# Structure-Aware Hierarchical Graph Pooling using Information Bottleneck

Kashob Kumar Roy, Amit Roy, A K M Mahbubur Rahman, M Ashraful Amin and Amin
Ahsan Ali  Artificial Intelligence and Cybernetics Lab, Independent University
Bangladesh
{kashobroy<EMAIL_ADDRESS>and {akmmrahman, aminmdashraful,
<EMAIL_ADDRESS>
###### Abstract

Graph pooling is an essential ingredient of Graph Neural Networks (GNNs) in
graph classification and regression tasks. For these tasks, different pooling
strategies have been proposed to generate a graph-level representation by
downsampling and summarizing nodes’ features in a graph. However, most
existing pooling methods are unable to capture distinguishable structural
information effectively. Besides, they are prone to adversarial attacks. In
this work, we propose a novel pooling method named as HIBPool where we
leverage the Information Bottleneck (IB) principle that optimally balances the
expressiveness and robustness of a model to learn representations of input
data. Furthermore, we introduce a novel structure-aware Discriminative Pooling
Readout (DiP-Readout) function to capture the informative local subgraph
structures in the graph. Finally, our experimental results show that our model
significantly outperforms other state-of-art methods on several graph
classification benchmarks and more resilient to feature-perturbation attack
than existing pooling methods111Source code at:
https://github.com/forkkr/HIBPool.

###### Index Terms:

Graph Neural Networks, Graph Pooling, Information Bottleneck, Graph
Classification.

##  I Introduction

Graph Neural Networks (GNNs) have seen tremendous success in learning
meaningful representations from graph-structured data for various downstream
tasks such as graph-classification [1, 2, 3, 4, 5, 6], graph-based regression
[7, 8], and node classification [9, 10].

In brief, graph classification aims to predict the label of an input graph by
utilizing the given graph structure and node features. Inspired by CNNs, graph
local pooling methods compute the graph-level representations through
downsampling and summarizing the local structures and node features
hierarchically. One line of research on pooling methods [5, 3, 11] follows
Top-$K$ node selection strategies. It involves setting different criteria to
select a subset of top $K$ nodes with high scores based on the criteria to
form a coarser graph for subsequent pooling layer (see Fig. 1). Another line
of research [12, 2, 13, 14, 15] focuses on capturing local substructures
explicitly through partitioning the graph into several communities/clusters
and finding one representative node for each community/cluster to form next
layer coarser graph (see Fig. 1).

However, recently, authors in [16] ran experiments on several existing pooling
methods raised questions about the efficacy of local pooling and concluded
that the accuracy of graph classification methods based on existing local
poolings are not superior to a simple global pooling method that computes a
graph-level summary representation from all node features through a simple
weighted summation or neural network directly. This paper investigates this
claim further. There could be several factors responsible for the lack of
performance improvement. Firstly, the existing methods might be incapable of
learning important local community/subgraph structures. Secondly, the simple
classification loss without regularization terms used in graph classification
may promote learning homogeneous node representations across the whole graph
that naturally impose challenges for simple local pooling functions to learn
informative local structures. Thirdly, irrelevant or redundant node features
may also produce more homogeneous representations. We elaborate on these
points below and describe the contributions of this paper.

Figure 1: Overview of pooling methods: HIBPool, DIFFPOOL (colors denotes local
communities/clusters); ASAP (color denotes selected top k local clusters);
Top-K based Pooling (colors indicates the set of selected top k nodes)

First, the ability to define proper assignments of nodes to local
community/clusters plays an essential role in learning important local
structural information. Communities in a graph are groups of nodes with denser
connections among them than connections with the rest of the graph, which may
share common properties or play similar roles, or represent functional modules
as a whole [17]. Community structures are ubiquitous and play essential roles
in the topology and function of various graph networks such as biological
networks [18], social networks [19] etc. For example, the class label of a
protein is enzyme or non-enzyme, which depends on not only features of amino-
acids but also the community structures in amino-acids interaction networks
[20] or motifs of amino-acids in its amino-acid sequence [21]. Besides, these
structures are important to identify the functional modules as well as to
capture the hierarchical feature information in graphs. Different local
pooling algorithms use different local substructures to pool from and thus
differ in their ability to capture local information and produce robust graph
representations. In recent years, few papers adapted different partitioning
approaches, which are very popular in complex network analysis. [15] proposes
pooling operation using clique i.e., each of its nodes is connected to all the
others. [14] introduces K-plex cover into graph pooling in which each node is
adjacent to all other nodes within a subgraph except at most k of them.
Authors in [13] leverage the formulation of the Min-Cut problem as a
regularization term with classification loss to compute cluster assignment of
nodes. However, these algorithms have several shortcomings. The methods take
the number of communities or their sizes as a parameter and only consider
internal cohesion within subgraphs [17, 22]. It is quite expected that Top-K
selection based pooling methods [5, 3, 11] might ignore the topological
structure of the graph. In Fig. 1 we can see that some nodes (w/o color) may
have low scores, but they have an important role in topological structures.
More structure-aware methods such as DIFFPOOL [12] hierarchically learns the
cluster assignment matrix. ASAP [2], on the other hand, defines all possible
local clusters consisting of a target node and its 1-hop neighborhood and
selects top $k$ local clusters depending on the feature-based fitness scores.
Further, we analyze the consistency of local clusters learned by DIFFPOOL and
ASAP with respect to the set of optimal communities discovered by Louvain
222Note that we never know the true community structures. Hence we choose the
most popular and widely accepted algorithm for detecting the communities
[23][23] from graph as baselines.

