# Topological Reconstruction of Particle Physics Processes using Graph Neural
Networks

Lukas Ehrkea<EMAIL_ADDRESS>Département de physique nucléaire et
corpusculaire, University of Geneva, Switzerland  John Andrew Rainea
<EMAIL_ADDRESS>Département de physique nucléaire et corpusculaire,
University of Geneva, Switzerland  Knut Zoch  Département de physique
nucléaire et corpusculaire, University of Geneva, Switzerland  Laboratory for
Particle Physics and Cosmology, Harvard University, Cambridge, 02138 MA, USA
Manuel Guth  Département de physique nucléaire et corpusculaire, University of
Geneva, Switzerland  Tobias Golling  Département de physique nucléaire et
corpusculaire, University of Geneva, Switzerland

###### Abstract

We present a new approach, the Topograph, which reconstructs underlying
physics processes, including the intermediary particles, by leveraging
underlying priors from the nature of particle physics decays and the
flexibility of message passing graph neural networks. The Topograph not only
solves the combinatoric assignment of observed final state objects,
associating them to their original mother particles, but directly predicts the
properties of intermediate particles in hard scatter processes and their
subsequent decays. In comparison to standard combinatoric approaches or modern
approaches using graph neural networks, which scale exponentially or
quadratically, the complexity of Topographs scales linearly with the number of
reconstructed objects.

We apply Topographs to top quark pair production in the all hadronic decay
channel, where we outperform the standard approach and match the performance
of the state-of-the-art machine learning technique.

††footnotetext: These authors contributed equally to this work

##  I Introduction

At the Large Hadron Collider (LHC) at CERN, beams of protons are collided
together at incredibly high energies to probe the underlying nature of the
universe. From the objects recorded by the detectors on the LHC, the
underlying physics processes in the collisions are attempted to be
reconstructed.

In processes with a high multiplicity of objects, reconstructing intermediate
particles, such as the two top quarks and $W$ bosons in top quark pair
production, is a crucial and challenging component in analyses [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. This is of particular interest
when measuring the mass of an intermediary particle, or its kinematic
properties as part of a cross section measurement.

In this work we introduce Topographs, a new machine learning approach for
reconstructing the full hypothesised decay chain from observed objects. It
leverages state-of-the-art machine learning (ML) with underlying priors from
particle physics, through the nature of particle decays, to reconstruct
underlying physics processes including the intermediary particles. The
computational complexity of Topographs scales linearly with object
multiplicity, whereas alternative methods scale quadratically [18, 19] or
exponentially [20, 21].

We apply Topographs to reconstruct the underlying processes in the production
of top quarks pairs in the all hadronic decay channel. However, the
architecture can be applied to any particle physics process and is not limited
to event level reconstruction.

##  II Motivation

In the case of top quark pair production, with each top quark decaying
hadronically $t\rightarrow Wb\rightarrow qq^{\prime}b$, each quark is expected
to initiate a shower in the detector which is reconstructed as a jet,
resulting in six jets. The number of combinations to match the reconstructed
jets to quarks from the top quark pair system is computationally intractable
and grows exponentially with additional jets reconstructed in the final state,
even when taking underlying symmetries into account. Solving the combinatorics
of this system is a key challenge in the measurement of the top quark and
$t\bar{t}$ pair, and one which is often computationally limited. The
combinatorics can be restricted by looking at event topologies where the high
momentum of the top quarks results in all three quarks being reconstructed in
a single large radius jet, however this restricts the phase space to such
topologies which represent only a small fraction of all events.

##  III Current Approaches

In top quark physics kinematic event reconstruction forms a key part of many
measurements. The $\chi^{2}$ method [4] and the Kinematic Likelihood fitter
(KLFitter) [20] have been employed in a large number of analyses [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In both approaches, all combinatorics of
jet matching to final state quarks and gluons (partons) in the $t\bar{t}$
final state are tested with kinematic constraints based on the masses of the
reconstructed $W$ bosons and top quarks as minimisation criteria. In the case
of KLFitter these are used in conjunction with with transfer functions and the
particle decay widths.

Although good performance can be achieved with such an approach, as the number
of jets in an event increases, as well as the multiplicity of final state
objects to be reconstructed, the number of combinations increases
exponentially. For example, in an all hadronic top quark pair event with 6
jets, there are 720 potential combinations. This is reduced to 90 by
exploiting underlying symmetries. However for events with 7 jets the
combinations increase to 630, and for 8 jets they increase further to 2520.
Furthermore, as the exact values of the mass of the top quark and $W$ boson
are used to test the likelihood of a combination, this leads to a biased
estimator which focusses on assigning jets which together are closest to the
hypothesised particles mass, rather than exploiting all the information about
the pairs or triplets of objects. It also assumes that in all cases the top
quarks and $W$ bosons are on-shell.

Building on the previous combinatoric approaches, simple approaches using
machine learning (ML) have been developed. Instead of finding the most
probable assignment using just the masses of intermediary particles, machine
learning discriminants are used to identify correct assignments, exploiting
more information from the event [21]. Nonetheless, these approaches still
suffer from the same problems as the KLFitter and $\chi^{2}$ methods, with
each combination needing to be tested to identify the most likely.

Another approach which uses more information from the event is the Matrix
Element Method (MEM) [22, 23, 15, 21]. The MEM not only attempts to match
objects to the final state objects in an event, but directly assesses the
likelihood of observing an event given the matrix element for a process. This
can be evaluated for each potential combination with the highest resulting
probability chosen as the correct assignment. However, it is extremely slow
and computationally intensive. To calculate the likelihood of an event, an
integral over the whole phase space of possible final state particle momenta
must be performed. It is also reliant on a transfer function, which is used to
convert the jets, charged leptons and missing transverse momentum recorded by
the detector to the partons, charged leptons and neutrinos before any
hadronisation and detector effects. As there is no accurate function to model
this, it is at best an approximation optimised by hand. Normalising flows
present a solution to the computational challenge and approximate functions
[24], however do not yet address the combinatoric solving.

### State of the art

The state of the art machine learning approach uses attention transformers
[25, 19, 26] to identify the indices of final state objects coming from
intermediate particles. In this approach no graph structure is used and only
the permutation invariant collection of objects are considered. The complexity
of the approach can be reduced by taking into account the symmetries, as
performed in Refs. [19, 26] (SPA-Net), corresponding to removing potential
solutions in the combinatoric approaches, which leads to an overall complexity
of $\mathcal{O}\left(N^{2}\right)$.

Graph Neural Networks [27, 28] are also employed in HEP to associate objects
to a common origin, for example in secondary vertex reconstruction [18] and
could similarly be applied to combinatoric solving at the event level. These
approaches have fully connected graphs with $N(N-1)$ edges.

In addition to their reduced computational complexity in comparison to
traditional approaches, both attention and GNN approaches also demonstrate
reduction in biases towards particle masses, as often seen in the combinatoric
approaches. However, in both GNNs and SPA-Net the target is to identify the
two triplets of objects which correspond to the decay of each top quark,
neglecting the structure of the decay, and the properties of the intermediary
particles.

Other approaches employ physics inspired layers in order to assign parton
labels [29] or try to predict the properties of intermediate particles
directly [30].

##  IV The Topograph

The use of GNNs in high energy physics applications [31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44] is a recent development which is gaining in
popularity. However, graphs have also long been used to describe underlying
processes occurring in particle physics in the form of Feynman diagrams.

