\section{Introduction}
In the past three decades, the ultrocold atom physics has been greatly developed with the progressive techniques to cool, trap and manipulate neutral atoms with electromagnetic fields\cite{RMP1999,PhysUsp}. The quantum nature of the ultracold atom gases will emerge after suppressing the thermal fluctuations, and these artificially controllable quantum systems provide the ideal platforms for realizing Bose-Einstein condensation, simulating quantum many-body phenomena, performing quantum computation, designing atomic interferometers, \emph{etc}. Two typical mechanisms to control the neutral atoms are the alternating-current Stark energy shift in the high-frequency optical field and the Zeeman energy shift in the inhomogeneous magnetic field. The former has been widely applied to construct optical lattices for ultracold atoms, which play the crucial role in design various quantum simulators\cite{OptLatt}; while the latter lays the foundation to design and develop atom chips\cite{AtomChip}. In fact, the two mechanisms are more often combined together to achieve the best level of control of the ultracold atoms.

Several strategies have been developed to produce the desirable magnetic field to control ultracold atoms. The most flexible approach is to fabricate current-carrying conductive microstructures to generate the Oersted field, which has been applied to guide and trap ultracold atoms in various configurations\cite{RMP2007}. This approach however suffers from the Johnson thermal noise and wire roughness for miniature devices. Another alternative way is to utilize the permanent ferromagnetic film with artifical patterns as the effective ``magnetization current"\cite{PRA2007,PRA2017}, which however is hard to be reconfigured and switched off. Besides, there also exist other theoretical proposals to generate magnetic lattice for ultracold atoms, such as the vortex array in the superconducting film\cite{NJP2010,PRL2013} and time-periodic magnetic field pulses\cite{NJP2015,PRL2016}.

In addition to the artificial patterns, noncollinear magnetic textures can also induce inhomogeneous configurations of magnetic field  near the surface of the ferromagnetic film. Indeed, West \emph{et al.} have demonstrated the ability to manipulate ultracold $^{87}$Rb atoms by magnetic domain walls in planar magnetic nanowires\cite{NL2012}. Recently, the chiral ferromagnetic materials with finite Dzyaloshinskii-Moriya interaction(DMI) resulting from the lack spatial inversion symmetry\cite{Dzya,Moriya} have attracted broad research interests\cite{NRM1,NRM2}, mainly due to the discovery of magnetic skyrmions therein\cite{PRL2001,Nature2006,Science2009,Nature2010,NatMat2010,NatPhys2011} and the promising applications to develop spintronics devices\cite{NatNano2013,Proc2016}. With the formation of magnetic skyrmions, the magnetic field distribution near the surface of chiral ferromagnetic film will also been modified. For example, the magnetic field produced by an isolated magnetic skyrmion has been measured with the nitrogen-vacancy center in diamond\cite{NC2018}. The magnetic field distributions generated by the magnetic skyrmion crystals (SkXs) with different helicity have also been investigated theoretically\cite{NJP2018}. Therefore, it will be interesting to examine the possibility to control the ultracold atoms with the magnetic field produced by the chiral ferromagnetic film.

In this paper, we show that the chiral ferromagnetic film can be utilized to design magnetic mirror, reflection grating, magnetic lattices, \emph{etc.}, which are the elementary devices to control the ultracold atoms. Considering the diversity of chiral ferromagnetic materials and the rapidly developing spintronics techniques, the approach proposed here could be crucial to develop ultracold atom physics in future.

\section{Basic Principle and Model}

The proposed device is schematically shown in Fig.~\ref{Fig1}(a). A chiral ferromagnetic film is placed in the $x$-$y$ plane with $z=0$~nm, which will generate the spatial distributions of magnetic field $\bm{\mathcal{B}}_{c}$. An uniform magnetic field $\bm{\mathcal{B}}_{0}$ is applied perpendicular to the film to create zero points at finite heights. Furthermore, a time-dependent magnetic field $\bm{\mathcal{B}}_{M}=\mathcal{B}_{M}(\sin\omega_{M}t,\cos\omega_{M}t,0)$ parallel to the film will be introduced to generate an effective time-orbiting potential in order to suppress the Majorana loss\cite{PRL1995,PRL2013}. Thus, the total magnetic field profile of such a device will be $\bm{\mathcal{B}}=\bm{\mathcal{B}}_{c}+\bm{\mathcal{B}}_{0}+\bm{\mathcal{B}}_{M}$.

\begin{figure}
\includegraphics[scale=0.5]{Fig1.pdf}
\caption{(Color online) (a) Schematic diagram for the device. The chiral ferromagnetic film with prepared magnetization configuration is placed at the $x$-$y$ plane with $z=0$~nm, which will produce a magnetic field $\bm{\mathcal{B}}_{c}$; a bias magnetic field $\bm{\mathcal{B}}_{0}$ is applied perpendicular to the film; a time-dependent magnetic field $\bm{\mathcal{B}}_{M}(t)$ is applied parallel to the film. The trapping potential for the ultracold atoms (blue balls) is a combination of the three magnetic fields. (b) Phase transition of the chiral ferromagnetic film defined by Eq.~(\ref{Ham}). The magnetization configurations of helical (H) phase and skyrmion crystal (SkX) phase are obtained at $\mathbf{B}=0$~T and $\mathbf{B}=-(0,0,0.3)$~T respectively. The arrows denote the magnetization direction at each site, and the amplitude of $m_{z}$ is shown with different color.}\label{Fig1}
\end{figure}

When an atom with magnetic dipole moment $\bm{\mu}$ is moving in the inhomogeneous magnetic field $\bm{\mathcal{B}}(\mathbf{r})$, it will experience the Stern-Gerlach force given by the potential energy $U(\mathbf{r})=-\bm{\mu}\cdot\bm{\mathcal{B}}(\mathbf{r})$. If the spatial change of the magnetic field felt by the atom is much slower than its Larmor precession, its magnetic moment will adiabatically follow the direction of the magnetic field, and the potential energy now depends only on the modulus of $\bm{\mathcal{B}(\mathbf{r})}$, which is given as $U(\mathbf{r})=m_{F}g_{F}\mu_{B}\mathcal{B}(\mathbf{r})$ for the hyperfine state $|F,m_{F}\rangle$\cite{AtomChip}. Here, $g_{F}$ is the Land\'{e} factor, $\mu_{B}$ is the Bohr magneton, $F$ is the total angular momentum quantum number, and $m_{F}$ is the magnetic quantum number of the atom. Therefore, the atom will be in the strong-field seeking state for $m_{F}g_{F}<0$ and in the weak-field seeking state for $m_{F}g_{F}>0$. Since no minimum can exist in the potential $U(\mathbf{r})$ for the strong-field seeking state according to the Earnshaw's theorem\cite{AtomChip}, only the atoms prepared in the weak-field seeking state can be magnetically trapped. For example, the $^{87}\text{Rb}$ and $^{7}\text{Li}$ atoms can be magnetically trapped when they stay at the state $|F=2,m_{F}=2\rangle$\cite{RMP2007,PRA2007-2}.

An external magnetic field $\mathbf{B}$ should be applied \emph{in advance} to tune the magnetization configuration of the chiral ferromagnetic film, which is described by the energy functional density
\begin{eqnarray}
\mathcal{E}[\mathbf{m}]=\frac{J}{2}(\nabla\mathbf{m})^{2}+D\mathbf{m}\cdot(\nabla\times\mathbf{m})-K\mathbf{m}_{z}^{2}-M_{s}\mathbf{B}\cdot\mathbf{m}.\nonumber\\\label{Ham}
\end{eqnarray}
Here, $\mathbf{m}(\mathbf{r})$ is the magnetization distribution of the film normalized by the saturation magnetization $M_{s}$; the four terms in Eq.~ (\ref{Ham}) are the Heisenberg exchange interaction, the DM interaction, the perpendicular magnetic anisotropy energy, and the Zeeman energy in presence of the applied magnetic field $\mathbf{B}$ respectively. Then the magnetization dynamics will be given by the Landau-Lifshitz-Gilbert(LLG) equation
\begin{eqnarray}
\frac{d\mathbf{m}}{dt}=-\gamma\mathbf{m}\times\mathbf{H}^{eff}+\alpha\mathbf{m}\times\frac{d\mathbf{m}}{dt}.\label{LLG}
\end{eqnarray}
Here, $\gamma$ is the gyromagnetic ratio, $\mathbf{H}^{eff}=-\frac{1}{M_{s}}\nabla_{\mathbf{m}}\mathcal{E}[\mathbf{m}]$ is the effective magnetic field, and $\alpha$ is the Gilbert damping coefficient. For a given set of parameters, the stable magnetization configuration of the chiral ferromagnetic film is achieved as the stationary solution of the LLG equation\cite{NJP2018}. As shown in Fig.~\ref{Fig1}(b), when the external magnetic field $\mathbf{B}$ is perpendicular to the film and is continuously increased from zero, the ground state of the chiral ferromagnetic film will evolve from the helical phase to the skyrmion crystal phase at first, and reach the ferromagnetic phase finally. During the micromagnetic simulations, the parameters are set as $J=15$~pJ/m, $D=3$~mJ/m$^{2}$, $K=0.7$~MJ/m$^{3}$, $M_{s}=580$~kA/m, $\alpha=0.3$, and the thickness of the film is assumed as $1$~nm. The film is subdivided into cubic grids with the size $1\times1\times1$~nm$^3$, and the time step for the simulations is set as $0.01$~ps. Besides, the temperature is linearly decreased from $1000$~K to $0$~K during $100$~ns to avoid possible local minima trap.