Datasets | ENZYMES | D&D | PROTEINS
---|---|---|---
Model | AMI | ARI | AMI | ARI | AMI | ARI
DIFFPOOL | 0.24 | 0.18 | 0.04 | 0.02 | 0.10 | 0.04
ASAP | 0.16 | 0.10 | 0.12 | 0.06 | 0.14 | 0.08
TABLE I: Consistency of learned communities w.r.t the communities generated by
Louvain${}^{\ref{louvain}}$ in original graphs: AMI - Adjusted Mutual
Information and ARI - Adjusted Rand Index. The higher value indicates better
consistency.

In Table I, we can observe that the values of Adjusted Mutual Information
(AMI) and Adjusted Rand Index (ARI) are very low, which demonstrates that they
fail to capture community structures in graphs. However, most state-of-the-art
community detection methods find communities based on the maximization of
modularity [23, 22, 24] and aim to identify the community structures of the
graph as a whole. We observed that the effectiveness of modularity-based
community detection algorithms to capture topological and functional
substructures in graph classification tasks had not been investigated well.

Second, simple pooling functions such as Sum, Mean, Max, Min, etc., fail to
obtain distinguishable representations of different substructures with
homogeneous node feature vectors (as shown in Fig. 2). Authors in [16] state
that multiple convolutions on node features encourage to learn over-smoothed
representation across their neighborhoods. [4, 25] discussed the limited
expressive power of simple aggregation functions to differentiate messages
from the neighborhood in the node classification context. Therefore, we
conduct rigorous experiments to investigate the limitations of simple pooling
functions and design a novel discriminative pooling readout (DiP-Readout)
function that addresses the limitations. Moreover, as the topological features
characterize the structures and dynamics of networks and are necessary to
identify different categories of structures [26], we incorporate the
topological features of nodes into the DiP-Readout function to distinguish the
different structures of local communities.

Figure 2: With identical feature vector for all same colored nodes simple
readout like Sum, Mean, Max, Min etc. can not distinguish the substructures G1
and G2. To discriminate G1 and G2, centrality measures e.g. clustering
coefficient can be incorporated to design sophisticated aggregation function
that we have discussed in our proposed section.

Third, Information Bottleneck (IB) suggests that an optimal learning method
should maximize the mutual information between latent representation and the
target label to make prediction accurate while minimizing the mutual
information between the latent representation and the input to promote
generalization ability [27, 28]. GIB [29] introduces IB to regularize the
structural and feature information in the node classification task. Similarly,
for graph classification, IB may promote the pooling function to learn the
informative pooled representation of local communities with minimal
redundancy, which would lead to distinct representations. Therefore, in this
work, we leverage IB for graph pooling, which encourages the pooled
representations for each community to be maximally informative about the
target graph label while discouraging to accumulate irrelevant information.
Hence, it helps to achieve minimal sufficient graph-level summary
representation.

In summary, our key contributions are as follows -

  * •

We propose a novel graph pooling (HIBPool) method to capture structural and
feature information from communities/local substructures in a hierarchical
manner to achieve more comprehensive graph-level representations.

  * •

We introduce more expressive structure-aware readout function DiP-Readout to
differentiate discriminative substructures/communities.

  * •

We leverage the IB principle into pooling method and demonstrate its
effectiveness in improving the pooling capability on graph classification.

  * •

We empirically show that our model HIBPool significantly outperforms the
state-of-art approaches on six graph classification benchmarks.

  * •

We empirically show that our model HIBPool is resilient to the feature
adversarial attacks.

##  II Background and Related works

###  II-A Graph Neural Networks (GNNs).

In recent years, a plethora of research has been conducted to learn either
node-level or graph-level meaningful representations through GNNs [30].
Several approaches [10, 9, 31, 4] follow a neighborhood aggregation scheme,
where the representations of a node is computed through iterative aggregation
and transformation of neighbor-node feature-information. They are also called
as Message-Passing Networks (MPN) since nodes aggregate messages from
neighbors through edges. Those are inherently flat and ineffective in learning
hierarchical structural information in a graph.

###  II-B Non-topological Graph Pooling:

The majority of graph pooling methods perform pooling to reduce the graph
structure based on node embeddings that hardly reflect graph topological
information. To learn hierarchical representations of graphs, DIFFPOOL [12]
aims to learn cluster assignment matrix from node features and map the nodes
to different clusters and thus generate coarser graph for the next layers. By
utilizing graph convolution SAGpool [3] determines self-attention scores, and
the top $k$ portion of the nodes are retained in the next pooling layer.
Similarly, inspired by the encoder-decoder based U-Net architectures widely
used for image augmentation in CNNs, Graph U-Nets [5] proposed pooling
operation gPool to adaptively select nodes based on their scalar projection
values on a trainable projection vector and inverse operation gUnpool for
upsampling. ASAP [2] proposed self-attention to determine the importance of
each node and learn the soft cluster assignment while GXN [11] introduce VI-
Pool to select the informative set of vertices that maximizes the mutual
information between node and neighborhood features. All of those approaches
are local pooling approach that pool graphs in a hierarchical way. In
contrast, global pooling methods use summation or neural networks to pool all
node embeddings into a global summary representation.