In a Feynman diagram, the vertices (nodes) represent the interactions between
particles and the edges represent the particles themselves. This
representation can equally be converted into a node-and-edge graph by
representing the particles as nodes, and defining edges based on the vertices.
An example of the Feynman digram and one such graph for top quark pair
production is shown in Fig. 1. In current graph based approaches this physical
inspired graph representation is not exploited, and instead fully connected
graphs are constructed from all objects recorded by the detector.

(a) Feynman diagram

(b) Node and edge graph

Figure 1: Two graph representations of top pair production, with one top
decaying semi-leptonically and the other hadronically. The production
mechanism in 1(a) is shown to be gluon fusion however in 1(b) it is
represented by the hashed circle. Time flows from left to right in both
graphs.

We introduce a new method, called the Topograph, which builds upon the
structure seen in Fig. 1(b) to define neural networks which can be used to
predict the correct edges from final state particles and the properties of
intermediate particles.

Firstly, the intermediate particles in a chosen physics process are injected
into the graph. Secondly, instead of connecting all objects to one another,
they are instead connected to all of their potential mother particles. From
this new graph, instead of identifying the edges based on a shared origin,
true edges are identified as being between a mother and daughter particle, and
the result is the reconstruction of the Feynman diagram as depicted in Fig. 1.

Secondly, as the intermediate particles are now represented by their own
nodes, during training the kinematic properties of the injected nodes are
predicted as auxiliary tasks with dedicated regression networks for each
particle type. Edges in GNNs are not only used to identify connections but
also to propagate information. By utilising the edges of the Topograph as
message passing layers to update the properties of the injected nodes,
properties of the intermediary particles are extracted from the graph and
regression networks can predict their kinematic properties. This is
advantageous both through improvements in the edge classification, but also
from the additional extracted information.

One leading advantage of Topographs over fully connected GNNs can be seen in
Fig. 2, showing the simple case of identifying the jets from the decay of a
single top quark. In comparison to the fully connected GNN, which has
$N\left(N-1\right)$ edges, the Topograph only requires
$\mathcal{O}\left(N\right)$, which scales linearly with the number of
intermediary particles; in this case there are $2N$ edges. For cases where
$N>M+1$, where $M$ is the number of intermediary particles, there are always
fewer edges associated to reconstructed objects in a Topograph than a fully
connected GNN.

(a) Fully connected graph

(b) Topograph

Figure 2: Comparison of the edges in 2(a) fully connected GNNs and 2(b)
Topographs, when identifying the three jets $j$ which originate from a top
quark decay. The blue nodes in 2(b) represent the injected nodes, a key
component of the Topograph, for both the top quark ($t$) and $W$ boson ($W$).
The true edges, which identify jets which 2(a) originate from the same top
quark or 2(b) reconstruct the decay chain, are green. All false edges in the
graphs are dashed red lines. The predefined connection between $t$ and $W$
boson is blue.

Furthermore, having a handle on the underlying kinematics of the intermediary
particles with auxiliary tasks should improve the resolution and accuracy of
current differential measurements, which are often a function of intermediate
particle kinematics. These properties could also be used in a likelihood test
of the process, as is done in the MEM, instead of only using objects recorded
by the detector.

With Topographs, complex underlying physics processes can be injected as
priors by changing the injected particles and their potential connections.
This enables additional information to be included when designing and training
the networks over standard approaches.

### Building blocks

Although the Topograph could be visualised as one large graph with injected
nodes and edges, we break them down into a simple set of building blocks. The
core building block of the Topograph is the particle block. It includes the
edge definitions between an input set of particles, or nodes, and the target
mother particle. In addition, it contains the regression network used to
predict the kinematic properties of the injected mother particle. If a
Topograph is visualised as a Feynman diagram of a process, a particle block is
the subcomponent which determines the correct connections to recreate a single
vertex alongside the properties of the incoming particle. The basic
representation of a particle block for a mother particle $M$ is depicted in
Fig. 3 alongside a representative Feynman diagram vertex. The injected mother
particle $M$ can be initialised with random values, or using information
extracted from all potential daughter particles. Its properties are learned
from message passing layers between itself and the input particles.

Figure 3: The particle block for a given mother particle $M$. All input
particles are connected to the mother particle by an edge; for illustrative
purposes the true edges are shown in green and the false edges are represented
by dashed red lines. The trapezoid $M_{\mathit{reg}}$ is the regression
network which predicts the kinematic properties of $M$. The inset is the
notation used to represent the whole particle block. Shown alongside the
Feynman diagram representing the process, with time running from bottom to
top.

Any hypothesised process can be described by combining multiple particle
blocks into a single network, connecting them to the input particles and to
one another. Edges between objects and particle blocks can also be predefined,
for example between the particle blocks for a $W$ boson and a top quark, as
shown in Fig. 4.

Figure 4: Particle block of a top quark $t$. A particle block for the $W$ is
nested within the $t$ block. A connection between the $W$ boson and the top
quark is predefined (shown in blue). Shown alongside the corresponding Feynman
diagram.

### Assembling a neural network with Topographs

For many processes, there will be more than one reconstructed object type or
final state particle in the chosen process. To address this a set of neural
networks $\phi_{p}$, one for each particle type $p$, can be incorporated into
the Topograph model as a series of particle embedding networks. Furthermore,
in order to maximise initial information exchange before the Topograph it may
be beneficial to include a normal message passing layer before the Topograph.
Options for this layer include attention transformers and standard message
passing GNN layers. Furthermore, in complex processes it is possible to define
which edges are to be predicted and which are fixed. For example, two leptons
in the production of a $Z$ boson in association with two top quarks could be
set to originate from the $Z$ boson, or one from each $W$ boson.

##  V Solving combinatorics in $\mathrm{t\bar{t}}$ events

For an initial application and for direct comparison with other state-of-the-
art methods we apply Topographs to top quark pair production with both tops
decaying hadronically. We compare the performance of our method to a benchmark
non-ML approach used in many top quark analyses, the $\chi^{2}$ method, and
the state-of-the-art ML approach, SPA-Net [26]. All models are trained and
evaluated on the same dataset.

20 million $t\bar{t}$ events with a centre of mass energy $\sqrt{s}=13$ TeV
are simulated using MadGraph5_aMC@NLO [45] (v3.1.0), with decays of top quarks
and $W$ bosons modelled with MadSpin [46], with both $W$ bosons decaying to
two quarks.111Dataset available at https://zenodo.org/record/7737248 [47] The
parton shower and hadronisation is performed with Pythia [48] (v8.243). The
detector response is simulated using Delphes [49] (v3.4.2) with a
parametrisation similar to the response of the ATLAS detector. Jet clustering
is performed using the anti-$k_{t}$ algorithm [50] with a radius parameter
$R=0.4$ using the FastJet [51] package. Jets originating from $b$-quarks
($b$-jets) are identified with a simple binary discriminant corresponding to
an inclusive 70% signal efficiency.

For training, events with at least six jets are selected, keeping up to 16
jets per event as ordered by their transverse momentum. Truth matching of jets
to the partons in the hard scatter is performed using a cone of $\Delta
R<0.4$. Events with partons matched to multiple jets, or jets matched to
multiple partons are discarded. Events are further required to have zero
reconstructed leptons, though no requirement on the number of $b$-jets.
Finally, events are required to be fully reconstructable, where jets are
matched to all six partons of the $t\bar{t}$ decay. After these requirements
there are 1,340,000 training events and 71,000 validation events.

