# Ab initio comparison of spin-transport properties in MgO-spaced
ferrimagnetic tunnel junctions based on Mn3Ga and Mn3Al

M. Stamenova<EMAIL_ADDRESS>School of Physics and CRANN, Trinity College
Dublin, Dublin 2, Ireland.  P. Stamenov  School of Physics and CRANN, Trinity
College Dublin, Dublin 2, Ireland.  N. Baadji  Laboratoire de Physique des
Matériaux et ses applications & Département de Physique, Faculté des Sciences,
Université Mohamed Boudiaf, M’sila, 28000, Algérie

###### Abstract

We report on first-principles spin-polarised quantum transport calculations
(from NEGF+DFT) in MgO-spaced magnetic tunnel junctions (MTJs) based on two
different Mn-based Heusler ferrimagnetic metals, namely Mn3Al and Mn3Ga in
their tetragonal DO22 phase. The former is a fully compensated half-metallic
ferrimagnet, while the latter is a low-moment high-spin-polarisation
ferrimagnet, both with a small lattice mismatch from MgO. In identical
symmetric and asymmetric interface reconstructions across a 3-monolayer thick
MgO barrier for both ferrimagets, the linear response (low-voltage) spin-
transfer torque (STT) and tunneling magneto-resistance (TMR) effects are
evaluated. A larger staggered in-plane STT is found in the Mn3Ga case, while
the STT in Mn3Al vanishes quickly away from the interface (similarly to STT in
ferromagnetic MTJs). The roles are reversed for the TMR, which is practically
100% in the half-metallic Mn3Al-based MTJs (using the conservative definition)
as opposed to 60% in the Mn3Ga case. The weak dependence on the exact
interface reconstruction would suggest Mn3Ga-Mn3Al solid solutions as a
possible route towards optimal trade-off of STT and TMR in the low-bias, low-
temperature transport regime.

††preprint: AIP/123-QED

##  I Introduction

Among the Mn-based Heusler ferrimagnets are a number of topical binary
materials, combining low moments with high Curie temperature and spin-
polarisation [1, 2, 3, 4], as well as high anisotropy with low Gilbert damping
[5, 6] – hence holding promise for the emerging field of Antiferromagnetic
Spintronics[7, 8]. Ferrimagnetic electrodes, and especially, the compensated
ones, in magnetic-tunnel junction (MTJ) devices lead to the reduction of the
demagnetizing field and, more importantly, the reduction of the critical
current needed to switch the magnetization together with the fast dynamics of
such switching, involving inter-sublattice exchange interactions. The possible
exploitation of this class of materials, however, depends on the ability to
produce high-frequency oscillations (in the 100s of GHz range) and the depth
of resistance modulation that junctions using these could support. Predictive
computational guidance for the magnitude and resilience of both the spin-
transfer torque (STT) and the tunneling magneto-resistance (TMR) effects,
towards the difficult-to-control experimentally barrier quality and interface
reconstruction, can help to speed-up the development of functional prototype
devices.

One such candidate is Mn3Ga, which in its tetragonal DO22 phase is a low-
moment ferrimagnet with a high spin-polarisation and anisotropy [1, 9, 2], but
also a low Gilbert damping and an established epitaxial relationship with
MgO(001) [10]. A large staggered long-ranged STT effect has been found
theoretically in MTJs based on DO22-phased Mn3Ga [11], present both in
Fe/MgO/Mn3Ga and in Mn3Ga/MgO/Mn3Ga tri-layers, and related to the mismatch of
the Fermi wavevectors of the majority and minority $\Delta_{1}$ symmetry band
in Mn3Ga in the direction of transport. Theoretically, the TMR effect in these
stacks reaches a few tens or percent and exhibits a sign change below 1V,
which is in accordance with experimental observations for similar
ferrimagnetic MTJs [12].

Another Mn-based Heusler Mn3Al has been shown to exhibit half-metallicity and
almost ideally fully compensated moment in its cubic DO3 phase [13], similarly
to Mn3Ga in this phase[3], and proposed MTJs with GaAs have shown large
theoretical TMR ratios [14]. A GGA-PBE geometry optimisation of Mn3Al reveals
a stable tetragonal DO22 solution with almost fully compensated moment and an
in-plane lattice constant commensurate with MgO. As geometrically the Mn3Ga
and Mn3Al stacks with MgO are very similar, but offer different placement of
the Fermi level with respect to the main band dispersions, a comparison of the
spin-transport in analogous MTJs could unveil further insights about the
Spintronic capacity of the two Heuslers. Here we first examine the electronic
structure properties of both materials in bulk (Section II) and then we
compare the spin-dependent transport properties of two pairs of MTJs, all with
3-monolayers (ML) thick MgO spacers, but featuring two different terminations
at one of the interfaces (Section III).

##  II Bulk properties

Figure 1:  Band structures and spin-polarised density of states of (a,c) Mn3Al
and (b,d) Mn3Ga, respectively in their depicted unit cell, calculated using
LDA-PW92 (for Mn3Al) and LDA-CA (for Mn3Ga) on the MgO-matching (strained)
geometries (black or red/blue filled curves, see text and insets). In all
panels a comparison is shown with a corresponding GGA-PBE calculation for the
relaxed unit cell (green curves, see text/legends for details). In (a,b) the
DO22 unit cells of both materials with the local spins of the Mn atoms shown
as arrows.

Intermetallic Heusler alloys X2YZ crystallize usually in the cubic L21
structure, especially at high temperature, and at low temperature they can
develop a DO22 tetragonal structure with a ratio $c/a$ around $\sqrt{2}$. The
larger departure of this ratio from $\sqrt{2}$ is a prelude for a higher
magneto-crystalline anisotropy [1]. Our calculated optimum lattice parameters,
using the GGA-PBE, of both compounds in their antiferromagnetic configuration
are $a=4.057$, $c=5.911$ Å and $a=3.78$, $c=7.1$ Å for Mn3Al and Mn3Ga,
respectively. The in-plane lattice constants are close to that of bulk MgO
($a_{\mathrm{MgO}}=4.21$ Å), which makes their integration in conventional
magnetic tunnel junctions feasible. The DO22 Mn3Al(Ga) structures are
constructed by alternating planes of Mn-Al(Ga) (Mn in $2b$ Wyckoff position:
MnI) and Mn-Mn (Mn in $4d$ Wyckoff position: MnII) coupled anti-
ferromagnetically, along the $z$-axis (see unit cell schematics in Fig. 1). In
such a lattice structure, Mn3Ga is metallic and both spins contribute to the
conductivity, while Mn3Al is a half-metal and only spin-up states contribute
to the conduction. Consequently, Mn3Al has a 100% spin polarization at the
Fermi level and therefore one can expect a high TMR for junctions based on
Mn3Al. Mn3Ga also shows a high spin polarization of 88% in other GGA-based
calculations [2]. Similar results for the Fermi level spin-polarisation are
obtained also when the transverse lattice constants are $a=b=4.1$ Å in both
materials (approaching that of MgO thin films), as shown in the density of
states presented in Fig. 1(c,d).

The corresponding band structure of both compounds is presented in Fig. 1(a,b)
and one can see that for Mn3Al the spin-down channel exhibits a gap of the
order of 0.4 eV when computed with GGA-PBE on the relaxed unit cell (green
points) and similarly about 0.25 eV, when calculated with LDA-PW92 (the black
points), as implemented in the Siesta code [15]. This gap can be tuned by
changing the lattice parameters. In the LDA case, we apply a longitudinal
tensile strain of 4% ($c=6.027$ Å) to open the gap and approximate the GGA
result (where $c=5.795$ Å). The impact of that on the layer-resolved magnetic
moments (calculated by Mulliken population analysis) is a small decrease by
about 7 % for both Mn sublattices compared to GGA. We have additionally
established that Mn3Al keeps its fully-compensated ferrimagnetic character for
applied strains ranging between -4% to 8% (so that the 100% polarization is
kept for such a strain). Note that the orbitals below the Fermi level are
mainly a hybridisation between d-$t_{2g}$ orbitals of the Mn atoms, while the
empty bands above the Fermi level are $e_{g}$ bands of the Mn occupying site
$4d$.

Figure 2: (a) Schematic of the four MTJs considered, including the directions
of the spin quantisation axes in the two leads at the 90° alignment for the
STT calculations. Note, that the corresponding ’asymmetric’ MTJs have one
layer of Mn removed from the right interface. (b,c) Self-consistently
calculated layer-resolved spin-components ($x$ and $z$) for the four MTJs at
equilibrium (see legend for the color code). (d,e) Corresponding layer-
resolved in-plane STT components calculated at the Fermi level in linear
response regime (so-called, torkance $\bm{\tau}=d\bm{T}(V)/dV$ at the limit
$V=0$, where $\bm{T}(V)$ is the STT, see Ref. 16 for precise definition) in
the four different MTJs (same colour code). $A=a^{2}=16.81$ Å2 is the cross-
sectional area of the junction (it is the same for all; note the periodic
boundary conditions in the $x-y$ plane).

For Mn3Ga we are using a longitudinal lattice constant $c=6.6$ Å, which has
been found to reproduce more reasonably the experimental values of the Mn
spins (as extracted from neutron-diffraction results [1]), within the LDA [see
Fig. 2(b,c) for the computed layer-resolved magnetic moments], compared to the
GGA-relaxed or the experimental lattice parameter values $c=7.1$ Å (larger by
about 7 %), for which the LDA-computed magnetic moments are larger by over 15
% with respect to the $c=6.6$ Å case and outside the experimental range. We
assume that, with these structural amendments (consistent also with Ref. 11
for Mn3Ga-based MTJs), the LDA, which is not currently replaceable by GGA in
our non-collinear-spin method for spin-transfer torque (STT)[11], captures the
essential Fermi-surface properties of the two materials relevant for the
linear-response regime investigated here and we continue exclusively with the
LDA and the described above lattice parameters of both materials in Section
III.

Consequently, for the just described geometries, both compounds are
ferrimagnetic with Mn3Al being fully compensated (total magnetic moment
$\simeq 0$ ), while Mn3Ga having a 2.6 $\mathrm{\mu_{B}}$ per cell (1.3
$\mathrm{\mu_{B}}$ per formula unit). The advantage of having a very low-spin-
moment lead is to reduce the demagnetizing field, but more importantly, to
reduce the critical current for a spin-transfer torque switching, which we
will discuss it in the next Section III. We should mention here that the
magneto-crystalline anisotropy calculated for Mn3Ga is much bigger than that
of Mn3Al and this is because of the much stronger spin-orbit interaction at
the Ga site compared to Al. The local spins (extracted by Mulliken population
analysis) on the two Mn sublattices in Mn3Al are nearly fully compensated with
3.31 $\mathrm{\mu_{B}}$ on the MnI site and -1.58 $\mathrm{\mu_{B}}$ on the
MnII site, respectively. In Mn3Ga the corresponding values are 3.53 and -2.46
$\mathrm{\mu_{B}}$ for MnI and MnII, respectively, which are consistent with
the measured moments[1].

