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82 R. Yoshida
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Fig. 4.12 Calibration of MD-calculated physical properties (specific heat capacity at constant pressure (C
), linear expansion coefficient, volume expansion coefficient). This figure is a reprint from
P
Hayashi et al. (2022) [
(top) were significantly improved by calibration using transfer learning (bottom)
9]. Bias and variations between the MD-calculated and experimental values
In the examples shown in Fig.
specific heat capacity, linear expansion coefficient, and volume expansion coefficient. As shown in Fig.
between the experimental and MD-calculated values, which stemmed from the presence or absence of quantum effects. The latter two had significantly large variations
even within the same polymer in both experimental and calculated properties. For
each property, the source task of transfer learning was defined to predict the MDcalculated properties, and the target task was to predict the experimental properties
in PoLyInfo. A prediction model defines a mapping from the fingerprinted chemical
structure of a given polymer repeating unit to the experimental or MD-calculated
properties. As shown in Fig.
showed significant improvements in predicting the experimental data compared with
the direct predictions from the MD calculations. The systematic bias in the specific
heat capacity has almost disappeared. Interestingly, for the linear and volume expansion coefficients, the transferred model not only corrected the systematic bias of
the MD-calculated properties but also significantly reduced the variability of the
experimental values.
No calculation conditions can be applied to a wide variety of polymers. Therefore, biases and variations always occur in the MD properties obtained from fully
4.12(b), the target properties to be predicted were the
4.12(a), the specific heat capacity exhibited a significant bias
4.12(b), for all three properties, the transferred models
4 Materials Informatics with Limited Data 83
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automated calculations. Biases and variations also occur in the experimental values
owing to the experimental and sample preparation conditions and the nature of
the measurement equipment. Transfer learning bridges the gap between complex
real-world systems and imperfect computer models.
Using RadonPy, we aim to create one of the world’s largest polymer property databases containing more than 100,000 molecular skeletons. Furthermore, in
October 2022, an industry-academia consortium was established for the joint development of RadonPy and the computational polymer property database. To date,
approximately 150 members from one national institute, three universities, and 29
companies have participated in the consortium. This project is supported by the
“Program for Promoting Research on the Supercomputer Fugaku” of the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) in Japan and is making
maximum use of the computational resources of one of the fastest supercomputers,
“Fugaku,” to produce and accumulate a vast amount of data on a daily basis. The
main computational targets were virtual polymers. We classified the polymer skeletons into 20 classes, including polyester, polyimide, and polyacrylate, and built a
molecular generator for each polymer class using the chemical language model of
Ikebata et al. (2017) [
models with the chemical structures of existing polymers and built structure generators that mimic the frequent patterns (e.g., fragmentation and bonding rules) that
appear in existing polymers. A comprehensive virtual library consisting of 1,778,039
candidate molecules with diverse molecular skeletons was created.
Currently, the calculations of the physical properties of 47,500 amorphous polymers have been completed. The joint distribution of multiple properties of a significant number of polymers has been clearly and comprehensively observed by
conducting computer experiments at this scale, as shown in Fig.
systematic knowledge of the location of the Pareto frontier formed by the tradeoffs of multiple properties and the structural features of the polymer groups constituting it was obtained. With the current experimental techniques for measurement
and synthesis, the physical properties and material space could not be comprehensively observed on such a scale. Furthermore, high-throughput thermal conductivity
calculations have identified novel polymers beyond the Pareto frontier. The thermal
conductivity of ordinary amorphous polymers i s approximately 0.2–0.3 W/(m・K) at
best; however, some of the calculated polymers had thermal conductivities exceeding
0.5 W/(m・K) (Fig.
revealed that the rigidity of the molecular backbone, presence of a high density of
hydrogen-bondable units, and mechanisms involving hydrogen bonding and dipole–
dipole interactions are responsible for the high thermal conductivity of amorphous
9
polymers [
This project aims to create a large map of polymer material properties. In particular, the project aims to create a systematic dataset of biodegradable plastics,
highly thermally conductive polymers, and thermosetting resins and to create new
materials that contribute to a decarbonized, recycling-oriented society and thermal
management.