An additional 298,000 events are reserved for evaluating the final
performance. These also contain events where not all partons have a jet
matched to them. In 76,000 of these events it is possible to associate a jet
to all six partons. After requiring at least 2 $b$-jets in the event, there
are 147,000 events of which 44,000 are fully reconstructable.

As inputs for both ML models the four momentum $(p_{x},p_{y},p_{z},E)$
together with a boolean flag showing whether the jet was $b$-tagged or not are
used. The three momenta are normalised subtracting the mean and dividing by
the standard deviation. The energy is normalised in the same way after
applying the logarithm. No normalisation or transformation is applied to the
$b$-tagging flag. As both models represent the single jets as nodes,
additional information per jet can easily be added for both models as
additional node features.

###  V.1 Topograph implementation

Before the particle blocks for the two top quarks, two fully connected
message-passing graph layers are used to provide an information exchange
between the jets in the event and update the jet features. From these updated
jets, the $W$ nodes are initialised using attention weighted pooling. Two
different networks are used to obtain two sets of attention weights, one for
each of the $W$ nodes. The top nodes are initialised in the same way from the
jets but with the corresponding $W$ node concatenated to the pooled jet
information. The regression targets of the $W$ and top nodes are the truth
level properties of the particles. In both cases the three momentum
$(p_{x},p_{y},p_{z})$ is chosen as the regression target. In our dataset the
truth mass is fixed to the Monte Carlo mass values. The network structure is
shown in Fig. 5.

Figure 5: Topograph network for the $t\bar{t}$ process comprising two $t$
blocks. The jets are passed through an information exchange layer, using a
fully connected graph message passible layer. All jets are connected to all
possible mother particles, as shown by the dashed edges. The information
exchange comprises multiple message passing graph layers and the node updates
with the Topograph are performed $N$ times before the edge values are used for
parton assignment and the properties for the top quarks and $W$ bosons are
extracted.

Within both the information exchange and Topograph blocks, message passing is
bidirectional with a separate edge for each direction. Shared weights are used
for calculating the attention pooling weights for each category of edge,
defined by the sending and receiving particle type. Edge features are formed
by concatenating the features of the sending node and the receiving node. Edge
features are persistent between updates, and after the first iteration the
current values are concatenated to the new features.

Four message passing steps in the Topograph update the jets, edges and
injected $W$ and top nodes. After the final message passing step, the
properties of the $W$ bosons and top quarks and particle matching scores are
extracted. There are four matching scores for each jet, from which six jets
are assigned to the six partons. The scores are calculated by passing the edge
properties of the jet to $W$ and top node through a classification network to
determine if it is a true edge.

#### Loss function

The loss function is calculated from both the edge classification task and the
regression tasks. Edges are classified as true if a jet originated from the
$b$-quark, in the case of the top nodes, or one of the two quarks from the
decay of the $W$ boson, for the $W$ nodes. Here, binary cross-entropy is used
for the loss function, and it is weighted using the number of true and false
edges across the dataset in order to improve convergence during training. For
each regression task, the mean absolute error (MAE) function is used between
the predicted values of the node and the true values of the top quark or $W$
boson.

Since it is not possible for the network to determine which top node
corresponds to the top quark or the anti-top quark, a symmetrised version of
the losses is calculated in which the loss is calculated for both possible
cases. The loss is then defined as the minimum of the two. Since the $W$ boson
nodes are directly connected to the top nodes, the loss terms for the $W$
bosons are included in the loss calculation without need for additional
symmetrisation. The full loss is given by

$\displaystyle\mathcal{L}_{\mathrm{symm}}=\min($
$\displaystyle\mathcal{L}(t_{1},t)+\mathcal{L}(W_{1},W^{+})$
$\displaystyle+\mathcal{L}(t_{2},\bar{t})+\mathcal{L}(W_{2},W^{-}),$
$\displaystyle\mathcal{L}(t_{1},\bar{t})+\mathcal{L}(W_{1},W^{-})$
$\displaystyle+\mathcal{L}(t_{2},t)+\mathcal{L}(W_{2},W^{+})),$

where $\mathcal{L}(p_{i},p)$ corresponds to the combined edge classification
loss and regression loss for each of the injected particles $p_{i}\in t_{1}$,
$t_{2}$, $W_{1}$ and $W_{2}$, with respect to the truth particle $p\in t$,
$\bar{t}$, $W^{+}$, $W^{-}$.

#### Parton assignment

Several options could be tested for assigning jets to the partons for the edge
score. For this initial study a simple iterative approach is chosen. First,
the edge with the highest score is labelled as a true edge, with the jet
assigned to this parton. Next all edges connected to the corresponding jet and
parton are removed, and the next highest edge is chosen. In the all hadronic
$t\bar{t}$ case, there is one parton per top quark, corresponding to the
$b$-quarks in the decay, and two per $W$ boson. This assignment procedure is
repeated until all six partons are assigned, and results in a solution where
neither a jet or parton can be assigned to multiple targets.

###  V.2 Reference methods

####  $\chi^{2}$ implementation

The $\chi^{2}$ score for $t\bar{t}$ decays used in this work is given by

$\displaystyle\chi^{2}=\frac{(m_{b_{1}q_{1}q_{2}}-m_{t})^{2}}{\sigma_{t}^{2}}+\frac{(m_{b_{2}q_{3}q_{4}}-m_{t})^{2}}{\sigma_{t}^{2}}$
$\displaystyle+\frac{(m_{q_{1}q_{2}}-m_{W})^{2}}{\sigma_{W}^{2}}+\frac{(m_{q_{3}q_{4}}-m_{W})^{2}}{\sigma_{W}^{2}},$

where $m_{b_{i}q_{j}q_{k}}$ is the invariant mass of the jets in the
permutation, $m_{t}$ and $m_{W}$ are the masses of the top quark and W boson,
and $\sigma_{t}$ and $\sigma_{W}$ are the widths of the top quark and W boson
in the dataset. The values for $m_{t}$ and $m_{W}$ are obtained by taking the
mean of the reconstructed invariant masses in our dataset, while $\sigma_{t}$
and $\sigma_{W}$ are set to the standard deviation. The calculated values are
$m_{t}=169.8\ \mathrm{GeV}$, $m_{W}=81.0\ \mathrm{GeV}$, $\sigma_{t}=29.0\
\mathrm{GeV}$, and $\sigma_{W}=18.5\ \mathrm{GeV}$. Jets associated to the
$b$-quark of the top decays are required to be $b$-tagged, while no
requirement is placed on the jets associated to the $W$ boson decays.

#### SPA-Net implementation

The implementation is taken from the github
repository222https://github.com/Alexanders101/SPANet/tree/v1.0. The
hyperparameters of the model are optimised for the dataset using fully matched
events, with the values in Ref. [19] using _v1.0_ resulting in the highest
overall efficiencies. All SPA-Net models are trained for 100 epochs, taking
the weights after the epoch with the lowest loss on the validation set.