##  III Spin-transport in Mn3Al and Mn3Ga junctions with MgO barriers

Four different junctions based on Mn3Al and Mn3Ga all sandwiching 3 MLs of MgO
have been investigated [Fig. 2a]. In the junctions, which we refer to as
’symmetric’, the two interfaces are the same, that is, in both cases the
interface is between the MgO and the MnII-plane of the DO22 lattice and
overall the junction is mirror-symmetric with respect to the central plane in
the MgO. In the ’asymmetric’ junctions we have removed one monolayer of Mn
from the right interface, but all distances, including the interface spacing,
are preserved. Note that these geometries have not been relaxed – the
interface distance we have chosen in the Mn3Al case is 2 Å, while in the Mn3Ga
junctions we have chosen 2.2 Å, as motivated in Ref. 11.

In Fig. 2 we compare the linear-response STT [16, 11] for a 90° misalignment
of the spin polarisations in the two leads, computed within the non-
equilibrium Green’s function (NEGF) open-boundaries method implemented in the
Smeagol code [17]. Panels (b) and (c) show the layer-resolved local moments in
the scattering region – in (b) the mirror symmetry is readily observed between
the $x$ and $z$ components of the spins on both sides of the junction. In
comparison, in (c) the asymmetry after the removal of one Mn plane at the
right interface is evident. It is worth noting that the spins in the left lead
across the MgO appear practically unaffected by this local structural
disturbance on the right interface, in both junctions.

The calculated layer-resolved in-plane STT for the symmetric junctions
[Fig.2(d)] displays a perfect left-to-right symmetry as well – this time an
inversion symmetry with respect to the central layer of MgO. For both
ferrimagnets the STT at the opposite interfaces has an opposite sign. There
is, however, a qualitative difference between the two ferrimagnets – if the
STT in the Mn3Al case is localised at the interface, in Mn3Ga it shows the
familiar long-range oscillatory decay with periodicity determined by the
difference of majority and minority spin wave-vectors of the
$\Delta_{1}$-symmetry band in Mn3Ga, as described in Ref. 11. This ideal
inversion symmetry of the STT across the barrier, however, will hinder the
switching of the such ideally symmetric junctions. For instance, a positive
$x$-component of STT will rotate anti-clockwise the MnII spins at the left
interface (aligned along $-z$), and also anti-clockwise the MnII spins aligned
along $-x$ at the right-hand-side interface. Hence such an alignment of the
STT cannot drive switching of one lead with respect to the other. It is worth
noting, however, that the STT changes sign in the next bi-layer – this will
act against the sub-lattice exchange coupling. This is would enable the
excitation of high frequency anti-phase modes and potentially lead to fasted
switching dynamics. It is likely that the ideal symmetry in our calculations
will be broken in real structures and there will be an STT imbalance leading
to the switching of one of the layers. In the Mn3Al junction the STT has
analogous symmetry, which does not lead to switching. It is, however, much
more localised at the interfacial layer and significantly lower in amplitude.

Figure 3: Energy-resolved properties at 0 V equilibrium for both the P and AP
states of the junctions (see text). (a,b) transmission coefficients and (c,d)
TMR ratio, defined as
$\mathrm{TMR}=(T_{\mathrm{P}}-T_{\mathrm{AP}})/(T_{\mathrm{P}}+T_{\mathrm{AP}})$
for the two symmetric and the two asymmetric MTJs, respectively.

To illustrate the effect of possible structural imperfection we consider the
’asymmetric’ versions of both junctions – with removed Mn-Mn monolayer from
the right interface, while keeping the interface spacing unchanged [Fig.
2(a)]. In this case the STT profiles change substantially [Fig.2(d,e)]. Now we
see both in the Mn3Al and Mn3Ga junctions the STT becomes asymmetric on both
sides of the barrier. This time it shows a tendency to rotate spins in
opposite directions, especially in the Mn3Al case, thus driving a switching of
one of the layers with respect to the other. In the asymmetric Mn3Al junction
the STT is significantly increased in magnitude from the symmetric case,
especially on the right-hand side, where we see a large STT also beyond the
interfacial layer.

We then examine the energy dependence of the total transmissions in the four
junctions in their collinear spin states – parallel (P) state in which the Mn
spin in each sublattice (MnI or MnII type) on the two sides of the barrier are
parallel, and AP state, when they are antiparallel. We clearly see the half-
metallicity of Mn3Al manifesting itself in the vanishing transmission in the
AP state around the Fermi level. This in turn drives a large TMR (practically
100% from the conservative definition with the sum of the transmission
coefficients in the denominator) for a wide range of energies around
$E_{\mathrm{F}}$ in both the symmetric and the asymmetric junctions [green
curve in Fig. 3 (c,d)]. In the case of Mn3Ga we find a TMR of about 60 % for
the symmetric junction, which drops to 30 % for the asymmetric case. We find a
significant enhancement of the TMR for energies higher than 0.2 eV above
$E_{\mathrm{F}}$, which is due to the $\Delta_{1}$ symmetry band edge and the
band gap opening above that energy for majority spin in the $\Gamma-Z$
(transport) direction [11] (note that the position of this band edge above
$E_{\mathrm{F}}$ is consistent between the LDA and GGA calculations in Fig.1).
The role of the interfacial asymmetry appears to be in reducing the
transmission in the P state, where in the asymmetric junctions the interfacial
spins are pointing in opposite directions, giving rise to additional
scattering at the interfaces.

Figure 4: Contour portraits of the total transmission coefficient (both spin
species) at the Fermi level decomposed over the transverse 2D-BZ [as a
function of $(k_{x},k_{y})$ on each panel, where $-\pi/a\leq
k_{x,y}\leq\pi/a$]. From left to right, the four different MTJs are depicted
(as indicated above), while the two rows correspond to the parallel (P) and
anti-parallel (AP) spin states of the junctions, respectively (see text).
Color-code bars for each panel are for the $T(k_{x},k_{y},E=E_{\mathrm{F}})$,
i.e. dimensionless transmission probabilities (see, e.g. Ref.16).

This can also be seen in the 2D portraits of the Fermi-level transmission
coefficients in the transverse 2D Brillouin zone (2D-BZ) (Fig. 4). In the
Mn3Ga case the transmission in all spin-states and junction geometries is
predominantly around the $\Gamma$ point. The P states show stronger
transmission in both junctions, while in the asymmetric case both spin-
polarised transmissions are somewhat suppressed with respect to the
corresponding ones in the symmetric MTJ. In contrast to Mn3Ga, the
transmission around the $\Gamma$ point is suppressed in the two Mn3Al
junctions in the P state, as in this material there is a band gap along
$\Gamma-Z$ direction for majority (up) spins [see Fig. 1(a)]. Interestingly,
despite the small quantitative changes, which the asymmetric interface
introduces in some regions of the transverse 2D-BZ, for the small range of
wavevector angles relevant to realistic specular transport around the $\Gamma$
point, the transmission appears relatively unaffected by the interfacial
detail. This effect is expected to be enhanced further for thicker MgO
barriers, where the transmission is more strongly filtered in the vicinity of
$\Gamma$ point and we expect the overall transmission in Mn3Al-based MTJs to
be even less sensitive to features of the interfaces. We also note that in the
AP state of the Mn3Al junctions the transmission is extremely small and at the
verge of computational accuracy, as expected from its halfmetallicity (see
Fig. 1).

##  IV Conclusions

We have compared linear-response spin-polarised transport properties in
ferrimagnetic tunnel junctions made from two similar Mn-based binary Heuslers
in their tetragonal phase sandwiching a MgO barrier. Junctions featuring Mn3Al
offer a different compromise between the magnitude of STT and TMR, when
compared to otherwise equivalent Mn3Ga-based ones. The overall transmission,
and therefore the expected differential conductivity, is higher in the
ferrimagnetic Mn3Ga case, together with the low-bias STT. The absence of one
type of spin states for Mn3Al-based junctions, leads to smaller overall
absorbed momentum (with STT acting only within a couple of lattice spacings
away from the interfaces) and a significantly suppressed low-bias differential
conductance. It can be argued, therefore that at least at low temperature and
applied bias, when compared on the basis of STT per unit dissipated power, the
Mn3Ga-based junctions would offer an edge over the otherwise better in their
TMR performance Mn3Al counterparts. The relatively low sensitivity of the
calculated parallel-state transmission (normal to the interface) and also of
the STT, although to a lesser extent, on the exact surface reconstruction in
both Mn3Al and Mn3Ga-based junctions, may further motivate the experimental
verification of spectroscopic TMR and STT features (at low temperatures),
hence, finite-bias calculations of the same would be well-warrantied. In view
of the above, aluminium-rich solid solutions of Mn3Al-Mn3Ga, with or without
Sn doping, could offer enhanced low-bias TMR performance, while preserving
tetragonality, at manageable growth-induced strain, in practical device
stacks.

###### Acknowledgements.

We gratefully acknowledge funding from the Science Foundation Ireland (SFI
Grant No. 18/SIRG/5515 and 18/NSFC/MANIAC). We thank the Irish Centre for
High-End Computing (ICHEC) and the Trinity Research IT Centre (TCHPC) for the
provision of computational facilities and support.

Copyright (2022) Authors. This article is distributed under a Creative Commons
Attribution (CC BY) License.

