\{z=x+iv_{0},|x|\leq r\\}$, $\mathcal{C}_{j}=\\{z=\pm
r+iy,|y|\leq v_{0}\\}$. In Section 5, we proved that for any
$z\in\mathcal{C}_{u}\cup\overline{\mathcal{C}_{u}}$,

$\displaystyle
M_{n}^{2}(z)-\mu_{n}(z)\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{p}}}}{{\rightarrow}}0.$

Above, $\mu_{n}(z)$ is defined in (11). Therefore, to complete the proof of
Lemma 2.3, it suffices to prove that

$\displaystyle\lim_{v_{0}\rightarrow
0}\limsup_{n}\int_{C_{j}}|M_{n}^{2}(z)|^{2}dz\rightarrow
0,\quad\lim_{v_{0}\rightarrow
0}\limsup_{n}\int_{C_{j}}|\mu_{n}(z)|^{2}dz\rightarrow 0.$

Our proof relies on the following results.

######  Lemma D.1.

Let $D(z)$, $D_{j}(z)$ be defined as in (28). For each of the functions $g$:

$\displaystyle
x_{j}^{T}T_{j}^{1/2}D_{j}^{-1}T_{j}^{1/2}x_{j},\quad\text{Tr}D_{j}^{-1}T_{j},\quad\mathbbm{E}\text{Tr}D_{j}^{-1}T_{j}\quad\mathbbm{E}\text{Tr}D^{-1}T_{j},$

we have

$\displaystyle\frac{1}{|1+g(z)|}\leq\frac{1}{2}\quad\text{a.s.}$

for all $|z|\geq 3C_{u}$, with $C_{u}$ defined in Lemma 4.3.

###### Proof.

Recall from the proof of Lemma B.2 that we can transform each of the functions
$g$ as (103). From Lemma E.5, we have for each $g$,

$\displaystyle|g(z)|\leq
k^{2}s_{L}^{2}s_{\Sigma}^{2}\left(\sum_{i=1}^{n}\frac{1}{|\lambda-z|^{2}}\right)^{1/2}.$

Together with Lemma 4.3, we obtain

$\displaystyle|g(z)|\leq\frac{k^{2}s_{L}^{2}s_{\Sigma}^{2}}{|z|-C_{u}}\leq\frac{1}{2}\quad\text{a.s.}.$

∎

######  Corollary D.1.1.

For all $z=\eta+vi\in C_{j}$, we have

$\displaystyle|\beta_{j}|,\quad|\tilde{\beta}_{j}|,\quad|b_{j}|,\quad|\psi_{j}|\leq
1/2\quad\text{a.s.},$ (129) $\displaystyle\lVert R^{-1}\rVert\leq
2|\eta|^{-1}$ (130)

for $\eta$ large enough.

###### Proof.

Applying Lemma D.1, we immediately have the boundedness results in (129).
Recall from (113) that

$\displaystyle R=$
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}|\psi_{j}|^{2}({1+\mathbbm{E}[\text{Tr}(U\text{diag}(\frac{\lambda_{i}-\eta}{(\lambda_{i}-\eta)^{2}+v^{2}})U^{T}T_{j})]})T_{j}-\eta
I$ $\displaystyle-
iv\left(I+\frac{1}{N}\sum_{j=1}^{N}|\psi_{j}|^{2}({1+\mathbbm{E}[\text{Tr}(U\text{diag}(\frac{1}{(\lambda_{i}-\eta)^{2}+v^{2}})U^{T}T_{j})]})T_{j}\right).$

As a result, $\lambda(R^{*}R)\geq\eta/2$ for $\eta$ large enough. ∎

######  Lemma D.2.

Let $z=\eta+iv$, with $v>n^{-2}$, then

$\displaystyle\mathbbm{E}\lVert D^{-1}\rVert\leq 2\eta^{-1}$

for $\eta$ large enough.

###### Proof.

Let $Q_{n}=\\{\lambda_{max}(S)\leq C_{u}+\epsilon\\}$. By Lemma 3.5, we have
that for any $\epsilon>0$ and positive integer $t$,

$\displaystyle\mathbbm{P}[Q_{n}^{c}]=o(n^{-t}).$

Thus

$\displaystyle\mathbbm{E}\lVert D^{-1}\rVert=\mathbbm{E}[\lVert
D^{-1}\rVert\mathbbm{1}_{Q_{n}}]+\mathbbm{E}[\lVert
D^{-1}\rVert\mathbbm{1}_{Q_{n}^{c}}]\leq 2\eta^{-1}$

for $\eta$ large enough. ∎

###  D.1 Bounding $\int_{C_{j}}|\mu_{n}(z)|^{2}dz$

Recall

$\displaystyle\mu_{n}(z)=\tilde{d}_{n0}(z)+n/N\sum_{r=1}^{k}\nu_{r}(z)\sum_{s=1}^{k}h_{N+1}^{rs}(z,z)\Xi_{0}^{s}(z),$
(131)

where the quantities $\tilde{d}_{n0}$, $\nu_{r}$, $\Xi_{0}^{a}$ are defined in
(91), (95) and (88). We proceed to bound each part individually.

Applying property (129), we have

$\displaystyle h_{N+1}^{ab}\leq s_{L}^{4},\quad h^{abc}\leq s_{L}^{6}.$

Applying property (130), we further have

$\displaystyle\Xi_{0}^{a},\quad\Xi_{1}^{ab},\quad\Xi_{2}^{ab},\quad\Xi_{3}^{abc}\leq
4s_{\Sigma}^{2}\eta^{-2}$ (132)

for $\eta$ large enough. Thus, for each $r=0,\ldots,k$,

$\displaystyle|\tilde{d}_{nr}|\leq C_{\eta,s_{L},s_{\Sigma}}$

by definition of $\tilde{d}_{nr}$ in (91), with $C_{\eta,s_{L},s_{\Sigma}}$
being a constant that only depends on $\eta$, $s_{L}$ and $s_{\Sigma}$.
Similarly, from the arbitrarily small upper bound established in (132), we
also have

$\displaystyle|\nu_{r}(z)|\leq C^{\prime}_{\eta,s_{L},s_{\Sigma}}.$

Collecting the terms yields

$\displaystyle|\mu_{n}(z)|\leq C^{\prime\prime}_{\eta,s_{L},s_{\Sigma}}.$

Hence

$\displaystyle\lim_{v_{0}\rightarrow
0}\limsup_{n}\int_{C_{j}}|\mu_{n}(z)|^{2}dz\rightarrow 0.$

###  D.2 Bounding $\int_{C_{j}}|M_{n}^{2}(z)|^{2}dz$

Our strategy involves further dividing $C_{j}$ into $C_{j1}=\\{z=\pm
r+iy,|y|\leq n^{-2}\\}$ and $C_{j2}=\\{z=\pm r+iy,n^{-2}\leq|y|\leq v_{0}\\}$

We begin by bounding the integral on $C_{j1}$. Recall that
$M_{n}^{2}(z)=\mathbbm{E}\text{Tr}(B_{n}-z)^{-1}-\text{Tr}(\tilde{B}_{n}-z)^{-1}$.
Let $z=\eta+vi$. From Lemma 4.3, it is straightforward to verify that for
$\eta$ and $n$ large enough, we have

$\displaystyle|M_{n}^{2}(z)|\leq 4n\eta^{-1}.$

Hence

$\displaystyle\lim_{v_{0}\rightarrow
0}\limsup_{n}\int_{C_{j1}}|M_{n}^{2}(z)|^{2}dz\rightarrow 0.$

As for the integral on $C_{j2}$, we follow similar arguments for bounding
$\mu_{n}(z)$ in the previous section. Recall from equation (93) that

$\displaystyle M_{n}^{2}(z)=$ $\displaystyle
d_{n0}+\sum_{r=1}^{k}n(\mathbbm{E}{g}_{2}^{(r)}-\tilde{g}_{2}^{(r)})\sum_{s=1}^{k}\frac{1}{N}\sum_{j=1}^{N}l_{sj}^{2}l_{rj}^{2}\tilde{b}_{j}(z)\psi_{j}(z)\frac{1}{n}\text{Tr}(R^{-1}\Sigma_{s}\tilde{R}^{-1}).$

Based on the calculations of $\mathcal{J}_{1}$, $\mathcal{J}_{2}$,
$\mathcal{J}_{3}$ in Section C.1, together with Lemma D.2, we obtain

$\displaystyle|d_{n0}|\leq C_{\eta,s_{L},s_{\Sigma}}.$

Similarly, for $r=1,\ldots,k$, we have

$\displaystyle|d_{nr}|\leq C^{\prime}_{\eta,s_{L},s_{\Sigma}}.$

As a result, it follows from the system of equations established in (94), the
arbitrarily small nature of $\lVert R^{-1}\rVert$ and the boundedness results
established in Corollary D.1.1 that

$\displaystyle|n(\mathbbm{E}{g}_{2}^{(r)}-\tilde{g}_{2}^{(r)})|\leq
C^{\prime\prime}_{\eta,s_{L},s_{\Sigma}}.$

Finally, we conclude that

$\displaystyle\lim_{v_{0}\rightarrow
0}\limsup_{n}\int_{C_{j2}}|M_{n}^{2}(z)|^{2}dz\rightarrow 0.$

##  Appendix E Useful results

######  Theorem E.1 (Theorem 35.12 of [Bil95]).

Suppose for each $n$, $Y_{n1}$,…,$Y_{nr_{n}}$ is a real martingale difference
sequence with respect to the increasing $\sigma$-field $\\{F_{nj}\\}$ having
second moments. If as $n\rightarrow\infty$,

$\displaystyle(i)$
$\displaystyle\sum_{j=1}^{r_{n}}\mathbbm{E}[Y_{nj}^{2}|F_{n,jk-1}]\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{p}}}}{{\rightarrow}}\sigma^{2}>0,$

and for each $\epsilon>0$,

$\displaystyle(ii)$
$\displaystyle\sum_{j=1}^{r_{n}}\mathbbm{E}[Y_{nj}^{2}\mathbbm{1}(|Y_{nj}|\geq\epsilon)]\rightarrow
0$

then

$\displaystyle\sum_{j=1}^{r_{n}}Y_{nj}\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{d}}}}{{\rightarrow}}N(0,\sigma^{2}).$

######  Lemma E.2 (Lemma 0.11 in [BLP19]).

Let $z\in\mathbbm{C}^{+}$ with $v=\text{Im}z$. Let $A$ and $B$ be $N\times N$
complex matrices, with $B$ a Hermitian matrix. Then for $\tau\in\mathbbm{R}$
and $r\in\mathbbm{C}^{N}$, we have

$\displaystyle\left|\text{Tr}\left((B-zI)^{-1}-(B+\tau
rr^{*}-zI)^{-1}\right)A\right|\leq\lVert A\rVert v^{-1}.$

###### Proof.

Let $C=B-zI$. From the Woodbury formula, we have

$\displaystyle(C+\tau rr^{*})^{-1}-C^{-1}=\tau\frac{C^{-1}rr^{*}C^{-1}}{1+\tau
r^{*}C^{-1}r}.$

Using the spectral decomposition $B=\sum\lambda_{j}q_{j}q_{j}^{*}$ with
$\lambda_{j}\in\mathbbm{R}$ and $r_{j}=q_{j}^{*}r$, we have

