Graph Segmentation in Scientific Datasets

Rajat Sahay
Department of Computer Science
Vellore Institute of Technology
Vellore, India
rajat.sahay2018@vitstudent.ac.in

Savannah Thais
Research Computing
Princeton University
Princeton, NJ 08544
sthais@princeton.edu

Abstract
Deep learning tools are being used extensively in a range of scientific domains; inparticular, there has been a steady increase in the number of geometric deep learning
solutions proposed to a variety of problems involving structured or relational
scientific data. In this work, we report on the performance of graph segmentation
methods for two scientific datasets from different fields. Based on observations, wewere able to characterize the individual impact each type of graph segmentation
methods has on the dataset and how they can be used as a precursors to deeplearning pipelines.

1

Introduction

Machine learning methods are increasingly employed in a range of scientific domains to complement
or even replace traditional statistical or data-driven solutions (1). Many physical science datasets
such as those from particle physics (2), physical chemistry (3), material science (4), and others, haveinherent geometric or relational structures that can be well represented by graphs. This has accelerated
the development and adaptation of graph-based deep learning solutions like Graph Neural Networks(GNNs) (5), Graph Convolutional Networks (GCNs) (6), and Graph Attention Transformers (GATs)(7), among others, in scientific research.
A common task in many of these datasets is clustering the raw data based on certain attributes inorder to obtain relevant contextual information or identify underlying geometric structures. In thispaper we study two such tasks: mapping the trajectories of charged particles at the Large Hadron
Collider (LHC) (2) and identifying ligand sites in proteins (8). We explore graph segmentation as aprecursor to a larger deep learning pipeline. By understanding more about the underlying geometricstructure of the data and, when possible, separating the input data into distinct sub-graphs, we areable to simplify the tasks of downstream graph-based deep learning architectures (9) and parallelize
inference in constrained computing environments (10).
Graph Segmentation Specifically, we assume G = (V, E) to be an undirected graph with vertices
vi ∈ V and edges (vi , vj ) ∈ E corresponding to pairs of neighbouring vertices. Vertex-basedgraph segmentation involves the construction of a segmentation S which is a partition of V into ncomponents or regions, C1 ...Cn ∈ S such that each region corresponds to a connected component
in a graph G0 = (V, E 0 ) where E 0 ⊆ E. In other words, any segmentation is induced by a subset
of edges in E. A good segmentation would ensure that elements in the same component would be
similar to each other and dissimilar from elements in different components.

2

Datasets

TrackML Track ML (2) is a dataset created by CERN to address new and developing softwareneeds for real-time preprocessing and filtering of data produced by proton-proton collisions at the35th Conference on Neural Information Processing Systems (NeurIPS 2021), Sydney, Australia.

LHC. The dataset consists 8850 simulated events (proton-protong collisions) interacting with arealistic detector (11). Each event can be defined as an unordered set of particle measurements (hits)
within the detector; each hit is a 3D measurement in Cartesian coordinates (x, y, z) of the location
where a particle interacted with the detector. All events are statistically independent and containdirectional information, unique identifiers for the true hit position and unique particle identifier aswell as representations of the final state of the particle.
The geometric learning task for the TrackML dataset is to identify groups of hits that correspond to
the trajectories (tracks) of individual charged particles produced from the collision. By implementing
graph segmentation methods we can begin to separate subgraphs that correspond to individual tracks
or groups of tracks that can be used as inputs to a more precise track finding algorithm.
sc-PDB The sc-PDB database (8) is a comprehensive collection of over seventeen thousand ligandable binding sites available in the Protein Data Bank (PDB). The database provides an all-atom
description of a protein, its ligands, binding sites and modes of binding. The steady development of
sc-PDB over the last 15 years has offered a wealth of information to assist in a variety of fields likecomputer-aided drug discovery applications.
Binding sites in sc-PDB can be represented as 3D shapes of cavities generated using VolSite (12).
A binding cavity is described by VolSite using a set of pharmacological properties laid out in a 3D
grid, based on the properties of the neighbouring proteins. For the purposes of this paper, the data
was split into 10 groups, based on the UniProt ID. This ensured that structures of a single protein
were present in the same group. VolSite was also used to describe a cavity structure for each ligand(apo structures were aligned to their holo counterparts) and the ligands were used to select pockets(orthosteric ligand binding-sites).