## References

  * Rode  _et al._ [2013] K. Rode, N. Baadji, D. Betto, Y. C. Lau, H. Kurt, M. Venkatesan, P. Stamenov, S. Sanvito, J. M. D. Coey, E. Fonda, E. Otero, F. Choueikani, P. Ohresser, F. Porcher, and G. André, Physical Review B 87, 184429 (2013).
  * Winterlik  _et al._ [2008] J. Winterlik, B. Balke, G. H. Fecher, C. Felser, M. C. Alves, F. Bernardi, and J. Morais, Physical Review B 77, 1 (2008).
  * Gao and Yao [2013] G. Y. Gao and K. L. Yao, Appl. Phys. Lett. 103, 232409 (2013).
  * Fan  _et al._ [2019] L. Fan, F. Chen, C.-m. Li, X. Hou, X. Zhu, J.-l. Luo, and Z.-Q. Chen, J. Magn. Magn. Mater. 497, 166060 (2019).
  * Mizukami  _et al._ [2016a] S. Mizukami, S. Iihama, Y. Sasaki, A. Sugihara, R. Ranjbar, and K. Z. Suzuki, Journal of Applied Physics 120, 10.1063/1.4961704 (2016a).
  * Mizukami  _et al._ [2016b] S. Mizukami, A. Sakuma, A. Sugihara, K. Z. Suzuki, and R. Ranjbar, Scripta Materialia 118, 70 (2016b).
  * Sinova  _et al._ [2017] J. Sinova, T. Jungwirth, and O. Gomonay, Phys. Status Solidi RRL 4, 1770322 (2017).
  * Jungwirth  _et al._ [2016] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016).
  * Kha [2014] J. Phys. Condens. Matt. 26, 126001 (2014).
  * Mizukami  _et al._ [2011] S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106, 117201 (2011).
  * Stamenova  _et al._ [2021] M. Stamenova, P. Stamenov, F. Mahfouzi, Q. Sun, N. Kioussis, and S. Sanvito, Physical Review B 103, 094403 (2021).
  * Borisov  _et al._ [2016] K. Borisov, D. Betto, Y. C. Lau, C. Fowley, A. Titova, N. Thiyagarajah, G. Atcheson, J. Lindner, A. M. Deac, J. M. Coey, P. Stamenov, and K. Rode, Appl. Phys. Lett. 108, 10.1063/1.4948934 (2016).
  * Jamer  _et al._ [2017] M. E. Jamer, Y. J. Wang, G. M. Stephen, I. J. McDonald, A. J. Grutter, G. E. Sterbinsky, D. A. Arena, J. A. Borchers, B. J. Kirby, L. H. Lewis, B. Barbiellini, A. Bansil, and D. Heiman, Physical Review Applied 7, 1 (2017).
  * Qiu  _et al._ [2021] M. Qiu, S. Ye, W. Wang, J. He, S. Chang, H. Wang, and Q. Huang, J. Phys. D: Appl. Phys. 54, 10.1088/1361-6463/abcf72 (2021).
  * Soler  _et al._ [2002] J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, J. Phys.: Cond. Matt. 14, 2745 (2002).
  * Rungger  _et al._ [2018] I. Rungger, A. Droghetti, and M. Stamenova, Non-equilibrium Green’s Function Methods for Spin Transport and Dynamics, Handbook of Materials Modeling , 1 (2018).
  * Rocha and Sanvito [2004] A. R. Rocha and S. Sanvito, Physical Review B 70, 094406 (2004), 0403351 .


# Cloning, deleting and hiding in modal quantum theory

Phillip Diamond and Benjamin Schumacher

###### Abstract

We examine the toy model of modal quantum theory (MQT), an analogue of actual
quantum theory based on finite fields. In particular, we investigate how
several essential “no-go” results (for cloning, deleting and hiding processes)
work in MQT. Cloning and deleting are still forbidden in MQT, though the
details of these results are somewhat different in the new context. However,
the information of a modal qubit can be completely hidden in the correlations
between two entangled modal qubits.

##  1 Introduction

No-go theorems show that a particular process or circumstance cannot occur
within the context of a given physical theory. Such theorems have proven to be
useful tools for understanding the essential content of quantum theory, and
they have played a key role in our understanding of quantum foundations and
quantum information theory. This paper continues a project of exploring
quantum no-go theorems by considering modal quantum theory (MQT), a
mathematical “toy model” of actual quantum theory based on finite fields. The
structure of MQT lacks the idea of probability, and the concepts of state,
time evolution and distinguishability are substantially different from actual
quantum theory.

We will find that some familiar no-go results have direct analogues in MQT,
while others do not. Even in the cases where MQT has a parallel no-go result,
the details can be quite dissimilar. By formulating our arguments within MQT,
we can shed light on the way no-go theorems work, and even arrive at new ideas
and arguments applicable to the quantum theory of our world.

##  2 Modal quantum theory

In modal quantum theory [6, 5], the space of states of a system is a finite-
dimensional vector space $\mathcal{V}$ over a general field $\mathcal{F}$,
usually taken to be finite. There is no inner product, and any non-null vector
$\left|\psi\right)$ represents a pure state. Given a basis set
$\\{\left|\phi_{i}\right)\\}$ for $\mathcal{V}$, we can write

$\left|\psi\right)=\sum_{i}\alpha_{i}\left|\phi_{i}\right),$ (1)

where $\alpha_{i}\in\mathcal{F}.$ The basis vectors $\left|\phi_{i}\right)$
are identified with the possible results of a simple “basic” measurement on
the system. But MQT does not predict probabilities for these results. Instead,
MQT only makes the “modal” distinction between possible and impossible
results. For state $\left|\psi\right)$, the result $\phi_{i}$ is possible if
$\alpha_{i}\neq 0$ and impossible if $\alpha_{i}=0$.

The time evolution of an isolated modal quantum system over a fixed time
interval is given by an invertible linear operator $T$ on $\mathcal{V}$. The
states of composite systems are non-null vectors in the tensor product space
$\mbox{$\mathcal{V}$}^{\mbox{\tiny(12)}}=\mbox{$\mathcal{V}$}^{\mbox{\tiny(1)}}\otimes\mbox{$\mathcal{V}$}^{\mbox{\tiny(2)}}$.

As in actual quantum theory, we may also consider mixed states. These arise in
two distinct situations. First, we may not have complete information about the
preparation of a system. Second, the system of interest may be part of a
larger composite system in an entangled state. In either case, MQT represents
a mixed state by a subspace $\mathcal{M}$ of $\mathcal{V}$. Suppose we have a
set $\\{\left|\psi\right)_{a}\\}$ of possible state vectors for the system.
Then the mixed state is

$\mathcal{M}=\left\langle\\{\left|\psi\right)_{a}\\}\right\rangle,$ (2)

where $\left\langle\cdots\right\rangle$ denotes the linear span of a set. A
measurement result $\left|\phi_{i}\right)$ is possible for $\mathcal{M}$
whenever it is possible for some $\left|\psi\right)\in\mathcal{M}$.

Alternatively, suppose we have a pure entangled state
$\left|\Psi^{\mbox{\tiny(12)}}\right)$ of a composition of systems 1 and 2. If
we choose a basis $\left|\phi_{i}^{\mbox{\tiny(1)}}\right)$ for system 1
states, we can write this as

$\left|\Psi^{\mbox{\tiny(12)}}\right)=\sum_{i}\left|\phi_{i}^{\mbox{\tiny(1)}}\right)\otimes\left|\psi_{i}^{\mbox{\tiny(2)}}\right).$
(3)

The measurement result $\left|\phi_{i}^{\mbox{\tiny(1)}}\right)$ for system 1
is possible when $\left|\psi_{i}^{\mbox{\tiny(2)}}\right)\neq 0$, and if this
result does occur then the conditional state of system 2 is
$\left|\psi_{i}^{\mbox{\tiny(2)}}\right)$. The mixed state of system 2 is just
$\mathcal{M}^{\mbox{\tiny(2)}}=\left\langle\\{\left|\psi_{i}^{\mbox{\tiny(2)}}\right)\\}\right\rangle$,
a subspace that is independent of the choice of basis
$\\{\left|\phi_{i}^{\mbox{\tiny(1)}}\right)\\}$ for
$\mbox{$\mathcal{V}$}^{\mbox{\tiny(1)}}$.

To summarize, modal quantum theory retains some of the structure of actual
quantum theory, while discarding other properties. The states are elements of
a vector space, (thus allowing for superpositions) but there is no inner
product on that space. Measurement results may be possible or impossible, but
no probabilities are entailed. Time evolution is linear and invertible, but
not unitary. Mixed states arise from incomplete information about system
preparation or as a subsystem state of an entangled composite state. These are
described by subspaces rather than density operators.

The concepts of actual quantum theory and modal quantum theory are compared in
Table 1.

| AQT | MQT
---|---|---
States |  $\left|\psi\right\rangle=\sum_{i}c_{i}\left|\phi_{i}\right\rangle$, $c_{i}\in\mathbb{C}$ |  $\left|\psi\right)=\sum_{i}\alpha_{i}\left|\phi_{i}\right)$, $\alpha_{i}\in\mbox{$\mathcal{F}$}$
Bases |  $\left|\phi_{i}\right\rangle$ orthonormal |  $\left|\phi_{i}\right\rangle$ linearly independent
Amplitudes | $P(\phi_{i})=\left|c_{i}\right|^{2}$ | $\begin{array}[]{l}\alpha_{i}\neq 0\Rightarrow\phi_{i}\mbox{ possible}\\\ \alpha_{i}=0\Rightarrow\phi_{i}\mbox{ impossible}\end{array}$
Normalization | $\left\langle\psi|\psi\right\rangle=\sum_{i}\left|c_{i}\right|^{2}=1$ |  $\left|\psi\right)\neq 0$ (at least one $\alpha_{i}\neq 0$)
Time evolution | $\left|\psi(t)\right\rangle=U\left|\psi(0)\right\rangle$ | $\left|\psi(t)\right)=T\left|\psi(0)\right)$
Composite states | $\left|\Psi^{\mbox{\tiny(12)}}\right\rangle\in\mathcal{H}^{\mbox{\tiny(1)}}\otimes\mathcal{H}^{\mbox{\tiny(2)}}$ | $\left|\Psi^{\mbox{\tiny(12)}}\right)\in\mathcal{V}^{\mbox{\tiny(1)}}\otimes\mathcal{V}^{\mbox{\tiny(2)}}$
Mixed states | $\rho=\sum_{a}p_{a}\left|\psi_{a}\right\rangle\\!\\!\left\langle\psi_{a}\right|$ | $\mathcal{M}=\left\langle\\{\left|\psi_{a}\right)\\}\right\rangle$
Table 1: Aspects of actual quantum theory (AQT) and their MQT analogues.

##  3 Three classical operations and their analogs

The three operations we will consider are suggested by operations in classical
reversible logic. Given a “control” bit $a$ and a “target” bit $b$, the
controlled-not (CNOT) operation is

$(a,b)\longrightarrow(a,a\oplus b),$ (4)

where $\oplus$ is addition mod 2. Thus,

$\begin{array}[]{ccc}(0,0)\rightarrow(0,0)&&(1,0)\rightarrow(1,1)\\\
(0,1)\rightarrow(0,1)&&(1,1)\rightarrow(1,0)\end{array}.$ (5)

This operation is clearly reversible, since it is its own inverse. We can use
the CNOT operation as the basis for several common tasks.

Cloning (or copying).

If the target bit $b$ is initially 0, then its final state will be an exact
copy of $a$:

$(a,0)\longrightarrow(a,a).$ (6)

Thus, we can make copies of a data $a$ into blank memory.

Deleting (or uncopying).

If the bits are initially equal, then the final state of the target bit will
be 0:

$(a,a)\longrightarrow(a,0).$ (7)

Thus, we can delete one of our two copies of the same information.

Hiding.