$\displaystyle\lVert C^{-1}r\rVert^{2}$
$\displaystyle=r^{*}(B-\bar{z}I)^{-1}(B-{z}I)^{-1}r=\sum_{j}\frac{r_{j}^{2}}{|\lambda_{j}-z|^{2}},$
$\displaystyle\text{Im}r^{*}C^{-1}r$
$\displaystyle=\sum_{j}r_{j}^{2}\text{Im}\frac{1}{\lambda_{j}-z}=(\text{Im}z)\sum_{j}\frac{r_{j}^{2}}{|\lambda_{j}-z|^{2}}.$

Hence

$\displaystyle\left|\text{tr}\left[C^{-1}-(C+\tau
rr^{*})^{-1}\right]A\right|=\left|\frac{\tau r^{*}C^{-1}AC^{-1}r}{1+\tau
r^{*}C^{-1}r}\right|\leq\frac{\lVert A\rVert\lVert
C^{-1}r\rVert^{2}}{|\text{Im}r^{*}C^{-1}r|}\leq\frac{\lVert
A\rVert}{\text{Im}z}.$

∎

######  Lemma E.3 (Lemma 0.13 from [BLP19]).

Let $M$ be a nonrandom matrix, for each $j=1,...,n$, we have

$\displaystyle\mathbbm{E}|\text{Tr}D_{j}^{-1}M-\mathbbm{E}\text{Tr}D_{j}^{-1}M|^{2}\leq
C\lVert M\rVert^{2}.$

###### Proof.

Following a similar argument as in (42), we have

$\displaystyle\text{Tr}D^{-1}M-\mathbbm{E}\text{Tr}D^{-1}M$
$\displaystyle=\sum_{j=1}^{N}\mathbbm{E}_{j}\text{Tr}D^{-1}M-\mathbbm{E}_{j-1}\text{Tr}D^{-1}M$
$\displaystyle=\sum_{j=1}^{N}(\mathbbm{E}_{j}-\mathbbm{E}_{j-1})\text{Tr}(D^{-1}-D_{j}^{-1})M,$
$\displaystyle=-\sum_{j=1}^{N}(\mathbbm{E}_{j}-\mathbbm{E}_{j-1})\beta_{j}(z)r_{j}^{T}D_{j}^{-1}MD_{j}^{-1}r_{j}.$

By the mutual independence of
$(\mathbbm{E}_{j}-\mathbbm{E}_{j-1})\beta_{j}(z)r_{j}^{T}D_{j}^{-1}MD_{j}^{-1}r_{j}$
and the basic algebraic fact that $(x+y)^{2}\leq 2(x^{2}+y^{2})$, we further
have

$\displaystyle\mathbbm{E}|\text{Tr}D^{-1}M-\mathbbm{E}\text{Tr}D^{-1}M|^{2}$
$\displaystyle=\sum_{j=1}^{N}\mathbbm{E}|(\mathbbm{E}_{j}-\mathbbm{E}_{j-1})\beta_{j}(z)r_{j}^{T}D_{j}^{-1}MD_{j}^{-1}r_{j}|^{2}$
$\displaystyle\leq
4\sum_{j=1}^{N}\mathbbm{E}|\beta_{j}(z)r_{j}^{T}D_{j}^{-1}MD_{j}^{-1}r_{j}|^{2}$
$\displaystyle\leq
8\sum_{j=1}^{N}\mathbbm{E}|\beta_{j}(z)|^{2}|r_{j}^{T}D_{j}^{-1}MD_{j}^{-1}r_{j}-N^{-1}\text{Tr}D_{j}^{-1}MD_{j}^{-1}T_{j}|^{2}$
$\displaystyle+8\sum_{j=1}^{N}\mathbbm{E}|N^{-1}\text{Tr}D_{j}^{-1}MD_{j}^{-1}T_{j}|^{2}.$

Together with Lemma 3.4 and the boundedness of $\beta_{j}$ established in
Lemma 4.1, we obtain

$\displaystyle\mathbbm{E}|\text{Tr}D^{-1}M-\mathbbm{E}\text{Tr}D^{-1}M|^{2}\leq
C\lVert M\rVert^{2},$

for some constant $C$. Applying the same arguments for $D_{j}(z)$,
$j=1,\ldots,N$, we conclude that

$\displaystyle\mathbbm{E}|\text{Tr}D_{j}^{-1}M-\mathbbm{E}\text{Tr}D_{j}^{-1}M|^{2}\leq
C\lVert M\rVert^{2}.$

∎

######  Lemma E.4 (Equation (1.15) in [BS04]).

Let $x_{1}=(x_{1i})_{1}^{n}$ be a vector with i.i.d. entries satisfying
$\mathbbm{E}x_{11}=0$, $\mathbbm{E}x_{11}^{2}=\sigma^{2}$,
$\mathbbm{E}x_{11}^{4}<\infty$, then for $n\times n$ matrices $A=(a_{ij})$ and
$B=(b_{ij})$, we have

$\displaystyle\mathbbm{E}(x_{1}^{T}Ax_{1}-\text{Tr}A)(x_{1}^{T}Bx_{1}-\text{Tr}B)$
$\displaystyle=$
$\displaystyle(\mathbbm{E}|x_{11}|^{4}-3\sigma^{4})\sum_{i=1}^{n}a_{ii}b_{ii}+\sigma^{4}(\text{Tr}AB+\text{Tr}AB^{T})+(\sigma^{2}-1)^{2}\text{Tr}A\text{Tr}B.$

Under Assumption 3.3, if additionally $B=B^{T}$, the above relationship
simplifies to

$\displaystyle\mathbbm{E}(x_{1}^{T}Ax_{1}-\text{Tr}A)(x_{1}^{T}Bx_{1}-\text{Tr}B)=2\text{Tr}AB+o(1)\sum_{i=1}^{n}a_{ii}b_{ii}.$

###### Proof.

Since $\mathbbm{E}x^{T}Ax=\sigma^{2}\text{Tr}A$, we get

$\displaystyle\mathbbm{E}(x_{1}^{T}Ax_{1}-\text{Tr}A)(x_{1}^{T}Bx_{1}-\text{Tr}B)=\mathbbm{E}(x_{1}^{T}Ax_{1})(x_{1}^{T}Bx_{1})-2\sigma^{2}\text{Tr}A\text{Tr}B+\text{Tr}A\text{Tr}B.$

It remains to study

$\displaystyle\mathbbm{E}(x_{1}^{T}Ax_{1})(x_{1}^{T}Bx_{1})=\sum_{ijkl}a_{ij}b_{kl}\mathbbm{E}x_{1i}x_{1j}x_{1k}x_{1l}.$

Denote each summand by $C_{ijkl}$, then by independence, the sum reduces to

$\displaystyle\mathbbm{E}(x_{1}^{T}Ax_{1})(x_{1}^{T}Bx_{1})=T_{ii}+T_{ij}+T_{ik}+T_{il},$

where

$\displaystyle
T_{ii}=\sum_{i=j=k=l}C_{ijkl}=\mathbbm{E}x_{1i}^{4}\sum_{i}a_{ii}b_{ii},$
$\displaystyle T_{ij}=\sum_{i=j\neq k=l}C_{ijkl}=\sigma^{4}\sum_{i\neq
k}a_{ii}b_{kk}=\sigma^{4}(\text{Tr}A\text{Tr}B-\sum_{i}a_{ii}b_{ii}),$
$\displaystyle T_{ik}=\sum_{i=k\neq j=l}C_{ijkl}=\sigma^{4}\sum_{i\neq
j}a_{ij}b_{ij}=\sigma^{4}(\text{Tr}AB^{T}-\sum_{i}a_{ii}b_{ii}),$
$\displaystyle T_{il}=\sum_{i=l\neq j=k}C_{ijkl}=\sigma^{4}\sum_{i\neq
j}a_{ij}b_{ji}=\sigma^{4}(\text{Tr}AB-\sum_{i}a_{ii}b_{ii}).$

∎

######  Lemma E.5.

For rectangular matrices $A$, $B$, $C$, $D$, we have

$\displaystyle|\text{Tr}(ABCD)|\leq\lVert A\rVert\lVert
C\rVert(\text{Tr}BB^{*})^{1/2}(\text{Tr}DD^{*})^{1/2}.$


# Characterizing bearing equivalence in directed graphs

Zhiyong Sun  Shiyu Zhao  Daniel Zelazo  Department of Electrical Engineering,
Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS> School of Engineering, Westlake
University, Hangzhou 310024, China (e-mail: zhaoshiyu@westlake.edu.cn).
Faculty of Aerospace Engineering, Technion-Israel Institute of Technology,
Haifa, Israel. (e-mail: dzelazo@technion.ac.il).

###### Abstract

In this paper, we study bearing equivalence in directed graphs. We first give
a strengthened definition of bearing equivalence based on the kernel
equivalence relationship between bearing rigidity matrix and bearing Laplacian
matrix. We then present several conditions to characterize bearing equivalence
for both directed acyclic and cyclic graphs. These conditions involve the
spectrum and null space of the associated bearing Laplacian matrix for a
directed bearing formation. For directed acyclic graphs, all eigenvalues of
the associated bearing Laplacian are real and nonnegative, while for directed
graphs containing cycles, the bearing Laplacian can have eigenvalues with
negative real parts. Several examples of bearing equivalent and bearing non-
equivalent formations are given to illustrate these conditions.

###### keywords:

Bearing rigidity; bearing equivalence; bearing-based formation; directed
graph.

††thanks: The work of Zhiyong Sun was partially supported by a starting grant
from Eindhoven Artificial Intelligence Systems Institute (EAISI), Eindhoven,
the Netherlands.

##  1 Introduction

Recent years have witnessed a growing interest in bearing-based distributed
control and estimation of networked multi-agent systems, such as bearing-based
formation control (Zhao and Zelazo (2015a); Tron et al. (2016); Karimian and
Tron (2021); Tang et al. (2022)) and network localization (Lin et al. (2016);
Arrigoni and Fusiello (2018)). The application of bearing measurements in
networked systems has been motivated by the advance of vision-based sensors,
which facilitate the sensing of relative directions between spatially
distributed agents. This is in contrast to relative position or distance
measurements, which tend to be more costly or unreliable.

One of the key graph concepts underpinning these bearing-based applications is
the bearing rigidity theory, which has been thoroughly discussed in Zhao and
Zelazo (2015b) with a focus on undirected graphs. The bearing rigidity theory
has provided a powerful framework for studying bearing-based formation control
of multi-agent systems, where the desired target shape is specified by
constant inter-agent bearings Zhao and Zelazo (2019); Tang et al. (2021).
Compared to displacement-based (Oh et al. (2015)) or distance-based (Sun et
al. (2017)) approaches, bearing-based formation control provides a natural
solution to the problem of formation scale control where the shape of the
formation is invariant but the inter-agent distance changes. One fundamental
problem of bearing-based formation control that has not been completely solved
is how to realize a desired bearing formation over directed graphs. Most
existing works on bearing-based formation or localization assume that the
underlying graph is undirected (Zhao and Zelazo (2019)) or satisfies special
directed graph structures (Tang et al. (2021)). Furthermore, the existing
bearing-based formation control laws may also become unstable for directed
graphs.