].
5] as described in Sect. 4.2.1. Here, we trained machine-learning
4.13 (a). As a result,
4.13(b)). Analysis of the structural features of these polymers
84 R. Yoshida
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Fig. 4.13 Polymer world map constructed by the high-throughput MD simulation using RadonPy.
This figure is a reprint from Hayashi et al. (2022) [
multiple physical properties (thermal conductivity, density, specific heat capacity at constant pressure (CP), volume expansion coefficient, linear expansion coefficient, refractive index) of polymer
materials. b Eight types of polymers that exhibited a high thermal conductivity of more than 0.4
W/(m・K) in the amorphous state
9]. a Joint distribution and Pareto frontier of
4.5 Concluding Remarks
This section provides an overview of MI in terms of forward and inverse problems.
Input/output variables in materials research can take various forms. Owing to this
diversity, methodologies and tools must be developed for each problem. The confluence of academic advances in data, computational, and experimental sciences with
this conventional workflow has produced new scientific methods and discoveries. In
recent years, the time lag between the confluence of cutting-edge technologies in
data science and applied fields has rapidly decreased.
The most important aspect of data-driven research is the data. Compared with
other applied fields of data science, the amount of data in materials research is far
less. Because a data-driven approach has not yet been fully introduced, the development of databases is still in its infancy. Additionally, because scientific outcomes
and industrial applications are closely linked, researchers are highly conscious of
information confidentiality, and in some areas, data sharing may not progress in the
future. In these areas, the barrier of limited data remains a problem. However, these
data are an endless source of knowledge. The volume and diversity of the data never
diminish, but rather increase monotonically. Simultaneously, a gap exists between
those who have data and those who do not. Power games are the essence of datadriven research. It is also important to examine the state of MI from a bird’s eye view
in an area where big data and small amounts of data coexist.
4 Materials Informatics with Limited Data 85
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Chapter 5
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Primer on Graph Machine Learning
Masatsugu Yamada and Mahito Sugiyama
5.1 Introduction
Designing de novo molecules for drugs and materials with desired properties is a
highly challenging task due to the massive number of candidates, that is, the search
space of molecules is exponentially huge with respect to the type of atoms and
bonds. To discover drug candidates, one of the most fundamental approaches is
using the quantitative structure-activity relationship (QSAR), a mathematical model
to explain relationships between biological activities and the structural properties
of molecules [
molecules, which is helpful to search datasets. However, to design new molecules,
one needs to solve an inverse problem of QSAR models, and it is challenging to solve
the optimization problem because a QSAR model loses graph topological structures.
Since molecules can be essentially represented as graphs with node and edge
attributes, graph mining methods and graph machine learning methods have been
studied to address this task [
in Sect.
mental approaches to treat graphs in machine learning and data mining. Then we
review graph neural networks (GNNs) in Sect.
approaches in graph machine learning. We also introduce some of the recent techniques of reinforcement learning that have been used for efficient search of molecules
in Sect.
1]. A QSAR model is often used for high-throughput screening of
2]. In this chapter, after reviewing basics of graph theory
5.2, we introduce graph kernels in Sect. 5.3, which is one of the funda-
5.4, which are now the most popular
5.5.
M. Yamada (B) · M. Sugiyama
National Institute of Informatics, Chiyoda-ku, Tokyo 101-8430, Japan
e-mail:
masatsugu-yamada@nii.ac.jp
M. Sugiyama
e-mail:
mahito@nii.ac.jp
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
H. Satoh et al. (eds.), Drug Development Supported by Informatics,
https://doi.org/10.1007/978-981-97-4828-0_5
87
88 M. Yamada and M. Sugiyama
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5.2 Graph Theory
First, we define terminology and notation of graphs. A graph is a tuple G = (V , E),
where V and E denote the set of nodes (or vertices)
E ={e1, e2,..., em}, respectively. Each edge is a pair of nodes (vi, vj) ∈ V × V .