Suppose the control bit is a cryptographic key $k$ and the target is a
plaintext message bit $p$. Then

$(k,p)\longrightarrow(k,c)$ (8)

where $c=k\oplus p$, the encrypted ciphertext. Neither $k$ nor $c$ alone
provides any information whatsoever about the input plaintext $p$, which can
only be recovered from the pair $(k,c)$. In other words, the $p$-information
is completely “hidden” in the correlations between $k$ and $c$.

In actual quantum theory, the analogues of these three classical operations
are forbidden by no-go theorems.

Quantum no-cloning theorem.

No process can duplicate an unknown input quantum state, taking
$\left|\psi\right\rangle$ as input and yielding $\left|\psi,\psi\right\rangle$
as output. [7, 3]

Quantum no-deleting theorem.

If a process deletes one of a duplicate pair of states—taking
$\left|\psi,\psi\right\rangle$ as input and yielding
$\left|\psi,0\right\rangle$ as output—then an isomorphic copy of state
$\left|\psi\right\rangle$ must be retained in the final state of the auxiliary
apparatus. [4]

Quantum no-hiding theorem.

The information of an unknown qubit state cannot be stored entirely in the
correlation between two qubits. [1]

In modal quantum theory, as we will see, the situation is more complicated.

##  4 Distinguishability, Cloning, and Deleting

In AQT, two unknown states are distinguishable if they are orthogonal, so that
there exists a measurement that always identifies one state from the other. If
$\left|\psi\right\rangle$ and $\left|\phi\right\rangle$ are distinct but not
orthogonal, they cannot be perfectly distinguished. Many copies of the same
states

$\underbrace{\left|\psi\right\rangle\otimes\left|\psi\right\rangle\cdots\left|\psi\right\rangle}_{\text{$n$
times}}\text{ and
}\underbrace{\left|\phi\right\rangle\otimes\left|\phi\right\rangle\otimes\cdots\otimes\left|\phi\right\rangle}_{\text{$n$
times}},$ (9)

may be approximately distinguished, since their inner product

$\big{(}\left\langle\psi\right|\otimes\left\langle\psi\right|\otimes\cdots\otimes\left\langle\psi\right|\big{)}\big{(}\left|\phi\right\rangle\otimes\left|\phi\right\rangle\otimes\cdots\otimes\left|\phi\right\rangle\big{)}=\left\langle\psi|\phi\right\rangle^{n}$
(10)

approaches 0 as $n\to\infty$. Nevertheless, perfect distinguishability is not
possible for finite $n$.

In MQT, two modal states $\left|\psi\right)$ and $\left|\phi\right)$ can be
distinguished as long as $\left|\phi\right)$ is not parallel to
$\left|\psi\right)$. In general, linearly independent sets of states in MQT
can be distinguished, and linearly dependent sets cannot. What about $n$-copy
states? We begin with a lemma.

Lemma. Given a set of $d+1$ distinct modal states spanning a $d$-dimensional
subspace, the set of $d+1$ two-copy states is distinguishable.

###### Proof.

Let $\mathcal{F}$ be the finite field of MQT state coefficients. Assume
$S=\\{\left|\psi_{1}\right),\left|\psi_{2}\right),...,\left|\psi_{d}\right),\left|\sigma\right)\\}$
is a set of $d+1$ vectors spanning a $d$-dimensional subspace, where

$\left|\sigma\right)=\sum_{k=1}^{d}\sigma_{k}\left|\psi_{k}\right)$ (11)

for some set $\\{\sigma_{k}\\}\in\mathcal{F}$ such that at least two
$\sigma_{k}$ are nonzero. Assume $\sigma_{1},\sigma_{2}\neq 0$. Now consider
the same set, but with an additional copy of each state:

$S^{(2)}=\\{\left|\psi_{1},\psi_{1}\right),\left|\psi_{2},\psi_{2}\right),...\,,\left|\psi_{d},\psi_{d}\right),\left|\sigma,\sigma\right)\\}.$
(12)

Since the original $\left|\psi_{k}\right)$’s are linearly independent, the
joint states $\left|\psi_{j},\psi_{k}\right)$ are also linearly independent.
The state $\left|\sigma,\sigma\right)$ is evidently not a linear combination
of the $\left|\psi_{k},\psi_{k}\right)$ states since it includes terms such as
$\sigma_{1}\sigma_{2}\left|\psi_{1},\psi_{2}\right)$. Hence, $S^{(2)}$ is a
linearly independent—and thus distinguishable—set of MQT states. ∎

Using this lemma inductively, we find that any finite, linearly dependent set
of $d$ distinct states can eventually be distinguished given a sufficient (but
finite) number of state copies. It suffices to double the number of state
copies repeatedly until every (previously) dependent state can be
distinguished.

This result has implications for the no-cloning and no-deleting theorems in
MQT. For any system in MQT with finite $\mathcal{F}$ and finite dimension
$\mathcal{V}$, there are finitely many distinct states. From the generalized
lemma, those states are distinguishable given sufficiently many copies. Let
$S$ be a set of states, and let $N$ denote the number of state copies
sufficient to distinguish every state in $S$. Since the $N$-copy states are
distinguishable, we can define an invertible linear map $T$ which clones from
$N\to N+1$ state copies and deletes (via $T^{-1}$) from $N+1\to N$ state
copies. Thus both cloning and deleting are possible with sufficient input
state copies.

Now let $M$ denote the minimum number of state copies necessary to distinguish
$S$. Cloning is not possible when the number of input state copies is less
than $M$ because the set of inputs will be linearly dependent. This can be
shown using an argument analogous to Wootters and Zurek [7]. Any machine that
clones linearly independent states will fail to clone their superposition.

Deleting is also not possible if the number of input state copies is less than
$M$, as the input states are linearly dependent. Like cloning, if a machine
deletes all linearly independent states in $S$, it will fail to delete
linearly dependent states and will instead map them linearly into the machine
state. This is analogous to the argument of Braunstein and Pati [4].
Interestingly, a form of deleting is still possible when deleting from $M\to
M-1$ state copies, in which the set of input states are distinguishable but
the output states are not. It is not possible to distinguish the outputs of
this deleting process; but since the $M$-copy inputs are distinguishable, a
deleting machine could store a classical identifier of the input state,
enabling the output state to be reconstructed.

##  5 Hiding

In a hypothetical hiding process, the unknown state of a single qubit is
mapped into the 4-dimensional state space of a pair of qubits. The quantum
information is hidden if the output states of the individual qubits are each
independent of the input. This can only be true if the output states are all
entangled and reduce to identical mixed states on the individual qubits. (If
the output states were product states, of course, at least one of the
subsystem states would have to depend on the input state.) This condition is
necessary in both actual and modal quantum theory.

We begin therefore by presenting an alternate proof of the no-hiding theorem
for qubits in AQT by proving that every 2D subspace of the 4D Hilbert space of
a qubit pair must contain at least one product state.

###### Proof.

We define an arbitrary transformation from the 2D subspace of one qubit to the
4D subspace of two qubits as follows:

$\displaystyle\left|0\right\rangle$ $\displaystyle\to$
$\displaystyle\left|\Psi_{0}\right\rangle=\sqrt{\lambda}\left|0,0\right\rangle+\sqrt{1-\lambda}\left|1,1\right\rangle$
$\displaystyle\left|1\right\rangle$ $\displaystyle\to$
$\displaystyle\left|\Psi_{1}\right\rangle=\left|0\right\rangle\otimes\big{(}C_{00}\left|0\right\rangle+C_{01}\left|1\right\rangle\big{)}+\left|1\right\rangle\otimes\big{(}C_{10}\left|0\right\rangle+C_{11}\left|1\right\rangle\big{)},$
(13)

where $\left|0\right\rangle$ and $\left|1\right\rangle$ are Schmidt basis
vectors for $\left|\Psi_{0}\right\rangle$, and the coefficients
$C_{ij}\in\mathbb{C}$. If $\left|\Psi_{0}\right\rangle$ is entangled, then
$0<\lambda<1$. The generic composite state $\left|\Psi_{1}\right\rangle$ may
or may not be entangled.

Applying the transformation on an arbitrary superposition of input states
yields

$\displaystyle\alpha\left|\Psi_{0}\right\rangle+\beta\left|\Psi_{1}\right\rangle$
$\displaystyle=$
$\displaystyle\left|0\right\rangle\otimes\big{(}(\alpha\sqrt{\lambda}+\beta
C_{00}\big{)}\left|0\right\rangle+\beta C_{01}\left|1\right\rangle\big{)}$
(14) $\displaystyle\qquad+\left|1\right\rangle\otimes\big{(}\beta
C_{10}\left|0\right\rangle+(\beta
C_{11}+\alpha\sqrt{1-\lambda})\left|1\right\rangle\big{)}.$

Under what circumstances is Equation 14 a product state? This will happen if
the conditional states of the second subsystem are complex scalar multiples of
each other. Explicitly,

$\displaystyle\alpha\sqrt{\lambda}+\beta C_{00}$ $\displaystyle=\beta C_{10}z$
(15) $\displaystyle\beta C_{01}$ $\displaystyle=\big{(}\beta
C_{11}+\alpha\sqrt{1-\lambda}\big{)}z,$ (16)

for some scalar $z\in\mathbb{C}$. Solving this system of equations yields a
quadratic polynomial in $z$:

$\left(C_{10}\sqrt{1-\lambda}\right)z^{2}+\left(C_{11}\sqrt{\lambda}-C_{00}\sqrt{1-\lambda}\right)z-C_{01}\sqrt{\lambda}=0$
(17)

Since $z\in\mathbb{C}$, this quadratic is solvable regardless of $\lambda$ and
$C_{ij}$. So given any $1\to 2$ qubit transformation (any $\lambda$ and
normalized set of coefficients $C_{ij}$), we can use the $z$ satisfying the
system of equations to solve for $\alpha$ and $\beta$: first set $\beta=1$,
then solve the system for $\alpha$ and re-normalize. The resulting
$\alpha\left|\Psi_{0}\right\rangle+\beta\left|\Psi_{1}\right\rangle$ and will
be a product state. Thus any transformation from the 2D subspace of one qubit
to the 4D subspace of two qubits will contain a product state, violating the
requirements for hiding. ∎

This algebraic proof, based on linearity but not unitarity, was suggested by
analysis of the analogous problem in MQT. In MQT, the system of equations that
determines whether hiding is possible yields the quadratic

$C_{10}k^{2}+(C_{11}-C_{00})k-C_{01}=0$

where $k\in\mathcal{F}$, some finite field, and the coefficients $C_{ij}$ are
defined as in the preceding proof (note the absence of the Schmidt parameter
$\lambda).$ The possibility of hiding is dependent on whether this arbitrary
quadratic is ever irreducible over $\mathcal{F}$. In fact, irreducible
quadratics exist over every finite field, so we conclude that hiding is always
possible in MQT.