###  V.3 Partial event trainings

It is also possible to train Topographs on events in which not all partons
from the $t\bar{t}$ decay are matched to jets. For these events, the same
network is used but the edge classification and regression loss terms are not
considered for the $W$ boson or top quarks which have partons not able to be
matched to jets. In the case of the $b$-quark from the top quark decay not
being matched to a jet, the $\mathcal{L}(W_{i},W^{+})$ term is still
considered. Where a quark from the $W$ boson decay is missing, both the top
quark and $W$ boson loss terms are not considered. At least one $W$ boson is
required to be fully reconstructable, with both quarks matched to jets in the
event.

As introduced in Ref. [26], SPA-Net can also be trained on non-fully
reconstructable events, however in comparison to Topographs this is only at
the level of each top quark. When a $b$-quark from the top quark is missing,
the corresponding $W$ boson is also not considered. For a fairer comparison
with both models trained on the same number of events, we compare models
trained only on fully reconstructable events. Results for the Topograph and
SPA-Net models trained with partial events are presented in Section A.3.

##  VI Results

Table 1: Event reconstruction efficiencies (%) for the $\chi^{2}$ method, the SPA-Net model and our Topograph model in different jet and $b$-jet multiplicities. The mean and standard deviation are calculated using five trainings, and the best model corresponds to the training with the highest efficiency for events with at least six jets and exactly two $b$-jets. The highest efficiency is highlighted in bold. |  | Mean | Best
---|---|---|---
$N_{jets}$ |  $N_{b-\mathrm{jets}}$ |  SPA-Net |  Topograph |  SPA-Net |  Topograph |  $\chi^{2}$
$\geq$$6$ |  $\geq$$2$ |  $81.43\pm 0.03$ |  $\mathbf{81.54\pm 0.12}$ |  $81.47\pm 0.31$ |  $\mathbf{81.70\pm 0.31}$ |  $72.73\pm 0.36$
$\geq$$6$ |  $\geq 2$ |  $79.45\pm 0.07$ |  $\mathbf{79.76\pm 0.14}$ |  $79.56\pm 0.30$ |  $\mathbf{79.95\pm 0.29}$ |  $70.94\pm 0.33$
$\geq$$7$ |  $\geq$$2$ |  $64.47\pm 0.08$ |  $\mathbf{65.46\pm 0.20}$ |  $64.69\pm 0.44$ |  $\mathbf{65.90\pm 0.44}$ |  $54.28\pm 0.46$
$\geq$$7$ |  $\geq 2$ |  $62.38\pm 0.12$ |  $\mathbf{63.51\pm 0.25}$ |  $62.86\pm 0.40$ |  $\mathbf{63.94\pm 0.40}$ |  $52.11\pm 0.41$
$\geq 6$ |  $\geq$$2$ |  $68.48\pm 0.06$ |  $\mathbf{69.12\pm 0.19}$ |  $68.65\pm 0.25$ |  $\mathbf{69.54\pm 0.25}$ |  $58.57\pm 0.26$
$\geq 6$ |  $\geq 2$ |  $65.76\pm 0.08$ |  $\mathbf{66.59\pm 0.22}$ |  $66.05\pm 0.23$ |  $\mathbf{67.03\pm 0.22}$ |  $55.90\pm 0.24$

For the evaluation of the performance only events with at least 2 $b$-tagged
jets are considered. This is a common requirement in physics analysis to
reduce the contribution from the multi-jet background. This requirement is not
imposed during training as it is found to reduce the performance as a result
of the reduction in training statistics.

###  VI.1 Jet parton assignment

Although Topographs predict the kinematics of injected particles, the most
important measure of performance is the association of jets to partons.
Efficiencies of reconstructing the whole event correctly are shown in Table 1
for varying jet and $b$-jet multiplicities. Only events which are fully
reconstructable are considered. Topographs and SPA-Net show very similar
performance across all jet multiplicities, with sub-percent differences in the
reconstruction efficiencies. Both clearly outperform the $\chi^{2}$ method by
around $10\%$ with larger differences at higher jet multiplicities.

Figure 6: Reconstructed $W$ boson invariant mass $m_{W}$ using jets assigned by the Topograph (solid), SPA-Net (dashed), and the $\chi^{2}$ method (dash-dot). Events are categorised into correct (green) and incorrect assignments (orange), with events missing a parton at reconstruction level labelled as impossible (blue). Figure 7: Reconstructed top quark invariant mass $m_{top}$ using jets assigned by the Topograph (solid), SPA-Net (dashed), and the $\chi^{2}$ method (dash-dot). Events are categorised into correct (green) and incorrect assignments (orange), with events missing a parton at reconstruction level labelled as impossible (blue).  Table 2: Percentage of events with at most $N$ incorrectly matched partons using the $\chi^{2}$ method, SPA-Net and Topographs. Events are categorised based on the number of partons matched to jets at truth level. The highest efficiency is highlighted in bold. $N_{partons}$ |  | Incorrectly matched partons [%]
---|---|---
matched |  Model |  0 |  $\leq 1$ |  $\leq 2$ |  $\leq 3$ |  $\leq 4$ |  $\leq 5$
3 |  SPA-Net |  $34.34\pm 0.65$ |  $74.88\pm 0.59$ |  $98.03\pm 0.19$ |  $100.00\pm 0.00$ |  - |  -
|  Topograph |  $\mathbf{36.57\pm 0.66}$ |  $77.80\pm 0.57$ |  $98.70\pm 0.15$ |  $100.00\pm 0.00$ |  - |  -
|  $\chi^{2}$ |  $34.30\pm 0.65$ |  $\mathbf{80.12\pm 0.54}$ |  $\mathbf{99.05\pm 0.13}$ |  $100.00\pm 0.00$ |  - |  -
4 |  SPA-Net |  $35.69\pm 0.28$ |  $64.53\pm 0.28$ |  $90.99\pm 0.17$ |  $99.41\pm 0.04$ |  $100.00\pm 0.00$ |  -
|  Topograph |  $\mathbf{37.74\pm 0.28}$ |  $\mathbf{67.06\pm 0.27}$ |  $\mathbf{92.21\pm 0.16}$ |  $99.53\pm 0.04$ |  $100.00\pm 0.00$ |  -
|  $\chi^{2}$ |  $31.01\pm 0.27$ |  $63.84\pm 0.28$ |  $92.19\pm 0.16$ |  $\mathbf{99.73\pm 0.03}$ |  $100.00\pm 0.00$ |  -
5 |  SPA-Net |  $44.74\pm 0.19$ |  $64.87\pm 0.18$ |  $86.92\pm 0.13$ |  $97.91\pm 0.05$ |  $99.94\pm 0.01$ |  $100.00\pm 0.00$
|  Topograph |  $\mathbf{46.57\pm 0.19}$ |  $\mathbf{66.88\pm 0.18}$ |  $\mathbf{88.58\pm 0.12}$ |  $\mathbf{98.45\pm 0.05}$ |  $\mathbf{99.96\pm 0.01}$ |  $100.00\pm 0.00$
|  $\chi^{2}$ |  $32.36\pm 0.18$ |  $58.27\pm 0.19$ |  $84.70\pm 0.14$ |  $98.21\pm 0.05$ |  $99.94\pm 0.01$ |  $100.00\pm 0.00$
6 |  SPA-Net |  $66.05\pm 0.23$ |  $74.43\pm 0.21$ |  $90.06\pm 0.14$ |  $96.22\pm 0.09$ |  $99.52\pm 0.03$ |  $1$$99.98\pm 0.01$
|  Topograph |  $\mathbf{67.03\pm 0.22}$ |  $\mathbf{75.71\pm 0.20}$ |  $\mathbf{91.33\pm 0.13}$ |  $\mathbf{97.03\pm 0.08}$ |  $\mathbf{99.70\pm 0.03}$ |  $\mathbf{100.00\pm 0.00}$
|  $\chi^{2}$ |  $55.90\pm 0.24$ |  $64.93\pm 0.23$ |  $84.14\pm 0.17$ |  $93.97\pm 0.11$ |  $99.43\pm 0.04$ |  $1$$99.98\pm 0.01$