###  II-C Topological Graph Pooling

Recent few works exploit graph topology to pool the graph. [15] pool the graph
using cliques, i.e., every node is connected with all other nodes in a group.
The hard constraint of clique potentially destroys the relationships among
nodes in a graph. Consequently, there are few relaxed concepts such as
$n$-cliques, $n$-clans, $k$-plexes, etc., to partition the graph. [14]
proposes a pooling method built on the concept of graph $k$-Plexes, i.e.,
pseudo-cliques where every node has direct ties to $n-k$ members in a group of
size $n$. In other words, a clique is equivalent to $1$-plex. Still, the
constraint of $k$-plex is too strict to find all interesting structural
relationships. It is naturally impossible to determine a fixed value of $k$ to
find out all structures. [13] utilize the formulation of the $k$-way Min-Cut
problem as a regularization term to guide the pooling function towards
capturing the local structures in the graph. This partitioning algorithm
depends on the value of hyper-parameter $k$. These limitations render the
above methods unsuitable for capturing community structures in a graph.

#### Community-based Graph Pooling

Community detection is of great importance in various disciplines such as
sociology, biology, computer science, etc., where systems are often
represented as graph networks [17, 22]. Communities uncover important
topological and functional features in graphs [24]. These topological and
functional feature information can enrich the embeddings of graphs. Newman-
Girvan modularity in Equation 1, the most popular global quality function, is
introduced to detect the community structures of the graph [32, 22] where the
high values of modularity indicate the good communities. The methods based on
the modularity maximization [33, 23] aims to identify groups of nodes that are
more densely connected than one would expect from a statistical null model of
the network. These methods do not require the number of communities or the
size of communities as predefined hyper-parameter and can balance the groups’
size in terms of their total connectivity. Louvain algorithm [23] is likely
the best method to find communities, especially in biological graphs [24], and
faster. However, the existing methods fail to explicitly leverage the
advantage of inherent community structure into graph pooling that is purely
topological, nonparametric, and more interpretable than non-topological
pooling methods as there are no learnable parameters. Therefore, in this
paper, we introduce community-based hierarchical pooling by using Louvain
algorithm [23] to detect the optimal set of communities.

###  II-D Preliminaries:

We consider a graph $\mathcal{G}=(V,A,X)$ with $N=|V|$ where $V$ and $A$ are
the set of nodes and adjacent matrix respectively. $X\in R^{N\times f}$
presents the node feature matrix. Graphs are associated with a class label $y$
from a defined set of labels, $Y=\\{1,\dots,c\\}$. The goal of graph
classification task is to learn a mapping function
$F(\mathcal{G}):\mathcal{G}\longrightarrow Y$ and predicts the labels of
unseen graphs.

Figure 3: To determine different substructures present in graph data, the
input graph is partitioned into communities where intra-community nodes are
represented with same color and they are densely connected than inter-
community nodes. After aggregating features from neighborhood using Messaging
Passing Network (MPN) we apply structure-aware DiP-Readout to obtain
distinguishable representations of communities with different substructures
while the pooled graph of next layer is created with the edges connecting
different communities. Summary representation is obtained by a readout
function after the final layer. Optimizing with IB principal ensure minimal
redundancy from graph data with sufficient information to predict the class
labels.

##  III HIBPool: Hierarchical IB-guided Pooling

We apply our proposed HIBPool on each layer graph
$\mathcal{G}^{(l)}=(V^{(l)},A^{(l)},X^{(l)})$ and form coarser pooled graph
$\mathcal{G}^{(l+1)}=(V^{(l+1)},A^{(l+1)})$ for next layer $(l+1)$. Here,
$V^{(l)}$ and $A^{(l)}$ are the set of nodes and adjacent matrix respectively,
$N^{(l)}=|V^{(l)}|$ at layer $l$. $\mathcal{G}^{(1)}$ denotes the given input
graph. Figure 3 shows the brief overview of HIBPool that is described
elaborately in following sections.

###  III-A Community Detection

Community structures unveil interesting relationships among nodes in a graph.
Intra-community nodes are densely interconnected and probably share common
properties and/or play similar roles within the graph. Analogous to standard
CNNs, we consider local communities in the graph as local patch and compute
pooled latent representation for each community (patch). We employ Louvain
algorithm [23] to partition the graph into communities by optimizing the
modularity $\mathcal{Q}$ (in Eq. 1) in a bottom up fashion where $d^{(l)}_{v}$
is degree of node $v$, $|A^{(l)}|$ is total number of edges, $\delta_{uv}=1$
when both nodes $u$ and $v$ are in the same community otherwise $0$.

$\mathcal{Q}(P^{(l)})=\frac{1}{2|A^{(l)}|}\sum_{u,v}[A^{(l)}_{uv}-\frac{d^{(l)}_{u}d^{(l)}_{v}}{2|A^{(l)}|}]\delta_{uv}$
(1)

We finds a set of communities, $P^{(l)}=\\{P^{(l)}_{1},P^{(l)}_{2},\dots\\}$
in graph $\mathcal{G}^{(l)}$ under the optimal partitioning that would
maximize the modularity as following,

$P^{(l)}=\operatorname*{arg\,max}_{\mathcal{P}_{i}\in\mathcal{P}}\mathcal{Q}(\mathcal{P}_{i})$
(2)

where the set of all possible partitioning is
$\mathcal{P}=\\{\mathcal{P}_{1},\mathcal{P}_{2},\dots\\}$. Afterwards,
community mapping matrix $M^{(l)}\in R^{N^{(l+1)}\times N^{(l)}}$ is computed
by following the Eq. 3 where $P^{(l)}=\\{P^{(l)}_{1},P^{(l)}_{2},\dots\\}$ and
$N^{(l+1)}=|P^{(l)}|$. In Figure 3, different colors indicate different
community structures.