As a key step towards solving the general problem of bearing-based formation
control for directed graphs, the paper Zhao and Zelazo (2015a) first studied
bearing persistence for bearing formations in directed graphs. The notion of
bearing persistence follows in spirit the work of distance rigidity and
persistence originally presented in Hendrickx et al. (2007); Anderson et al.
(2008). In Zhao and Zelazo (2015a), however, the definition of bearing-based
persistence involves the kernel equivalence between the bearing rigidity
matrix (for undirected graphs) and the bearing Laplacian matrix (for directed
graphs). To differentiate between these notions, in this work we term the
condition as bearing equivalence. However, different to the full development
of distance persistence and its application in distance-based multi-agent
control, conditions for bearing equivalence are not well understood and its
characterization still remains open. In this paper, following Zhao and Zelazo
(2015a), we aim to provide several conditions to characterize bearing
equivalence in directed graphs. These conditions involve the spectrum and null
space of the associated bearing Laplacian matrix, which are separately
presented for bearing-based directed graphs with or without directed cycles.

The remaining parts of this paper are organized as follows. In Section 2 we
review key concepts and preliminaries on graph theory, bearing rigidity and
useful matrix results. Section 3 introduces the notion of bearing equivalence
and derives a useful formula for bearing Laplacian. Section 4 and Section 5
present several key conditions to characterize bearing equivalence for
directed graphs without cycles and with cycles, respectively. Section 6
concludes this paper.

##  2 Preliminaries

###  2.1 Notations

The notations in this paper are fairly standard. We use
$\mathrm{diag}(A_{i})\triangleq\mathrm{blkdiag}\\{A_{1},\dots,A_{n}\\}\in\mathbb{R}^{np\times
nq}$ to denote block diagonal matrix with given $A_{i}\in\mathbb{R}^{p\times
q}$ for $i=1,\dots,n$. Let $\mathrm{Null}(\cdot)$ and $\mathrm{Range}(\cdot)$
be the null space and range space of a matrix, respectively, and $\dim(\cdot)$
be the dimension of a linear space. Denote $I_{d}\in\mathbb{R}^{d\times d}$ as
the identity matrix, and $\mathbf{1}_{n}\triangleq[1\,\cdots\,1]^{\mathrm{T}}$
(when the subscript of $\mathbf{1}$ is omitted, the vector dimension should be
clear from the context). Let $\|\cdot\|$ be the Euclidean norm of a vector or
the spectral norm of a matrix, and $\otimes$ be the Kronecker product. For any
nonzero vector $x\in\mathbb{R}^{d}$ ($d\geq 2$), we define the projection
matrix $P:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times d}$ as

$\displaystyle P(x)\triangleq
I_{d}-\frac{x}{\|x\|}\frac{x^{\mathrm{T}}}{\|x\|}.$ (1)

For notational simplicity, we also denote $P_{x}=P(x)$.

###  2.2 Preliminaries on graph theory

Consider a directed graph with $n$ vertices and $m$ edges, denoted by
$\mathcal{G}=(\mathcal{V},\mathcal{E})$. The vertex set
$\mathcal{V}=\\{1,2,\ldots,n\\}$ represents the index of $n$ agents in the
group, and the edge set $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$
indicates the interconnection or neighboring relationship of $n$ agents. If
$(i,j)\in\mathcal{E}$, we say agent $i$ can “see” agent $j$, which means agent
$i$ can access the relative information of agent $j$. For agent $i$, its
neighbor set $\mathcal{N}_{i}$ is defined as
$\mathcal{N}_{i}:=\\{j\in\mathcal{V}:(i,j)\in\mathcal{E}\\}$. The incidence
matrix $H=\\{h_{ij}\\}\in\mathbb{R}^{m\times n}$ for a directed graph
$\mathcal{G}$ is defined by $h_{ki}=1$ if the $k$th edge sinks at node $i$, or
$h_{ki}=-1$ if the $k$th edge leaves node $i$, or $h_{ki}=0$ otherwise.
Furthermore, its Laplacian matrix
$L(\mathcal{G})=\\{L_{ij}\\}\in\mathbb{R}^{n\times n}$ is defined as
$L_{ii}=|\mathcal{N}_{i}|$, $L_{ij}=-1$ if $(i,j)\in\mathcal{E}$ and
$L_{ij}=0$ if $(i,j)\notin\mathcal{E}$. It is well known that
$\text{rank}(L(\mathcal{G}))=n-1$ with 0 being a simple eigenvalue of
$L(\mathcal{G})$ if and only if the directed graph $\mathcal{G}$ contains a
(directed) spanning tree (see e.g., Mesbahi and Egerstedt (2010)).

###  2.3 Preliminaries on bearing rigidity

Bearing rigidity theory plays a key role in the analysis of bearing-based
distributed formation control and network localization problems. In this
section, we review key notions and results in the bearing rigidity theory
presented in Zhao and Zelazo (2015b).

Given a finite collection of $n$ points $\\{p_{i}\\}_{i=1}^{n}$ in
$\mathbb{R}^{d}$ ($n\geq 2$, $d\geq 2$), denote
$p=[p_{1}^{\mathrm{T}}\,\cdots\,p_{n}^{\mathrm{T}}]^{\mathrm{T}}\in\mathbb{R}^{dn}$.
A _formation_ in $\mathbb{R}^{d}$, denoted as $\mathcal{G}(p)$, is a directed
graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ together with $p$, where vertex
$i\in\mathcal{V}$ in the graph is mapped to the point $p_{i}$. For a formation
$\mathcal{G}(p)$, define the _edge vector_ and the _bearing_ , respectively,
as

$\displaystyle e_{ij}\triangleq p_{j}-p_{i},\quad g_{ij}\triangleq
e_{ij}/\|e_{ij}\|,\quad\forall(i,j)\in\mathcal{E}.$

The bearing $g_{ij}$ is a unit vector.

######  Definition 1 (Bearing Equivalent Formations)

Two formations $\mathcal{G}(p)$ and $\mathcal{G}(p^{\prime})$ are _bearing
equivalent_ if $P_{g_{ij}}g^{\prime}_{ij}=0$ for all $(i,j)\in\mathcal{E}$.

By Definition 1, bearing equivalent formations have parallel inter-neighbor
bearings. Suppose $|\mathcal{E}|=m$ and index all the directed edges from $1$
to $m$. Re-express the edge vector and the bearing as $e_{k}$ and
$g_{k}\triangleq{e_{k}}/{\|e_{k}\|}$, $\forall k\in\\{1,\dots,m\\}$. Let
$e=[e_{1}^{\mathrm{T}}\,\cdots\,e_{m}^{\mathrm{T}}]^{\mathrm{T}}$ and
$g=[g_{1}^{\mathrm{T}}\,\cdots\,g_{m}^{\mathrm{T}}]^{\mathrm{T}}$. Note $e$
satisfies $e=\bar{H}p$ where $\bar{H}=H\otimes I_{d}$ and $H$ is the incidence
matrix of the graph $\mathcal{G}$. Define the _bearing function_
$F_{B}:\mathbb{R}^{dn}\rightarrow\mathbb{R}^{dm}$ as

$\displaystyle
F_{\text{\scriptsize{B}}}(p)\triangleq[g_{1}^{\mathrm{T}},\cdots,g_{m}^{\mathrm{T}}]^{\mathrm{T}}.$

The bearing function describes all the bearings in the formation. The _bearing
rigidity matrix_ is defined as the Jacobian of the bearing function,

$\displaystyle R_{\text{\scriptsize{B}}}(p)\triangleq\frac{\partial
F_{\text{\scriptsize{B}}}(p)}{\partial p}\in\mathbb{R}^{dm\times dn}.$ (2)

Let $\delta p$ be a variation of $p$. If $R_{\text{\scriptsize{B}}}(p)\delta
p=0$, then $\delta p$ is called an _infinitesimal bearing motion_ of
$\mathcal{G}(p)$.

######  Definition 2 (Infinitesimal Bearing Rigidity)

A formation is _infinitesimally bearing rigid_ if all the infinitesimal
bearing motions of the formation are trivial (i.e., translation and scaling of
the entire formation).

######  Lemma 1 (Zhao and Zelazo (2015b))

For any formation $\mathcal{G}(p)$, the bearing rigidity matrix defined in (2)
satisfies

  1. 1.

$R_{\text{\scriptsize{B}}}(p)=\mathrm{diag}\left({P_{g_{k}}}/{\|e_{k}\|}\right)\bar{H}$;

  2. 2.

$\mathrm{rank}(R_{\text{\scriptsize{B}}})\leq dn-d-1$ and
$\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}\subseteq\mathrm{Null}(R_{\text{\scriptsize{B}}})$.

######  Theorem 1 (Zhao and Zelazo (2015b))

For any formation $\mathcal{G}(p)$, the following statements are equivalent:

  1. 1.

$\mathcal{G}(p)$ is infinitesimally bearing rigid;

  2. 2.

$\mathcal{G}(p)$ can be uniquely determined up to a translation and a scaling
factor by the inter-neighbor bearings;

  3. 3.

$\mathrm{rank}(R_{\text{\scriptsize{B}}})=dn-d-1$;

  4. 4.

$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$.

###  2.4 Useful matrix results

The following matrix results will be frequently used in this paper to
characterize bearing rigidity/equivalence and stability of bearing-based
formations.

###### Lemma 2

(Null space of matrix product, Sun et al. (2017)) Consider two matrices
$A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times k}$ and the matrix
product $C=AB$. Then there holds
$\dim(\mathrm{Null}(C))=\dim(\mathrm{Null}(B))+\dim((\mathrm{Null}(A)\cap\mathrm{Range}(B))$.
In particular, it holds $\mathrm{Null}(C)=\mathrm{Null}(B)$ if and only if
$\mathrm{Null}(A)\cap\mathrm{Range}(B)=\\{0\\}$.

###### Lemma 3

(Block triangular matrix, Harville (2008)) Consider a real-valued block upper
triangular square matrix $A$, with the $i$-th diagonal square block denoted by
$A_{ii}$. Then the eigenvalues of $A$ are the union of the set of eigenvalues
of each diagonal block $A_{ii}$; i.e., it holds
$\lambda(A)=\bigcup_{i=1}^{n}\lambda(A_{ii})$.

###### Lemma 4

(Properties of projection matrices, Bernstein (2018)) For a projection matrix
$P_{x}$ defined in (1), it holds $P_{x}^{\mathrm{T}}=P_{x}$,
$P_{x}P_{x}=P_{x}$, and $P_{x}$ is positive semi-definite. Moreover,
$\mathrm{Null}(P_{x})=\mathrm{span}\\{x\\}$ and the eigenvalues of $P_{x}$ are
$\\{0,1^{(d-1)}\\}$. Any two nonzero vectors $x,y\in\mathbb{R}^{d}$ are
parallel if and only if $P_{x}y=0$ (or equivalently $P_{y}x=0$).

###### Lemma 5

(Sum of projection matrices) Consider a set of projection matrix,
$P_{i}(x_{i})$ defined in (1) associated with nonzero vector
$x_{i}\in\mathbb{R}^{d}$ ($d\geq 2$). Then the following holds:

  1. 1.