Figures 6 and 7 show the reconstructed $W$ boson and top quark invariant
masses. The distributions are shown separately for events with the correct and
incorrect jet assignments, as well as events which are not possible to be
fully reconstructed due to not all partons being matched to reconstructed jets
(impossible). Events where the correct triplet is assigned to the top quark
but with the incorrect matching account for 5% of the incorrect events for
Topographs and SPA-Net, and 3% for $\chi^{2}$. All methods are able to
reconstruct candidates closest to the particle mass, with a broader
distribution for events without the corresponding partons matched to jets in
the event. Normalised and stacked distributions can be found in Section A.6.
SPA-Net and Topographs have very similar distributions showing no difference
in potential sculpting of the mass distributions. The $\chi^{2}$ method has
distributions which are shifted towards slightly higher values compare to SPA-
Net and Topographs.

For events which do not contain all partons in the final state, it is of
interest to see how many of the partons are correctly matched to jets. In
Table 2 the matching efficiencies of only the partons which are present in the
event are compared for the three approaches. The Topograph performs slightly
better than SPA-Net, with higher efficiencies for correctly matching all
available partons or only one incorrectly matched parton. Both Topographs and
SPA-Net are substantially better than the $\chi^{2}$ at correctly identifying
all available partons. It should be noted that the perfect reconstruction in
this case is substantially lower than the fully reconstructable events in
Table 1.

In Table 3 we break down the matching efficiencies of jets in incorrect events
by the truth flavour of the jet. The jet truth flavour is assigned using
$\Delta R$ matching to $B$\- and $C$-hadrons. We observe that Topographs and
SPA-Net have similar jet matching efficiencies for all jet flavours.
Topographs have a slightly higher efficiency for $b$-jets, which correspond to
the jets coming from the $b$-quarks in the top quark decays as well as in
matching light jets to the quarks in the $W$ boson decay. The $\chi^{2}$
method has the largest efficiency drop for truth flavour $c$-jets. This
results from $c$-jets being mis-identified by the $b$-tagging, which in the
$\chi^{2}$ method allows them to be matched to the $b$-quarks from the top
decays.

Table 3: Efficiencies (%) of correctly associating a jet to its corresponding parton for the $\chi^{2}$ method, SPA-Net and Topographs. Jets are categorised into their truth flavour using parton matching. Only jets coming from the $t\bar{t}$ decays in events with at least one incorrect assignment are considered. The highest efficiency is highlighted in bold. Correct |  SPA-Net |  Topograph |  $\chi^{2}$
---|---|---|---
$b$-jets |  $58.94\pm 0.29$ |  $\mathbf{59.99\pm 0.29}$ |  $57.68\pm 0.25$
$c-$jets |  $55.08\pm 0.39$ |  $\mathbf{55.81\pm 0.39}$ |  $50.38\pm 0.34$
light jets |  $70.42\pm 0.22$ |  $\mathbf{72.01\pm 0.22}$ |  $68.51\pm 0.20$
Table 4: Efficiencies (%) of correctly associating a jet to the corresponding parton for events with at least one incorrect assignment for the $\chi^{2}$ method, SPA-Net and Topographs. Top quarks are classified by the momentum of the corresponding $b$-quark. The quarks from the $W$ boson decays are ordered by their $p_{\mathrm{T}}$. The highest efficiency is highlighted in bold.  |  Correctly identified |  SPA-Net |  Topograph |  $\chi^{2}$
---|---|---|---|---
top quark (leading $b$) |  $b$-quark |  $61.83\pm 0.40$ |  $\mathbf{62.94\pm 0.40}$ |  $59.17\pm 0.35$
|  Leading quark ($W$) |  $67.79\pm 0.38$ |  $\mathbf{68.92\pm 0.39}$ |  $61.80\pm 0.35$
|  Subleading quark ($W$) |  $63.22\pm 0.40$ |  $\mathbf{64.70\pm 0.40}$ |  $65.09\pm 0.34$
top quark (subleading $b$) |  $b$-quark |  $56.05\pm 0.41$ |  $\mathbf{57.03\pm 0.41}$ |  $56.20\pm 0.36$
|  Leading quark ($W$) |  $67.49\pm 0.38$ |  $\mathbf{68.22\pm 0.39}$ |  $61.30\pm 0.35$
|  Subleading quark ($W$) |  $66.41\pm 0.39$ |  $\mathbf{68.31\pm 0.39}$ |  $65.94\pm 0.34$

The matching efficiencies of partons in incorrect events are shown in Table 4.
Here we see that the $\chi^{2}$ method mostly performs worse for matching jets
to the quarks in the $W$ boson decay. Topographs and SPA-Net show similar
behaviour, with the slightly better performance in Topographs for incorrect
events evenly distributed across all partons.

###  VI.2 Interpreting edge scores

Due to the individual edge scores, the confidence of the combinatoric
assignment from Topographs can be obtained by aggregating the edge scores.
This could be used to filter events as likely to be incorrect, as well as
identify events for which it is not possible to match jets to all partons.

Figure 8: Event matching score for events with the correct assignment (green),
incorrect assignment (orange) and non fully-reconstructable events
(impossible, blue). Figure 9: Event matching score for non-fully
reconstructable events, categorised by the number of correctly assigned jets.

In Fig. 8 the product of assigned jet edge scores is used as an event matching
score. There is reasonable separation between events with a correct assignment
and those with the incorrect and impossible assignments. However, a shoulder
towards higher values is visible for the impossible events.

In Fig. 9 the impossible events are categorised into the number of jets which
are correctly assigned. It can be seen that the score is still high for events
where there is a single parton missing, but all remaining partons are
correctly matched to jets. Whether using this score to remove events will
benefit other down-stream applications, such as top quark mass measurements
would need to be tested.

###  VI.3 Regression performance

Instead of reconstructing the top quark and $W$ boson kinematics from the
matched jets, Topographs are trained to predict their properties directly.
Figure 10 compares the resolution of the $W$ boson $p_{\mathrm{T}}$ and Fig.
11 the top quark $p_{\mathrm{T}}$ from the Topograph regression networks and
the invariant system of the assigned jets. The predictions are shown for both
correct and incorrect jet assignments.

Figure 10: Resolution of the reconstructed $p_{\mathrm{T}}$ of the $W$ boson
from the Topograph. Comparing the prediction from the invariant system of the
assigned jets (solid line) and the Topograph regression network (dashed line)
for correct assigned events (green) and incorrect assigned events (orange).
Figure 11: Resolution of the reconstructed $p_{\mathrm{T}}$ of the top quark
from the Topograph. Comparing the prediction from the invariant system of the
assigned jets (solid line) and the Topograph regression network (dashed line)
for correct assigned events (green) and incorrect assigned events (orange).

