Topological Reconstruction of Particle Physics Processes using Graph Neural Networks

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.


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 this work we introduce Topographs, a new machine learning approach for reconstructing the full hypothesised decay chain from observed objects.It lever- * lukas.ehrke@unige.ch† john.raine@unige.cha These authors contributed equally to this work ages 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 → W b → qq ′ 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 arXiv:2303.13937v5[hep-ph] 13 Oct 2023 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 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 χ 2 method [4] and the Kinematic Likelihood fitter (KLFitter) [20] have been employed in a large number of analyses [1-14].In both approaches, all combinatorics of jet matching to final state quarks and gluons (partons) in the t 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 KL-Fitter 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 χ 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) [15,[21][22][23].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 [19,25,26] 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 O N 2 .
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 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.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.
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.

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 ϕ 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 t 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 χ 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 t events with a centre of mass energy √ s = 13 TeV are simulated using Mad-Graph5_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. 1 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 1 Dataset available at https://zenodo.org/record/7737248[47] 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 ∆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 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 momen- where L(p i , p) corresponds to the combined edge classification loss and regression loss for each of the injected particles p i ∈ t 1 , t 2 , W 1 and W 2 , with respect to the truth particle p ∈ t, 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 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.

B. Reference methods
The χ 2 score for t t decays used in this work is given by where m biqj q k is the invariant mass of the jets in the

SPA-Net implementation
The implementation is taken from the github repository 2 .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.

C. Partial event trainings
It is also possible to train Topographs on events in which not all partons from the t 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 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 2 https://github.com/Alexanders101/SPANet/tree/v1.0 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 Appendix A 3.

VI. RESULTS
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.

A. 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 I 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 χ 2 method by around 10% with larger differences at higher jet multiplicities.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 II 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 χ 2 at correctly identifying all available partons.It should be noted that the perfect reconstruction in this case is substan-  tially lower than the fully reconstructable events in Table I.
In Table III   The matching efficiencies of partons in incorrect events are shown in Table IV.Here we see that the χ 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.

B. 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.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  For events with all jets correctly assigned, using the p 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.

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 [9] CMS Collaboration, Measurement of the top quark mass with lepton+jets final states using p p collisions at

Example Topograph
A complete Topograph network is shown in Fig. 12 for reconstruction of t tW 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 tW ), 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.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.

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 alterna-tive sample consists of all-hadronic t 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 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 χ 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 χ 2 method suffers from a substantial drop in efficiency, especially at higher jet multiplicities.

In aFIG. 2 :
FIG. 1: Two graph representations of top pair production, with one top decaying semi-leptonically and the other hadronically.The production mechanism in (a) is shown to be gluon fusion however in (b) it is represented by the hashed circle.Time flows from left to right in both graphs.

p 1 p 3 M
FIG. 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 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.

FIG. 4 :
FIG. 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.

FIG. 5 :
FIG. 5: Topograph network for the t 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.
permutation, m t and m W are the masses of the top quark and W boson, and σ t and σ 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 σ t and σ W are set to the standard deviation.The calculated values are m t = 169.8GeV, m W = 81.0GeV, σ t = 29.0GeV, and σ W = 18.5 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.

Figures 6 and 7 FIG. 6 :
Figures6 and 7show 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

FIG. 7 :
FIG.7: Reconstructed top quark invariant mass m top using jets assigned by the Topograph (solid), SPA-Net (dashed), and the χ 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).

FIG. 10 :
FIG. 10: Resolution of the reconstructed p 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).

FIG. 11 :
FIG.11: Resolution of the reconstructed p 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).
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.CRSII5_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.[1] ATLAS Collaboration, Measurements of normalized differential cross-sections for t t production in pp collisions at √ s = 7 TeV using the ATLAS detector, Phys.Rev. D 90, 072004 (2014), arXiv:1407.0371[hep-ex].[2] CMS Collaboration, Measurement of the top quark mass using proton-proton data at √ s = 7 and 8 TeV, Phys.Rev. D 93, 072004 (2016), arXiv:1509.04044[hep-ex].[3] ATLAS Collaboration, Determination of the top-quark pole mass using tt + 1-jet events collected with the ATLAS experiment in 7 TeV pp collisions, JHEP 10, 121, arXiv:1507.01769[hep-ex].[4] ATLAS Collaboration, Top-quark mass measurement in the all-hadronic t t decay channel at √ s = 8 TeV with the ATLAS detector, JHEP 09, 118, arXiv:1702.07546[hep-ex].[5] ATLAS Collaboration, Measurements of top-quark pair differential cross-sections in the lepton+jets channel in pp collisions at √ s = 13 TeV using the ATLAS detector, JHEP 11, 191, arXiv:1708.00727[hep-ex].[6] CMS Collaboration, Measurement of differential cross sections for top quark pair production using the lep-ton+jets final state in proton-proton collisions at 13 TeV, Phys.Rev. D 95, 092001 (2017), arXiv:1610.04191[hepex].[7] CMS Collaboration, Measurement of the top quark mass in the all-jets final state at √ s = 13 TeV and combination with the lepton+jets channel, Eur.Phys.J. C 79, 313 (2019), arXiv:1812.10534[hep-ex].[8] CMS Collaboration, Measurement of differential cross sections for the production of top quark pairs and of additional jets in lepton+jets events from pp collisions at √ s = 13 TeV, Phys.Rev. D 97, 112003 (2018), arXiv:1803.08856[hep-ex].

Fig. 14
Fig.14shows 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

Figs. 16 and 17
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.

Figures 18 FIG. 19 :FIG. 20 :FIG. 21 :
Figures 18 and 19 show the regression results for the η and ϕ coordinate.The difference between the value of the predicted η or ϕ and the true parton property is shown.For the predicted properties, the parton is either recon-

TABLE I :
Event reconstruction efficiencies (%) for the χ 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.

TABLE II :
Percentage of events with at most N incorrectly matched partons using the χ 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.

TABLE III :
Efficiencies (%) of correctly associating a jet to its corresponding parton for the χ 2 method, SPA-Net and Topographs.Jets are categorised into their truth flavour using parton matching.Only jets coming from the t t decays in events with at least one incorrect assignment are considered.The highest efficiency is highlighted in bold.

TABLE IV :
Efficiencies (%) of correctly associating a jet to the corresponding parton for events with at least one incorrect assignment for the χ 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 T .The highest efficiency is highlighted in bold.
[52]eVshows 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 V :
Hyper parameters used for the training of

TABLE VI :
Event reconstruction efficiencies (%) for the χ 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.

TABLE VII :
Percentage of events with at most N incorrectly matched partons using the χ 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.