For the matrix sum $P_{i}(x_{i})+P_{j}(x_{j})$, if the vectors $x_{i}$ and
$x_{j}$ are parallel (i.e., linearly dependent), then
$\mathrm{Null}(P_{i}(x_{i})+P_{j}(x_{j}))=\mathrm{span}\\{x_{i}\\}$ and
$P_{i}(x_{i})+P_{j}(x_{j})$ is positive semi-definite.

  2. 2.

Otherwise, if the vectors $x_{i}$ and $x_{j}$ are linearly independent, then
$\mathrm{Null}(P_{i}(x_{i})+P_{j}(x_{j}))=\\{0\\}$ and the matrix sum
$P_{i}(x_{i})+P_{j}(x_{j})$ is positive definite.

  3. 3.

For the matrix sum $P_{i}(x_{i})+P_{j}(x_{j})+\cdots+P_{k}(x_{k})$, if all
vectors $x_{i},x_{j},\cdots,x_{k}$ are parallel, then

$\mathrm{Null}(P_{i}(x_{i})+P_{j}(x_{j})+\cdots+P_{k}(x_{k}))=\mathrm{span}\\{x_{i}\\}.$

  4. 4.

The matrix sum $P_{i}(x_{i})+P_{j}(x_{j})+\cdots+P_{k}(x_{k})$ is positive
definite if there exist at least two vectors in the list that are non-
parallel.

The proofs of the above lemmas will be provided in the journal version of this
paper.

##  3 Notion of bearing equivalence

The development of bearing equivalence is motivated by the bearing-based
formation control in directed graphs. The bearing-based formation control
problem specifies a set of desired inter-agent bearings, $g_{ij}^{*}$, and the
objective is to design a distributed control using only relative position
measurements from neighboring agents to drive the agents to the formation
specified by the $g_{ij}^{*}$’s, i.e.,
$\lim_{t\to\infty}g_{ij}(t)=g_{ij}^{*}$. Zhao and Zelazo (2015a) proposed the
following control scheme,

$\displaystyle\dot{p}_{i}(t)=-\sum_{j\in\mathcal{N}_{i}}P_{g_{ij}^{*}}\left(p_{i}(t)-p_{j}(t)\right),\quad
i\in\mathcal{V},$ (3)

where $P_{g_{ij}^{*}}=I_{d}-(g_{ij}^{*})(g_{ij}^{*})^{\mathrm{T}}$. The
stability and convergence of the above distributed and linear formation
control system depends on the spectrum and null space of the bearing Laplacian
$L_{\text{\scriptsize{B}}}\in\mathbb{R}^{dn\times dn}$, whose $ij$th block
submatrix is defined as below

$\displaystyle\left\\{\begin{array}[]{ll}[L_{\text{\scriptsize{B}}}]_{ij}=0_{d\times
d},&i\neq j,(i,j)\notin\mathcal{E},\\\
{[L_{\text{\scriptsize{B}}}]_{ij}}=-P_{g_{ij}^{*}},&i\neq
j,(i,j)\in\mathcal{E},\\\
{[L_{\text{\scriptsize{B}}}]_{ii}}=\sum_{j\in\mathcal{N}_{i}}P_{g_{ij}^{*}},&i\in\mathcal{V}.\\\
\end{array}\right.$

In this way, the compact matrix expression of the control law (3) is

$\displaystyle\dot{p}(t)=-L_{\text{\scriptsize{B}}}p(t).$ (4)

It becomes apparent that the matrix $L_{\text{\scriptsize{B}}}$ can also be
interpreted as a matrix-weighted graph Laplacian with orthogonal projection
matrix weights. The bearing Laplacian $L_{\text{\scriptsize{B}}}$, together
with its spectrum and null space, are jointly determined by the topological
structure of the underlying graph and bearing information of the formation.

For an undirected graph, it is easy to show that

$\displaystyle
L_{\text{\scriptsize{B}}}=\bar{H}^{\mathrm{T}}\mathrm{diag}{(P_{g_{k}})}\bar{H}$
(5)

which immediately leads to that, for an undirected formation $\mathcal{G}(p)$,
the bearing Laplacian $L_{\text{\scriptsize{B}}}$ is symmetric positive semi-
definite and satisfies
$\mathrm{Null}(L_{\text{\scriptsize{B}}})=\mathrm{Null}(R_{\text{\scriptsize{B}}})$.

However, when the graph is directed, the bearing Laplacian is not symmetric
and the above kernel equivalence relationship would not hold in general. The
notion of bearing equivalence is defined first in Zhao and Zelazo (2015a)
(where it was termed ‘bearing persistence’) based on the kernel equivalence
condition as
$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{Null}(L_{\text{\scriptsize{B}}})$.
In this paper, we shall introduce a strengthened definition of bearing
equivalence as below.

######  Definition 3 (Bearing Equivalence)

A directed formation $\mathcal{G}(p)$ is _bearing kernel equivalent_ (in
short, bearing equivalent) if
$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{Null}(L_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$.

The problem of characterizing favourable properties of bearing Laplacian
$L_{\text{\scriptsize{B}}}$ and bearing equivalence is motivated by the
bearing-based formation system (3) in directed graphs. In particular,

  * •

The spectrum of $L_{\text{\scriptsize{B}}}$ determines the stability
properties of the formation system (3). Preferably, we aim to find conditions
to guarantee that $L_{\text{\scriptsize{B}}}$ has all eigenvalues with non-
negative real parts such that the formation system (3) is stable.

  * •

The null space of $L_{\text{\scriptsize{B}}}$ determines the converged
formation shape of the system (3). Preferably, we aim to find conditions to
guarantee
$\mathrm{Null}(L_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$ such that the converged formation shape is bearing equivalent to
the target formation.

Graph conditions to guarantee bearing equivalence for a bearing formation
still remain open, though some partial solutions were presented in Zhao and
Zelazo (2015a). In this paper, we will revisit these conditions from Zhao and
Zelazo (2015a) and present more (necessary and/or sufficient) conditions to
characterize bearing equivalence. In particular, we will give several graph
topological conditions to characterize the spectrum and null space of
$L_{\text{\scriptsize{B}}}$, which underpin certain key requirements to ensure
the formation convergence by the bearing control law (3).

###  3.1 A useful formula for $L_{\text{\scriptsize{B}}}$

As a counterpart of the matrix expression (5) of bearing Laplacian for
undirected graphs, we first derive the following expression of bearing
Laplacian for directed graphs. For a bearing formation with a directed graph
$\mathcal{G}$, the associated bearing Laplacian can be expressed by

$\displaystyle
L_{\text{\scriptsize{B}}}=\bar{J}^{\mathrm{T}}\mathrm{diag}{(P_{g_{k}})}\bar{H}$
(6)

where $\bar{J}=J\otimes I_{d}$ and the matrix $J$ is obtained by replacing the
‘$-1$’ entry of the incidence matrix $H$ by ‘ $0$’ from the directed graph.

The bearing Laplacian formula of (6) follows from the formula of the
conventional Laplacian matrix $L$ for a directed graph: $L=J^{\mathrm{T}}H$.
By augmenting the Kronecker product and the matrix weight (in terms of the
projection matrix $P_{g_{k}}$ associated to each directed edge), one can
obtain (6).

The formula of (6) gives the following set inclusion of null spaces (Theorem 4
of Zhao and Zelazo (2015a)).

###### Lemma 6

For a directed formation $\mathcal{G}(p)$, the bearing Laplacian
$L_{\text{\scriptsize{B}}}$ satisfies

$\displaystyle\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}\subseteq\mathrm{Null}(R_{\text{\scriptsize{B}}})\subseteq\mathrm{Null}(L_{\text{\scriptsize{B}}}).$
(7)

This lemma is proved in Zhao and Zelazo (2015a), and here we give a shorter
proof based on the formula (6). First we define
$\tilde{R}_{B}=\mathrm{diag}(\|e_{k}\|)R_{\text{\scriptsize{B}}}$ where
$R_{\text{\scriptsize{B}}}$ is the bearing rigidity matrix. Then it holds
$\mathrm{Null}(\tilde{R}_{B})=\mathrm{Null}(R_{\text{\scriptsize{B}}})$ and
therefore $\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}\subseteq\mathrm{Null}(\tilde{R}_{B})$. Note that
$L_{\text{\scriptsize{B}}}=\bar{J}^{\mathrm{T}}\tilde{R}_{B}$ and one has
$\mathrm{Null}(\tilde{R}_{B})\subseteq\mathrm{Null}(L_{\text{\scriptsize{B}}})$,
which concludes the set inclusion of (7).

We remark that, by the property of projection matrix in Lemma 4, an
alternative formula for $L_{\text{\scriptsize{B}}}$ is given as below

$\displaystyle
L_{\text{\scriptsize{B}}}=\bar{J}^{\mathrm{T}}\mathrm{diag}{(P_{g_{k}})}\bar{H}=\underbrace{\bar{J}^{\mathrm{T}}\mathrm{diag}{(P_{g_{k}}^{\mathrm{T}})}}_{\tilde{J}_{B}^{\mathrm{T}}}\underbrace{\mathrm{diag}{(P_{g_{k}})}\bar{H}}_{\tilde{R}_{B}}.$
(8)

Based on this formula (8), we give a necessary and sufficient condition to
guarantee bearing equivalence.

###### Theorem 2

For a directed formation $\mathcal{G}(p)$, the equality

$\mathrm{Null}(L_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$

holds if and only if the following two conditions are both satisfied

  1. I.

$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$ (i.e., the formation is infinitesimally bearing rigid), and

  2. II.

$\mathrm{Null}({\tilde{J}_{B}^{\mathrm{T}}})\cap\mathrm{Range}({\tilde{R}_{B}})=\\{0\\}$.

This necessary and sufficient condition follows from the condition of
infinitesimally bearing rigidity in Theorem 1 and the null space property of
$L_{\text{\scriptsize{B}}}=\tilde{J}_{B}^{\mathrm{T}}\tilde{R}_{B}$ as in (8)
by applying Lemma 2. It is not clear at this stage how to use the second
condition, or what it means. In the following sections, we will derive more
concrete conditions to characterize the spectrum and null space of
$L_{\text{\scriptsize{B}}}$.

##  4 Bearing equivalence in
directed acyclic graphs

In this section, we focus on directed acyclic graphs and present several
conditions on bearing equivalence.

###  4.1 Spectrum of bearing Laplacian

The first result characterizes the spectrum property of
$L_{\text{\scriptsize{B}}}$ for directed acyclic graphs.

###### Proposition 1

For directed acyclic graphs, the eigenvalues of the bearing Laplacian
$L_{\text{\scriptsize{B}}}$ are real and nonnegative.

For directed acyclic graphs, the bearing Laplacian can always be reconstructed
(with permutation of vertices and edges) as a block triangular matrix, while
the $i$-th diagonal block consists of a projection matrix $P_{g^{*}_{ij}}$ (if
the associated vertex $i$ has only one out-going edge $(i,j)$), or a sum of
projection matrix $\sum_{j\in\mathcal{N}_{i}}P_{g^{*}_{ij}}$ (if the
associated vertex $i$ has multiple out-going edges $(i,j)$), or is a zero
block (if the associated vertex $i$ has no out-going edge).