When one considers an undirected graph, an edge does not have a direction and it is
treated as a set. The number of nodes and edges of G, the cardinality of V and E,are
represented as
via label functions
label domain
is a triple
lV and lE, respectively. For two graphs G = (V , E) and G'= (V', E'),wesay
from
G'is a subgraph of G, denoted by G'⊑ G,if V'⊆ V and E'⊆ (V'× V') ∩ E.
that
|V | and |E|, respectively. Nodes and edges can have labels (attributes)
lV : V → ∑V for nodes and lE : E → ∑E for edges with some
∑V and ∑E, which can be any set such as Z and Rd . A labeled graph
G = (V , E, L), where L is the set of node labels and edge labels obtained
Two nodes vi and vj in a graph G = (V , E) are said to be adjacent if an edge
e
= (vi, vj) exists in E. The neighboring information of a graph is represented by
ij
an adjacency matrix. The adjacency matrix
A
=
ij
A ∈ R
, vj) ∈ E,
1if (v
i
0 otherwise.
The set of neighborhood nodes with respect to a node vi ∈ V is denoted as
V ={v1, v2,..., vn} and edges
|V |×|V |
of a graph is defined as
(5.1)
N (vi) ={vj ∈ V | (vi, vj) ∈ E}.
The degree or valency of a node vi inagraph G is the number of neighboring nodes,
that is,
deg(vi) =|N (vi)|. (5.2)
A graph can be generalized as a hypergraph H = (X , E), where X is a s et of
vertices and E is hyper edges that have multiple edges rather than single pair of
nodes. Examples of a graph and a hypergraph are shown in Fig.
in the figure shows a simple graph
G = (V , E), where V ={v1, v2, v3, v4} and
5.1. The left graph
E ={e1, e2, e3, e4} with e1 = (v1, v2), e2 = (v1, v3), e3 = (v2, v3), and e4 =
, v4). The right graph in the figure shows a hypergraph H = (X , E) defined as
(v
3
X ={v1, v2, v3, v4} and E ={e1, e2, e3} with e1 ={v1, v2, v3}, e2 ={v1, v3}, and
e3 ={v3, v4}. This notation is convenient when considering graph generation by
combining two subgraphs.
A graph can be decomposed into a junction tree composed of the set of subgraphs
or cliques C. For a graph
G = (V , E),atree T with subsets C1, ... , Cn of V is called
a junction tree for G if it satisfies the following properties:
5 Primer on Graph Machine Learning 89
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Fig. 5.1 An undirected graph and an undirected hypergraph. Left: An undirected graph that has
4 vertices and 4 edges. Right: An undirected hypergraph that has 4 vertices and 3 edges. An edge
e1 is overlapped with half-toned. Edges e1 and e2 are duplication of vertices (v1, v3), but can have
different edge labels
1. The union of all sets C1, ... , Cn equals to V; that is,
i
Ci = V .
2. For every edge (u, v) ∈ E, there exists Ci ∈ V such that u ∈ Ci and v ∈ Ci.
3. If Ck is on a path from Ci to Cj in T , Vi ∩ Vj ⊆ Vk.
An example of a junction tree is shown in Fig. 5.2. A graph G is decomposed
into a junction tree
, v3, v4}, C3 ={v3, v5}, and C4 ={v5, v6}. The intersection of nodes becomes
{v
2
T . Cliques in G include four cliques: C1 ={v1, v2}, C2 =
edges of the original graph G in reconstruction from the junction tree. This notation
is helpful when computing marginalization in general graphs in Bayesian networks
and generating graphs being retained with cycles.
Fig. 5.2 An undirected graph and its corresponding junction tree. Left: An undirected graph with
a cycle that has 6 vertices and 6 edges. Right: A junction tree that has 4 cliques encircled with blue.