3

Methodology

3.1

DBSCAN

Density-based clustering (13) provides a general and rigorous probabilistic framework in which theclustering task is well-defined and amenable to statistical analysis. The Density Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm (14) is an extremely popular methodology
for flat clustering. Based on a set of points, DBSCAN groups together points that are close to eachother based on a distance measurement and a minimum number of points.
(14) defines a simpler version of the DBSCAN algorithm where k is fixed. In (14), the parameters
defined as h and k are known as Eps and MinPts which define the neighbourhood radius and thenumber of points required to seed a cluster respectively.
3.2

Spectral Clustering

Spectral Clustering (15) is a technique with roots in graph theory, where the primary approachis to identify communities in nodes of the graph based on the edges connecting them. It usuallyoutperforms traditional clustering algorithms such as the k-means algorithm (16) and is also solved
efficiently by most linear algebra software applications.
We assume that our data consists of n arbitrary ’points’, x1 ...xn . We leverage a non-negative andsymmetric similarity function to measure pairwise similarities sij and construct a similarity matrix S.
S can be mathematically defined as: S = (sij )i,j=1...n . This gives rise to a (n, n)-sized matrix. For
the purposes of our experiments, we use the method proposed by (17), where the entry (i, j) is the
Euclidean distance between i and j. However, we also leverage the Eigengap Heuristic approach todetermine the number of clusters the data should be partitioned into.
Eigengap Heuristic There usually exists a well-defined criteria to choose the number of clusters
in a model based setting. This might be based on the log-likelihood of the data which can be treatedin a Bayesian way (18). The Eigengap Heuristic is a procedure designed specifically for Spectral
Clustering. Here, the goal is to choose k such that all the eigenvalues λ1 , ...λk are small but λk+1
is relatively larger. An explanation for this can be provided by spectral graph theory, where manygeometric invariants of the graph can be expressed with the help of the first eigenvalues of the graphLaplacian. The sizes of the cuts are related closely to the size of the first eigenvalues.
2

3.3

Dynamic KNNs

k-Nearest Neighbours (kNNs) has been widely been used in areas like data mining due to its simplicity,
effectiveness and robustness. It is an instance-based learning algorithm and a typical example oflazy learning; since it stores training instances to delay learning till classification. The classification
accuracy of a kNN clustering approach is sensitive to the value of k. A simple method to increase
this classification accuracy would be to learn the best k-value amongst different measurements of k.
Dynamic kNN (DkNN) (19) works on the basis of this principle to choose an optimal value of k in
the training datasets dynamically. DkNN uses an individual observation from the original sample
for validation purposes and treats the remaining observations as training data. This is recursively
repeated in a way such that each observation from the original sample is used for validation once.
3.4

Gaussian Mixture Models

Gaussian Mixture Models (20) are probabilistic models that use a soft-clustering approach to
distribute data points into different clusters. The objective function of a GMM is to maximize
the likelihood value of data X, defined as p(X). By assuming a mixture of G Gaussians, p(X)can be formulated as the marginalized property, summed up over all G clusters. Mathematically:PGp(Xi ) = g=1 p(Xi |cg )p(cg ).
Gaussian Mixture Models work on a two-step algorithm called Expectation-Maximization (EM).
When given the number of clusters, the EM algorithm tries to figure out the parameters of these
Gaussian distribution in two distinct steps: The Estimation Step guesses the parameters of the method
based on available data while the Maximization step updates cluster parameters based on resultsfrom the Estimation step. This is repeated until convergence is reached. To calculate the number ofclusters, we make use of Bayesian Gaussian Mixture Models, which allow us to infer an approximate
posterior distribution over the parameters of a Gaussian Mixture Distribution.