According to Lemma 3, the set of eigenvalues of a block triangular matrix is
the union of eigenvalues of each diagonal block. In this bearing Laplacian,
each diagonal block is either a zero block or a positive semi-definite matrix
(as a single projection or a sum of projection matrices). Therefore, according
to Lemma 5, the eigenvalues of the bearing Laplacian for directed acyclic
graphs are real and nonnegative.

###  4.2 Conditions for bearing equivalence

The following conditions are presented to characterize the null space property
of $L_{\text{\scriptsize{B}}}$ for directed acyclic graphs. Note that any
directed acyclic graph contains a vertex with zero out-going edge, which is
often termed as the “leader agent” in formation control.

###### Proposition 2

For a bearing formation modeled by a directed acyclic graphs, any of the
following conditions will result in $\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}\subset\mathrm{Null}(L_{\text{\scriptsize{B}}})$, leading to a
bearing non-equivalent formation.

  1. I.

There are at least two vertices that have zero out-going edge;

  2. II.

There are at least two vertices with only one out-going edge;

  3. III.

There are at least two vertices with collinear out-going edges.

Due to the block triangular structure of $L_{\text{\scriptsize{B}}}$ of
directed acyclic graphs, any of the above conditions will introduce additional
null spaces for $L_{\text{\scriptsize{B}}}$ in addition to
$\mathrm{span}\\{\mathbf{1}\otimes I_{d},p\\}$. The detailed proof is omitted
here due to space limit.

A special class of acyclic directed graphs is the _leader-first-follower_
(LFF) graph; see Trinh et al. (2018).

######  Definition 4 (Leader-first-follower graph)

A _leader-first-follower_ (LFF) graph is a directed graph on $n>1$ nodes such
that

  * i)

One vertex (called the leader) has zero out-going edge.

  * ii)

One vertex (called the first follower) has one out-going edge and the
corresponding edge is incident to the leader.

  * iii)

Every other vertex has at least two out-going edges and their out-going edges
are not collinear.

We now give a sufficient condition for bearing equivalence, which
characterizes the leader-first-follower (LFF) structure in directed graphs.

###### Theorem 3

Directed formations over leader-first-follower graphs are bearing equivalent.

The proof of Theorem 3 again follows from the block triangular structure of
$L_{\text{\scriptsize{B}}}$, and the detail is omitted here due to space
limit. We also remark that this theorem can be seen as an extension of the
distance persistence (Hendrickx et al. (2007), Anderson et al. (2008)) to
bearing equivalence in directed LFF graphs.

##  5 Bearing equivalence in directed graphs containing cycles

In this section, we focus on directed graphs containing cycles and derive
several necessary and/or sufficient conditions on bearing Laplacian and
bearing equivalence.

###  5.1 Spectrum of bearing Laplacian

For directed graphs that contain cycles, the associated bearing Laplacian
(which is asymmetric) cannot be written in a block triangular structure, and
thus its eigenvalues are often complex.

The following conjecture on the bearing Laplacian spectrum was proposed in
(Zhao and Zelazo, 2015a).

###### Conjecture 1

The eigenvalues of the bearing Laplacian $L_{\text{\scriptsize{B}}}$ of a
directed formation have nonnegative real parts.

This conjecture is not true. For counterexamples, see the formation graphs in
Fig. 1 which will be discussed later. Note that the entries of the bearing
Laplacian (and thus its eigenvalue locations) depend on the configurations
(agents’ positions). For a given bearing Laplacian associated with a cyclic
directed graph and under some special positions, eigenvalues with negative
real parts can occur.

###  5.2 Conditions for bearing equivalence

We first generalize a sufficient condition for characterizing
$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{Null}(L_{\text{\scriptsize{B}}})$
in $\mathbb{R}^{2}$ (from (Zhao and Zelazo, 2015a, Proposition 1) to bearing
formations in general-dimensional spaces.

###### Proposition 3

For a directed formation $\mathcal{G}(p)$ in $\mathbb{R}^{d}$ ($d\geq 2$), if
each agent has at most two non-collinear out-going edges, then the formation
satisfies
$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{Null}(L_{\text{\scriptsize{B}}})$.

The proof follows from the proof of (Zhao and Zelazo, 2015a, Proposition 1)
and is omitted here. We note that this condition is not necessary. For a
counterexample, see the graph (c) in Fig. 1. In this example, agent 2 has
three out-going edges while the bearing formation of Fig. 1(c) satisfies
$\mathrm{Null}(R_{\text{\scriptsize{B}}})=\mathrm{Null}(L_{\text{\scriptsize{B}}})$
and still is bearing equivalent.

Now we provide a necessary condition for bearing equivalence in directed
graphs.

###### Proposition 4

For a directed bearing formation $\mathcal{G}(p)$, if the associated bearing
Laplacian satisfies
$\mathrm{Null}(L_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$, then the underlying directed graph contains a directed spanning
tree.

If the underlying directed graph does not contain a spanning tree, then the
Laplacian matrix $L=J^{T}H$ will have additional null vector besides
$\\{\mathbf{1}\\}$, implying that the augmented Laplacian matrix
$\bar{L}=L\otimes I_{d}=\bar{J}^{T}\bar{H}$ will have additional null space
besides $\mathrm{span}\\{\mathbf{1}\otimes I_{d}\\}$, i.e.,
$\mathrm{span}\\{\mathbf{1}\otimes I_{d}\\}\subset\mathrm{Null}(\bar{L})$.
According to Lemma 2, since the block diagonal matrix
$\mathrm{diag}{(P_{g_{k}})}$ is always singular that leads to the null space
$\mathrm{span}\\{p\\}$ of $L_{\text{\scriptsize{B}}}$, there holds

$\mathrm{Null}(\bar{L})=\mathrm{Null}(\bar{J}^{\mathrm{T}}\bar{H})\subset\mathrm{Null}(\bar{J}^{\mathrm{T}}\mathrm{diag}{(P_{g_{k}})}\bar{H})=\mathrm{Null}(L_{\text{\scriptsize{B}}}),$

which implies $\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}\subset\mathrm{Null}(L_{\text{\scriptsize{B}}})$. Thus, to ensure
that
$\mathrm{Null}(L_{\text{\scriptsize{B}}})=\mathrm{span}\\{\mathbf{1}\otimes
I_{d},p\\}$, the underlying directed graph must contain a spanning tree.

Proposition 4 provides a necessary condition for bearing equivalence that
holds for both acyclic and cyclic directed graphs. In particular, Condition
(1) in Proposition 2 violates the spanning tree condition, and therefore any
bearing formation with two leader agents (i.e., two vertices with zero out-
degree) are not bearing equivalent.

###  5.3 Growing bearing equivalent formations

A full characterization of bearing equivalence in directed graphs containing
cycles still remains an open problem. We now present an alternative
characterization to grow bearing equivalent formations, to more number of
agents or to a higher-dimensional space.

###### Proposition 5

Consider a bearing equivalent formation $\mathcal{G}(p)$ in $\mathbb{R}^{d}$
with $n$ agents modelled by a directed graph $\mathcal{G}$. Suppose an
additional vertex (agent) $p^{\prime}$ is added to $\mathcal{G}(p)$ with at
least two out-going non-collinear edges incident to existing vertices in
$\mathcal{G}(p)$. Then this augmented formation
$\mathcal{G}^{\prime}(p,p^{\prime})$ is bearing equivalent. In particular, if
$\mathrm{span}\\{\mathbf{1}_{n}\otimes
I_{d},p\\}=\mathrm{Null}(L_{\text{\scriptsize{B}}}(\mathcal{G}))$, then
$\mathrm{span}\\{\mathbf{1}_{n+1}\otimes
I_{d},(p,p^{\prime})\\}=\mathrm{Null}(L_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime}))$.

Without loss of generality we assign the index ‘$n+1$’ to the new vertex
$p^{\prime}$ in the augmented graph $\mathcal{G}^{\prime}$. The augmented
bearing Laplacian $L_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime})$ with the
augmented graph $\mathcal{G}^{\prime}$ can be expressed by

$\displaystyle
L_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime}(p,p^{\prime}))=\left[\begin{array}[]{
*{4}{c} }&&&0\\\ &&&\vdots\\\
\lx@intercol\hfil\raisebox{11.5pt}[0.0pt][0.0pt]{\scalebox{1.2}{$L_{\text{\scriptsize{B}}}(\mathcal{G}(p))$}}\hfil\lx@intercol&0\\\
\cdots&-P_{g_{(n+1)j}}&\cdots&\sum_{j\in\mathcal{N}_{n+1}}P_{g_{(n+1)j}}\end{array}\right]$
(13)

The condition that the new vertex $p^{\prime}$ has at least two out-going non-
collinear edges connected with existing vertices guarantees that no additional
null vector is introduced in $L_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime})$
with the edge addition. Therefore, if
$\mathrm{rank}(R_{\text{\scriptsize{B}}}(\mathcal{G}))=\mathrm{rank}(L_{\text{\scriptsize{B}}}(\mathcal{G}))$,
then it holds that
$\mathrm{rank}(R_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime}))=\mathrm{rank}(L_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime}))$.
In particular, with the matrix structure in (13), it is straightforward to
show that if $\mathrm{span}\\{\mathbf{1}_{n}\otimes
I_{d},p\\}=\mathrm{Null}(L_{\text{\scriptsize{B}}}(\mathcal{G}))$, then
$\mathrm{span}\\{\mathbf{1}_{n+1}\otimes
I_{d},(p,p^{\prime})\\}=\mathrm{Null}(L_{\text{\scriptsize{B}}}(\mathcal{G}^{\prime}))$.
This proposition holds for both acyclic and cyclic directed formations, and
therefore can be used to analyze complex bearing formations if they can be
decomposed by simple sub-graphs consisting of vertices with non-collinear out-
going edges.

The following statement shows that a bearing equivalent formation in a lower-
dimensional space will remain bearing equivalent in a higher-dimensional
space.

###### Proposition 6

(Dimensional invariance) Bearing equivalence is invariant to space dimensions.

This proposition can be seen as a generation of the dimensional invariance
property of infinitesimal bearing rigidity discovered in Zhao and Zelazo
(2015b).

###  5.4 Examples

Fig. 1 shows several examples of bearing equivalent formations modelled by
directed graphs with cycles. As a consequence of the cyclic structure in these
graphs, their bearing Laplacians have complex eigenvalues. For some special
agents’ positions, the associated bearing Laplacian can have eigenvalues with
negative real parts.

Take the graph (a) in Fig. 1 as an example. We randomly choose agents’
positions and the bearing Laplacian of the graph (a) often delivers
eigenvalues with negative real parts. One such position is

$\displaystyle p_{1}$
$\displaystyle=[5.8009,0.1698]^{\mathrm{T}},p_{2}=[1.2086,8.6271]^{\mathrm{T}},$
$\displaystyle p_{3}$
$\displaystyle=[4.84308.4486]^{\mathrm{T}},p_{4}=[2.0941,5.5229]^{\mathrm{T}}.$

The spectrum of the bearing Laplacian is computed as below

$\displaystyle\lambda_{1,2,3}$ $\displaystyle=0,\lambda_{4,5}=1.6400\pm
0.7564i,$ $\displaystyle\lambda_{6,7}$ $\displaystyle=0.8879\pm
0.3799i,\lambda_{8}=-0.0559,$

which gives only one eigenvalue with negative real part. 111Matlab code for
evaluating bearing equivalence in directed bearing formation examples is
available at https://www.dropbox.com/sh/5qzk9lqab0s448k/AACGEOr1r4riQVg-
RqjmNDO4a?dl=0.