The external magnetic field $\mathbf{B}$ will be withdrawn once the desirable magnetization pattern is stably reached. Then the magnetic field $\bm{\mathcal{B}}_{c}(\mathbf{r})$ generated by the magnetization configuration $\mathbf{m}(\mathbf{r})$ of the chiral ferromagnetic film can be directly calculated as\cite{NJP2018}
\begin{eqnarray}
\bm{\mathcal{B}}_{c}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int d\mathbf{r}'\frac{3(\mathbf{m}(\mathbf{r}')\cdot\hat{\bm{\mathcal{R}}})\hat{\bm{\mathcal{R}}}-\mathbf{m}(\mathbf{r}')}{\mathcal{R}^{3}},\label{StrayField}
\end{eqnarray}
where $\bm{\mathcal{R}}=\mathbf{r}-\mathbf{r}'$. Below, we will study how to control the ultracold atoms in the weak-field seeking state with the magnetic field $\bm{\mathcal{B}}_{c}(\mathbf{r})$ when the chiral ferromagnetic film is prepared in the helical phase and skyrmion crystal phase respectively.

\section{Helical Phase}
\begin{figure}
\includegraphics[scale=0.5]{Fig2.pdf}
\caption{(Color online) Profiles of magnetic field amplitude $\mathcal{B}(\mathbf{r})$ for the chiral ferromagnetic film at helical phase. (a) Magnetic mirror when $\mathcal{B}_{0}=0$~mT and $\mathcal{B}_{M}=0$~mT. Top : isosurfaces of $\mathcal{B}(\mathbf{r})$ with three values $8$~mT, $16$~mT, and $24$~mT; middle and bottom : cross sections of $\mathcal{B}(\mathbf{r})$ when $z=7$~nm, $x=21$~nm, and $y=42$~nm respectively. (b) One-dimensional magnetic lattice when $\mathcal{B}_{0}=20$~mT and $\mathcal{B}_{M}=10$~mT. Top : isosurfaces of $\mathcal{B}(\mathbf{r})$ with three values $13.5$~mT, $17.0$~mT, $20.5$~mT; middle and bottom : cross sections of $\mathcal{B}(\mathbf{r})$ when $z=8$~nm, $x=21$~nm, and $y=42$~nm respectively. }\label{Fig2}
\end{figure}

As shown in Fig.~\ref{Fig1}(b), the helical phase can be achieved as the ground state of the chiral ferromagnetic film when $\mathbf{B}=0$, where the magnetization configuration is periodically modulated in one direction and mimics the structure of a magnetic mirror for ultracold atoms fabricated from the permanent ferromagnetic materials\cite{AtomChip,JPD1999}. Nevertheless, it is advantageous to utilize the chiral ferromagnetic film in helical phase as a magnetic mirror, since the helical phase is spontaneously formed as the ground state and can also be easily switched off by applying an external magnetic field. Here, the complex fabrication processes are not required in comparison to the ferromagnetic film with artificial patterns\cite{AtomChip,JPD1999}. Besides, the period of the helical phase can be tuned by engineering the DMI\cite{NatNano2013-2,NatNano2015}, which thereby makes it flexible to control ultracold atoms.

The magnetic field amplitude $\mathcal{B}_{c}(\mathbf{r})$ solely generated by the chiral ferromagnetic film in the helical phase in Fig.~\ref{Fig1}(b) has been numerically calculated based on Eq.~(\ref{StrayField}) and is demonstrated in Fig.~\ref{Fig2}(a). Here, the bias magnetic field $\bm{\mathcal{B}}_{0}$ and the time-dependent field $\bm{\mathcal{B}}_{M}$ have not been switched on yet. We see that the isosurfaces of $\mathcal{B}_{c}(\mathbf{r})$ will be nearly parallel to the $x$-$y$ plane. In fact, a weak periodic modulation of $\mathcal{B}_{c}(\mathbf{r})$ will happen at fixed height $z$. For example, $\mathcal{B}_{c}(\mathbf{r})$ will periodically vary from $21$~mT to $24$~mT when $z=7$~nm. On the other hand, $\mathcal{B}_{c}(\mathbf{r})$ will decay from $81$~mT to $2$~mT when the height $z$ increases from $0$~nm to $25$~nm. Therefore, the effective distance for the chiral ferromagnetic film to control the ultracold atoms will be tens of nanometers with the given simulation parameters here.

The simulated results in Fig.~\ref{Fig2}(a) can be further understood considering that the magnetization configuration of the helical phase can be approximated as a single-$\mathbf{Q}$ state
$\mathbf{m}_{h}(\mathbf{r})=\mathbf{m}_{0}\delta(z)+\mathcal{A}[\hat{\mathbf{e}}_{z}\cos(\mathbf{Q}\cdot\mathbf{r})+\hat{\mathbf{e}}\sin(\mathbf{Q}\cdot\mathbf{r})]\delta(z)$. Here, $\hat{\mathbf{e}}_{z}$ is the unit vector normal to the film and $\mathbf{Q}$ defines the wavevector of the helical state; the unit vector $\hat{\mathbf{e}}$ is determined as $\hat{\mathbf{e}}=\hat{\mathbf{e}}_{z}\times\hat{\mathbf{Q}}$ for the Bloch-type DMI given in Eq.~ (\ref{Ham}); $\mathcal{A}$ gives the amplitude of the modulated magnetization configuration of the helical state. The delta function $\delta(z)$ here implies that the thickness of the film is neglected. Then the magnetic field distribution $\bm{\mathcal{B}}_{h}(\mathbf{r})$ generated by the single-$\mathbf{Q}$ state $\mathbf{m}_{h}(\mathbf{r})$ is analytically obtained as\cite{NJP2018}
\begin{eqnarray}
\bm{\mathcal{B}}_{h}(\mathbf{r})=\frac{\mathcal{A}Q}{2}e^{-Q|z|}[\text{sgn}(z)\hat{\mathbf{Q}}\sin(\mathbf{Q}\cdot\mathbf{r})+\hat{\mathbf{e}}_{z}\cos(\mathbf{Q}\cdot\mathbf{r})],\nonumber\\\label{mirror}
\end{eqnarray}
and its modulus is $\mathcal{B}_{h}(\mathbf{r})=\frac{\mathcal{A}Q}{2}e^{-Q|z|}$. Therefore, the chiral ferromagnetic film described by the single-$\mathbf{Q}$ state will establish an exponential repulsive potential to reflect the ultracold atoms at weak-field seeking state. The decay length of the potential is proportional to the period of the helical phase, which is usually about a few tens of nanometers or even smaller for typical chiral ferromagnetic materials\cite{NatNano2013}. Since the period and decay length of the artifical microstructures are usually in the order of magnitude of micrometers\cite{AtomChip}, a much harder magnetic mirror for ultracold atoms can be realized with the chiral ferromagnetic film. Note that $\mathcal{B}_{h}(\mathbf{r})$ here is constant at given height $z$, which is different from the numerical result in Fig.~\ref{Fig2}(a). This minor deviation origins from the magnetic anisotropy energy in Eq.~(\ref{Ham}), where the single-$\mathbf{Q}$ state becomes an oversimplified and inaccurate description of the helical phase. It will be important to choose the materials with smaller magnetic anisotropy to get more smooth mirrors.