For events with all jets correctly assigned, using the $p_{\mathrm{T}}$ values
from the Topograph regression networks leads to a narrower peak than the
reconstructed quantities, demonstrating a more accurate prediction. They also
show a reduced bias compared to the values reconstructed solely using the
jets. For events with incorrectly assigned jets, no difference is observed
between the two predictions. This shows that the Topograph is not learning to
guess the $p_{\mathrm{T}}$ from the training data, but instead is using the
associated jets. Accurate reconstruction of the intermediate particles has a
strong dependence on whether the model has correctly assigned jets to partons.
There are no biases observed using either method for the top quark and $W$
boson, and There is very little difference for reconstruction of the top quark
$\eta$ and $\phi$ coordinates. However reconstructing the $W$ boson $\eta$ and
$\phi$ coordinates using the assigned jets performs better than the prediction
from the network. In all cases, the Topograph is trained to predict the
momentum components $p_{x},p_{y},p_{z}$ rather than
$p_{\mathrm{T}},\eta,\phi$.

##  VII Conclusion

In this work we have introduced Topographs, a novel approach for solving the
combinatorics and reconstructing the topology of a particle physics process
from final state objects reconstructed by a detector. The performance matches
the current state-of-the-art technique using symmetry preserving attention
transformers, and surpasses the standard approach commonly used in analyses,
with a computational complexity which scales only linearly with increasing
final state object multiplicity.

The edge scores from Topographs can be combined into discriminants to assign a
confidence to the jet-parton assignments, which could be useful in downstream
applications. Furthermore, the additional regression tasks included in
Topographs demonstrate good predictive power with similar accuracy but reduced
bias compared to using only the jets assigned to intermediate particles.
However, in both cases there remains room for improvement.

There are several other areas open for further optimisation. Due to the
message passing layers used to define the Topograph, it was found that fully
connected graph layers between all jets for information exchange lead to
faster convergence whilst training, and also resulted in requiring fewer
learnable parameters in the model. This causes the complexity of the network
in this work to scale quadratically with the number of jets, and not linearly
as can be achieved by only using the particle blocks in Topographs. By moving
to other architectures such as Transformer Encoders with cross-attention, the
need for information exchange layers could be mitigated. Alternative
approaches for assigning jets to partons based on the edge scores could also
improve the overall performance.

Applications of Topographs are not limited to the case study presented in this
paper, and due to their modular nature Topographs can be generalised to almost
all particle physics processes. Their applications are also not limited to
matching final state objects to an underlying physics process, but could also
find use in jet identification, reconstructing displaced vertices from heavy
flavour hadron decays or using the constituents in large radius jets in
boosted topologies.

## Acknowledgements

The authors would like to acknowledge funding through the SNSF Sinergia grant
called "Robust Deep Density Models for High-Energy Particle Physics and Solar
Flare Analysis (RODEM)" with funding number CRSII$5\\_193716$, the SNSF
project grant 200020_212127 called "At the two upgrade frontiers: machine
learning and the ITk Pixel detector", and the Alexander von Humboldt
foundation Feodor Lynen fellowship programme.

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##  Appendix A Appendix

###  A.1 Hyperparameters

Table 5 shows the hyper parameters used for the training of the Topograph
models presented in this paper. The models were trained using Tensorflow v2.10
[52].

Table 5: Hyper parameters used for the training of Topographs Hyper parameter | Value
---|---
Optimizer | AdamW
Epochs | 100
n original message passing | 2
n topograph updates | 4
lr schedule | cosine annealing
initial learning rate | 0.001
decay steps | 2 epochs
batch size | 256
pooling | attention
W initialisation | attention pooling
Top initialisation | attention pooling
attention units | [32, 32, 1]
activation | gelu
normalisation | layer norm
regression units | [64, 64, 3]
edge classification units | [128, 128, 128, 1]
graph processing units | [256, 256, 64]
persistent edges | true
classification loss | weighted binary crossentropy
regression loss | mean absolute error

###  A.2 Example Topograph

A complete Topograph network is shown in Fig. 12 for reconstruction of
$t\bar{t}W$ in events containing exactly one lepton, one neutrino and multiple
jets. For the production of a top quark pair in association with a $W$ boson
($t\bar{t}W$), two top quark blocks and one $W$ block are connected to the
input particles, but not to one another. The Topograph network is trained to
identify the edges of the true daughter particles of each particle in the
process, and predicts the kinematics of the two top quarks and the three $W$
bosons. As Topographs are defined to represent an underlying physics process,
it is also a choice whether to predefine whether the lepton and neutrino
originate from a $W$ boson coming from a top decay, or the additional $W$
boson. Figure 13 shows a Topograph where the lepton is required to come from a
top decay.

Figure 12: Topograph network for the $t\bar{t}W$ process comprising two $t$
blocks and a $W$ block. The input particles are first passed through an
embedding network $\phi$ unique to each type of particle; either a jet $j$,
lepton $\ell$ or neutrino $\nu$. Embedded particles are then passed through an
(optional) information exchange layer. All particles are connected to all
possible mother particles, as shown by the dashed edges.

Figure 13: Topograph network for the $t\bar{t}W$ process comprising two $t$
blocks and a $W$ block where the $\ell$ and $\nu$ are predefined to connect to
the $W$ in one top block.

###  A.3 Comparisons with partial event trainings

Instead of training using only complete events, both the Topograph and SPA-Net
models can be trained including partial events, that is, events where some
partons are not able to be matched to reconstructed jets. In the training
events, for Topographs at least one $W$ boson, and for SPA-Net at least one
top quark are required to be fully reconstructable. For these models the
efficiencies of fully reconstructable events change slightly, with Topographs
having a slight reduction in efficiencies and SPA-Net a slight increase, as
shown in Table 6. However, for most selections, the efficiencies are still
within the uncertainties of models trained only on fully reconstructable
events. Table 7 shows the parton matching efficiencies for events categorised
based on how many partons can be matched to reconstructed jets. Here the
benefit of training on partial events can be seen, with both Topographs and
SPA-Net having higher efficiencies of correctly matching jets to the all
available partons.