Further, we can use an irreducible quadratic over a finite field to construct
a hiding transformation in MQT. Consider MQT with
$\mathcal{F}=\mathbb{Z}_{3}$. The polynomial $x^{2}+x+1$ is irreducible over
$\mathbb{Z}_{3}$, so we define $C_{10}=1$, $1=C_{11}-C_{00},$ and $-C_{01}=1$.
We assume $C_{00}=1$. Plugging the $C$’s into the same transformation
described in the AQT proof (without normalization) yields

$\left|0\right)\rightarrow\left|0,0\right)+\left|1,1\right)$

and

$\left|1\right)\rightarrow\left|0\right)\otimes\big{(}\left|0\right)-\left|1\right)\big{)}+\left|1\right)\otimes\big{(}\left|0\right)+2\left|1\right)\big{)}.$

Taking an arbitrary superposition of the transformation acting on
$\left|0\right)$ and $\left|1\right)$, then requiring the result to simplify
to a product state yields the system of equations

$\displaystyle a+b$ $\displaystyle=kb$ $\displaystyle-b$
$\displaystyle=ka+2kb,$

leading to the irreducible quadratic

$k^{2}+k+1=0.$

Thus, the transformation

$\displaystyle\left|0\right)\rightarrow\left|0,0\right)+\left|1,1\right)$
$\displaystyle\left|1\right)\rightarrow\left|0\right)\otimes\big{(}\left|0\right)-\left|1\right)\big{)}+\left|1\right)\otimes\big{(}\left|0\right)+2\left|1\right)\big{)}$

Will hide any input modal qubit, since there is no $k$ satisfying the system
of equations that determines whether the transformation will simplify into a
product state.

##  6 Remarks

To summarize, the MQT toy model has both no-cloning and no-deleting theorems.
However, if we begin with sufficiently many copies of the input states, both
cloning and deleting are possible, since these $n$-copy states are perfectly
distinguishable.

In contrast to actual quantum theory, on the other hand, in MQT we can always
hide the information in a modal qubit in the correlations between two
entangled qubits. The possibility of qubit hiding in MQT arises from the
existence of irreducible quadratic polynomials in a finite field
$\mathcal{F}$. Yet there is more to be said. A key property of MQT is that all
entangled states of a pair of qubits have the same subsystem states:
$\mathcal{V}$, the entire two-dimensional state space. In actual quantum
theory, entangled qubits yield subsystem states described by density
operators. Even if two density operators are both supported on the entire
state space, they may differ in the probabilities of different measurement
effects. Since modal quantum theory has no probabilities, all mixed qubit
states are identical.

In a separate paper [2], we examine another MQT analogue to a basic no-go
theorem of actual quantum theory: the no-broadcasting theorem, the mixed state
generalization of the no-cloning theorem. These and other results help
illuminate the structure of quantum theory by showing how its basic concepts
change—or remain the same—in the simpler “foil” theory of MQT.

We would like to thank Michael D. Westmoreland for many useful conversations
and suggestions during our work, and we gratefully acknowledge the support of
the Kenyon Summer Science Scholar program in the summer of 2022.

## References

  * [1] Samuel L. Braunstein and Arun K. Pati  “Quantum Information Cannot Be Completely Hidden in Correlations: Implications for the Black-Hole Information Paradox”  In _Physical Review Letters_ 98.8, 2007, pp. 080502
  * [2] Phillip Diamond, Benjamin Schumacher and Michael D. Westmoreland  “A no-broadcasting theorem for modal quantum theory” arXiv:2304.13561
  * [3] Dennis Dieks  “Communication by EPR devices”  In _Physics Letters A_ 92.6, 1982, pp. 271–272
  * [4] A.. Pati and S.. Braunstein  “Impossibility of Deleting an Unknown Quantum State”  In _Nature_ 404, 2000, pp. 164
  * [5] Benjamin Schumacher and Michael D. Westmoreland  “Almost Quantum Theory”  In _Quantum Theory: Informational Foundations and Foils_ , 2015, pp. 45–81
  * [6] Benjamin Schumacher and Michael D. Westmoreland  “Modal Quantum Theory”  In _Foundations of Physics_ 42, 2012, pp. 918–925
  * [7] William Wootters and Wojciech Zurek  “A Single Quantum Cannot Be Cloned”  In _Nature_ 299, 1982, pp. 802–803


#  Nonlinear network identifiability: The static case

Renato Vizuete and Julien M. Hendrickx *This work was supported by F.R.S.-FNRS
via the _KORNET_ project and via the Incentive Grant for Scientific Research
(MIS) _Learning from Pairwise Comparisons_ , and by the _RevealFlight_
Concerted Research Action (ARC) of the Fédération Wallonie-Bruxelles.R.
Vizuete and J. M. Hendrickx are with ICTEAM institute, UCLouvain, B-1348,
Louvain-la-Neuve, Belgium. R. Vizuete is a FNRS Postdoctoral Researcher - CR.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We analyze the problem of network identifiability with nonlinear functions
associated with the edges. We consider a static model for the output of each
node and by assuming a perfect identification of the function associated with
the measurement of a node, we provide conditions for the identifiability of
the edges in a specific class of functions. First, we analyze the
identifiability conditions in the class of all nonlinear functions and show
that even for a path graph, it is necessary to measure all the nodes except by
the source. Then, we consider analytic functions satisfying $f(0)=0$ and we
provide conditions for the identifiability of paths and trees. Finally, by
restricting the problem to a smaller class of functions where none of the
functions is linear, we derive conditions for the identifiability of directed
acyclic graphs. Some examples are presented to illustrate the results.

##  I Introduction

Networked systems composed by nodes or subsystems interacting with each other
are ubiquitous [1]. In several of these systems, the knowledge of the dynamics
associated with edges is essential for the analysis of the system and design
of control algorithms. However, the identification of the networked systems
from partial measurements without disconnections of some parts of the network
can be really challenging since a measured signal depends on the combination
of the dynamics of potentially many edges.

There has been some recent works in the linear case on the conditions of
identifiability: when is it possible to unambiguously recover local dynamics
from a set of measured nodes? This question is important in order to design
experiments and position sensors and excitations [2, 3, 4]. This depends
mainly on the topology of the network and on the position of the excitation
and measured signal. Graph theoretical conditions are available in full
measurement case or full excitation [5], but not in the general case yet [6,
7, 8, 9]. However, most actual systems of interest are nonlinear, including
many different research fields like coupled oscillators [10], gene regulatory
networks [11], biochemical reaction networks [12], social networks [13], among
others. While linear systems usually provide a local approximation of
nonlinear phenomena, no one to the best of our knowledge has studied the
identifiability question for nonlinear systems.

The identification of a nonlinear system is itself a challenging problem due
to the variety of potential models (e.g., Hammerstein, Wiener, Volterra
series) and the constant arising of new formulations for particular
applications. Depending on the type of nonlinearities and its location (i.e.,
at the level of inputs, outputs or in the middle of interactions), certain
models can be more suited for specific applications, while others could not
give a good description of some systems [14, 15, 16]. In addition to the
complexity of a single nonlinear model, a network involves several nonlinear
systems associated with the edges, which generates complex collective
behaviors and increase considerably the difficulties of the identification
problem.

In the nonlinear case, the conditions for identifiability of networks do not
depend only on the network topology but also on the types of nonlinear
functions. For instance, trigonometric functions in coupled oscillators [10]
are very different from the activation functions in neural networks that can
be nondifferentiable [17]. Furthermore, in heterogeneous networks, different
types of functions could be associated with edges in the same network. In
addition, the class of functions considered for the problem of identifiability
could be determinant. It is clear that if we restrict the problem to a small
class of functions, the conditions for identifiability of a network could be
relaxed, but the functions in the class could not fit real models. Moreover,
properties of functions such as continuity, differentiability, analyticity,
etc., could play an important role in the determination of conditions for the
identifiability of networks.

We study here the question of identifiability in the nonlinear setting,
assuming in this first work that the local dynamics have very simple structure
(i.e., the output of a node is entirely determined by static interactions with
the neighbors). We show that, surprisingly, the conditions for identifiability
in directed acyclic graphs are weaker than in the linear case, provided that
the dynamics are indeed not linear, and do not involve constant output
component (when they do, the problem is indeed unsolvable). We explain this by
the fact that in the linear case, the loss of identifiability often results
from ambiguities made possible by the superposition principle/superposition of
signal, which is no longer possible in the nonlinear case.

In this work, we provide a formulation of the network identifiability problem
in the nonlinear case, by considering a static model in the edges. By
restricting the problem to a specific class of functions, we provide
identifiability conditions for paths and trees. Furthermore, by considering a
smaller class of functions, we derive conditions for the identifiability of
directed acyclic graphs.

##  II Problem formulation

###  II-A Model class

Since the type of nonlinear dynamics in a network can be really complex, in
this preliminary work, we will consider a static additive model to focus on
the effect of the nonlinearities. Therefore, we exclude dynamical processes
that involve any memory at the level of nodes or edges. Our objective is to
generalize the results of this paper to more complex dynamical models in
future works.

For a network composed by $n$ nodes, we consider that the output of each node
$i$ is given by:

$y_{i}^{k}=\sum_{j\in\mathcal{N}_{i}}f_{i,j}(y_{j}^{k-1})+u_{i}^{k-1},\;\;\text{for
all }i\in\\{1,\ldots,n\\},$ (1)

where the superscripts denote the value of the inputs and outputs at the
specific time instant, $f_{i,j}$ is a nonlinear function, $\mathcal{N}_{i}$ is
the set of in-neighbors of node $i$, and $u_{i}$ is an external excitation
signal. The node $i$ is not included in $\mathcal{N}_{i}$, since it would
imply a dynamical process at the level of the node. The model (1) corresponds
to a nonlinear static version of the model considered in [5, 6, 7], where the
nonlinearities are located in the edges. In this case, the output of a node
$i$ is determined by its own excitation signal $u_{i}$, and the outputs of the
neighbors $y_{j}$ affected by a nonlinear function $f_{i,j}$ associated with
the edge that connects the neighbor. Notice that when the functions $f_{i,j}$
in (1) are linear, the conditions for identifiability of linear networks
derived in [5, 6, 7] also hold. In this work, we will consider analytic
functions with a Taylor series that converges to the function for all
$x\in\mathbb{R}$. This representation as power series will allow us to derive
conditions for the identifiability of nonlinear networks.

Model (1) corresponds to the full excitation case where all the nodes are
excited. The nonzero functions $f_{i,j}$ between the agents define the
topology of the network $G$, forming the set of edges $E$. In this work, we do
not consider multi-edges between two nodes.

###### Assumption 1

The topology of the network is known, where the presence of an edge implies a
nonzero function.