The profile of magnetic field amplitude will be modified significantly when the bias magnetic field $\bm{\mathcal{B}}_{0}$ and the time-dependent magnetic field $\bm{\mathcal{B}}_{M}$ are turned on\cite{AtomChip,JPD1999}. Fig.~\ref{Fig2}(b) shows the distribution of total magnetic field amplitude $\mathcal{B}(\mathbf{r})$ for the device when $\bm{\mathcal{B}}_{0}=(0,0,20)$~mT and $\mathcal{B}_{M}=10$~mT. As shown in Fig.~\ref{Fig2}(b), the minima of $\mathcal{B}(\mathbf{r})$ in this case will locate at the parallel lines in the plane $z=8$~nm, and the appearing one-dimensional magnetic lattice is able to trap the ultracold atoms in the weak-field seeking state. We can define the ``recoil energy'' of the lattice as $E_{R}=\pi^{2}\hbar^{2}/(2m_{a}a^{2})$, where $m_{a}$ is the atom mass and $a$ is the lattice constant\cite{PRL2013}. For the helical phase with $a=42$~nm here, the recoil energy $E_{R}$ is estimated as $1.3$~neV for $^{87}\text{Rb}$ atom or $16$~neV for $^{7}\text{Li}$ atom respectively.

The numerical results in Fig.~\ref{Fig2}(b) can also be understood with the help of the single-$\mathbf{Q}$ state. Assuming that $\bm{\mathcal{B}}_{0}=\frac{\mathcal{A}Q}{2}(0,0,B_{0})$ and $\mathcal{•}mathcal{B}_{M}=\frac{\mathcal{A}Q}{2}B_{M}$, the modulus of the total magnetic field $\bm{\mathcal{B}}(\mathbf{r})$ will be
\begin{eqnarray}
\mathcal{B}(\mathbf{r})=\frac{\mathcal{A}Q}{2}\sqrt{B_{0}^{2}+B_{M}^{2}+e^{-2Q|z|}+2B_{0}e^{-Q|z|}\cos(\mathbf{Q}\cdot\mathbf{r})}.\nonumber\\\label{grating}
\end{eqnarray}
Therefore, such an operation will create an one-dimensional periodic potential for the ultracold atoms in the plane parallel to the film. This in fact forms an effective reflection grating for the incident atoms\cite{JPD1999}, which will attain the period in the range of nanometers and hence greatly extend the border of current technologies. Furthermore, the modulus $\mathcal{B}(\mathbf{r})$ will achieve its minimal value $\mathcal{B}_{\text{min}}=\mathcal{B}_{M}$ along the parallel lines defined by $\cos(\mathbf{Q}\cdot\mathbf{r}_{\text{min}})=-1,|z_{\text{min}}|=-\frac{1}{Q}\ln B_{0}$ in the case $B_{0}<1$. Thus, the magnetic field around each line will become a guide to trap the ultracold atoms, where the trapping height $|z_{\text{min}}|$ depends on the applied magnetic field. When $\mathcal{B}_{0}$ is continuously increased, the guides will get close to the film and finally disappear at $B_{0}=1$. This trend has also been verified in the numerical calculations, as shown in Fig.~\ref{Fig3}(a). When the magnetic field $\mathcal{B}_{0}$ varies from $5$~mT to $40$~mT, the trapping height of the one-dimensional magnetic lattice will decay from about $17$~nm to $3$~nm.

In addition to the trapping height, the potential barrier for the atoms to escape from the minima locations $z_{\text{min}}$ to the infinitely far region $z\rightarrow\infty$ can also been tuned by the magnetic field $\bm{\mathcal{B}}_{0}$. The difference between the magnetic field amplitudes $\mathcal{B}(\infty)$ and $\mathcal{B}(z_{\text{min}})$ will be $\delta\mathcal{B}=\sqrt{\mathcal{B}_{0}^{2}+\mathcal{B}_{M}^{2}}-\mathcal{B}_{M}$, and the potential barrier will be $U_{\text{barr}}=m_{F}g_{F}\mu_{B}\delta\mathcal{B}$. When $\mathcal{B}_{M}$ is fixed as $10$~mT and $\mathcal{B}_{0}$ increases from $5$~mT to $40$~mT, the potential barrier for the $^{87}\text{Rb}$ and $^{7}\text{Li}$ atoms at $|F=2,m_{F}=2\rangle$ state will vary from $0.068$~$\mu$eV to $1.81$~$\mu$eV, which corresponds to the temperature range from $0.8$~mK to $21$~mK. In Fig.~\ref{Fig3}(a), the dependence of $U_{\text{barr}}$ on $\mathcal{B}_{0}$ is explicitly shown in the unit of the recoil energy $E_{R}$ for $^{87}\text{Rb}$ atom.

\begin{figure}
\includegraphics[scale=0.60]{Fig3.pdf}
\caption{(Color online) Field-dependent parameters of the one-dimensional magnetic lattice for $^{87}\text{Rb}$ atom. (a) trapping height $z_{\text{min}}$ and potential barrier $U_{\text{barr}}$ in the unit of the recoil energy $E_{R}$; (b) trapping frequency $\omega_{T}/2\pi$; (c) Marojana loss rate $\Gamma/2\pi$. Here, $\mathcal{B}_{0}$ varies from $5$~mT to $40$~mT, while $\mathcal{B}_{M}$ is fixed as $10$~mT. }\label{Fig3}
\end{figure}

The trapping frequency $\omega_{T}$ of the atoms near the minima of the one-dimensional magentic lattice can be calculated as $\omega_{T}=\sqrt{m_{F}g_{F}\mu_{B}\mathcal{B}''_{\text{min}}/m_{a}}$, where $\mathcal{B}''_{\text{min}}$ is the magnetic field curvature at the minima. Fig.~\ref{Fig3}(b) shows the trapping frequencies along $x$ and $z$ directions for $^{87}\text{Rb}$ atom at $|F=2,m_{F}=2\rangle$ state when $\mathcal{B}_{M}=10$~mT and $\mathcal{B}_{0}\in [5,40]$~mT, where $\omega_{T,x}/2\pi\in[9.5,61.3]$~MHz and $\omega_{T,z}/2\pi\in[9.7,60.9]$~MHz respectively. Then the energy of the $^{87}\text{Rb}$ atom will be in the range $[0.04,0.25]$~$\mu$eV, which corresponds to the temperature range $[0.46,2.9]$~mT. Since the mass of $^{7}\text{Li}$ atom is about one-twenlfth of $^{87}\text{Rb}$ atom mass, its trapping frequency will be about $3.5$ times of the estimated value above.

The Majorana loss rate near the minima of the lattice potential can be estimated as $\Gamma/2\pi=\omega_{T}\text{exp}(-4\omega_{L}/\omega_{T})$\cite{PRL2013,PRA2006}, where $\omega_{L}=m_{F}g_{F}\mu_{B}\mathcal{B}_{M}/\hbar$ is the Larmor precession frequency at the minima. As shown in Fig.~\ref{Fig3}(c), $\Gamma/2\pi$ for $^{87}\text{Rb}$ atom will increase from $10^{-18}$~Hz to $10^{4}$~Hz when the magnetic field $\mathcal{B}_{0}$ varies from $5$~mT to $40$~mT. Therefore, for given $\mathcal{B}_{M}$, the magnetic field $\mathcal{B}_{0}$ will give higher potential barrier $U_{\text{barr}}$ and larger Majorana loss rate $\Gamma/2\pi$ simultaneously, which should be optimally chosen during practical applications. Besides, the Majorana loss rate  for $^{7}\text{Li}$ atom will be larger because of its higher trapping frequency in the lattice.

More complex potential profiles for ultracold atoms can be constructed based on the helical phase in chiral ferromagnetic film. For example, if the applied magnetic field is along the magnetic stripes of the helical phase, the reflection plane of the magnetic mirror will be shifted toward the film\cite{JPD1999}. Then a vibrating magnetic mirror for the ultracold atoms can be realized if the applied field is harmonically oscillated\cite{JPD1999}. Furthermore, a ``moving grating" or ``conveyor belt"\cite{JPD1999} can be obtained if a magnetic field is rotating in the plane perpendicular to the magnetic stripes. Therefore, we expect that the helical phase in chiral ferromagnetic film can have widespread applications to develop and design various elementary devices for atom optics and atom chip.