Intersection in the junction tree is colored orange. Graphs and trees can be converted interchangeably
90 M. Yamada and M. Sugiyama
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5.3 Graph Kernels
The graph kernel is a kernel function that computes the similarity between graphs
by the inner product of features obtained from graphs, which can be plugged into
any kernel-based machine learning methods like SVM and kernel PCA [
essential idea for considering structured objects like graphs has been established as
R-convolutional kernels in the seminal paper by Haussler [
4], where we decompose
each object into sub-parts and count the common sub-parts to measure the similarity
between them. The most basic and natural instance of R-convolutional kernels for
graphs is known to be all-subgraph kernels, which is defined as
k
subgraph
(G, G') =
S⊑GS'⊑G
k
isomorphism
'
(S, S'), (5.3)
where
k
isomorphism
(S, S') =
1if S and S
0 otherwise.
'
are isomorphism,
Although it gives a canonical way of computing the similarity between graphs,
Gärtner et al. show that this computation is NP-hard [
5], hence it is not practical.
To date, a number of graph kernels have been proposed to efficiently and effec-
], and libraries for computing graph
tively compute the similarity between graphs [
kernels are widely available [7]. A kernel function k(x, x
larity between x and
features, that is,
x'of features. Every kernel function must be symmetric about
k(x, x') = k(x', x) and semi-definite [8]. The feature f computed
6
'
) is a measure of simi-
from a graph is mainly composed of graph specific structures. In the following, we
introduce two representative graph kernels, the vertex histogram kernel, which is one
of the simplest graph kernel, and the Weisfeiler–Lehman graph kernel, which is the
most popular kernel.
3]. The
5.3.1 The Vertex Histogram Kernel
Given a pair of graphs G = (V , E) and G'= (V', E'), the vertex histogram kernel
counts the joint occurrences of node labels in them [
of node labels, that is,
generality. The node label histogram
is given as
The vertex histogram kernel kV between G and G'is defined as
9]. Let d be the number of types
d =|∑v|, and assume that ∑v ={1, 2,..., d } without loss of
f = (f1,..., fd ) ∈ Nd of a graph G = (V , E)
fi =|{v ∈ V | lv = i}|. (5.4)
5 Primer on Graph Machine Learning 91
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d
kV (G, G') =⟨f, f'⟩=
'
fif
. (5.5)
i
i=1
The time complexity of computing the vertex histogram kernel is O(|V |+|V'|).
5.3.2 The Weisfeiler–Lehman Graph Kernel
Although the vertex histogram kernel is efficient, it does not take any graph topological structure into account. To consider the subgraph structures, a number of graph
kernels have been proposed. Among them, one of the most powerful graph kernels is
the Weisfeiler–Lehman (WL) subtree kernel [
Lehman test of isomorphism [
11]. Fig. 5.3 shows the procedure of the Weisfeiler–
Lehman algorithm at the second iteration with respect to v
simplicity, the compression for relabeling is not shown in Fig.
label is determined after finishing iteration. The WL procedure is to aggregate node
labels in the neighboring nodes and count their occurrence. For example, let us take
the node label of
v0 and v
'
. In the first iteration, the neighboring node of v0 is v1, and
0
they are aggregated as a new label [0, 3]. In order to compress the representation, the
new node label is replaced as
with that of
labels in G and
v0 at first iteration. After completing iteration over entire vertices, new
G'are shown in Fig. 5.4.
5 := l([0, 3]) in this case. The label of v
A new label obtained by the WL procedure represents the subtree structure of a
graph as shown in Fig.
5.4. At the end of the iteration, feature vector representation
is obtained as a series of counts of original and compressed node labels, resulting in
the WL kernel
10], which is based on the Weisfeiler–
'
and v
0
. Note that, for
0
5.3. Each compressed
'
is the same
0
where φ is the vertex histogram of node labels after iterations.
5.3.3 Extend Connectivity Fingerprints
Extend connectivity fingerprints (ECFPs) are a class of topological fingerprints for
molecular characterization [
related to the WL graph kernel procedure.
The algorithm of ECFPs is basically the same process of aggregating neighboring
node labels, such as atom labels. After obtaining a new node label, it is relabeled by
a hash function that converts it to unique integer values to represent the subgraph
kWL(G, G') =⟨φ(G), φ (G')⟩, (5.6)
12
]. Although it is not a rigorous graph kernel, it is highly
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