Assumption 1 implies that we know which nodes are connected by nonzero
functions. The objective is to determine which nodes need to be measured to
identify all the nonlinear functions in the network.

Similarly to [5, 6, 7], for the identification process we assume that in an
ideal scenario the relations between excitations and outputs of the nodes have
been perfectly identified. In this work, we restrict our attention to networks
that do not contain any cycle (i.e., directed acyclic graphs). This implies
that when we measure a node $i$, we identify the function $F_{i}^{k}$:

$\\!\\!\\!\\!\\!y_{i}^{k}\\!=\\!u_{i}^{k-1}\\!+F_{i}^{k}(u_{1}^{k-2},\ldots,u_{1}^{k-m_{1}},\ldots,u_{n_{i}}^{k-2},\ldots,u_{n_{i}}^{k-m_{n_{i}}}),\\\
1,\dots,n_{i}\in\mathcal{N}_{i}^{p},$ (2)

where $\mathcal{N}_{i}^{p}$ denotes the set of nodes that have a path to the
measured node $i$. The function $F_{i}^{k}$ is implicitly defined by (1) and
only depends on a finite number of inputs due to the absence of memory on the
edges and nodes, and the absence of cycles. With a slight abuse of notation,
we use the superscript in the function $F_{i}^{k-s}$ to indicate that all the
inputs in (2) are delayed by $s$.

123$f_{2,1}$$f_{3,2}$$f_{3,1}$$u_{1}$$u_{2}$$u_{3}$ Figure 1: The function
$F_{3}^{k}$ associated with the measurement of the node 3 depends on the past
inputs of the nodes 1 and 2 that have a path to the node 3.

###### Example 1

Let us consider the graph in Fig. 1 where the measurement of the node 3
provides the output:

$\displaystyle y_{3}^{k}$ $\displaystyle=u_{3}^{k-1}+F_{3}^{k}$
$\displaystyle=u_{3}^{k-1}+f_{3,2}(u_{2}^{k-2}+f_{2,1}(u_{1}^{k-3}))+f_{3,1}(u_{1}^{k-2}).$
(3)

We can observe that the function $F_{3}^{k}$ depends on the inputs of the
nodes 1 and 2 that have a path to the node 3.

###  II-B Identifiability

The identifiability problem is related to the possibility of identifying the
functions $f_{i,j}$ based on several measurements. For this, we introduce the
following relationship between the measurements and the functions $f_{i,j}$.

######  Definition 1 (Set of measured functions)

Given a set of measured nodes $\mathcal{N}^{m}$, the set of measured functions
$F(\mathcal{N}^{m})$ associated with $\mathcal{N}^{m}$ is given by:

$F(\mathcal{N}^{m}):=\\{F_{i}^{k}\;|\;i\in\mathcal{N}^{m}\\}.$

We say that a function $f_{i,j}$ associated with an edge satisfies
$F(\mathcal{N}^{m})$ if $f_{i,j}$ can lead to $F(\mathcal{N}^{m})$ through
(1).

For completely arbitrary functions, the identifiability problem can be really
challenging or even unrealistic. For this reason, we restrict the
identifiability problem to a certain class of functions $\mathcal{F}$, which
implies that the functions associated with the edges belong to $\mathcal{F}$
and that the identifiability is considered only among the functions belonging
to $\mathcal{F}$. The different classes of functions will be specified
depending on the results.

######  Definition 2 (Edge identifiable)

In a network $G$, an edge $f_{i,j}$ is identifiable in a class $\mathcal{F}$
if given a set of measured functions $F(\mathcal{N}^{m})$, every set of
functions in $\mathcal{F}$ leading to $F(\mathcal{N}^{m})$ has the same
$f_{i,j}$.

######  Definition 3 (Network identifiable)

A network $G$ is identifiable in a class $\mathcal{F}$ if all the edges are
identifiable in the class $\mathcal{F}$.

The function $F_{i}^{k}$ in (2) is the most complete information that we can
obtain when we measure a node $i$. This implies that if it is not possible to
identify the functions $f_{i,j}$ with $F_{i}^{k}$, these edges are
unidentifiable. On the contrary, if the functions $f_{i,j}$ are identifiable,
it seems reasonable that under some conditions, the function $F_{i}^{k}$ can
be well approximated after sufficiently long experiments, which could allow us
to identify the functions $f_{i,j}$ approximately.

###  II-C First results

We provide a result about the information that we can obtain with the
measurement of sinks and sources111A source is a node with no incoming edges.
A sink is a node with no outgoing edges..

######  Proposition 1 (Sinks and sources)

The measurement of the sources is never necessary for the identifiability of
the network. The measurement of all the sinks is necessary for the
identifiability of the network.

###### Proof:

First, the measurement of any source $j$ generates the output
$y_{j}^{k}=u_{j}^{k-1},$ which does not provide any information about
functions associated with edges in the network. Next, let us consider a sink
$i$ with $m$ incoming edges. The measurement of this sink provides an output:

$y_{i}^{k}=u_{i}^{k-1}+f_{i,1}(y_{1}^{k-1})+\cdots+f_{i,m}(y_{m}^{k-1}),$ (4)

and it is the only way of obtaining information of the functions
$f_{i,1},\ldots,f_{i,m}$. Thus, the measurement of all the sinks is necessary.
∎

The following lemma provides a result about the structure of the function
$F_{i}^{k}$ associated with the measurement of a node $i$ with respect to the
excitation signals of the in-neighbors.

###### Lemma 1

Let $j$ be an in-neighbor of a measured node $i$ and the function $F_{i}^{k}$.
Then, by assuming all the variables but $u_{j}^{k-2}$ constant, we have:

$F_{i}^{k}=\alpha+f_{i,j}(u_{j}^{k-2}+\beta),$

where $\alpha$ and $\beta$ are constants with respect to $u_{j}^{k-2}$.

###### Proof:

According to (2), the function $F_{i}^{k}$ of a measured node $i$ is given by:

$\displaystyle F_{i}^{k}$
$\displaystyle=\sum_{\ell=1}^{m}f_{i,\ell}(y_{\ell}^{k-1})$
$\displaystyle=\sum_{\ell=1}^{m}f_{i,\ell}(u_{\ell}^{k-2}+F_{\ell}^{k-1}),$
(5)

where $m$ is the number of in-neighbors of the node $i$. All the functions
$F_{\ell}^{k-1}$ depend on inputs delayed by 1, which implies that no
$F_{\ell}^{k-1}$ depends on $u_{j}^{k-2}$. Finally, no $f_{i,p}$ with $p\neq
j$ can be a function of $u_{j}^{k-2}$ since there are no multi-edges. ∎

Lemma 1 implies that $f_{i,j}$ in (5) is the only function that depends on
$u_{j}^{k-2}$, and $F_{j}^{k-1}$ does not depend on $u_{j}^{k-2}$.

##  III Paths and trees

###  III-A Strong requirements for general nonlinear functions

Since the conditions for the identifiability of linear networks are based on
the existence of paths in the network that carry information from the excited
nodes to the measured nodes [5, 7], we first focus on the conditions for the
identifiability of a path graph 222A path graph is a graph that can be drawn
so that all the nodes and edges lie on a single straight line. in the
nonlinear case.

In the linear case, for this graph topology we only need to measure the sink
to identify all the transfer functions of the network thanks to the
superposition principle [5]. However, this is not true for the nonlinear case.

######  Example 2 (Path graph)

Fig. 2 presents a simple path graph with 3 nodes where the measurement of the
sink is not enough to identify the network when general nonlinear functions
are considered.

######  Proposition 2 (General nonlinear functions)

For identifiability of a path graph in the class of general nonlinear
functions, it is necessary to measure all the nodes except by the source.

###### Proof:

Let us consider a path graph with $n>2$ nodes and a node $i$ in the middle,
which is neither the source nor the sink. The output of the node $i+1$ is
given by:

$\displaystyle y_{i+1}^{k}$ $\displaystyle=u_{i+1}^{k-1}+F_{i+1}^{k}$ (6)
$\displaystyle=u_{i+1}^{k-1}+f_{i+1,i}(u_{i}^{k-2}+f_{i,i-1}(u_{i-1}^{k-3}+F_{i-1}^{k-2})).$

If the node $i$ is not measured and we consider the functions
$\tilde{f}_{i,i-1}(x)=f_{i,i-1}(x)+\gamma$ and
$\tilde{f}_{i+1,i}(x)=f_{i+1,i}(x-\gamma)$ with $\gamma\neq 0$, the function
$F_{i+1}^{k}$ in (6) is the same, which implies that the path graph cannot be
identified.

On the other hand, if we measure all the nodes, we know the function
$F_{j}^{k}$ associated with any node $j$ in the network. Let us consider a
node $i$ with an in-neighbor and we set all the inputs to 0 except
$u_{i}^{k-1}$. Then, the measurement of the node $i$ gives us:

$\displaystyle y_{i}^{k}$ $\displaystyle=u_{i}^{k-1}+F_{i}^{k}$ (7)
$\displaystyle=u_{i}^{k-1}+f_{i,i-1}(u_{i}^{k-2}+F_{i-1}^{k-1}(0))$

If there is another function $\tilde{f}_{i,i-1}$ satisfying $F_{i}^{k}$ in
(7), we would have for all $u_{i}^{k-2}\in\mathbb{R}$:

$f_{i,i-1}(u_{i}^{k-2}+F_{i-1}^{k}(0))=\tilde{f}_{i,i-1}(u_{i}^{k-2}+F_{i-1}^{k}(0)),$

which implies that $f_{i,i-1}=\tilde{f}_{i,i-1}$ and we can identify
$f_{i,i-1}$. Following a similar approach for the other nodes, we can identify
all the nonlinear functions in the path graph. Finally, by Proposition 1, it
is never necessary to measure the source. ∎

123$f_{3,2}(x)$$f_{2,1}(x)$123$\tilde{f}_{3,2}(x)=f_{3,2}(x-\gamma)$$\tilde{f}_{2,1}(x)=f_{2,1}(x)+\gamma$
Figure 2: A path graph with 3 nodes and different nonlinear functions that
satisfy $F_{3}^{k}=\tilde{F}_{3}^{k}$. For any $\gamma\neq 0$, the measurement
of the sink is not enough for the identification of the network.

Notice that in the proof of Proposition 2 we do not use properties of analytic
functions and the results are also valid for nonlinear functions that are not
analytic. Proposition 2 shows that even in a simple graph topology like a path
graph, which is the key for the identification of more complex network
topologies, the identification of general nonlinear functions cannot be
performed by only measuring the sink. This is due to the constant factor
associated with the static behavior of the function.

###  III-B Identifiability conditions for functions with no constant effect

We restrict the identifiability problem to a smaller class of functions
without a static component.