\section{Skyrmion Crystal Phase}

\begin{figure*}
\includegraphics[scale=0.45]{Fig4.pdf}
\caption{(Color online) Profiles of magnetic field amplitude $\mathcal{B}(\mathbf{r})$ for the chiral ferromagnetic film at SkX phase. (a) Two-dimensional reflection gating when $\mathcal{B}_{0}=0$~mT and $\mathcal{B}_{M}=0$~mT. Top : isosurfaces of $\mathcal{B}(\mathbf{r})$ with three values $3$~mT, $6$~mT, and $9$~mT; middle and bottom : cross sections of $\mathcal{B}(\mathbf{r})$ when $z=5$~nm, $x=31$~nm, and $y=35$~nm respectively. (b) Honeycomb magnetic lattice when $\mathcal{B}_{0}=10$~mT and $\mathcal{B}_{M}=10$~mT. Top : isosurfaces of $\mathcal{B}(\mathbf{r})$ with three values $10.2$~mT, $10.4$~mT, $10.6$~mT; middle and bottom : cross sections of $\mathcal{B}(\mathbf{r})$ when $z=9$~nm, $x=21$~nm, and $y=35$~nm respectively. (c) Kagome magnetic lattice when $\mathcal{B}_{0}=10$~mT and $\mathcal{B}_{M}=10$~mT. Top : isosurfaces of $\mathcal{B}(\mathbf{r})$ with three values $10.2$~mT, $10.4$~mT, $10.6$~mT; middle and bottom : cross sections of $\mathcal{B}(\mathbf{r})$ when $z=8$~nm, $x=31$~nm, and $y=35$~nm respectively. (d) Triangular magnetic lattice when $\mathcal{B}_{0}=-10$~mT and $\mathcal{B}_{M}=10$~mT. Top : isosurfaces of $\mathcal{B}(\mathbf{r})$ with three values $11$~mT, $12$~mT, $13$~mT; middle and bottom : cross sections of $\mathcal{B}(\mathbf{r})$ when $z=12$~nm, $x=31$~nm, and $y=17$~nm respectively. }\label{Fig4}
\end{figure*}

The chiral ferromagnetic film can be driven into the SkX phase by a strong magnetic field $\mathbf{B}$, as shown in Fig.~\ref{Fig1}(b). Due to its topologically protected nature, the SkX phase will exist as a metastable state after withdrawing the applied magnetic field\cite{NatPhys2016}. The lifetime of the metastable SkX phase there can be as long as $10^{4}$~s at the temperature $23$~K\cite{NatPhys2016}, which is long enough to manipulate the ultracold atoms. The magnetic field distributions generated by Bloch-type and N\'{e}el-type SkXs have both been investigated thoroughly\cite{NJP2018}, which can be used to construct two-dimensional magnetic lattices for ultracold atoms. Here, we will focus on the Bloch-type SkXs given by Eq.~(\ref{Ham}), and the N\'{e}el-type SkXs can be treated in the same way.

The magnetic field amplitude $\mathcal{B}_{c}(\mathbf{r})$ generated by the SkX phase in Fig.~\ref{Fig1}(b) without the external magnetic field $\mathbf{B}$ is shown in Fig.~\ref{Fig4}(a). Here, $\mathcal{B}_{c}(\mathbf{r})$ will decrease from $105$~mT to $1$~mT when the height $z$ increases from $0$~nm to $25$~nm; while in the plane at height $z=5$~nm, $\mathcal{B}_{c}(\mathbf{r})$ will periodically varies from $18$~mT to $40$~mT. Specially, the minima of $\mathcal{B}_{c}(\mathbf{r})$ at fixed height form a Kagome lattice, while its maxima at the same height form a triangular lattice. Therefore, the chiral ferromagnet film in the SkX phase can be viewed as a two-dimensional ``reflection grating" for the ultracold atoms.

When the bias magnetic field $\bm{\mathcal{B}}_{0}$ and the time-dependent magnetic field $\bm{\mathcal{B}}_{M}$ are switched on, the minima of the total magnetic field amplitude $\mathcal{B}(\mathbf{r})$ will appear at finite height, which will form two-dimensional magnetic lattices to trap the ultracold atoms. For example, the distribution of $\mathcal{B}(\mathbf{r})$ when $\mathcal{B}_{0}=\mathcal{B}_{M}=10$~mT is shown in Fig.~\ref{Fig4}(b) and (c). Here, one group of minimal points appear at the height $z_{\text{min}}=9$~nm and locate at the centers of the triangles determined by any three nearest skyrmions, which thus form a honeycomb magnetic lattice. Another group of minimal points locate at the middle points between any two nearest skyrmions at the height $z_{\text{min}}=8$~nm, which thus form a Kagome magnetic lattice. When the direction of the bias magnetic field is reversed with $\mathcal{B}_{0}=-10$~mT, a triangular magnetic lattice will be formed at the height $z_{\text{min}}=12$~nm, and the minimal points of $\mathcal{B}(\mathbf{r})$ will locate at the center of the skyrmions, as shown Fig.~\ref{Fig4}(d). The recoil energy $E_{R}$ now will be $1.9$~neV for $^{87}\text{Rb}$ atom and $24$~neV for $^{7}\text{Li}$ atom, where the lattice constant for SkX phase is about $35$~nm.

The magnetization configuration of the Bloch-type SkX is approximately described by a triple-$\mathbf{Q}$ state\cite{NatNano2013,NJP2018}
$\mathbf{m}_{skx}(\mathbf{r})=\mathbf{m}_{0}(\mathbf{r})\delta(z)
+\mathcal{A}\sum\limits_{i=1}^{3}[\hat{\mathbf{e}}_{z}\cos(\mathbf{Q}_{i}\cdot\mathbf{r})+\hat{\mathbf{e}}_{i}\sin(\mathbf{Q}_{i}\cdot\mathbf{r})]\delta(z)$. Without loss of generality, the wavevectors $\mathbf{Q}_{i}=Q\hat{\mathbf{Q}}_{i}$ here are set as $\hat{\mathbf{Q}}_{1}=(1,0,0),\hat{\mathbf{Q}}_{2}=(-\frac{1}{2},\frac{\sqrt{3}}{2},0),\hat{\mathbf{Q}}_{3}=(-\frac{1}{2},-\frac{\sqrt{3}}{2},0)$, which then give the set of unit vectors $\hat{\mathbf{e}}_{i}=\hat{\mathbf{e}}_{z}\times\hat{\mathbf{Q}}_{i}$. The spatial distribution of magnetic field generated by Bloch-type SkX is explicitly expressed as\cite{NJP2018}
\begin{eqnarray}
&&\bm{\mathcal{B}}_{skx}(\mathbf{r})=\frac{\mathcal{A}Q}{2}e^{-Q|z|}\nonumber\\&&\times\left(\begin{array}{c}\text{sgn}(z)(\sin(Qx)+\sin(\frac{1}{2}Qx)\cos(\frac{\sqrt{3}}{2}Qy))\\\text{sgn}(z)\sqrt{3}\cos(\frac{1}{2}Qx)\sin(\frac{\sqrt{3}}{2}Qy)\\\cos(Qx)+2\cos(\frac{1}{2}Qx)\cos(\frac{\sqrt{3}}{2}Qy)
\end{array}\right).\label{Bskx}
\end{eqnarray}
Its modulus is
$\mathcal{B}_{skx}(\mathbf{r})=\frac{\mathcal{A}Q}{2}e^{-Q|z|}\sqrt{\mathcal{F}(x,y)}$, where $\mathcal{F}(x,y)=1+3\cos^{2}(\frac{1}{2}Qx)+\cos^{2}(\frac{\sqrt{3}}{2}Qy)+4\cos^{3}(\frac{1}{2}Qx)\cos(\frac{\sqrt{3}}{2}Qy)$. We see that $\mathcal{B}_{skx}(\mathbf{r})$ will decay exponentially along with the height $|z|$. Besides, $\mathcal{F}(x,y)$ will reach it minimal value $1$ on a Kagome lattice and reach its maximal value $9$ on a triangular lattice. Therefore, the analytical results based on the triple-$\mathbf{Q}$ state here coincide with the features revealed numerically in Fig.~\ref{Fig4}(a).

\begin{figure}
\includegraphics[scale=0.60]{Fig5.pdf}
\caption{(Color online) The sites of three types of two-dimensional magnetic lattices, which are determined by the three sets of solutions of Eq.~(\ref{Beq0}). (a) Honeycomb lattice; (b) Kagome lattice; (c) triangular lattice.}\label{Fig5}
\end{figure}