$M^{(l)}=\\{M^{(l)}_{(i,v)}=1|\forall v\in P^{(l)}_{i},P^{(l)}_{i}\in
P^{(l)},v\in V^{(l)}\\}$ (3)

We use the mapping $M^{(l)}$ to compute the latent representations for each
community and to construct coarser graph for next pooling layer. In Figure 3,
each colored node in the pooled graph $\mathcal{G}^{(l+1)}$ corresponds to a
particular community in $\mathcal{G}^{(l)}$.

###  III-B Feature Propagation

Before applying DiP-Readout to compute the representations of each community,
we apply a message passing network (MPN) to propagate features across neighbor
nodes as follows:

$Z^{(l)}=ReLU(X^{(l)}*W^{(l)}_{1}+A^{(l)}X^{(l)}W^{(l)}_{2})\in
R^{N^{(l)}\times h}$ (4)

where $W_{1}^{(l)}$, $W_{2}^{(l)}$ are weights at layer $l$ and $h$ \- output
dimension.

Figure 4: Overview of A) MHIBPool with varying the value of resolution
parameter $\alpha$; B) Multi-layer HIBPool

###  III-C Discriminative Pooling Readout, (DiP-Readout)

In Figure 1, we show that simple readout functions fail to distinguish
different structures. Thus, the simple readout functions limit the
generalization ability of a model for graph classification. For the node
classification tasks, Corso et al. [25] demonstrates that a hybrid aggregator
PNA consisting of four simple aggregators with three degree-scalers can
effectively increase the ability to capture different neighborhood structures.
Nevertheless, it is to be noted that directly incorporating PNA as a readout
function is not enough to discriminate different substructures. We can see
that the degree-based scaling of PNA could not differentiate two substructures
in Figure 2. Therefore, we utilize the topological features such as degree,
clustering coefficient, and betweenness centralities, etc., to design a
structure-aware pooling readout that generates pooled representation for each
community while preserving different substructures. To define the
discriminative pooling readout (DiP-Readout), let $C\in
R^{|V^{(l)}|\times\kappa}$ where $\kappa$ is the number of centralities that
we consider here. Then we normalize the centralities as follows,

$\forall v\in
V^{(l)},i=\delta_{v},\widehat{C}[v,j]=\frac{exp(C[v,j])}{\sum_{u\in
P^{(l)}_{i}}exp(C[u,j])}$ (5)

where, $\delta_{v}$ denotes the community membership of node v. Next, we
compute scaled-embeddings of nodes by multiplying centralities with the
embeddings $Z^{(l)}$. We get different scaled-embeddings of nodes for each
centrality. We concatenate $\kappa$ scaled-embeddings with no scaled-
embeddings in following,

$\widehat{Z}^{(l)}=Z^{(l)}||(\widehat{C}\odot Z^{(l)})\in R^{|V^{(l)}|\times
h(\kappa+1)}$ (6)

where $\odot$ \- broadcasted hadamard product, $||$ \- concatenation,
$|\cdot|$ \- cardinality.

It is to be noticed that each community is a multiset of node feature vectors.
In order to discriminate between multisets of features, we compute a statistic
over node features within a community such as Sum, Mean, Max, Min as follows,

$\begin{split}&Z_{sum}=M^{(l)}\widehat{Z}^{(l)}\in R^{|P^{(l)}|\times
h(\kappa+1)}\\\ &Z_{mean}=D^{-1}M^{(l)}\widehat{Z}^{(l)}\in R^{|P^{(l)}|\times
h(\kappa+1)}\\\ &Z_{max}[i,:]=\max_{j\in p_{i}}(\widehat{Z}^{(l)}_{j})\in
R^{|P^{(l)}|\times h(\kappa+1)}\\\ &Z_{min}[i,:]=\min_{j\in
p_{i}}(\widehat{Z}^{(l)}_{j})\in R^{|P^{(l)}|\times h(\kappa+1)}\end{split}$
(7)

where $D$ is diagonal degree matrix of $M^{(l)}$, $h$ \- output dimension,
$||$ \- concatenation, $|\cdot|$ \- cardinality.

After computing the statistics for communities, we concatenate and pass them
through MLP to compute discriminative representations of each community as,

$H^{(l)}=MLP(Z_{sum}||Z_{mean}||Z_{max}||Z_{min})\in R^{|P^{(l)}|\times 2h}$
(8)

Afterwards, we compute the summary representation of the graph at pooling
layer $l$ as follows,

$S^{(l)}=\frac{1}{|P^{(l)}|}\sum_{i=1}^{|P^{(l)}|}H^{(l)}[i,:h]\in R^{1\times
h}$ (9)

###  III-D Pooled Graph

We compute $\mathcal{G}^{(l+1)}=(A^{(l+1)},X^{(l+1)})$ and pass it as an input
for next pooling layer as following,

$\begin{split}A^{(l+1)}&=M^{(l)}\widehat{A}^{(l-1)}M^{(l)T}\in
R^{|P^{(l)}|\times|P^{(l)}|}\\\ X^{(l+1)}&=H^{(l)}[,:h]\end{split}$ (10)

where $\widehat{A}^{(l)}=A^{(l)}+I$, T denotes transpose operation. Equation
10 ensures the connectivity of pooled graph where any two pooled node i and j
are connected if there is any edge between node pair $u\in P^{(l)}_{i}$ and
$v\in P^{(l)}_{j}$.