4

Experiments and Results

TrackML The graph segmentation algorithms we discuss in this paper are responsible for thegraph construction in a a full ML-based track reconstruction pipeline. Events present in TrackML are
converted into hitgraphs and individual hits are assigned to graph nodes after conditional filtering.
The algorithms cluster these hit nodes and decide whether an edge eij should be extended between
hits xi and xj . We present 4 different quantitative measurements taken for TrackML, measured over6 different values of minimum particle momentum, pmin
. We measured Truth Efficiency; the fractionTof true track segments represented as graph edges, Edge Efficiency; the ratio of true edges to the totaledges in the graph, and the Number of Nodes and Edges for each method. A high Truth Efficiency isnecessary to ensure all tracks can be reconstructed while a high Edge Efficiency means fewer false
edges must be removed by a downstream edge-classifying network. The number of Nodes and Edges
describe the size of the graphs and the potential for these methods to be used in resource constrained
computing environments like those at the LHC.
From Table 1, we see that GMMs generally achieve high combined Truth Efficiency and Edge
Efficiency across the full range of pmin
. High Truth Efficiency indicates that hits belonging to theTsame track are likely to be clustered into the same component, while high Edge Efficiency indicatesthat distinct tracks are likely to be separated into different components. With these partially segmented
graphs we can separate the raw input into distinct subgraphs. This can help accelerate graph-based
deep learning solutions for particle tracking which are being run on distributed architectures (10).
sc-PDB Binding sites for proteins in the sc-PDB were generated using the SiteHopper create tool.
The default SiteHopper PatchScore represents a summation of Tanimoto similarity coefficients (21)weighted 3:1 in favor of color similarity over shape similarity. SiteHopper made use of the fpocket 1
surface protein atoms as pseudo-ligands and was able to isolate binding site patches from the original
protein structures. fpocket is a freely available binding site detection tool capable of operating inlarge molecular and genomics datasets. Each graph segmentation method, when applied to thesc-PDB dataset generates a map with potential binding sites clustered together. The clusters vary
1

fpocket: https://bioserv.rpbs.univ-paris-diderot.fr/services/fpocket/

3

Method

DBSCAN

Spectral Clustering

Dynamic kNN

GMM

pmin
T
0.5
0.6
0.75
1.0
1.5
2.0
0.5
0.6
0.75
1.0
1.5
2.0
0.5
0.6
0.75
1.0
1.5
2.0
0.5
0.6
0.75
1.0
1.5
2.0

Truth Efficiency
0.972 ± 0.03
0.974 ± 0.05
0.979 ± 0.09
0.981 ± 0.10
0.983 ± 0.14
0.982 ± 0.15
0.968 ± 0.003
0.972 ± 0.004
0.979 ± 0.004
0.981 ± 0.005
0.981 ± 0.006
0.981 ± 0.008
0.972 ± 0.003
0.974 ± 0.003
0.977 ± 0.005
0.979 ± 0.005
0.983 ± 0.008
0.984 ± 0.010
0.966 ± 0.002
0.974 ± 0.003
0.979 ± 0.006
0.982 ± 0.006
0.983 ± 0.008
0.982 ± 0.010

Edge Efficiency
0.046 ± 0.00
0.051 ± 0.00
0.091 ± 0.01
0.14 ± 0.03
0.21 ± 0.05
0.25 ± 0.09
0.042 ± 0.01
0.108 ± 0.03
0.108 ± 0.03
0.180 ± 0.13
0.399 ± 0.17
0.719 ± 0.15
0.028 ± 0.00
0.074 ± 0.00
0.081 ± 0.00
0.119 ± 0.01
0.253 ± 0.03
0.375 ± 0.04
0.115 ± 0.00
0.151 ± 0.01
0.207 ± 0.02
0.207 ± 0.03
0.361 ± 0.05
0.452 ± 0.05

Number of Nodes
1.41 × 105
1.02 × 105
8.68 × 104
5.93 × 104
2.19 × 104
1.02 × 104
1.59 × 105
8.268 × 104
7.97 × 104
6.351 × 103
3.742 × 104
1.924 × 104
1.621 × 105
8.491 × 104
8.699 × 104
7.372 × 104
4.206 × 104
2.519 × 104
8.491 × 104
7.916 × 104
6.582 × 104
5.218 × 104
2.625 × 104
9.350 × 103

Number of Edges
2.35 × 106
1.80 × 106
1.17 × 106
7.99 × 105
9.27 × 104
5.09 × 104
2.382 × 105
9.358 × 104
9.27 × 104
7.591 × 104
4.260 × 104
1.834 × 104
3.209 × 105
5.372 × 105
9.973 × 104
6.138 × 104
4.528 × 104
1.972 × 104
5.372 × 105
4.684 × 105
2.461 × 105
9.207 × 104
6.948 × 104
9.958 × 103