After turning on the magnetic fields $\mathcal{B}_{0}=\frac{\mathcal{A}Q}{2}B_{0}$ and $\mathcal{B}_{M}=\frac{\mathcal{A}Q}{2}B_{M}$, which are weak enough that the SkX phase is not destroyed, the modulus of the total magnetic field will be $\mathcal{B}(\mathbf{r})=\frac{\mathcal{A}Q}{2}\sqrt{\mathcal{G}(\mathbf{r})}$, where $\mathcal{G}(\mathbf{r})=B_{0}^{2}+B_{M}^{2}+2B_{0}e^{-Q|z|}[\cos(Qx)+2\cos(\frac{1}{2}Qx)\cos(\frac{\sqrt{3}}{2}Qy)]+e^{-2Q|z|}\mathcal{F}(x,y)$. Hence, $\mathcal{B}(\mathbf{r})$ will be finite and homogeneously distributed as $|z|\rightarrow\infty$, while the minimal value $\mathcal{B}_{\text{min}}=10$~mT of $\mathcal{B}(\mathbf{r})$ can be achieved at periodic arrays of points at finite height $z_{\text{min}}$. The minimal points of $\mathcal{B}(\mathbf{r})$ will be determined by
\begin{eqnarray}
\left\{
\begin{aligned}
\sin(\frac{1}{2}Qx)(2\cos(\frac{1}{2}Qx)+\cos(\frac{\sqrt{3}}{2}Qy))&=&0,\\
\cos(\frac{1}{2}Qx)\sin(\frac{\sqrt{3}}{2}Qy)&=&0,\\
e^{-Q|z|}(\cos^{2}(\frac{1}{2}Qx)-\sin^{2}(\frac{1}{2}Qx)\quad\quad&&\\
+2\cos(\frac{1}{2}Qx)\cos(\frac{\sqrt{3}}{2}Qy))+B_{0}&=&0.
\end{aligned}\right.\label{Beq0}
\end{eqnarray}

\begin{figure*}
\includegraphics[scale=0.60]{Fig6.pdf}
\caption{(Color online) The dependence of trapping height $z_{\text{min}}$, potential barrier $U_{\text{barr}}$, trapping frequency $\omega_{T}/2\pi$, and Marojana loss rate $\Gamma/2\pi$ for $^{87}\text{Rb}$ atom on the bias magnetic field. Here, $E_{R}$ is the recoil energy of $^{87}\text{Rb}$ atom, $\mathcal{B}_{0}$ varies from $2.5$~mT to $20$~mT, and $\mathcal{B}_{M}$ is fixed as $10$~mT. (a) Honeycomb lattice; (b) Kagome lattice; (c) triangular lattice.}\label{Fig6}
\end{figure*}

Three sets of solutions are found for Eq.~(\ref{Beq0}), which then define three types of two-dimensional magnetic lattices for the ultracold atoms:

(I) Honeycomb lattice if $0<B_{0}<3/2$. The lattice sites are defined by $\sin(\frac{1}{2}Qx)=\pm\frac{\sqrt{3}}{2},\quad\cos(\frac{1}{2}Qx)=-\frac{1}{2},\quad\sin(\frac{\sqrt{3}}{2}Qy)=0,\quad\cos(\frac{\sqrt{3}}{2}Qy)=1,\quad e^{-Q|z|}=\frac{2}{3}\mathcal{B}_{0}$, or $\sin(\frac{1}{2}Qx)=\pm\frac{\sqrt{3}}{2},\quad\cos(\frac{1}{2}Qx)=\frac{1}{2},\quad\sin(\frac{\sqrt{3}}{2}Qy)=0,\quad\cos(\frac{\sqrt{3}}{2}Qy)=-1,\quad e^{-Q|z|}=\frac{2}{3}B_{0}$, as shown in Fig.~\ref{Fig5}(a). The height of these points is $|z_{\text{min}}|=-\frac{1}{Q}\ln(\frac{2}{3}B_{0})$, which will decrease with increased $B_{0}$ and vanish at $B_{0}=3/2$. The honeycomb optical lattices have been proposed\cite{PRA2009} and realized to trap the ultracold atoms, which then are able to simulate the exotic phenomena including the superfluid-to-Mott-insulator transition\cite{NatPhys2011-2}, Dirac points\cite{Nature2012}, quantum anomalous Hall effect\cite{Nature2014}, \emph{etc}. Therefore, similar physical phenomena of the ultracold atoms are expected when they are trapped by the honeycomb magnetic lattice here.

(II) Kagome lattice if $0<B_{0}<1$. The lattice sites are defined by
$\sin(\frac{1}{2}Qx)=\pm 1,\quad\cos(\frac{1}{2}Qx)=0,\quad\sin(\frac{\sqrt{3}}{2}Qy)=\pm 1\quad\cos(\frac{\sqrt{3}}{2}Qy)=0,\quad e^{-Q|z|}=B_{0}$, or
$\sin(\frac{1}{2}Qx)=0,\quad\cos(\frac{1}{2}Qx)=1,\quad\sin(\frac{\sqrt{3}}{2}Qy)=0,\quad\cos(\frac{\sqrt{3}}{2}Qy)=-1,\quad e^{-Q|z|}=B_{0}$, or
$\sin(\frac{1}{2}Qx)=0,\quad\cos(\frac{1}{2}Qx)=-1,\quad\sin(\frac{\sqrt{3}}{2}Qy)=0,\quad\cos(\frac{\sqrt{3}}{2}Qy)=1,\quad e^{-Q|z|}=B_{0}$, as shown in Fig.~\ref{Fig5}(b). The height of these points is $|z_{\text{min}}|=-\frac{1}{Q}\ln B_{0}$, which will also decrease with increased $B_{0}$ and vanish at $B_{0}=1$. Therefore, the Kagome lattice is more close to the film surface than the honeycomb lattice when the two coexist, which is also revealed in Fig.~\ref{Fig4}(b)(c). Kagome lattice has the intriguing ability to host flat band states of ultracold atoms, which is an ideal platform to study quantum many-body physics. Hence, the chiral ferromagnetic film provides a novel platform to realize the Kagome lattice for ultracold atoms in addition to the optical method\cite{PRL2009,PRL2012}.

(III) Triangular lattice if $-3<B_{0}<0$. The lattice sites are defined by $
\sin(\frac{1}{2}Qx)=0,\quad\cos(\frac{1}{2}Qx)=1,\quad\sin(\frac{\sqrt{3}}{2}Qy)=0,\quad\cos(\frac{\sqrt{3}}{2}Qy)=1,\quad e^{-Q|z|}=-\frac{1}{3}B_{0}$,
or $\sin(\frac{1}{2}Qx)=0,\quad\cos(\frac{1}{2}Qx)=-1,\quad\sin(\frac{\sqrt{3}}{2}Qy)=0,\quad\cos(\frac{\sqrt{3}}{2}Qy)=-1,\quad e^{-Q|z|}=-\frac{1}{3}B_{0}$, as shown in Fig.~\ref{Fig5}(c). The height of these points is $|z_{\text{min}}|=-\frac{1}{Q}\ln(-\frac{1}{3}B_{z})$, which will decrease with increased $B_{0}$ and vanish at $B_{0}=-3$. Similar to the case of optical lattice, the triangular magnetic lattice here can also be useful to simulate geometricaly frustrated magnetism with ultracold atoms\cite{Science2011}.

The properties of the two-dimensional magnetic lattices are further investigated for the chiral ferromagnetic film described by Eq.~(\ref{Ham}). As shown in Fig.~\ref{Fig6}, the trapping height of each lattice will decrease when the bias magnetic field $\mathcal{B}_{0}$ is continuously increased with fixed $\mathcal{B}_{M}=10$~mT, which agree with the analytical results above. Meanwhile, the potential barrier $U_{\text{barr}}$ for the same type of atom will be the same as the one-dimensional magnetic lattice discussed in Section III, since $\delta\mathcal{B}$ only depends on $\mathcal{B}_{0}$ and $\mathcal{B}_{M}$. The trapping height and potential barrier will be the same for $^{7}\text{Li}$ atom at the state $|F=2,m_{F}=2\rangle$.

The ultracold atoms will be confined in three directions by the two-dimensional magnetic lattices, and the trapping frequencies $\omega_{T}$ for $^{87}\text{Rb}$ atoms are given in Fig.~\ref{Fig6} when $\mathcal{B}_{M}=10$~mT and $|\mathcal{B}_{0}|$ varies from $2.5$~mT to $20$~mT. Similar to the one-dimensional magnetic lattice, stronger bias magnetic field will induce higher trapping frequency here. Besides, the inequivalence of trapping frequencies along different direction suggest the anisotropy of the trapping potential near the minima of the two-dimensional magnetic lattices. The Majorana loss rates $\Gamma/2\pi$ for $^{87}\text{Rb}$ atom in the two-dimensional magnetic lattices are also presented in Fig.~\ref{Fig6}. As expected, $\Gamma/2\pi$ will be enhanced with increasing bias magnetic field $\mathcal{B}_{0}$, and has the same anisotropic feature as the trapping frequency $\omega_{T}$. Similar to the one-dimensional magnetic lattice, $^{7}\text{Li}$ atom at the state $|F=2,m_{F}=2\rangle$ will attain higher trapping frequency and larger Majorana loss rate due to its smaller mass.