######  Definition 4 (Class of functions $\mathcal{F}_{Z}$)

Let $\mathcal{F}_{Z}$ be the class of functions $f:\mathbb{R}\to\mathbb{R}$
with the following properties:

  1. 1.

$f$ is analytic in $\mathbb{R}$.

  2. 2.

$f(0)=0$.

Thus, we consider that for any function in $\mathcal{F}_{Z}$, the Taylor
series at 0 converges to the function for all $x\in\mathbb{R}$. Since the
power series is unique, there is no loss of generality by considering the
Taylor series at 0 and not centered at a different point in $\mathbb{R}$.

Notice that the class $\mathcal{F}_{Z}$ encompasses numerous nonlinear
functions [18], including polynomial functions which are used for the
approximation of continuous functions through the Weierstrass Approximation
theorem [19]. Also, there is no loss of generality with the class
$\mathcal{F}_{Z}$ since the results are valid for the variable part of the
functions.

###### Lemma 2

For identifiability in the class $\mathcal{F}_{Z}$, the measurement of a node
provides the identification of all the incoming edges of the node.

###### Proof:

Let us consider a node $i$ with $m$ incoming edges. The output of the node $i$
is given by:

$y_{i}^{k}=u_{i}^{k-1}+F_{i}^{k}(u_{1}^{k-2},\ldots,u_{m}^{k-2},\ldots),$ (8)

where $F_{i}^{k}$ is determined by the set of functions $\\{f\\}$ associated
with the edges of the network. Let us assume that there exists another set of
functions $\\{\tilde{f}\\}\neq\\{f\\}$ such that:

$F_{i}^{k}(u_{1}^{k-2},\ldots,u_{m}^{k-2},\ldots)=\tilde{F}_{i}^{k}(u_{1}^{k-2},\ldots,u_{m}^{k-2},\ldots),$

where $\tilde{F}_{i}^{k}$ is composed by the functions in the set
$\\{\tilde{f}\\}$. Let us choose a point $(u_{j}^{k-2},0,\ldots,0)$ with
$j=1,\ldots,m$, such that all the inputs are set to zero except one of the
inputs of the incoming edges of the node $i$. Then, since each function is in
$\mathcal{F}_{Z}$ and by Lemma 1 we have:

$F_{i}^{k}=f_{i,j}(u_{j}^{k-2})\quad\text{and}\quad\tilde{F}_{i}^{k}=\tilde{f}_{i,j}(u_{j}^{k-2}).$

Since we assume that $F_{i}^{k}=\tilde{F}_{i}^{k}$, it yields:

$f_{i,j}(u_{j}^{k-2})=\tilde{f}_{i,j}(u_{j}^{k-2}),\text{ for all
}u_{j}^{k-2}\in\mathbb{R},$

which implies that $f_{i,j}=\tilde{f}_{i,j}$. Following a similar argument for
each incoming edge of node $i$, we prove that all the functions associated
with the incoming edges of the node $i$ are unique and can be identified. ∎

Notice that due to other possible paths from an in-neighbor $j$ of the node
$i$, additional terms of the form $u_{j}^{k-r},r>2$ could appear in (8).
However, they will always be delayed by virtue of Lemma 1.

The following lemmas involve properties of analytic functions that will be
used in the proof of the results in this section.

###### Lemma 3

(Periodic functions) If for some $p_{0}\in\mathbb{R}$, an analytic function
$f:\mathbb{R}\to\mathbb{R}$ is periodic for periods
$p\in[p_{0}-\epsilon,p_{0}+\epsilon]$ with $\epsilon>0$, then the function $f$
is constant.

###### Proof:

The proof is left to Appendix -A. ∎

###### Lemma 4

Given three non-zero analytic functions $f:\mathbb{R}\to\mathbb{R}$ and
$g,\tilde{g}:\mathbb{R}^{m}\to\mathbb{R}$ satisfying $g(0)=\tilde{g}(0)=0$. If
for all $x\in\mathbb{R}$, $y\in\mathbb{R}^{m}$, the functions $f$, $g$ and
$\tilde{g}$ satisfy:

$f(x+g(y_{1},\ldots,y_{m}))=f(x+\tilde{g}(y_{1},\ldots,y_{m})),$

then either $g=\tilde{g}$ or $f$ is constant.

###### Proof:

The proof is left to Appendix -B. ∎

###### Corollary 1

Under the same conditions as in Lemma 4, if $f(0)=0$, then

$g=\tilde{g}.$

######  Proposition 3 (Paths)

For identifiability of a path graph in the class $\mathcal{F}_{Z}$, it is
necessary and sufficient to measure the sink.

###### Proof:

Let us consider a path with $n$ nodes. The measurement of the sink gives us
the output:

$\displaystyle y_{n}^{k}$ $\displaystyle=u_{n}^{k-1}+F_{n}^{k}$
$\displaystyle=u_{n}^{k-1}+f_{n,n-1}(u_{n-1}^{k-2}+F_{n-1}^{k-1}).$ (9)

Let us assume that there is a set $\\{\tilde{f}\\}\neq\\{f\\}$ such that
$F_{n}^{k}=\tilde{F}_{n}^{k}$, which by (9) implies:

$f_{n,n-1}(u_{n-1}^{k-2}+F_{n-1}^{k-1})=\tilde{f}_{n,n-1}(u_{n-1}^{k-2}+\tilde{F}_{n-1}^{k-1}).$

By Lemma 2, we can guarantee that $f_{n,n-1}=\tilde{f}_{n,n-1}$, and we have:

$f_{n,n-1}(u_{n-1}^{k-2}+F_{n-1}^{k-1})=f_{n,n-1}(u_{n-1}^{k-2}+\tilde{F}_{n-1}^{k-1}).$

Then, we use Corollary 1 to guarantee that
$F_{n-1}^{k-1}=\tilde{F}_{n-1}^{k-1}$. Notice that now the identifiability
problem is equivalent to having measured the node $n-1$ and by following a
similar approach, we can continue with the identification of all the edges and
guarantee that $\\{f\\}=\\{\tilde{f}\\}$, such that all the path can be
identified. ∎

######  Proposition 4 (Trees)

For identifiability of a tree in the class $\mathcal{F}_{Z}$, it is necessary
and sufficient to measure all the sinks.

###### Proof:

From Proposition 1, it is necessary to measure all the sinks. Let us consider
an arbitrary tree and the measurement of a sink $i$. Let us assume that there
are $m$ in-neighbors of the sink $i$ and there is a set
$\\{\tilde{f}\\}\neq\\{f\\}$ such that $F_{i}^{k}=\tilde{F}_{i}^{k}$, which
implies:

$\sum_{\ell=1}^{m}f_{i,\ell}(u_{\ell}^{k-2}+F_{\ell}^{k-1})=\sum_{\ell=1}^{m}\tilde{f}_{i,\ell}(u_{\ell}^{k-2}+\tilde{F}_{\ell}^{k-1}).$
(10)

Since in a tree, the functions $F_{\ell}^{k-1}$ do not have common inputs
because they come from different branches, we can select a in-neighbor $j$ and
set to zero the inputs of all the nodes that do not have a path to $j$, such
that we have:

$f_{i,j}(u_{j}^{k-2}+F_{j}^{k-1})=\tilde{f}_{j,j}(u_{j}^{k-2}+\tilde{F}_{j}^{k-1}).$

Then, by using Lemma 2 and Corollary 1 we can guarantee that
$f_{i,j}=\tilde{f}_{i,j}$ and $F_{j}^{k-1}=\tilde{F}_{j}^{k-1}$ for all
$j=1,\ldots,m$, which is equivalent to having measured the in-neighbors of
$i$. Then, we can continue with the identification of each branch
independently and by following the same approach we can identify all the paths
that finish in the sink $i$. Finally, by measuring the other sinks and
following a similar approach, we can identify all the edges in the tree. ∎

######  Remark 1 (Linear functions)

Notice that Propositions 3 and 4 are also valid if all or some of the edges in
the network contain pure linear functions. In the next section, we will
provide stronger results in the identification of nonlinear networks when
linear functions are excluded.

##  IV Directed acyclic graphs

Directed acyclic graphs encompass a large number of graph topologies that
present specific characteristics that can be used for the derivation of
conditions for identifiability [20]. Unlike a tree, in a directed acyclic
graph, the functions $F_{\ell}^{k-1}$ in (10) can have common variables due to
several possible paths between two nodes with the same length, which makes
impossible the application of Corollary 1. In order to obtain a result similar
to Corollary 1 that allow us to identify a directed acyclic graph, we consider
a smaller class of functions.

######  Definition 5 (Class of functions $\mathcal{F}_{Z,NL}$)

Let $\mathcal{F}_{Z,NL}$ be the class of functions $f:\mathbb{R}\to\mathbb{R}$
with the following properties:

  1. 1.

$f$ is analytic in $\mathbb{R}$.

  2. 2.

$f(0)=0$.

  3. 3.

The associated Taylor series $f(x)=\sum_{n=1}^{\infty}a_{n}x^{n}$ contains at
least one coefficient $a_{n}\neq 0$ with $n>1$.

The third property of the functions in $\mathcal{F}_{Z,NL}$ implies that none
of the functions is linear.

Clearly $\mathcal{F}_{Z,NL}$ is a subclass of $\mathcal{F}_{Z}$ and all the
results of the previous section for functions in $\mathcal{F}_{Z}$ are also
valid for functions in $\mathcal{F}_{Z,NL}$.

###### Lemma 5

Given the non-zero analytic functions $f_{i}:\mathbb{R}\to\mathbb{R}$ and
$g_{i},\tilde{g}_{i}:\mathbb{R}^{m}\to\mathbb{R}$ satisfying
$f_{i}(0)=g_{i}(0)=\tilde{g}_{i}(0)=0$ for $i=1,\ldots,n$. Let us assume that
none of the functions $f_{i}$ is linear. If for all $x\in\mathbb{R}$,
$y\in\mathbb{R}^{m}$, the functions $f_{i}$, $g_{i}$ and $\tilde{g}_{i}$
satisfy:

$\sum_{i=1}^{n}f_{i}(x_{i}+g_{i}(y_{1},\ldots,y_{m}))=\sum_{i=1}^{n}f_{i}(x_{i}+\tilde{g}_{i}(y_{1},\ldots,y_{m})),$

then $g_{i}=\tilde{g}_{i}$ for all $i=1,\ldots,n$.

###### Proof:

The proof is left to Appendix -C. ∎

Notice that when $n=1$, Lemma 5 is also covered by Corollary 1.

###### Proposition 5

For the functions in $\mathcal{F}_{Z,NL}$, in a directed acyclic graph, the
measurement of a node provides the identification of all the nonlinear
functions of any path that finishes in the measured node.