For the example of graph (c) in Fig. 1, it can be constructed by adding agent
2 with three out-going edges to agents 1-3-4 in a cyclic triangle formation
(which is bearing equivalent). Therefore, the formation of Fig. 1(c) is
bearing equivalent according to Proposition 5. For all the bearing equivalent
formations evaluated in the 2D space in Fig. 1, they remain bearing equivalent
when agents’ positions are lifted in the 3-D or higher-dimensional space
according to the dimensional invariance property in Proposition 6.

$3$$1$$2$$4$

(a)

$3$$1$$2$$4$

(b)

$3$$1$$2$$4$

(c)

Figure 1: Examples of bearing _equivalent_ formations modelled by directed
graphs with cycles.

##  6 Conclusion

In this paper, motivated by the problem of bearing-based formation control in
directed graphs, we present several conditions to characterize bearing
equivalence for directed bearing formations. The notion of bearing equivalence
is defined by the kernel equivalence of bearing rigidity matrix and bearing
Laplacian. These conditions for bearing equivalence are divided into two
cases: bearing formations with acyclic directed graphs (that do not contain
any cycle) and bearing formations with cyclic directed graphs (that contain at
least one cycle). Several necessary and/or sufficient conditions are derived
to reveal the properties of the spectrum and null space of the associated
bearing Laplacian matrix.

The notion of bearing equivalence emerges as one of the key properties to
critically affect stability and convergence of bearing-based formation systems
modelled by directed graphs. In our future work, we will aim to present a
complete characterization of bearing equivalence and apply the obtained
conditions to solve the bearing-based formation control and network
localization problems underpinned by directed graphs.

## Acknowledgment

The authors would like to thank Dr. Minh Hoang Trinh for helpful discussions
and inputs to this work.

## References

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# The Very Early Soft X-ray Plateau of GRB 230307A: Signature of an Evolving
Radiative Efficiency in Magnetar Wind Dissipation?

Shu-Qing Zhong School of Astronomy and Space Science, University of Science
and Technology of China, Hefei 230026, China<EMAIL_ADDRESS>Long Li
School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China<EMAIL_ADDRESS>Di Xiao Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, China  Hui Sun Key
Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China  Bin-Bin
Zhang School of Astronomy and Space Science, Nanjing University, Nanjing
210023, China  Zi-Gao Dai School of Astronomy and Space Science, University of
Science and Technology of China, Hefei 230026, China<EMAIL_ADDRESS>
###### Abstract

Very recently, a particularly long gamma-ray burst (GRB) 230307A was reported
and proposed to originate from a compact binary merger based on its host
galaxy property, kilonova, and heavy elements. More intriguingly, a very early
plateau followed by a rapid decline in soft X-ray band was detected in its
light curve by the Lobster Eye Imager for Astronomy, indicating strong
evidence of the existence of a magnetar as the merger product. This work
explores that the Magnetar Wind Internal Gradual MAgnetic Dissipation (MIGMAD)
model, in which the radiative efficiency evolves over time, successfully fits
it to the observed data. Our results reinforce the notion that the X-ray
plateau serves as a powerful indicator of a magnetar and imply that an
evolving efficiency is likely to be a common feature in X-ray plateaus of GRB
afterglows. In addition, we also discuss the explanations for the prompt
emission, GRB afterglows, as well as kilonova, and predict possible kilonova
afterglows in a magnetar central engine.

Compact binary stars (283); Gamma-ray bursts (629); Magnetars (992)

##  1 Introduction

According to the observed bimodal distribution of duration (Kouveliotou et
al., 1993), it is widely accepted that gamma-ray bursts (GRBs) are generally
classified as short GRBs (SGRBs) originating from the merger of a binary
compact object, and long GRBs (LGRBs) invoking the core collapse of a massive
star. However, this classification is unreliable for a few apparently SGRBs
with a collapsar origin, e.g., GRBs 090426 (Levesque et al., 2010) and 200826A
(Ahumada et al., 2021; Zhang et al., 2021), and some apparently LGRBs with a
merger origin, e.g., GRBs 060614 (Gal-Yam et al., 2006; Della Valle et al.,
2006; Fynbo et al., 2006; Gehrels et al., 2006), 211227A (Lü et al., 2022),
211211A111Recent simulations (Siegel et al., 2019) and case study (Li et al.,
2023a) suggested that a “collapsar” supernova associated with a LGRB may also
produce r-process elements which likely give rise to a kilonova-like
component, but Barnes & Metzger (2023) showed that a collapsar origin for GRB
211211A is just barely possible because of the requirements of high kinetic
energy and an unexpected pattern of 56Ni enrichment. (Rastinejad et al., 2022;
Yang et al., 2022; Troja et al., 2022; Zhang et al., 2022b; Mei et al., 2022),
and the most recent case GRB 230307A (Levan et al., 2023; Sun et al., 2023;
Gillanders et al., 2023; Yang et al., 2023).

GRB 230307A is most likely hosted by a bright spiral galaxy at $z\sim 0.065$,
contains a kilonova similar to AT2017gfo, and has a spectroscopic evidence of
tellurium emission line (Levan et al., 2023; Gillanders et al., 2023; Yang et
al., 2023). Its own prompt emission contains a long-duration main burst with
two breaks in decay phase (Sun et al., 2023) and a possible precursor
(Dichiara et al., 2023), followed by optical and radio afterglows (Levan et
al., 2023) as well as a soft X-ray plateau in afterglow light curves (Sun et
al., 2023).

X-ray plateaus in GRB afterglows are usually classified as “external plateaus”
and “internal plateaus” (e.g., Troja et al., 2007; Lü & Zhang, 2014). External
plateaus are interpreted as the decelerating external forward shock by
introducing: continuous energy injection from the central engine (Dai & Lu,
1998a, b; Zhang & Mészáros, 2001; Rees & Mészáros, 1998; Sari & Mészáros,
2000; Zhang et al., 2006), a two-component jet (Racusin et al., 2008), a
reverse shock (Uhm & Beloborodov, 2007; Genet et al., 2007), evolution of
microphysical parameters (Ioka et al., 2006), or slightly misaligned viewing
angle to the GRB jet (Eichler & Granot, 2006; Beniamini et al., 2020).
Internal plateaus result from internal energy dissipation in the wind of a
central magnetar (Coroniti, 1990; Usov, 1994) or in the fallback accretion of
the stellar envelope into a black hole (BH; Kumar et al., 2008a, b). The very
early soft X-ray plateau followed by a decline phase with a slope of $-2$ in
GRB 230307A afterglows, detected by the Lobster Eye Imager for Astronomy
(LEIA; Zhang et al., 2022a; Ling et al., 2023), shows a chromatic behavior
with the optical afterglows from Levan et al. (2023). Moreover, its spectrum
exhibits less significant evolution within the first 100 s (Sun et al., 2023).
The former feature rules out the external forward shock by introducing energy
injection, evolution of microphysical parameters, or slightly misaligned
viewing angle to the GRB jet. While the latter feature usually disfavors the
external forward shock by introducing a two-component jet or a reverse shock.
As a result, these features suggest that the soft X-ray plateau is most likely
to belong to an internal plateau: the said plateau dominated by internal
dissipation emission but with a slope of $<3$ in decline phase (see
Conclusions and Discussion in Lü & Zhang, 2014). Internal X-ray plateaus in
SGRB afterglows are typically attributed to internal dissipation of the
magnetar wind rather than the fallback accretion into a BH 222The fallback
accretion of the stellar envelope into a BH can interpret internal X-ray
plateaus in LGRBs (Kumar et al., 2008a, b), but it usually cannot explain
internal X-ray plateaus in SGRBs since the required fallback mass is too large
for SGRBs with a merger origin..

Figure 1: Left panel: The best-fit results with an evolving radiative
efficiency via MIGMAD model (black solid line) and with a constant efficiency
(orange dashed line). Right panel: The corresponding evolving and constant
efficiencies. The time-averaged value $<\log_{10}(\eta_{X})>=-2.06$ of the
evolving efficiency is plotted with a black dashed line.

The X-ray plateau in GRB 230307A afterglows, which has a similar shape with
the X-ray transient CDF-S XT2 (Xue et al., 2019), therefore is naturally
ascribed to internal dissipation of the magnetar wind. While the numerical
modeling with a constant radiative efficiency during the wind dissipation
successfully reproduces the X-ray data, it is not good enough especially for
the decay phase, as shown by Figure 4 in Sun et al. (2023). This may suggest
an evolving efficiency in the wind dissipation, which is exactly the focus of
the paper.

Table 1: The Best-fit Results for the X-ray Plateaus of GRB 230307A

Parameters | Values
---|---
(catalog ) | MIGMAD | Const $\eta_{X}$
(catalog ) | Priors | Results | Priors | Results
$\log_{10}(\eta_{X})$ (catalog ) |  | Evolvinga | [$-6.0$, $0.0$] | $-1.61^{+0.41}_{-0.42}$
$\log_{10}(B_{\rm p}/{\rm G})$ (catalog ) | [12.0, 17.0] | $16.04^{+0.05}_{-0.04}$ | [12.0, 17.0] | $16.23^{+0.23}_{-0.21}$
$\log_{10}(P_{0}/{\rm s})$ (catalog ) | [$-3.1$, $-1.0$] | $-3.03^{+0.05}_{-0.04}$ | [$-3.1$, $-1.0$] | $-2.81^{+0.22}_{-0.20}$
$\log_{10}(\lambda/\epsilon)$ (catalog ) | [7.0, 10.0] | $7.69^{+0.23}_{-0.18}$ |  |
$\epsilon_{e}$ (catalog ) | [0.001, 0.5] | $0.41^{+0.05}_{-0.07}$ |  |
$\log_{10}(\Gamma_{\infty})$ (catalog ) | [1.0, 4.0] | $3.25^{+0.04}_{-0.05}$ |  |
$p$ (catalog ) | [1.0, 4.0] | $2.74^{+0.15}_{-0.16}$ |  |
$\chi^{2}$/dof (catalog ) |  | 33/16=2.1 |  | 73/19=3.8
BIC valueb (catalog ) |  | 52 |  | 82

aafootnotetext:  The efficiency in the MIGMAD model is not a free parameter,
which relies on other free parameters as well as evolves with time and has a
time-averaged value $<\log_{10}(\eta_{X})>=-2.06$ from the MCMC results.

bbfootnotetext:  The Bayesian information criterion (BIC) value is calculated
by ${\rm BIC}=\ln(n)k-2\ln(L)$ in which $n$ is the number of data point, $k$
is the number of model parameters, and $L$ is the likelihood function.

Figure 2: The spectral evolution of LEIA X-ray data plotted from the Extended
Data Table 1 of Sun et al. (2023).