###  III-E Deriving Information Bottleneck for Pooling

To incorporate IB principle [29] into pooling, it requires pooled
representations $S^{(l)}$ to minimize the redundant information from
($A^{(l)},X^{(l)}$) and maximize the information to predict class label $Y$.
The intractability of general IB formulation leads us to approximate our
objective in Equation 11 with some additional assumptions. Here, we rely on
the community-dependence assumption: each pooled representation $H^{(l)}_{i}$
only depends on respective community structure, $P^{(l)}_{i}$ and intra-
community node features, $\\{X_{v}^{(l)}|v\in P^{(l)}_{i}\\}$ while being
independent of the rest of the graph.

$IB_{\beta}(A^{(l)},X^{(l)},Y;S^{(l)})=[-I(Y;S^{(l)})+\beta
I(A^{(l)},X^{(l)};S^{(l)})]$ (11)

With our community-depedence assumption, we can simply estimate
$I(A^{(l)},X^{(l)};S^{(l)})$ as follows,

$\small
I(A^{(l)},X^{(l)};S^{(l)})\longrightarrow\sum^{|P^{(l)}|}_{k=1}I(A^{(l)},X^{(l)};H^{(l)}_{k})$
(12)

To instantiate the objective, following GIB [29], we set an upper bound of
$I(A^{(l)},X^{(l)};H^{(l)}_{k})$, and estimate the upper bound by using the
sampled $H^{(i)}_{X,j}\sim\mathcal{N}(\mu_{i,j},\sigma^{2}_{i,j})$ where
$\mu_{i,j}=H^{(i)}_{j}[j,:h]$ and $\sigma^{2}_{i,j}=H^{(i)}_{j}[j,h:]$ as,

$\begin{split}&I(A^{(l)},X^{(l)};H^{(l)}_{k})\rightarrow\log\frac{P(H^{(l)}_{X,k}|A^{(l)},X^{(l)})}{Q(H^{(l)}_{X,k})}\\\
&=\log\Phi(H^{(l)}_{X,k};\mu_{l,j},\sigma_{l,j}^{2})\\\ &-\log(\sum_{v\in
P_{i}}\Phi(H^{(l)}_{X,k};\mu_{l-1,v},\sigma^{2}_{l-1,v}))\\\ \end{split}$ (13)

To estimate the upper bound of $I(A^{(l)},X^{(l)};H^{(l)}_{k})$, we assume for
any node $v$, the given input feature
$X_{v}\sim\mathcal{N}(\mu_{0,v}=0,\sigma^{2}_{0,v}=1)$.

Again, to estimate $I(Y;S^{(l)})$, we use the cross-entropy loss i.e.,

$\small I(Y;H^{(l)})\longrightarrow-\sum_{i=1}^{|Y|}Y_{i}\log(S^{(l)}W_{out})$
(14)

Finally, plugging the Equation LABEL:eq:xib and Equation 14 into Equation 11,
we utilize the benefit of IB principle to learn minimally redundant maximally
informative graph-level representations through training model by optimizing
the objective function.

###  III-F Multi-layer HIBPool

Similar to Figure 4(B), we stack $L$ HIBPool layers to compute the graph-level
representation hierarchically. For each pooling layer $l$, we execute HIBPool
to compute the summary representation $S^{(l)}$. Afterward, we concatenate the
summary representations of all layer and take the weighted summation as,

$S=\sum_{i=1}^{L}W_{f}S^{(l)}$ (15)

$S$ is considered as the final representation and is used to predict the label
of input graph.

###  III-G MHIBPool: Multiresolution HIB Pooling

We can see in Equation 1 that only node pairs belonging to the same community
contribute to the sum, so we can group these contributions of intra-community
nodes together and rewrite the sum over the node pairs as a sum over the
community, as follows,

$Q(P^{(l)})=\frac{1}{2}\sum^{N^{(l)}}_{c=1}[\frac{e^{(l)}_{c}}{|A^{(l)}|}-(\frac{(d^{(l)}_{c})^{2}}{2(|A^{(l)}|)^{2}})]$
(16)

where $N^{(l)}$ is the total number of communities, $e^{(l)}_{c}$ is the total
number of intra-community edges, and $d^{(l)}_{e}$ is the sum of the degree of
intra-community nodes.

To address the Louvain algorithm’s resolution limitation, we use multi-scale
modularity [34] to detect the multiresolution communities. The multi-scale
modularity can be defined as,

$Q(P^{(l)})=\sum^{N^{(l)}}_{c=1}[\alpha\frac{e^{(l)}_{c}}{|A^{(l)}|}-(1-\alpha)(\frac{d^{(l)}_{c}}{2(|A^{(l)}|)})^{2}]$
(17)

In Equation 17, $0<\alpha<1$ is the resolution parameter. We see that for
$\alpha=\frac{1}{2}$, the equation is equivalent to 16. Therefore, we can find
multiresolution communities by varying the value of $\alpha$ (see Figure
4(A)). Similar to multi-layer HIBPool in Figure 4(B), the multi-layer MHIBPool
consists of multple consecutive layers of MHIBPool.

##  IV Experimental Analysis

In this section, we validate the efficacy of our model by conducting extensive
experimental analysis. We present the datasets and experimental setup,
comparison with baselines, the effectiveness of the DiP-Readout function, and
the effect of adding the IB principle. Finally, we show the resilience of our
model against feature perturbations.

###  IV-A Datasets and Experimental Setup

To investigate our research findings mentioned in the introduction section and
evaluate our proposed model’s efficacy, we have performed experiments on six
popular graph classification benchmark datasets: ENZYMES, DD, PROTEINS, NCI1,
NCI109, and FRANKENSTEIN [2]. Those datasets are bioinformatics datasets and
used for performance comparison in several state-of-the-art pooling methods
e.g. DIFFPOOL [12], SAGPool [3], Graph U-Net [5], and GXN [11] etc. In all the
experiments, we followed 80%, 10%, and 10% split for training, validation, and
test set, and then performed 10-fold cross-validation, and reported the
average accuracy for evaluation. We perform fine-tuning for the hyperparameter
of IB term $\beta=0.01$. For all the experiments, we use Adam optimizer and
set the learning rate as 0.01. We set $L=2$ for all experiments. For our
baseline models, we follow the exact experiment setup that is mentioned in
their manuscript.