\section{Discussions and Conclusion}
Although the chiral ferromagnetic film has shown attractive features to control the ultracold atoms, there are still several issues to be addressed for practical performances. First of all, one needs to transfer the ultracold atoms to the magnetic lattices near the film surface for further operations. This might be accomplished by loading the atoms from an external dimple trap\cite{DimpleTrap} by adiabatically tuning the trapping height with the bias field $\mathcal{B}_{0}$, which in fact has been proposed for the magnetic lattice created by a superconducting film\cite{PRL2013}. Secondly, imperfections will always be introduced to the magnetic field by the unavoidable defects in chiral ferromagnetic film, such as the bent helical stripes, irregular array of skyrmions, or even their mixtures\cite{NatNano2013}. In order to get ideal magnetic lattices, it will be critical to improve the quality of chiral ferromagnetic film and get perfect magnetic patterns. Finally, the trapping height of the ultracold atoms will be tens of nanometers, which is so close that the Casimir-Polder force between the atoms and the film surface will be significantly enhanced\cite{AtomChip}. Therefore, the dynamic behavior of ultracold atoms interacting with the chiral ferromagnetic film should be highly different from the case of optical lattice, which needs to be explored further.

In conclusion, we have investigated the possibility to control ultracold atoms with the magnetic field produced by the chiral ferromagnetic film. We demonstrate how to realize magnetic mirror, reflection grating, one-dimensional and two-dimensional magnetic lattices in the proposed device. Compared with current top-down techniques to produce magnetic fields for ultracold atoms, the strategies based on chiral ferromagnetic film here belong to the bottom-up category and will be more flexible and controllable. With the benefit of the continuous advances of the spintronics technology and material science, the interactive ultracold atoms and chiral ferromagnetic film can be a promising platform to demonstrate exotic physics phenomena and develop novel quantum techniques.

\section*{Acknowledgements}
This work has been supported by NSFC Project No. 61674083 and No. 11604162, and ``the Fundamental Research Funds for the Central Universitie'' Nankai University (7540).


\section{Introduction}

In light of the perceived importance of interdisciplinary research, many studies have been conducted that quantify article interdisciplinarity in an effort to identify relevant research trends and to explore their impact. The most widely-accepted methods for measuring interdisciplinarity assess \emph{knowledge integration} or \emph{knowledge diffusion} using citation information, thus measuring interdisciplinarity as some function of the balance, diversity, and dissimilarity of the disciplines identified in an article's cited papers \cite{porter2009science,rafols2010diversity} or citing papers \cite{porter1985indicator,van2015interdisciplinary}.
To implement these metrics, each paper in a research corpus must be assigned to an explicit research topic or subject category, for which sources are numerous, inconsistent, and sometimes unavailable. Subject categories are most commonly assigned to papers according to the journals in which they are published. However, these assignments rarely agree with underlying citation community structure \cite{porter2009science}.
There is evidence that interdisciplinary research can be identified in a corpus according solely to the citation structure. Specifically, it has been shown that frameworks that encode the \emph{structural role} of articles in a citation graph can predict interdisciplinary interactions more accurately than those that encode only the proximity between papers \cite{cunningham2022assessing}. In light of this, we explore the potential for modern graph learning methods to identify the citation structures associated with interdisciplinary research.

Numerous approaches have been proposed in the literature for the task of structural role discovery on graphs, where nodes on a graph are divided into classes of \emph{structurally equivalent} nodes \cite{rossi2014role}. Early approaches in this area relied on graph sub-structures known as \emph{graphlets} or \emph{motifs} \cite{milo2002network}. In large graphs, where we wish to identify higher-order structural features, counting large graphlets is very expensive. The majority of recent approaches employ a representation learning technique known as node embedding, which substitutes graphlets with alternative structural features like degree distributions \cite{ribeiro2017struc2vec} and diffusion wavelet patterns \cite{donnat2018learning}. These methods are designed to learn higher-order structural relationships than those that can be discovered by small graphlets. However, in many cases, these alternatives approaches come at the cost of interpretability. When applied to graphs that are too large to be visualised reasonably, it is difficult to understand the substantive meaning of a given set of structural roles.

While embedding methods for role analysis have previously been shown to be capable of grouping nodes into known roles or structures (such as those in synthetic graphs or transport networks \cite{donnat2018learning,ribeiro2017struc2vec}), it remains unclear as to how these roles should be interpreted or validated when applied to real-world graphs with unknown roles. Moreover, different role discovery methods learn different sets of structural roles, depending on the application many or none of these clusterings may be valid. As such, it is critical that we can compare the roles discovered by different methods.

The core contribution of this work is a new framework for explaining a set of discovered roles  using graphlet \emph{orbits}, as described in detail in Section \ref{sec:methods}. Later in Section \ref{sec:application} we apply this framework to a large, multidisciplinary citation network to extract and interpret sets of structural roles. In the case of each paper, we compute Rao-Stirling diversity scores to indicate an articles interdisciplinarity. Crucially, in addition to interpreting a set of candidate roles, we explore the distributions of IDR scores assigned to papers in different clustering to assess if any of the candidate clusterings have grouped papers according to their interdisciplinarity.

\section{Related Work}
\subsection{Measuring Interdisciplinarity}

\emph{Interdisciplinary research} is most commonly defined as research activity which integrates expertise, data or methodologies from two or more distinct disciplines. Accordingly, many studies assign an interdisciplinary research (IDR) score to a research article, which is calculated as a function of the balance, diversity and dissimilarity of the disciplines identified in the articles reference list \cite{porter2009science,rafols2010diversity}. Alternatively, some studies compute a similar score on the disciplines identified in an article's \emph{citing} literature, instead measuring IDR according to an articles impact/influence across multiple disciplines \cite{porter1985indicator,van2015interdisciplinary}. A popular function for measuring IDR is the Rao-Stirling Diversity index \cite{stirling2007general}
\begin{equation}
\label{eq:r_s_diversity}
    D = \sum_{i,j (i\neq j)}p_ip_jd_{ij}
\end{equation}
where IDR is measured as a pairwise summation over all pairs of disciplines identified in a set of articles cited by (or citing) some focal paper. Here $p_i$ and $p_j$ represent the proportion of references to disciplines $i$ and $j$ respectively, while $d_{ij}$ refers to some precomputed distance between the disciplines. This metric, and its many variants, are reliant on explicit topic or discipline categories for research papers, such as those provided by Web of Science, Scopus, or Microsoft Academic. Such explicit categorisations for research papers, especially those assigned at a journal level, are problematic \cite{abramo2018comparison,milojevic20science}. Moreover, the inconsistencies evident across many of these subject taxonomies \cite{shen2019node2vec} may confirm that no singular, correct categorisation exists.  However, recent graph learning methods may be capable of identifying the citation structures associated with interdisciplinary research, without any knowledge of explicit discipline categorisation.

\subsection{Local Structure and Role Embeddings}

Research developed in social science studied local graph structures using small graph patterns such as triads, cycles and stars \cite{moreno1934shall}. More recent research in computational science further developed these methods and proposed the term `motif' -- a subgraph pattern (or \emph{graphlet}) which is significantly over-represented in a graph \cite{milo2002network}. Motifs, and graphlets represent a powerful tool for expressing graph structure, and have been employed in graph learning tasks such as node classification and anomaly detection \cite{cunningham2013motifs}. Figure \ref{fig:graphlets} illustrates a subset of graphlets with 2, 3, 4 and 5 nodes, and includes each of the distinct orbits on these graphlets, as they were enumerated by \cite{prvzulj2007biological}.

\begin{figure}[!b]
    \centering
\includegraphics[width=\linewidth]{graphlets_2.png}
    \caption{Graphlets with 2, 3, 4, and 5 nodes. Graphlets and orbits are enumerated according to \cite{prvzulj2007biological}}
    \label{fig:graphlets}
\end{figure}

\emph{Role discovery} is the task of grouping nodes which share similar structural patterns in a graph into distinct classes \cite{rossi2014role}. Many modern approaches to role discovery rely on graph embedding, where nodes are transformed into low-dimensional vector representations \cite{ahmed2019role2vec,henderson2012rolx,ribeiro2017struc2vec,donnat2018learning}. Specifically, graph embedding methods for the purpose of role discovery (often called ``role embedding'') learn dense vector representations for nodes such that nodes that are structurally similar will be embedded nearby in the embedding space (i.e., will have similar representations). A clustering of the role embedding space thus represents a set of discovered roles. However, if this network cannot be visualised, it is difficult to interpret the roles. Moreover, with numerous approaches to role embedding (e.g. \cite{ahmed2019role2vec,donnat2018learning,henderson2012rolx,ribeiro2017struc2vec}), and many possible clusterings of each embedding space, we require some approach to explain a set of discovered roles so that they can be compared and validated.

\subsection{Explanation via Surrogate Models}

In recent years, much research has been conducted in the field of \emph{explainable artificial intelligence} (XAI) towards a goal of understanding and explaining so-called `\emph{black box}' models \cite{adadi2018peeking}. One popular approach is to use a global surrogate model: ``an interpretable model that is trained to approximate the predictions of a black box model'' \cite{molnar2020interpretable}. As such, the decisions made by a previously uninterpretable system can be explained through interpretations coming from the surrogate model.