##  2 Magnetar Wind Dissipation with an Evolving Radiative Efficiency

In the modeling result for the X-ray plateau by the Markov chain Monte Carlo
(MCMC) algorithm in Sun et al. (2023), the gravitational wave (GW) torque to
the magnetar spin-down is found to be negligible, so the spin-down is
dominated by the magnetic dipole torque. In this case, the magnetar rotational
energy loss rate can be written as (e.g., Shapiro & Teukolsky, 1983)

$\dot{E}=I\Omega\dot{\Omega}=-\frac{B_{\mathrm{p}}^{2}R^{6}\Omega^{4}}{6c^{3}},$
(1)

where $B_{\rm p}$, $\Omega=2\pi/P$, and $R$ are the surface dipole magnetic
field strength, spin angular velocity, and radius of the magnetar, $P$ is its
spin period, $c$ is the speed of light, $I=\frac{2}{5}MR^{2}$ is its moment of
inertia in which $M$ is the mass, throughout the paper $R=10^{6}$ cm and
$M=2.0M_{\odot}$ are adopted. One can solve above equation and obtain the form
of $\Omega(t)$. Further the observed X-ray plateau luminosity can be obtained
from the magnetic dipole luminosity after introducing a radiative efficiency
$\eta_{X}$

$L_{\mathrm{X}}(t)=\eta_{X}L_{\mathrm{sd}}=\eta_{X}\frac{B_{\mathrm{p}}^{2}R^{6}\Omega^{4}(t)}{6c^{3}}.$
(2)

Figure 3: The corner plot for the X-ray plateau of GRB 230307A.

Considering the radiative process from the magnetar spin-down to the observed
X-ray radiation via the magnetar wind internal gradual magnetic dissipation
(MIGMAD) model (Beniamini & Giannios, 2017; Xiao & Dai, 2017; Xiao et al.,
2018), as done in Xiao & Dai (2019) for X-ray plateaus in a few GRB afterglows
and in Xiao et al. (2019) for CDF-S XT2, the efficiency strongly depends on
the injected luminosity and the wind’s saturation Lorentz factor. Therefore,
the radiative efficiency should evolve with time rather than being a constant.
The basic picture of the MIGMAD model is illuminated here. A newborn
millisecond magnetar serves as the central engine and powers an initial
Poynting-flux dominated outflow with a “striped wind” magnetic field
configuration. Then the magnetic energy in the outflow is dissipated via
internal gradual reconnection process and partially converted to high-energy
emission that consists of a thermalized component and a non-thermal component
from synchrotron radiation of accelerated electrons. There are several free
parameters in the model: the surface dipole magnetic field strength $B_{\rm
p}$, the initial spin period $P_{0}$ of the magnetar, the wavelength of the
magnetic field in the striped wind configuration $\lambda$ (Coroniti, 1990;
Spruit et al., 2001; Drenkhahn, 2002; Drenkhahn & Spruit, 2002), the ratio of
reconnection velocity to the speed of light $\epsilon$ (Guo et al., 2015; Liu
et al., 2015), the fraction of dissipated energy into electrons
$\epsilon_{e}$, the bulk Lorentz factor of the wind at the saturation radius
$\Gamma_{\infty}$, and the electron distribution with an index $p$ accelerated
by reconnection (Sironi & Spitkovsky, 2014; Guo et al., 2015; Kagan et al.,
2015; Werner et al., 2016). For further details one can refer to Xiao et al.
(2018). Through the MCMC algorithm, we have a best reproduction for the
observed LEIA X-ray data with parameter values $\log_{10}(B_{\rm p}/{\rm
G})=16.04^{+0.05}_{-0.04}$, $\log_{10}(P_{0}/{\rm s})=-3.03^{+0.05}_{-0.04}$,
$\log_{10}(\lambda/\epsilon)=7.69^{+0.23}_{-0.18}$,
$\epsilon_{e}=0.41^{+0.05}_{-0.07}$,
$\log_{10}(\Gamma_{\infty})=3.25^{+0.04}_{-0.05}$, and
$p=2.74^{+0.15}_{-0.16}$, which are listed in Table 1 and plotted in Figure 3
with a corner. The corner plot shows some degenerate parameters. For example,
$B_{\rm p}$ and $P_{0}$ is degenerate because of the total luminosity of the
wind $L_{\rm sd}$ (Equation (1)), $P_{0}$ and $\lambda/\epsilon$ because of
the typical wavelength of the field $\lambda=cP$ in the striped wind
configuration (Coroniti, 1990; Spruit et al., 2001; Drenkhahn, 2002; Drenkhahn
& Spruit, 2002), as well as $\lambda/\epsilon$ and $\Gamma_{\infty}$ because
of the bulk Lorentz factor of the wind at the saturation radius given by
$r_{\infty}=\lambda\Gamma_{\infty}^{2}/(6\epsilon)$ (Beniamini & Giannios,
2017). From the MCMC results, the model prediction of spectral evolution from
$F_{\nu}\propto\nu^{-0.5}$ to $F_{\nu}\propto\nu^{-(p-1)/2}$ (Xiao & Dai,
2019) obtained by the electron distribution index $p\sim 2.74$ generally
matches the transition of the observed X-ray photon spectral index from an
average value of $\sim 1.4$ before $\sim 100$ s to $\sim 2.1$ after $\sim 100$
s, see Figure 2 plotted from the Extended Data Table 1 of Sun et al. (2023).
Note that the efficiency in the MIGMAD model is not a free parameter, which
relies on other free parameters and evolves with time. It has a time-averaged
value $<\log_{10}(\eta_{X})>=-2.06$, smaller than the value
$\log_{10}(\eta_{X})=-1.61$ in the constant efficiency scenario.

The fitting results are exhibited in Figure 1, in which the best fit with a
constant efficiency is also displayed as a comparison. As one can see, the
best fit with an evolving efficiency very well reproduces the observed data.
It is much better than the fit with a constant efficiency, especially for the
decay phase of the X-ray data. This also can be directly seen due to the
goodness comparison between these two fits with an evolving efficiency and
with a constant efficiency, from both the $\chi^{2}$/dof difference (2.1 vs.
3.8) and the BIC difference ($\Delta{\rm BIC}=-30$), see Table 1. The result
comparison with and without an evolving efficiency suggests that a specific
wind dissipation process with evolving efficiency should present in the
conversion from the magnetar spin-down power to the observed X-ray radiation.
The requirement for such an evolving efficiency is likely to be prevalent in
X-ray plateaus of GRB afterglows.

Figure 4: The best joint fitting for the multi-wavelength afterglow and
kilonova light curves of GRB 230307A by combining the standard external
forward shock model and the kilonova model. The early X-ray (1keV) data before
300 s is scaled from the LEIA data and the late is scaled from the Swift XRT,
XMM-Newton, and Chandra data. The early X-ray data as a plateau is fitted by
the MIGMAD model (black dashed-dotted line), same as the left panel of Figure
1. The data points denoted by an inverted triangle are upper limits. The GRB
afterglow (AG) components are denoted by dashed lines.

##  3 Prompt Emission, GRB Afterglows, Kilonova, and Kilonova Afterglows in a
Magnetar Central Engine

Prompt Emission. In general, there are two types of models to generate a
prompt emission if the central engine is a magnetar. One extract uniform
rotation of magnetar and invoke an internal dissipation of magnetar wind
(Usov, 1992; Metzger et al., 2011; Beniamini & Giannios, 2017), the other
extract its differential rotation and invoke magnetic bubbles (Kluźniak &
Ruderman, 1998; Dai & Lu, 1998b; Spruit, 1999; Ruderman et al., 2000; Dai et
al., 2006; Zhong et al., 2023). In this work, we hold the opinion that the
prompt emission of GRB 230307A is likely produced by the differential-
rotation-induced magnetic bubbles. This is because: (1) we have considered
that the X-ray plateau of GRB 230307A is relevant to an internal dissipation
of magnetar wind; (2) the process to the prompt emission should be different
from that to the X-ray plateau, which is supported by the joint spectral
fittings and the light curve behaviors of GECAM and LEIA data, see Sun et al.
(2023). In this case, the information of the prompt emission from the magnetar
differential rotation such as its energy and duration is not required to
compare with the energy reservoir and duration of the magnetar uniform
rotation but its differential rotation. As known by the observed properties of
the prompt emission (see Table 1 in Sun et al., 2023) and the fitting for the
GRB multi-wavelength afterglows below (Table 2), the total isotropic energy of
the jet is $E_{\rm jet,iso}=E_{\gamma,\rm iso}+E_{\rm K,\rm iso}=3.08\times
10^{52}+5.62\times 10^{50}\approx 3.14\times 10^{52}$ erg333The extremely high
radiative efficiency $\eta_{\gamma}\approx 98\%$ of the jet is consistent with
the view that the jet is Poynting-flux dominated, suggested by Yi et al.
(2023) and Du et al. (2024). This efficiency and its inferred Poynting-flux
dominated jet are naturally consistent with the jet origin from differential-
rotation-induced magnetic bubbles suggested in our work.. The maximum kinetic
energy of a differentially rotating newly-born magnetar up to $3\times
10^{53}$ erg (Cook et al., 1994) is certainly adequate for the jet energetics.
The uniform rotation energy of the magnetar
$E_{\Omega}=\frac{1}{2}I\left(\frac{2\pi}{P_{0}}\right)^{2}\approx 2\times
10^{52}$ erg inferred from Table 1 is also certainly adequate for the X-ray
plateau energetics ($2.47\times 10^{50}$ erg in Table 1 of Sun et al., 2023),
with a time-averaged radiative efficiency $<\log_{10}(\eta_{X})>=-2.06$. This
small efficiency implies that the lost of the uniform roration energy in the
X-ray plateau is nearly negligible. As will be shown in the following, the
rest of the uniform rotation energy should be poured into the kilonova ejecta
and ultimately the circumburst medium (CSM), and make its radio afterglow much
brighter.

GRB Afterglows and Kilonova. In a magnetar central engine, the transition from
internal to external dissipation in the X-ray afterglow should occur post the
LEIA data but prior to the Swift XRT data, as shown by the best joint fitting
for the multi-wavelength GRB afterglow and kilonova light curves in Figure 4.
Within which the early X-ray (1keV) data before 300 s are scaled from the LEIA
(Sun et al., 2023) while the late are collected from the Swift XRT, XMM-
Newton, and Chandra (Yang et al., 2023), the optical and radio data are
collected from Levan et al. (2023). The best joint fitting is obtained by the
MCMC technique, within the framework of combining the standard external
forward shock model (Sari et al., 1998; Huang et al., 1999) and the kilonova
model used in Xiao et al. (2017). Its corresponding parameter results are
listed in Table 2. Within which there are seven free parameters for the
standard external forward shock model: the jet half-opening angle $\theta_{\rm
j}$, the isotropic kinetic energy $E_{\rm K,iso}$, the initial Lorentz factor
$\Gamma_{0}$, the ratio of shock energy to the magnetic field $\epsilon_{B}$,
the ratio of shock energy to the electron $\epsilon_{e,\rm f}$, the
interstellar medium (ISM) density $n_{\rm ISM}$, and the power-law index of
the electron distribution $p_{\rm f}$, while four free parameters for the
kilonova model: the velocity $v_{\rm ej}$, the polar opening angle
$\theta_{\rm ej}$, the mass $M_{\rm ej}$, and the opacity $\kappa$ of the
ejecta.