Model Dataset | ENZYMES | D&D | PROTEINS | NCI1 | NCI109 | FRANKENSTEIN
---|---|---|---|---|---|---
DIFFPOOL [12] | 62.53 ± 2.74 | 66.95 ± 2.41 | 68.20 ± 2.02 | 62.32 ± 1.90 | 61.98 ± 1.98 | 60.60 ± 1.62
KPLEXPOOL [14] | 39.67 ± 7.52 | 77.76 ± 2.92 | 75.11 ± 2.80 | 79.17 ± 1.73 | - | -
SAGPool [3] | 43.99 ± 4.23 | 76.45 ± 0.97 | 71.86 ± 0.97 | 67.45 ± 1.11 | 67.86 ± 1.41 | 61.73 ± 0.76
Graph U-Net [5] | 50.21 ± 5.53 | 81.34±3.23 | 76.78 ± 4.23 | - | - | -
MinCutPool [13] | - | 80.8±2.3 | 76.5±2.6 | - | - | -
ASAP [2] | 29.43 ± 2.34 | 76.87 ± 0.7 | 74.19 ± 0.79 | 71.48 ± 0.42 | 70.07 ± 0.55 | 66.26 ± 0.47
GXN [11] | 57.50 ± 6.1 | 82.68 ± 4.1 | 79.91 ± 4.1 | - | - | -
| HPool (Mean) w/o IB
---
60.0 ± 2.3 | 81.32 ± 2.12 | 77.12 ± 0.34 | 74.32 ± 0.85 | 76.02 ± 1.94 | 62.09 ± 1.51
| HPool (DiP-Readout) w/o IB
---
61.33 ± 1.66 | 79.23 ± 1.43 | 78.03 ± 2.14 | 77.37 ± 0.73 | 75.05 ± 1.69 | 64.66 ± 2.39
| GlobalPool w/ IB
---
63.3 ± 1.95 | 80.66 ± 0.98 | 78.45 ±1.5 | 76.18±1.34 | 75.1 ± 1.24 | 63.78 ± 1.21
| RandomPool w/ IB
---
62.0 ± 2.34 | 77.27 ± 3.3 | 76.28 ± 2.21 | 74.20 ± 1.95 | 72.39 ± 2.5 | 66.12 ± 2.43
| HIBPool (Mean)
---
71.67 ± 1.12 | 82.95 ± 1.56 | 81.98 ± 0.48 | 79.80 ± 0.69 | 77.73 ± 1.73 |  | 71.42 ± 1.43
---
| HIBPool
---
73.33 ± 1.8 | 83.05 ± 0.34 | 82.88 ± 0.23 | 83.45 ± 0.73 | 79.66 ± 0.72 |  | 73.50 ± 0.23
---
| MHIBPool
---
75.0 ± 1.20 | 83.20 ± 1.27 | 80.35 ± 0.31 | 83.48 ± 0.90 | 79.93 ± 1.43 | 71.42 ± 0.53
TABLE II: Comparison with Baselines: ’-’ denotes that results are not publicly
available.

###  IV-B Other Variants of HIBPool

  * •

HPool (Mean) w/o IB - It finds communities and apply Mean readout to compute
one higher-order representation for each community through optimizing simple
classification loss.

  * •

HPool (DiP-Readout) w/o IB - It employs DiP-Readout instead of Mean readout
function.

  * •

GlobalPool w/ IB - It applies one-layer MPN and then takes the mean of all
node embeddings to compute the graph-level representation. It utilizes the IB
principle to learn parameters.

  * •

RandomPool w/ IB - It replaces the community assignment of HIBPool by random
matrix $M\sim\mathcal{N}(0,1)$.

  * •

HIBPool (Mean) - It replaces Dip-Readout function of HIBPool by Mean pooling
function.

###  IV-C HIBPool Consistently Outperforms Baselines

We compare HIBPool and its variants to recent state-of-arts methods including
DIFFPOOL [12], KPLEXPOOL [14], SAGPool [3], Graph U-Net [5], MinCutPool [13],
ASAP [2] and GXN [11]. Table II demonstrates the superior performance of our
model HIBPool on all benchmark datasets for graph classification. We can see
that simple HPool performs better than all baselines on 3 out of 6 datasets
and very close to GXN (on D&D, PROTEINS) and ASAP(on FRANKENSTEIN). It
indicates that still community/local subgraph structural information promotes
learning more informative graph-level representation. In contrast, learned
assignment matrix [12] and Top-K selection strategies could not capture
structural information explicitly. Again, the impressive performance of
GlobalPool with IB over state-of-art methods on some benchmarks indicates that
IB leads to learning more informative graph-level representation with minimal
redundancy. Even HIBPool (Mean), which has the limited potential to capture
local structural information, still outperforms all existing methods, HPool,
and GlobalPool by a reasonable margin. This clearly indicates that both
community structural information and IB regularization lead to attaining
better accuracy. Furthermore, the structure-aware DiP-Readout function in
HIBPool has more expressive potential to learn distinguishable representation
for communities, even with homogeneous node features. Therefore we can
conclude that several key factors, such as i) preserving structural
information explicitly, ii) capturing discriminative community structures, and
iii) minimizing the redundancy, promote HIBPool to learn sufficient
informative representations, which leads to outstanding performance on all
benchmarks.