Some classification models (such a logistic regression models or decision tree-based classifiers) are interpretable by definition, as any input feature's effect on the models classification can be measured (for example, using regression coefficients). However, many model-agnostic methods of interpretation have also been developed. Generally, these methods propose different means of perturbing the input data, and exploring its effect on the models output. For example, \emph{Partial Dependence Plots} (PDPs) \cite{friedman2001greedy} offer a graphical representation of a models prediction as a function of its arguments or inputs. Broadly speaking, a PDP shows how a feature influences the prediction of a model, on average, for every value that it can take. The \emph{Accumulated Local Effects} plot (ALE) \cite{apley2020visualizing} provides a similar illustration of a features effect, except it accounts for possible correlations between features, and as a result is more robust than the PDP in some applications. In 2021, global surrogate models were employed to provide feature-based explanations for communities detected on graphs \cite{sadler2021selecting}. Specifically, this work assessed the importance of different graph features using \emph{permutation importance} \cite{breiman2001random}, where the values for a feature are permuted throughout the dataset and the effect on model performance is measured to indicate its importance.

\section{Methods}
\label{sec:methods}

In this section we outline a general framework for uncovering and evaluating structural roles on a graph. The complete process is illustrated in Figure \ref{fig:workflow}.

\begin{figure}[!t]
    \centering
\includegraphics[width=\linewidth]{workflow.png}
    \caption{Overview of the workflow for the proposed framework.}
    \label{fig:workflow}
\end{figure}

We begin with a graph $G = (V,E)$, from which we wish to identify a set of discrete structural roles. We employ a role embedding algorithm to map each node $v \in V$ to an embedding space $X_{emb} = \mathbb{R}^{128}$, thus a clustering of $X_{emb}$ is considered a set of discovered roles $y_{role}$. Additionally, we represent the same nodes in the graphlet space $X_{graphlet} = \mathbb{Z}^{72}$ derived from $G$, where each node $v$ is represented by a \emph{bag-of-orbits} vector $x^v = \{x_0^v, x_1^v, ..., x_{72}^v\}$, with $x_i^v$ denoting the number of times node $v$ is involved in induced graphlet orbit $i$. We use a vocabulary of 72 orbits, which we count on graphlets of size 2 to 5 using the ORCA method \cite{hovcevar2017combinatorial}. We refer to graphlets and orbits according to the enumeration by \cite{prvzulj2007biological}. The graphlet space is first used to validate any set of roles we identify in the graph. By clustering the embedding space, we group nodes into $k$ discrete roles, which we can evaluate using cluster validity metrics calculated on the graphlet space. Employing various role embedding algorithms and clustering methods, we identify a set of candidate clusterings (or roles) according to the separation they achieve in the graphlet space.

Graphlet-orbit counts are a powerful language for describing the structural features of nodes on a graph \cite{cunningham2013motifs,prvzulj2007biological}. As such, a clustering (according to node embeddings) which is valid in the graphlet-orbit space should offer a reasonable set of structural roles. Additionally, we employ graphlet orbits to interpret a set of discovered roles. For a candidate clustering $y_{role}$, we fit a surrogate model $f:X_{graphlet}\mapsto y_{role}$. By modelling the role assignment in the graphlet space, we can explore the feature importance and effect of the different graphlet orbits in role assignments, according to many \emph{model-agnostic} explanation techniques from the field of XAI \cite{molnar2020interpretable}. In Section \ref{sec:application}, we demonstrate graphlet-based explanation using permutation importance \cite{breiman2001random} and accumulated local effects plots \cite{apley2020visualizing}. As we will show, highlighting important or discriminatory orbits can offer a visual means of understanding the structure of a role in the graph.

\section{Application}
\label{sec:application}

In this section we apply the methods described in Section \ref{sec:methods} to extract and interpret sets of structural roles in a citation network. In the case of each paper, we compute Rao-Stirling diversity scores to indicate an articles interdisciplinarity and thus explore the distributions of IDR scores assigned to papers in different roles. We identify a set of roles which has grouped papers according to their interdisciplinarity. Finally, we use graphlets to interpret the structure of these more interdisciplinary roles, and highlight certain citation structures which are specific to interdisciplinary research.

\subsection{Data}
In order to discover the citation structures of interdisciplinary research, we require a large, dense citation network that contains research from a diverse set of disciplines. In addition, we require that each paper can be assigned to a subject category or discipline, according to an established taxonomy.
We construct a novel citation network using Microsoft Academic Graph \cite{sinha2015overview} citation data from a seed set of journal papers. This sets consist of samples of articles from Scopus indexed journals, stratified according to their All Science Journal Categories (ASJC). The graph contains samples of 1,500 articles published between 2017 and 2018 in Scopus indexed journals with the ASJCs \emph{`Computer Science', `Mathematics', `Medicine', `Chemistry', `Social Sciences', `Neuroscience', `Engineering'}, and \emph{`Biochemistry, Genetics and Molecular Biology'}. We maximise the completeness of the graph by including all available referenced articles that are published in Scopus indexed journals. In this manner, we produce a dense, multidisciplinary citation network, such that each article can be categorised according to the ASJC of the journal in which it was published. Later, these discipline categories can be used to compute article interdisciplinarity according to Rao-Stirling diversity of disciplines identified in both an articles citing and cited papers. In total, the citation graph contains 41,895 papers (nodes) and 129,159 citations (undirected edges).

\subsection{Role Discovery}
For each article in the citation graph, we learn role embeddings according to 4 approaches: (i) \emph{Role2Vec} \cite{ahmed2019role2vec}, (ii) \emph{Struc2Vec} \cite{ribeiro2017struc2vec}, (iii) \emph{RolX} \cite{henderson2012rolx}, (iv) \emph{GraphWave} \cite{donnat2018learning}, and we cluster each embedding space using $k$-means clustering for values of $k$ between 2 and 19. Articles clustered according to their role embeddings represent a set of structural roles.

\subsection{Role Interpretation}

\subsubsection{Validation.}
Figure \ref{fig:silhouettes} shows the cluster validity of the roles discovered according to the 4 role embedding methods. Each set of roles represents a $k$-means clustering of the embedding space, which is then transformed to the graphlet space, where we measure the validity of the clustering using silhouette score \cite{rousseeuwW1987silhouettes}. We plot the silhouette score for each embedding for values of $k$ in the range $[2,19]$. Silhouette scores can take a value in the range [-1,1]. A high score represents dense, separate clusters, a score of 0 indicates an overlapping clustering, while a negative score indicates an incorrect clustering.
According to these scores, we choose 3 candidate roles to demonstrate interpretation and explanation: (i) Struc2Vec ($k = 6$) which is an outlier in the Struc2vec roles and achieves an overlapping clustering, (ii) RolX ($k = 3$) which has the highest silhouette score for all approaches with more than 2 clusters, and (iii) GraphWave ($k = 3$) which achieves a positive score.
\begin{figure}[h!t]
    \centering
\includegraphics[width=0.8\linewidth]{silhouettes.png}
\vskip -1em
    \caption{Silhouette score for $k$-means clusterings of the different embedding spaces. Silhouettes are calculated according the clusterings if we describe each node by their bag-of-orbits vector, rather than their embedding.}
    \label{fig:silhouettes}
\end{figure}

\subsubsection{Surrogate models.}
\vskip-2em
We fit a random forest classifier to model role assignments according to the bag-of-orbits vectors for each node. That is, we learn a function $f:X_{graphlet}\mapsto y_{role}$ for each of the candidate roles $y_{role}$. We choose a random forest classifier as we anticipate that structural roles may be non-linear in the graphlet space and may rely on combinations and interactions of features to model higher-order structures. As the orbit counts follow a power-law distribution, we log-transform all features in the graphlet space. Table \ref{tab:permutation_importance} reports the 5 most informative features for each model according to permutation importance. In the case of the Struc2Vec ($k = 6$) roles, the overlap between clusters in the graphlet space is evident. The only informative features (with non-zero permutation importance) are small, local orbits -- the approach is blind to deeper, more complex structures. The RolX ($k = 3$) approach, which best separates the nodes in the graphlet space, has grouped nodes according to larger, simple structures. The most informative orbits: 15, 4, and 1, each refer to chains of varying length. Finally, GraphWave ($k = 3$) appears to have grouped the nodes according to more complex, higher-order structures. Many of the features important to role classification in the GraphWave case (27, 24, and 18) contain combinations of chains, stars and cliques. Depending on the domain or application in which we employ role discovery, any one of these sets of roles may be the most valid or useful. However, without modelling the role assignments in the graphlet space, we are unable to understand which structures are being grouped in the discovered roles. We will use the GraphWave roles to demonstrate further explanation and explore the structure of interdisciplinary papers.