Table 1: Quantitative Evaluation of Track Finding Efficiencies with different Graph Segmentation
Techniquesin size, indicating that some of the cavities are more conservative than others. For all proteins, thelargest cluster corresponds to the conserved, orthosteric ligand-binding site. The PatchScore can beconsidered a quantitative evaluation of these clusters. Each value in the Table 2 is associated with the
maximum PatchScore exhibited by the members of the two binding site clusters.
One major disadvantage introduced by cavity detection is that it introduces noise due to non-conserved
potential binding sites. Table 2 shows that Dynamic kNNs display higher values compared to othergraph segmentation methods while clustering. This suggests that clustering methods like DkNNs
can be considered an appropriate method for the generation of clustered binding sites as well as
mitigating the noise introduced to the dataset by the fpocket cavity detection.
MethodDBSCAN

Spectral Clustering

Dynamic kNN

GMM

Protein
Cationic Trypsin
BRD-4
CDK2
Estrogen Receptor
HIV-1 Protease
Prothrombin
Cationic Trypsin
BRD-4
CDK2
Estrogen Receptor
HIV-1 Protease
Prothrombin
Cationic Trypsin
BRD-4
CDK2
Estrogen Receptor
HIV-1 Protease
Prothrombin
Cationic Trypsin
BRD-4
CDK2
Estrogen Receptor
HIV-1 Protease
Prothrombin

Cationic Trypsin
0.03
0.09
0.08
0.17
0.16
0.15
0.14
0.02
0.15
0.11
0.19
0.16
0.10
0.15
0.19
0.07
0.01
0.11
0.05
0.18

BRD-4
0.04
0.13
0.18
0.01
0.05
0.06
0.15
0.18
0.02
0.11
0.04
0.19
0.18
0.15
0.28
0.04
0.08
0.18
0.11
0.24

CDK2
0.07
0.08
0.04
0.19
0.24
0.17
0.18
0.16
0.08
0.13
0.15
0.35
0.22
0.21
0.31
0.10
0.17
0.09
0.17
0.07

Estrogen Receptor
0.03
0.11
0.06
0.18
0.11
0.13
0.09
0.07
0.07
0.22
0.11
0.14
0.15
0.17
0.23
0.08
0.14
0.04
0.06
0.04

HIV-1 Protease
0.03
0.05
0.11
0.12
0.09
0.08
0.12
0.14
0.03
0.05
0.27
0.08
0.19
0.18
0.17
0.16
0.11
0.13
0.11
0.09

Prothrombin
0.08
0.13
0.18
0.08
0.31
0.20
0.06
0.05
0.06
0.16
0.19
0.13
0.22
0.09
0.24
0.09
0.03
0.19
0.07
0.02
-

Table 2: A quantitative evaluation of clustering and mapping of potential binding sites between
different proteins in the sc-PDB dataset.
In order to better understand the results displayed in Table 1 and Table 2, we also show the effectiveness of each graph clustering approach in comparison to the others. We define two values, eT rackM L
and esc−P DB to measure how well each clustering approach works on either dataset. eT rackM L canbe quantitatively defined as the ratio of segmented subgraphs to the total number of tracks in theevent, which is equal to the number of particles produced. esc−P DB is the ratio of the total number of4

Dataset
DBSCAN
Spectral Clustering
Dynamic kNN
GMM

MethodTrackML
sc-PDB
TrackML
sc-PDB
TrackML
sc-PDB
TrackML
sc-PDBeT rackM L ↑
0.579
0.602
0.513
0.735
-

esc−P DB ↑
0.481
0.517
0.594
0.408

χT rackM L ↑0.7424
0.59680.50790.8194-χsc−P DB ↑0.28630.42620.50380.3920

Table 3: Effectiveness of each graph segmentation method on both datasets

clustered binding sites to the total isolated binding sites identified by fpocket. From Table 3 we noticethat Spectral Clustering and GMM perform well on TrackML while DkNN is the most effective andclustering potential binding sites in sc-PDB.
To get a deeper understanding of two e efficiency variables, we also look at how useful eachsegmentation measure is for downstream tasks. This is quantitatively measured by defining two
variables, χT rackM L which is the ratio of the number of clusters containing more than 50% particles
from the same track and the total number of tracks. We also define χsc−P DB as the number ofobserved clusters which actually correspond to a distinct ligand binding site and the total number ofobserved clusters. We see that GMMs perform best on the TrackML dataset and is able to separate
over 80% of tracks into distinct subgraphs. Dynamic kNN performs best on the sc-PDB dataset,however only 50% of segmented subgraphs correspond to true individual ligand binding sites.