###  IV-D Simple Pooling function is Not Enough

State-of-art methods apply multiple convolutions to learn node embedding by
utilizing graph structures. [16] showed that applying multiple convolutions
may result in homogeneous node embeddings. It limits the effectiveness of
simple pooling functions such as Sum, Mean, Max, Min, etc., to learn
meaningful local community/subgraph structures (as shown in Fig. 2). In Table
II HIBPool consistently shows greater average accuracy than HIBPool (Mean) on
all six datasets. It implies that DiP-Readout is more expressive to learn
meaningful local community structures than simple Mean function. However,
HPool (Mean) shows competitive performance to HPool (DiP-Readout), which
indicates that under simple classification loss, the DiP-Readout function
suffers from redundant information. It may overshadow the expressive power of
DiP-Readout. On the other hand, IB in Eq. 11 promotes DiP-Readout to minimize
redundant information by imposing constraints on the preserved information of
input data while maximizing the preserved relevant information to predict the
target.

###  IV-E IB Principle Strengthens Local Pooling.

To demonstrate the efficacy of Local Pooling to learn graph-level
representation, we compare HIBPool to two other variants GlobalPool where the
global mean pooling is performed, and RandomPool, where the random cluster
assignment has been used ignoring intrinsic community structures in the graph.
All three approaches are optimized with the supervised loss along with the IB
principle. [16] concludes that graph convolutions is the main reason behind
the success of local pooling approaches and shows that pooling does not play a
significant role in performance improvement over global pooling. Table II
demonstrates that HIBPool outperforms both GlobalPool and RandomPool. These
results indicate that the ability to capture the local/community structural
information through local pooling is essential to produce better informative
and distinguishable representations of graphs for classification. Here, IB
promotes our local pooling method to capture this important structural
information, resulting in superior performance.

Models |  | Perturb.
---
Ratio
NCI1 | D&D | PROTEINS
DIFFPOOL | 0.0 | 62.32 ± 1.90 | 66.95 ± 2.41 | 68.20 ± 2.02
0.5 | 49.48 ± 0.01 | 46.95 ± 0.06 | 50.84 ± 0.03
1.0 | 42.72 ± 0.02 | 46.27 ± 0.02 | 40.51 ± 0.02
1.5 | 39.70 ± 0.02 | 45.93 ± 0.02 | 37.96 ± 0.023
2.0 | 35.09 ± 0.03 | 45.83 ± 0.06 | 36.44 ± 0.05
ASAP | 0.0 | 71.48 ± 0.42 | 76.87 ± 0.7 | 74.19 ± 0.79
0.5 | 51.45±0.02 | 66.78 ± 0.06 | 57.41 ± 0.04
1.0 | 50.28±0.02 | 64.11 ± 0.03 | 49.89 ± 0.03
1.5 | 49.87±0.01 | 63.82 ± 0.02 | 45.00 ± 0.04
2.0 | 48.17±0.01 | 61.28 ± 0.03 | 44.25 ± 0.04
HIBPool | 0.0 | 83.45 ± 0.73 | 83.05 ± 0.34 | 82.88 ± 0.23
0.5 | 81.50 ± 0.24 | 82.48 ± 1.59 | 81.38 ± 0.42
1.0 | 81.17 ± 0.09 | 81.35 ± 0.00 | 81.08 ± 0.00
1.5 | 79.73 ± 2.25 | 80.92 ± 0.42 | 81.08 ± 0.00
2.0 | 79.35 ± 1.90 | 80.5 ± 0.68 | 81.08 ± 0.00

TABLE III: Feature Perturbation Analysis

###  IV-F Robustness Against Feature Perturbations

To observe the resilience of the model in feature attack, we perturb the input
feature. We add gaussian noises to each dimension of node features as follows,

$\widehat{X^{(l)}}=X^{(l)}+\gamma\cdot r\cdot\epsilon$ (18)

where r is the mean of maximal value of each node’s features and
$\epsilon\sim\mathcal{N}(0,1)$ and $\gamma$ is noise-ratio. We test models’
performance with $\gamma\in\\{0.5,1,1.5,2.0\\}$. As reported in Table III, the
performance of DIFFPOOL and ASAP degrade significantly when the input feature
is perturbed compared to our model. These state-of-the-art models can not
distinguish the noisy information from the input features, resulting in
inferior performance when the input is perturbed.

###  IV-G Analysis on Multiresolution HIBPool (MHIBPool)

We varied the value of $\alpha$ and observed results in Table II. MHIBPool
shows the best performance in the ENZYME, D&D, NC1, and NC109 datasets, and in
the rest of the datasets, its performance is competitive to HIBPool.

To evaluate how much the stability of communities discovered by the Louvain
algorithm affects our model’s performance, we detect communities once for each
run. Then we take the average of the accuracy of all runs. The smaller
standard deviations of accuracies across multiple runs indicate that our model
consistently performs with the discovered communities for all graph datasets.

##  V Conclusion

We introduce HIBPool, a novel community structure-aware pooling method for
graph classification. We leverage the Information Bottleneck principle into a
novel DiP-Readout function to learn graph-level representation hierarchically
that is minimal sufficient information for the task but maximal informative to
target. IB helps HIBPool to attain resilience to feature perturbations.
Empirical results exhibit the efficacy of HIBPool on all benchmark datasets.
Further extensive analysis validates our findings, i.e., i) local pooling that
is capable of important community structures are superior to global pooling,
ii) a sophisticated pooling function is required to preserve distinguishable
structures even it is indispensable for graphs with homogeneous node features,
and iii) IB principle promotes pooling function to learn distinct
representations. In future works, there are scopes to work on the limitations
of community detection algorithms such as resolution limit, stability, etc.