\begin{table}[]
    \centering
    \begin{tabular}{|>{\centering}p{0.5cm}|>{\raggedleft}p{3.5cm}|>{\raggedleft}p{3.5cm}|>{\raggedleft\arraybackslash}p{3.5cm}|}
\hline
{} & \centering Struc2Vec ($k = 6$) & \centering RolX ($k = 3$) & \multicolumn{1}{c|}{GraphWave ($k = 3$)} \\
\hline
1 &   0 (0.111 ±0.0005) &  15 (0.113 ±0.0016) &  27 (0.022 ±0.0006) \\
2 &   2 (0.010 ±0.0001) &   4 (0.007 ±0.0002) &  15 (0.014 ±0.0002)  \\
3 &   3 (0.006 ±0.0001) &   1 (0.006 ±0.0003) &  17 (0.013 ±0.0006)  \\
4 &   5 (0.002 ±0.0001) &  27 (0.005 ±0.0002) &  24 (0.013 ±0.0004) \\
5 &  16 (0.000 ±0.0000) &  19 (0.004 ±0.0003) &  18 (0.013 ±0.0004) \\
\hline
\end{tabular}
\vskip 0.5em
    \caption{Top 5 most important features for each surrogate model. Orbits are ranked by permutation importance, which is included in parentheses.}
    \label{tab:permutation_importance}
\end{table}

\subsubsection{Roles and their structure.}
\vskip-3.5em
The GraphWave method clusters 35,136 papers into role 0, 16,453 papers into role 1, and 306 papers into role 2. Figure \ref{fig:ale} contains Accumulated Local Effects (ALE) \cite{apley2020visualizing} plots for 3 features/orbits and their effect on classification to each of the 3 GraphWave roles. We illustrate the ALE of orbits 27 (the end of a chain adjacent to a clique) and 17 (the middle of a long chain) as two of the most important structures (according to permutation importance). We also include the ALE of orbit 0 (node degree) as a valuable reference; it is useful to confirm that the roles are indeed separating nodes according to more complex features and not simply the number of edges. The ALE plot for orbit 27 shows that for low-to-mid values of that orbit count, a node will be classified as role 0. However, if a node's count for orbit 27 exceeds a threshold, it will be classified as role 1. We suppose 2 scenarios when a focal node's count for orbit 27 (a chain adjacent to a clique) will become large: (i) the node is adjacent to a large community -- each triangle in which the node at position 30 participates will increase the count, (ii) the node exists at the center of a barbell graphlet, i.e. on a longer chain between two or more communities communities. We illustrate these scenarios in Figure \ref{fig:graphlet_explanation}. There should exist some threshold value of 27 beyond which a node must exist on the chain between two communities. For example, if a focal node has a count for orbit 27 that is greater than the count of triangles (orbit 3) for the node at position 30, then the focal node must be adjacent to a second community (scenario (iii)). This threshold will be represented by the greatest value of orbit 3 in the graph. We include this value for reference in Figure \ref{fig:ale}. Beyond this threshold, a node is more likely to be classified in role 1. Accordingly, we conclude that a node/paper that is on the end of a chain adjacent to a community will be assigned to role 0, while a node that exists on a bridge between two communities will be assigned to role 1.

\begin{figure}[!t]
    \centering
    \includegraphics[width=\linewidth]{graphwave_k_3_ale.png}
    \vskip -0.3em
    \caption{Accumulated Local Effect \cite{apley2020visualizing} plots for a surrogate model which classifies nodes to GraphWave roles according to graphlet orbit counts. The figure shows the effect of 3 features: orbits 27 and 17; the most important features as measured by permutation importance, and orbit 0 (node degree); which we include for reference. In the case of orbit 27, we highlight the maximum value of orbit 3 that was observed in the graph.}
    \label{fig:ale}
\end{figure}

\begin{figure}[h!t]
    \centering
\includegraphics[width=\linewidth]{graphlet_explanation.png}
    \caption{Higher order structures containing orbit 27. Here (i) and (ii) represent likely structures for nodes with high counts for orbit 27. When the count for orbit 27 exceeds a threshold, we infer structure (iii). }
    \label{fig:graphlet_explanation}
\end{figure}

In order to identify the structures specific to the smallest role (role 3), we can fit another surrogate model on only the nodes in cluster 1 and 2. ALE plots for this model are included in Figure \ref{fig:ale_2}. In this case we find that orbit 27 does not meaningfully distinguish between the two roles. Instead, orbit 18 (the end of a chain adjacent to a star) is the most informative feature, and, for very high values of this orbit count, nodes will be assigned to role 2. Such nodes likely represent the centre of a bridge between large communities that are less densely connected (i.e., containing many open triads). We conclude this to be an important structure for role 3.

\begin{figure}[!t]
    \centering
    \includegraphics[width=\linewidth]{graphwave_k_3_ale_2.png}
    \vskip -0.3em
    \caption{Accumulated Local Effect \cite{apley2020visualizing} plots for a surrogate model which classifies nodes to GraphWave roles 1 or 2 according to graphlet orbit counts. The figure plots the effect of 3 features: orbits 0; node degree and 27; previously the most important feature for the global model, and orbit 18; the most important feature in the surrogate model.}
    \label{fig:ale_2}
\end{figure}

\subsection{Interdisciplinary Roles}

Figure \ref{fig:idr} plots the IDR scores for the papers in each of the GraphWave roles. IDR is calculated according to the Rao-Stirling diversity of the ASJC categories identified in an article's citing papers. As Rao-Stirling IDR scores may be biased according to the number of articles in the summation, we bin nodes by degree, and plot IDR distributions for each role, within each bin. Specifically, we log-transform the node degrees and group nodes into 10 bins of equal width, within which we plot IDR distributions for each role if the bin contains more than 50 papers from each role. According to these plots, we note that the structural roles identified by GraphWave have grouped papers into clusters with different IDR distributions. Even when we account for node degree (a potential bias of the Rao-Stirling IDR score), papers assigned to structural roles 1 and 2 have consistently greater IDR distributions than those assigned to role 0. We recall some of the important structures that were identified for roles 1 and 2: (1) a bridge between densely connected communities, and (2) a bridge between large, sparsely connected communities. We conclude these to be citation important structures associated with interdisciplinary research.

\begin{figure}[!t]
    \centering
\includegraphics[width=\linewidth]{graphwave_k_3_idr.png}
    \caption{Interdisciplinarity scores (IDR) computed for papers in each of the 3 GraphWave clusters. IDR is computed as the Rao-Stirling \cite{stirling2007general} diversity of the research disciplines identified in an articles citing papers. We bin papers/nodes according to the log of their degree, and compare distributions within each bin.}
    \label{fig:idr}
\end{figure}

\section{Discussion and Conclusions}

Many modern methods for role discovery in graphs rely on node embeddings \cite{rossi2020proximity}. We have applied a number of these methods to learn structural roles on a research citation network. While these methods have previously been shown to be capable of grouping nodes into known roles (e.g., in synthetic graphs or transport networks \cite{ribeiro2017struc2vec,donnat2018learning}), it remains unclear how roles could be understood or validated when applied to graphs with unknown roles. In this work we outlined a framework for interpreting a set of discovered roles using graphlets and orbits. We leveraged methods from the field of explainable AI to explore the subgraph structures that are specific to discovered roles. We demonstrated our approach on a citation network, where we identified important structures specific to interdisciplinary research. It is evident in our analysis that different role discovery methods learn different sets of structural roles. In different applications many or all of these clusterings may be valid, but it is critical that we can compare the roles discovered by different methods. While our framework is general, and applicable to explanation and validation in all role discovery tasks, we highlight the utility of structural role embeddings in mapping interdisciplinary research.

For the task of identifying and mapping IDR interactions, structural paper embeddings could be augmented by considering additional, non-structural information, such as article or abstract text. This could provide a richer paper representation, without imposing a predefined or static disciplinary classification on the graph. There is also scope for improving upon our proposed framework. Many \emph{model-agnostic} approaches have been developed for explaining surrogate models \cite{molnar2020interpretable}, which could be applied to interpret the role assignments in the graphlet space. For example, second-order effects of pairs of features can be calculated in a similar manner to the ALE analysis we have included \cite{apley2020visualizing}. Combinations of graphlets could be highly effective in modelling higher-order, more complex graph structures. One possible limitation of our current framework is the number of correlated features in the graphlet space. In future applications it may be possible to reduce the set of graphlet orbits to a minimal set of uncorrelated features via traditional feature selection techniques.

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