Kilonova Afterglows. Due to the significantly small uniform rotation energy
lost in the X-ray plateau, the energy poured into the kilonova ejecta can be
the total uniform rotation energy $E_{\Omega}$. According to the estimate of
Metzger & Bower (2014), the kilonova afterglow in radio band $\nu=1.4$ GHz
should have a peak flux density given by

$\displaystyle F_{\rm peak}\approx$ $\displaystyle 3~{}{\rm
mJy}\left(\frac{E_{\Omega}}{10^{52}\rm erg}\right)\left(\frac{n_{\rm
ISM}}{{\rm cm}^{-3}}\right)^{0.83}\left(\frac{\epsilon_{B,\rm
ej}}{0.1}\right)^{0.83}$ (3) $\displaystyle\times\left(\frac{\epsilon_{e,\rm
ej}}{0.1}\right)^{1.3}\beta_{0}^{2.3}d_{28}^{-2}$

at the deceleration time

$t_{\rm dec}\approx 1.2~{}{\rm yr}\left(\frac{E_{\Omega}}{10^{52}{\rm
erg}}\right)^{1/3}\left(\frac{n_{\rm ISM}}{{\rm
cm}^{-3}}\right)^{-1/3}\beta_{0}^{-5/3},$ (4)

where $\epsilon_{B,\rm ej}$ and $\epsilon_{e,\rm ej}$ are the ratios of shock
energy to the magnetic field and to the electron in the ejecta-related shock,
$d=10^{28}d_{28}$ cm is the luminosity distance, and $\beta_{0}\sim
1.0\left(\frac{E_{\Omega}}{10^{52}\rm erg}\right)^{1/2}\left(\frac{M_{\rm
ej}}{10^{-2}M_{\odot}}\right)^{-1/2}$ is the accelerated velocity of the
matter in which the magnetar wind couples its energy to the ejecta. Since
$E_{\Omega}$, $n_{\rm ISM}$, and $M_{\rm ej}$ have been obtained (see Table
2), if given $\epsilon_{B,\rm ej}=0.1$ and $\epsilon_{e,\rm ej}=0.1$, one
would have $F_{\rm peak}\approx 0.66$ mJy at $t_{\rm dec}\approx 15$ yr. This
predicted late-time radio emission may be used as a critical way to test
whether the central engine of GRB 230307A is a millisecond magnetar.

Table 2: The Best Parameter Distributions for Jointly Modeling the Multi-wavelength Afterglows and Kilonova of GRB 230307A by Combining the Standard External Forward Shock Model and the Kilonova Model Parameters | Priors | Values
---|---|---
Standard Afterglow Model (catalog ) |  |
$\theta_{\rm j}$ (rad) (catalog ) | [0.001, 0.5] | $0.08^{+0.02}_{-0.01}$
$\log_{10}(E_{\rm K,iso}/{\rm erg})$ (catalog ) | [48.0, 55.0] | $50.76^{+0.26}_{-0.28}$
$\log_{10}(\Gamma_{0})$ (catalog ) | [2.0, 4.0] | $2.77^{+0.28}_{-0.27}$
$\log_{10}(n_{\rm ISM}/{\rm cm}^{-3}$) (catalog ) | [$-5.0$, $-2.0$] | $-4.43^{+0.42}_{-0.39}$
$\log_{10}(\epsilon_{B})$ (catalog ) | [$-7.0$, $-0.5$] | $-0.87^{+0.25}_{-0.54}$
$\log_{10}(\epsilon_{e,\rm f})$ (catalog ) | [$-6.0$, $-0.5$] | $-0.59^{+0.06}_{-0.07}$
$p_{\rm f}$ (catalog ) | [2.0, 3.0] | $2.83^{+0.05}_{-0.03}$
Kilonova Model (catalog ) |  |
$\log_{10}(v_{\rm ej}/c)$ (catalog ) | [$-2.0$, $-0.3$] | $-1.27^{+0.01}_{-0.01}$
$\theta_{\rm ej}$ (rad) (catalog ) | [$\pi/8$, $3\pi/8$] | $0.75^{+0.27}_{-0.25}$
$\log_{10}(M_{\rm ej}/M_{\odot})$ (catalog ) | [$-3.0$, $-1.3$] | $-2.27^{+0.03}_{-0.04}$
$\log_{10}(\kappa/{\rm cm^{2}~{}g^{-1}})$ (catalog ) | [$-1.0$, $2.0$] | $1.31^{+0.14}_{-0.16}$

##  4 Summary and Discussion

We have presented numerical fittings for the X-ray plateau of GRB 230307A with
and without MIGMAD model. The comparative fitting results imply a signature of
an evolving efficiency in the magnetar wind dissipation from the magnetar
spin-down power to the observed X-ray plateau. Moreover, such an evolving
efficiency in magnetar wind dissipation is likely to be ubiquitous in those
X-ray plateaus of afterglows for either SGRBs or LGRBs have been fitted by
many authors using an empirical smoothly broken power-law function (e.g.,
Rowlinson et al., 2013; Lü & Zhang, 2014; Lü et al., 2015; Tang et al., 2019;
Zou et al., 2019, 2021). We have also discussed the explanations for the
prompt emission, GRB afterglows, as well as kilonova, and also predicted
possible kilonova afterglows in a magnetar central engine.

In this work, one caveat related to the X-ray data should be kept in mind,
that is, the first data point is lower than the second one. This can be
explained by, during the initial ten seconds after the birth of proto-
magnetar, the magnetar wind that is baryon-loaded due to the strong neutrino
heating and thus has a small $\Gamma_{\infty}$, as already noticed in Metzger
et al. (2011). This therefore leads to an inefficient dissipation of the
magnetar wind and a very low value of $\eta_{X}$ via the MIGMAD model.

Another caveat is the possibility that magnetar wind powers internal X-ray
plateaus in non-collapsar GRBs is challenged by the distributions of energies
and durations of internal plateaus (see Beniamini & Lu, 2021), the ought-to-be
brighter kilonova boosted by magnetar wind (mergernova; Yu et al., 2013; Gao
et al., 2013), as well as by the possible absent of forthcoming late-time
radio emission of SGRBs due to the interaction between the merger ejecta and
CSM (Metzger & Bower, 2014; Fong et al., 2016; Horesh et al., 2016; Ricci et
al., 2021). The second inconsistency between the mergernova expectation and
the not-too-bright kilonova of GRB 230307A (Levan et al., 2023; Yang et al.,
2023) could be mediated by a quite low heating efficiency (Ai et al., 2022).
But a more recent work by Wang et al. (2024) discussed a counter-argument to
this low heating efficiency suggestion. While the last may be mediated by a
much lower circumburst median density (Metzger & Bower, 2014), a relatively
lower $\epsilon_{B,\rm ej}$ (Liu et al., 2020), or an incomplete sweeping on
the interstellar medium of kilonova ejecta (Li et al., 2023b), though we have
predicted a bright late-time radio emission peaking around 0.66 mJy at $\sim
15$ yr in Section 3.

While most of current observations seem to favor a compact binary merger for
the progenitor of GRB 230307A, it is still on debate that what the compact
binary is. If the final object is a magnetar as required in our work for GRB
230307A, similar to GRB 211211A, the progenitor system could only be neutron
star$-$neutron star (NS$-$NS) or neutron star$-$white dwarf binaries(NS$-$WD;
Yang et al., 2022; Zhong et al., 2023). For an NS$-$NS merger, however, how to
explain the particularly long duration of GRB 230307A may be a challenge.
While for an NS$-$WD merger, the production of a kilonova is an open question.

Yang et al. (2022) suggested that the NS might merge into the center of the WD
and induce the collapse of the WD, then kilonova could be produced by the
ejecta consisting of a neutron-poor component and a neutron-rich component.
The former component is from the materials of the WD neutronized and mixed
with the materials of the pre-merger NS during the WD collapsing, while the
latter is from the disk wind after a proto-magnetar forms. This idea is novel
and its key assumption is that the WD is not totally disrupted by the NS.
However, this assumption is contrast to the conventional view that the WD is
completely tidally disrupted by the NS (Metzger, 2012; Fernández & Metzger,
2013; Margalit & Metzger, 2016; Bobrick et al., 2017, 2022; Fernández et al.,
2019; Zenati et al., 2019, 2020; Kaltenborn et al., 2023; Morán-Fraile et al.,
2024). Moreover, whether the pre-merger NS can induce the collapse of the WD
is unknown.

In the model of the tidally-disrupted NS$-$WD merger of Zhong et al. (2023) in
which the WD is completely tidally disrupted by the NS, the kilonova-like
emission of GRB 211211A might actually be a fast-evolving transient like SN
2005ek (Drout et al., 2013) or SN 2018kzr (McBrien et al., 2019; Gillanders et
al., 2020) powered by 56Ni radioactive decay adding an energy injection from
central engine. While for GRB 230307A, r-process heavy elements are found in
the kilonova-like emission. In a tidally-disrupted NS$-$WD merger, these
r-process elements could be produced by the neutron-rich material away from
the newly formed postmerger magnetar. For which the neutron-rich material is
accelerated by a magnetic jet. This picture should be similar to that
summarized in Siegel (2022) for magnetorotational or “jet-driven” core-
collapse supernovae. That is, in a competition between neutrino irradiation
and acceleration by the magnetic jet, a fraction of the ejecta may evade the
strong protonizing neutrino irradiation from the postmerger hot magnetar to
remain relatively neutron-rich. Production of r-process elements beyond the
second peak requires a magnetar-strength magnetic field and a millisecond
rotation period to trigger a prompt jet-like ejection of neutron-rich matter
(Winteler et al., 2012; Nishimura et al., 2017; Halevi & Mösta, 2018; Mösta et
al., 2018; Reichert et al., 2021). These requirements seem to be met in the
tidally-disrupted NS$-$WD merger model of Zhong et al. (2023). The possibility
that the tidally-disrupted NS$-$WD merger model accounts for all the
observations of GRB 230307A, particularly the r-process heavy elements, will
be explored elsewhere.

We are very grateful to thank the referee for the careful and thoughtful
suggestions that have helped us improve this manuscript substantially. We
thank Ruo-Yu Liu for helpful discussions. This work is supported by the
National SKA Program of China (grant No. 2020SKA0120300) and National Natural
Science Foundation of China (grant No. 12393812). S.Q.Z. acknowledges support
from the National Natural Science Foundation of China (grant No. 12247144) and
China Postdoctoral Science Foundation (grant Nos. 2021TQ0325 and 2022M723060).
L.L. is supported by the National Natural Science Foundation of China (grant
No. 12303050), China Postdoctoral Science Foundation (grant No. 2023M743397),
and Fundamental Research Funds for the Central Universities. D.X. is supported
by the National Natural Science Foundation of China (grant No. 12373052). H.S.
is supported by the National Natural Science Foundation of China (grant No.
12103065). BBZ acknowledges the support by the National Key Research and
Development Programs of China (2022YFF0711404, 2022SKA0130102), the National
SKA Program of China (2022SKA0130100), the National Natural Science Foundation
of China (Grant Nos. 11833003, U2038105, 12121003), the science research
grants from the China Manned Space Project with NO.CMS-CSST-2021-B11, the
Fundamental Research Funds for the Central Universities, and the Program for
Innovative Talents and Entrepreneur in Jiangsu.

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