###### Proof:

Let us assume an arbitrary directed acyclic graph. The measurement of a node
$i$ provides an output of the type:

$\displaystyle y_{i}^{k}$ $\displaystyle=u_{i}^{k-1}+F_{i}^{k}$
$\displaystyle=u_{i}^{k-1}+\sum_{j=1}^{m}f_{i,j}(u_{j}^{k-2}+F_{j}^{k-1}),$

where $m$ is the number of in-neighbors of $i$. Let us assume that there is a
set $\\{\tilde{f}\\}\neq\\{f\\}$ such that $F_{i}^{k}=\tilde{F}_{i}^{k}$,
which implies:

$\sum_{j=1}^{m}f_{i,j}(u_{j}^{k-2}+F_{j}^{k-1})=\sum_{j=1}^{m}\tilde{f}_{i,j}(u_{j}^{k-2}+\tilde{F}_{j}^{k-1}),$

By applying Lemma 2 we have $f_{i,j}=\tilde{f}_{i,j}$ for all $j=1,\ldots,m$
and:

$\sum_{j=1}^{m}f_{i,j}(u_{j}^{k-2}+F_{j}^{k-1})=\sum_{j=1}^{m}f_{i,j}(u_{j}^{k-2}+\tilde{F}_{j}^{k-1}),$

and by using Lemma 5 we guarantee:

$F_{j}^{k-1}=\tilde{F}_{j}^{k-1}\quad\text{for all }j=1,\ldots,m.$

Notice that the identification of each $F_{j}^{k-1}$ is equivalent to having
measured the node $j$ and can be treated in a similar way to the node $i$,
independently of other paths corresponding to the other in-neighbors of $i$.
By following a similar approach, we can guarantee that
$\\{f\\}=\\{\tilde{f}\\}$ for every path that ends in the node $i$. ∎

######  Theorem 1 (Directed acyclic graph)

For identifiability of a directed acyclic graph in the class
$\mathcal{F}_{Z,NL}$, it is necessary and sufficient to measure all the sinks.

###### Proof:

From Proposition 1, it is necessary to measure all the sinks. In a directed
acyclic graph, we can always find a path from any node $i$ to some sink [21].
Therefore, according to Proposition 5, it is sufficient to measure the sinks
to identify all the paths in a directed acyclic graph. ∎

Unlike the linear case, where the measurement of the sinks is not enough to
guarantee identifiability of directed acyclic graphs [5], Theorem 1 provides
weaker conditions for the identifiability in the nonlinear case when linear
functions are excluded.

1243$f_{2,1}$$f_{4,2}$$f_{3,1}$$f_{4,3}$ Figure 3: Nonlinear network that can
be identified by only measuring the sink. In the linear case, the measurement
of the sink is not enough to identify this network.

######  Example 3 (Directed acyclic graph)

Let us consider the graph in Fig. 3. In the linear case, this network cannot
be identified by only measuring the sink since the functions $f_{2,1}$ and
$f_{3,1}$ cannot be distinguished. However, in the nonlinear case, the
measurement of the sink is enough to identify all the network.

##  V Conclusions and future work

We have derived identifiability conditions for a network characterized by
nonlinear interactions through a static model. We showed that in a path graph
it is necessary to measure all the nodes, except by the source, when the
nonlinear functions have a static component. Then, by restricting the
identifiability problem to a specific class of functions, we showed that the
measurement of the sinks is necessary and sufficient to identify all the edges
in paths and trees. Finally, by considering a smaller class of functions, we
showed that the measurement of the sinks is necessary and sufficient for the
identifiability of directed acyclic graphs. This simple model of nonlinear
interactions allowed us to highlight fundamental differences with respect to
the linear case.

For future work, it would be interesting to extend the results to the case of
general digraphs with cycles where the function $F_{i}^{k}$ in (2) depends on
an infinite number of inputs. Also, it would be important to consider
dynamical models that include past inputs. In this case, the Volterra series
seems to be the more adequate model since it only depends on past inputs, and
many of our results could still hold.

## References

  * [1] F. Bullo, _Lectures on Network Systems_ , 1.6 ed. Kindle Direct Publishing, 2022.
  * [2] K. R. Ramaswamy, P. M. J. Van den Hof, and A. G. Dankers, “Generalized sensing and actuation schemes for local module identification in dynamic networks,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_ , 2019, pp. 5519–5524.
  * [3] X. Bombois, K. Colin, P. M. J. Van den Hof, and H. Hjalmarsson, “On the informativity of direct identification experiments in dynamical networks,” _Automatica_ , vol. 148, p. 110742, 2023.
  * [4] E. M. M. Kivits and P. M. J. Van den Hof, “Identifiability of diffusively coupled linear networks with partial instrumentation,” in _22nd World Congress of the International Federation of Automatic Control (IFAC 2023 World Congress)_ , 2023.
  * [5] J. M. Hendrickx, M. Gevers, and A. S. Bazanella, “Identifiability of dynamical networks with partial node measurements,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 6, pp. 2240–2253, 2019.
  * [6] A. Legat and J. M. Hendrickx, “Local network identifiability with partial excitation and measurement,” in _2020 59th IEEE Conference on Decision and Control (CDC)_ , 2020, pp. 4342–4347.
  * [7] ——, “Path-based conditions for local network identifiability,” in _2021 60th IEEE Conference on Decision and Control (CDC)_ , 2021, pp. 3024–3029.
  * [8] A. S. Bazanella, M. Gevers, and J. M. Hendrickx, “Network identification with partial excitation and measurement,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_ , 2019, pp. 5500–5506.
  * [9] X. Cheng, S. Shi, I. Lestas, and P. M. J. Van den Hof, “A necessary condition for network identifiability with partial excitation and measurement,” _IEEE Transactions on Automatic Control_ , 2023.
  * [10] F. Dörfler and F. Bullo, “Synchronization in complex networks of phase oscillators: A survey,” _Automatica_ , vol. 50, no. 6, pp. 1539–1564, 2014\.
  * [11] W. Pan, Y. Yuan, J. Gonçalves, and G.-B. Stan, “Reconstruction of arbitrary biochemical reaction networks: A compressive sensing approach,” in _2012 IEEE 51st IEEE Conference on Decision and Control (CDC)_. IEEE, 2012, pp. 2334–2339.
  * [12] A. Aalto, L. Viitasaari, P. Ilmonen, L. Mombaerts, and J. Gonçalves, “Gene regulatory network inference from sparsely sampled noisy data,” _Nature communications_ , vol. 11, no. 1, p. 3493, 2020.
  * [13] A. Bizyaeva, A. Franci, and N. E. Leonard, “Nonlinear opinion dynamics with tunable sensitivity,” _IEEE Transactions on Automatic Control_ , vol. 68, no. 3, pp. 1415–1430, 2023.
  * [14] A. Janczak, _Identification of Nonlinear Systems Using Neural Networks and Polynomial Models_. Springer, 2004.
  * [15] O. Nelles, _Nonlinear System Identification_. Springer, 2020.
  * [16] J. Paduart, L. Lauwers, J. Swevers, K. Smolders, J. Schoukens, and R. Pintelon, “Identification of nonlinear systems using polynomial nonlinear state space models,” _Automatica_ , vol. 46, no. 4, pp. 647–656, 2010.
  * [17] C. C. Aggarwal, “Neural Networks and Deep Learning,” _Springer_ , 2018.
  * [18] M. Abramowitz and I. Stegun, _Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables_ , ser. Applied mathematics series. Dover Publications, 1965.
  * [19] J. G. Llavona, _Approximation of Continuously Differentiable Functions_. North-Holland Mathematics Studies, 1986.
  * [20] E. Mapurunga, M. Gevers, and A. S. Bazanella, “Excitation and measurement patterns for the identifiability of directed acyclic graphs,” in _2022 IEEE 61st Conference on Decision and Control (CDC)_ , 2022, pp. 1616–1621.
  * [21] J. Bang-Jensen and G. Z. Gutin, _Digraphs: Theory, Algorithms and Applications_. Springer Science & Business Media, 2008.
  * [22] S. G. Krantz and H. R. Parks, _A Primer of Real Analytic Functions_. Springer Science & Business Media, 2002.

###  -A Proof of Lemma 3

Let us choose an arbitrary point $\hat{x}$. Since $f$ is periodic for
$p\in[p_{0}-\epsilon,p_{0}+\epsilon]$, we have that for all
$y\in[\hat{x}-\epsilon+p_{0},\hat{x}+\epsilon+p_{0}]$:

$f(y)=f(\hat{x}),$

which implies that the function $f$ is constant and its derivative
$f^{\prime}$ is zero in $[\hat{x}-\epsilon+p_{0},\hat{x}+\epsilon+p_{0}]$.
Since $f$ is analytic, its derivative $f^{\prime}$ is also analytic and due to
the Principle of isolated zeros for 1-dimensional real analytic functions
[22], the derivative $f^{\prime}$ is zero for all $x$, which implies that $f$
is constant for all $x$.

###  -B Proof of Lemma 4

Let us assume that there exists a point $\hat{y}\in\mathbb{R}^{m}$ such that
$g(\hat{y})=a$ and $\tilde{g}(\hat{y})=b$ with $a\neq b$. Then, we would have:

$f(x+a)=f(x+b),\text{ for all }x\in\mathbb{R},$

which is equivalent to

$f(z)=f(z+b-a),\text{ for all }z\in\mathbb{R},$

implying that $f$ is periodic with period $b-a$. Since $g(0)=\tilde{g}(0)=0$,
and $g$ and $\tilde{g}$ are continuous, the function $\tilde{g}-g$ is also
continuous and all the values between 0 and $b-a$ belong to its range. Thus,
$f$ should be periodic in the interval $[0,b-a]$. But by virtue of Lemma 3,
the function $f$ should be constant. If $f$ is not constant, we have a
contradiction which implies that $g=\tilde{g}$.

###  -C Proof of Lemma 5

Let us take the derivative with respect to only one variable $x_{j}$ where
$j=1,\ldots,n$. Then, we have:

$f^{\prime}_{j}(x_{j}+g_{j}(y))=f^{\prime}_{j}(x_{j}+\tilde{g}_{j}(y)),\text{
for all }x_{j}\in\mathbb{R},\text{ }y\in\mathbb{R}^{m}.$

Since the function $f_{j}$ is analytic, its derivative $f^{\prime}_{j}$ is
also analytic, and by Lemma 4, we can have two cases: either $f^{\prime}_{j}$
is constant or $g_{j}=\tilde{g}_{j}$. If $f^{\prime}_{j}$ is constant, then
$f_{j}$ is linear or constant (i.e., $f_{j}=0$), which is a contradiction.
Therefore $g_{j}=\tilde{g}_{j}$. Following the same procedure for the other
variables $x_{j}$, we complete the proof.