Table 6: Event reconstruction efficiencies (%) for the $\chi^{2}$ method, the SPA-Net model and our Topograph model in different jet and $b$-jet multiplicities. Both models were trained on complete and partial events. The highest efficiency is highlighted in bold. $N_{jets}$ |  $N_{b-\mathrm{jets}}$ |  SPA-Net |  Topograph |  $\chi^{2}$
---|---|---|---|---
$\geq$ $6$ |  $\geq$ $2$ |  $\mathbf{81.03\pm 0.32}$ |  $80.77\pm 0.32$ |  $72.73\pm 0.36$
$\geq$ $6$ |  $\geq 2$ |  $\mathbf{79.11\pm 0.30}$ |  $78.93\pm 0.30$ |  $70.94\pm 0.33$
$\geq$ $7$ |  $\geq$ $2$ |  $\mathbf{65.35\pm 0.44}$ |  $65.18\pm 0.44$ |  $54.28\pm 0.46$
$\geq$ $7$ |  $\geq 2$ |  $\mathbf{63.40\pm 0.40}$ |  $63.13\pm 0.40$ |  $52.11\pm 0.41$
$\geq 6$ |  $\geq$ $2$ |  $\mathbf{68.93\pm 0.25}$ |  $68.75\pm 0.25$ |  $58.57\pm 0.26$
$\geq 6$ |  $\geq 2$ |  $\mathbf{66.33\pm 0.23}$ |  $66.21\pm 0.23$ |  $55.90\pm 0.24$
Table 7: Percentage of events with at most $N$ incorrectly matched partons using the $\chi^{2}$ method, SPA-Net and Topographs. The SPA-Net model was trained on complete and partial events. Events are categorised based on the number of partons matched to jets at truth level. The highest efficiency is highlighted in bold. $N_{partons}$ |  | Incorrectly matched partons [%]
---|---|---
matched |  Model |  0 |  $\leq 1$ |  $\leq 2$ |  $\leq 3$ |  $\leq 4$ |  $\leq 5$
3 |  SPA-Net |  $36.68\pm 0.66$ |  $76.94\pm 0.57$ |  $98.55\pm 0.16$ |  $100.00\pm 0.00$ |  - |  -
|  Topograph |  $\mathbf{37.27\pm 0.66}$ |  $77.89\pm 0.57$ |  $98.77\pm 0.15$ |  $100.00\pm 0.00$ |  - |  -
|  $\chi^{2}$ |  $34.30\pm 0.65$ |  $\mathbf{80.12\pm 0.54}$ |  $\mathbf{99.05\pm 0.13}$ |  $100.00\pm 0.00$ |  - |  -
4 |  SPA-Net |  $37.78\pm 0.28$ |  $66.29\pm 0.28$ |  $91.78\pm 0.16$ |  $99.46\pm 0.04$ |  $100.00\pm 0.00$ |  -
|  Topograph |  $\mathbf{38.08\pm 0.28}$ |  $\mathbf{67.53\pm 0.27}$ |  $\mathbf{92.57\pm 0.15}$ |  $99.62\pm 0.04$ |  $100.00\pm 0.00$ |  -
|  $\chi^{2}$ |  $31.01\pm 0.27$ |  $63.84\pm 0.28$ |  $92.19\pm 0.16$ |  $\mathbf{99.73\pm 0.03}$ |  $100.00\pm 0.00$ |  -
5 |  SPA-Net |  $48.14\pm 0.19$ |  $67.24\pm 0.18$ |  $88.18\pm 0.12$ |  $98.14\pm 0.05$ |  $99.95\pm 0.01$ |  $100.00\pm 0.00$
|  Topograph |  $\mathbf{48.22\pm 0.19}$ |  $\mathbf{69.27\pm 0.18}$ |  $\mathbf{89.52\pm 0.12}$ |  $\mathbf{98.68\pm 0.04}$ |  $\mathbf{99.96\pm 0.01}$ |  $100.00\pm 0.00$
|  $\chi^{2}$ |  $32.36\pm 0.18$ |  $58.27\pm 0.19$ |  $84.70\pm 0.14$ |  $98.21\pm 0.05$ |  $99.94\pm 0.01$ |  $100.00\pm 0.00$
6 |  SPA-Net |  $\mathbf{66.33\pm 0.23}$ |  $74.38\pm 0.21$ |  $89.86\pm 0.14$ |  $96.18\pm 0.09$ |  $99.54\pm 0.03$ |  $100.00\pm 0.00$
|  Topograph |  $66.20\pm 0.23$ |  $\mathbf{74.87\pm 0.21}$ |  $\mathbf{91.11\pm 0.14}$ |  $\mathbf{97.04\pm 0.08}$ |  $\mathbf{99.73\pm 0.02}$ |  $100.00\pm 0.00$
|  $\chi^{2}$ |  $55.90\pm 0.24$ |  $64.93\pm 0.23$ |  $84.14\pm 0.17$ |  $93.97\pm 0.11$ |  $99.43\pm 0.04$ |  $1$$99.98\pm 0.01$

###  A.4 Impact of systematic variations

For applications in high energy physics, it is crucial that any new approach
is not sensitive to changes under systematic variations. In particular with
machine learning approaches, it would be problematic if methods were sensitive
to underlying and non-physical effects arising from the simulated samples on
which they were trained. Other sources of variation come from differences in
the calibration or reconstruction of physics objects between simulation and
data.

To test the dependence on the simulated samples used to train the Topograph,
we evaluate the best performing model trained on the nominal MadGraph data on
an alternative independent dataset. This alternative sample consists of all-
hadronic $t\bar{t}$ events simulates both the hard interactions and parton
shower are with Pythia8 (v8.307), using the Monash tuned set of parameters
[53] at leading order accuracy.

To test the dependence on reconstruction effects, we apply a shift or gaussian
smearing to the energy of reconstructed jets in the events.

The absolute change in performance arising from the systematic variations is
summarised in Table 8. The impact is compared for Topographs, SPA-Net, and
$\chi^{2}$. Evaluating on the alternative sample results in a slightly reduced
overall efficiency for all three approaches. This effects Topographs and SPA-
Net slightly more than $\chi^{2}$, however, the overall gain in performance
remains similar. Both Topographs and SPA-Net are robust under systematic
shifts or reduced jet energy resolution, whereas the $\chi^{2}$ method suffers
from a substantial drop in efficiency, especially at higher jet
multiplicities.

Table 8: Difference in reconstruction efficiencies when evaluating the different methods on systematic variations. The difference is taken with respect to Table 1. The systematic variations include: evaluating on a dataset produced with a Pythia for the matrix element generation and the parton shower, evaluating on the nominal data set scaling all jet energies by 2.5%, and evaluating on the nominal data set smearing all jet energies by 5%. |  | Pythia | Jet Scale | Jet Resolution
---|---|---|---|---
$N_{jets}$ |  $N_{b-\mathrm{jets}}$ |  SPA-Net |  Topograph |  $\chi^{2}$ |  SPA-Net |  Topograph |  $\chi^{2}$ |  SPA-Net |  Topograph |  $\chi^{2}$
$\geq$ $6$ |  $\geq$ $2$ |  $-2.13$ |  $-2.00$ |  $-0.94$ |  $+0.04$ |  $\pm 0.00$ |  $\vphantom{1}-2.82$ |  $+0.08$ |  $\pm 0.00$ |  $-12.79$
$\geq$ $6$ |  $\geq 2$ |  $-2.28$ |  $-2.13$ |  $-1.33$ |  $+0.07$ |  $+0.03$ |  $\vphantom{1}-3.63$ |  $+0.04$ |  $+0.01$ |  $-14.30$
$\geq$ $7$ |  $\geq$ $2$ |  $-2.04$ |  $-2.14$ |  $-1.66$ |  $\pm 0.00$ |  $\pm 0.00$ |  $-12.30$ |  $-0.05$ |  $-0.06$ |  $-21.52$
$\geq$ $7$ |  $\geq 2$ |  $-2.36$ |  $-2.37$ |  $-1.62$ |  $-0.02$ |  $+0.04$ |  $-12.69$ |  $-0.10$ |  $-0.07$ |  $-21.63$
$\geq 6$ |  $\geq$ $2$ |  $-1.64$ |  $-1.53$ |  $-0.72$ |  $+0.04$ |  $+0.01$ |  $\vphantom{1}-8.23$ |  $+0.02$ |  $+0.02$ |  $-17.18$
$\geq 6$ |  $\geq 2$ |  $-1.86$ |  $-1.89$ |  $-0.94$ |  $+0.06$ |  $+0.05$ |  $\vphantom{1}-9.02$ |  $-0.01$ |  $+0.03$ |  $-17.96$