5

Conclusion and Future Work

We take a look at how different types of graph segmentation approaches work on scientific datasetsand how they could be used as a precursor for deep learning pipelines with graph-based data. We
conduct comprehensive evaluations over two scientific datasets used in separate fields and show how
graph segmentation would be able to point towards factors that would inevitably help speed-up or
improve the accuracy of the overall pipeline it is fitted into. From these initial studies we demonstratethat relatively graph segmentation approaches can effectively identify and separate unique graphcomponents. However, the most effective algorithm for graph segmentation depends on the specificdataset of interest. In future work we would like to examine other datasets in the particle physics
and proteins and expand to other domains in physical sciences to understand if these conclusions aredataset specific or generalizable across datasets in a particular subfield.

6

Impact Statement/Ethical Considerations

This work may be used for a variety of purposes in different fields. Based on the datasets included inthis paper, this work has the potential to be used for improve the efficiency of particle tracking at the
LHC. It may also be used as a starting point for obtaining more information about newly discovered
protein structures. The results of these tasks would yield many benefits to science and society. This
work could also reasonably be used to improve processing of other structured scientific dataset andenable a number of additional learning tasks. We encourage any reseachers applying these methods to
new dataset carefully design a quantitative measurement of accurate segmentation for the individual
dataset.
We are also aware that this work exists in the broader context of computer vision and geometricmachine learning. This can be exploited for many negative purposes such as surveillance and targeting.
We are committed to continuing to learn more about the broader use of machine learning and support
our communities in advocating against harmful uses.

Acknowledgments and Disclosure of Funding
This work is supported by IRIS-HEP through the U.S. National Science Foundation (NSF) under
Cooperative Agreement OAC-1836650.
5

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A

Appendix

For the sake of brevity, the following tables and figures were not included in the main paper. However,
we feel that they may be vital to some readers in order to get a deeper understanding and a betteroverall view of the topics discussed in the paper.
We take a look at the hitgraph generated by Spectral Clustering for an event in the η − φ space.

Figure 1: Progressively increasing number of false edges with a decrease in pminin the η − φ space
T
Table 4 lists out the UniProt ID, among other features, of the proteins that were selected for our
experiments. Here, n defines the number of predicted binding sites according to fpocket.
ProteinN
UniProt IDnCationic Trypsin
315
P00918968
Bromodomain-Containing Protein 4 (BRD4)
93
O60885
192
Cyclin-Dependent Kinase 2 (CDK2)
148
P24941490
Estrogen Receptor (ER)52
P03372
241
Human Immunodeficiency Virus-1 (HIV-1) Protease 335
N/A*
481
Prothrombin
142
P00734336
Table 4: A list of proteins studied with their UniProtID∗
HIV-1 Protease entries were defined by a 90% sequence similarity search using the consensus B
protease sequence retrieved from HIVdb.

We also list out the hyperparameters that were used for the graph segmentation algorithms that we
discussed in the paper. For DBSCAN, we refer to the implementation done by (9) and accordingly
alter the value of  and MinPts according to the value of pmin
. We also look at n for Gaussian Mixture
T
Models. It defines the number of initializationsto perform, and the result with the highest lower value
bond on the likelihood is used. All experiments were run on on an NVIDIA GTX1080Ti.
MethodDBSCANDBSCAN
GMM

GMM

pmin

MinPts
T
0.5
0.22
3
0.6
0.18
3
TrackML 0.75
0.1
3
1.0
0.08
3
1.5
0.06
3
2.0
0.05
3
sc-PDB7
10
0.5
0.6
TrackML 0.75
1.0
1.5
2.0
sc-PDB
Table 5: Hyperparameters
Dataset

8

n
3
3
3
3
3
3
9