###  A.5 Studying edge scores

Fig. 14 shows the distribution of edge scores of the jets to the $W$ helper
node which is decided to be the $W^{+}$ based on the loss. Fig. 14(a) includes
all jets in the event. The distribution for the jets which originate from the
$W^{+}$ peak at one, whereas the distributions of all other jet types peak at
zero. A small peak at one can be seen for the $b$-jets originating from the
top quark. To investigate this peak, the scores of the $b$-jets from both the
top and the anti-top quark are shown in Fig. 14(b). They are further split
based on their $b$-tagging score. The small peak at one originates from
$b$-jets from the top quark which are not $b$-tagged. Furthermore, the impact
of the $b$-tagging on the score can be seen. The $b$-jets from the anti-top
quark which are not $b$-tagged have on average higher scores than the $b$-jets
from the top quark which are $b$-tagged. This could point to the $b$-tagging
result being more important for the association to the $W$ nodes than
kinematics.

Fig. 15 shows the same distributions but to the top helper node which is
decided to be the top quark. Again, good separation of the true edges from the
false edges can be seen. Considering only $b$-jets and splitting them based on
their $b$-tagging score, it can be seen, that the $b$-tagged jets originating
from the anti-top quark have on average lower scores than the non $b$-tagged
jets originating from the top quark. So, the $b$-tagging decision is not as
important for the association to the top helper nodes.

This reliance on the $b$-tagging result is not unexpected. With a $b$-tagging
efficiency of $70\%$, around $30\%$ of the $b$-jets will not be tagged as a
$b$-jet, leaving a large contribution of true edges to the top helper nodes
which are not $b$-tagged. The observed mis-tag efficiency is around $5\%$.
Therefore, only a small fraction of the true edges to the $W$ helper nodes is
$b$-tagged.

Figs. 16 and 17 show the same plots for the $W^{-}$ and anti-top. No
qualitative differences can be observed to the plots for the $W^{+}$ and the
top.

(a)

(b)

Figure 14: Edge scores of jets to the helper node representing the $W^{+}$.
The decision which $W$ node is the $W^{+}$ is taken by choosing the minimum of
the loss under both hypotheses. 14(a) shows all jets, while 14(b) only shows
the $b$-jets from the two tops. They are further split into whether the jet
was tagged as a $b$-jet or not. No requirement is placed on the number of
$b$-tags.

(a)

(b)

Figure 15: Edge scores of jets to the helper node representing the $t$. The
decision which $t$ node is the $t$ is taken by choosing the minimum of the
loss under both hypotheses. 15(a) shows all jets, while 15(b) only shows the
$b$-jets from the two tops. They are further split into whether the jet was
tagged as a $b$-jet or not. No requirement is placed on the number of
$b$-tags.

(a)

(b)

Figure 16: Edge scores of jets to the helper node representing the $W^{-}$.
The decision which $W$ node is the $W^{-}$ is taken by choosing the minimum of
the loss under both hypotheses. 16(a) shows all jets, while 16(b) only shows
the $b$-jets from the two tops. They are further split into whether the jet
was tagged as a $b$-jet or not. No requirement is placed on the number of
$b$-tags.

(a)

(b)

Figure 17: Edge scores of jets to the helper node representing the $\bar{t}$.
The decision which $t$ node is the $\bar{t}$ is taken by choosing the minimum
of the loss under both hypotheses. 17(a) shows all jets, while 17(b) only
shows the $b$-jets from the two tops. They are further split into whether the
jet was tagged as a $b$-jet or not. No requirement is placed on the number of
$b$-tags.

###  A.6 Additional figures

Figures 18 and 19 show the regression results for the $\eta$ and $\phi$
coordinate. The difference between the value of the predicted $\eta$ or $\phi$
and the true parton property is shown. For the predicted properties, the
parton is either reconstructed from the jets that the model associates to the
parton or the regression result is taken. ‘Correct’ and ‘incorrect’ events are
shown in separate distributions. For ‘incorrect’ events an additional
prediction is shown by taking the true jets to reconstruct the parton
properties. For ‘correct’ events, the regression has for both quantities a
wider distribution than the model association. For ‘incorrect’ events, the
regression and the model association perform worse, however using the true
jets also has a worse resolution for these events. Especially for the $W$
boson, the regression and the model association have a worse performance.
However, for the top quark ‘incorrect’ labelled events can still contain the
correct three jets but with a wrong assignment by switching one of the
$W$-jets with the $b$-jet.

(a)

(b)

Figure 18: Resolution of the reconstructed $W$ boson $\eta$ and $\phi$
coordinates from the Topograph. Comparing the prediction from the invariant
system of the assigned jets (solid line) and the Topograph regression network
(dashed line) for correct assigned events (green) and incorrect assigned
events (orange).

(a)

(b)

Figure 19: Resolution of the reconstructed top quark $\eta$ and $\phi$
coordinates from the Topograph. Comparing the prediction from the invariant
system of the assigned jets (solid line) and the Topograph regression network
(dashed line) for correct assigned events (green) and incorrect assigned
events (orange).

Figure 20 shows the distribution of the reconstructed $W$ mass split into
‘correct’, ‘incorrect’, and ‘impossible’ events as stacked histograms as an
alternative representation compared to Fig. 6. Similarly, Fig. 21 is an
alternative representation of Fig. 7 showing the reconstructed top quark mass.

(a)

(b)

(c)

Figure 20: Stacked distributions of the reconstructed $m_{W}$ using 20(a)
Topographs, 20(b)SPA-Net, and 20(c) $\chi^{2}$. The single histograms split
the candidates into ‘correct’, ‘incorrect’, and ‘impossible’ candidates.

(a)

(b)

(c)

Figure 21: Stacked distributions of the reconstructed $m_{top}$ using 21(a)
Topographs, 21(b)SPA-Net, and 21(c) $\chi^{2}$. The single histograms split
the candidates into ‘correct’, ‘incorrect’, and ‘impossible’ candidates.

Figures 22 and 23 show the same distributions for the reconstructed $W$ and
top quark mass, respectively, but every distribution is normalised to unity.

(a)

(b)

(c)

Figure 22: Normalised distributions of the reconstructed $m_{W}$ using 22(a)
Topographs, 22(b)SPA-Net, and 22(c) $\chi^{2}$. The single histograms split
the candidates into ‘correct’, ‘incorrect’, and ‘impossible’ candidates.

(a)

(b)

(c)

Figure 23: Normalised distributions of the reconstructed $m_{top}$ using 23(a)
Topographs, 23(b)SPA-Net, and 23(c) $\chi^{2}$. The single histograms split
the candidates into ‘correct’, ‘incorrect’, and ‘impossible’ candidates.

Figure 24 shows the same distributions but with the different models combined
in single plots.

(a)

(b)

Figure 24: Distributions of the reconstructed 24(a) $m_{W}$ and 24(b)
$m_{top}$. Each histogram is normalised to an area of one. The solid lines
show the distributions for the Topograph, the dashed lines show the
distributions for the SPA-Net, and the dashed-dotted lines show the
distributions for the $\chi^{2}$ method. The different colours show the
different types of events based on the assignment of the model: correct,
incorrect, impossible.