##  VI Acknowledgements

This project is supported by a grant from the Independent University
Bangladesh and ICT Division of Bangladesh Government.

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# Fractional Dimensional Approach to Dielectric Tuning Effects on Excitonic
Parameters in 2D semiconductor materials

Lakshminarayan Sharma  Carlos Rodriguez-Fernandez  Humeyra Caglayan  Faculty
of Engineering and Natural Sciences, Tampere University, Tampere 33880 Finland
<EMAIL_ADDRESS>
###### Abstract

We demonstrated the potential of the fractional dimensional approach to
understand exciton parameters in the exemplary atomically thin semiconductor
material, a monolayer of WS2. This approach has proved to be successful in
finding the exciton binding energy and quasiparticle bandgap for the WS2
monolayer in varying dielectric environments. A tuning of the quasiparticle
bandgap and binding energy by 141 meV and 188 meV, respectively, has been
achieved by varying the dielectric of the environment from 1.52 to 8.1. The
approach is justified by comparing the changes in the binding energy with the
computational results from the Quantum Electrostatic Heterostructures model.
The fractional dimension found through the excitonic Rydberg series is close
to 2.8 for WS2 monolayer in all different dielectric surroundings. Thus, this
approach provides a rapid and robust method for determining the binding energy
of excitons in 2D semiconductors independent of the particular dielectric
environment.

###### keywords:

2D materials, Binding energy, Excitons, Fractional dimension, Dielectric
Tuning

##  1 Introduction

Two-dimensional transition metal dichalcogenides (2D-TMDCs) exhibit a plethora
of remarkable and intriguing properties. Notable among these is the transition
from an indirect to a direct bandgap in monolayers, a substantial magnitude of
spin‐orbit splitting, the phenomenon of valley polarization, and the emission
of single photons 1, 2, 3, 4, 5, 6. The presence of tightly bound
electron‐hole pairs, known as excitons, is central to the optical
characteristics of these atomically thin materials. The excitons found in
these materials possess considerable binding energies. This notable attribute
arises from the combination of reduced dielectric screening in two dimensions
and the quantum confinement effects inherent to such systems 7. One
particularly fascinating aspect is the pronounced sensitivity of the exciton
binding energy to the spatial dielectric environment 8, 9. The electrical
manipulation of these excitonic states enables the tuning of light‐matter
response 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 6, 7 and creation of
atomically thin devices such as transistors 4, 21, photodetectors 22, 23, p-n
diode 24, 25, 26, LEDs 25, 17, 27 and solar cells 25, 28, 27.

While the concept of dielectric engineering in 2D materials is well-
established, the intricate relationship between the dielectric environment and
specific excitonic parameters, such as the exciton binding energy (BE) and
quasiparticle bandgap (QBG), has remained a subject of intense investigation
8, 29, 30, 9, 31, 32, 33, 34, 35, 36, 37. The quantitative knowledge of both
BE and QBG is necessary for optimizing 2D materials for various electronic and
photonics applications. Addressing this challenge conventionally involves
employing integer-dimensional models, but these models frequently prove
inadequate in capturing the unique attributes of 2D TMDCs. These
characteristics encompass the monolayer’s finite thickness in the out-of-plane
direction and the abrupt potential changes that manifest at the interfaces
within the monolayer. To overcome this limitation, new approaches based on
integrating a quasi-2D Coulomb potential (such as the Ohno Potential 38) or
truncating the Coulomb potential 39 have been proposed. However, these all
require extensive numerical solutions. Yet, the fractional dimensional
approach (FDA), which emerges as a powerful method, precisely accounts for the
finite out-of-plane extension’s influence. However, these all require
extensive numerical solutions. Yet, the fractional dimensional approach (FDA),
which emerges as a powerful method, precisely accounts for the finite out-of-
plane extension’s influence by not restricting the system to an absolute 2
dimensions.

The FDA is a formalism based on the non-integer (Hausdorff) dimension -
$\alpha$ in which the principal quantum number ‘n’ remains a valid quantum
number for the Coulomb Potential 40, 41, 42. The fractional dimension results
from the exciton’s constrained motion in an atomically thin material due to
the exciton-lattice interaction 41, 42, 43. Thus, the fractional dimension
manifests the exciton dynamics 41, 44. Since its inception in 1990 45, 44, 46,
the FDA has been applied to the study of the excitons in quantum well 42, 43,
47, 48, 49, 50, quantum wires 42, 43, exciton-phonon interaction 50,
biexcitons 51 and exciton-exciton interaction 47.

In contrast to the numerical approach for solving the Schr$\ddot{o}$dinger
equation in 2D dimension with Rytova-Keldysh potential, the FDA provides an
analytical and systematic solution to the problem. This approach represents a
lightweight analytical model with efficient and excellent tunability as the
fractional dimension $\alpha$ inherently encompasses the anisotropy caused by
the environment and the influence of the perturbative potential interacting
with the exciton 43, 42. Along with this, when the first and second excited
states of the exciton transition energies are known, the method enables the
direct calculation of the BE 48.

In this study, we employ a combined approach of linear reflectance
spectroscopy and the FDA to effectively measure exciton BE and QBG while
considering dielectric environment effects. We show, for the first time as of
our knowledge, the experimental application of this method to the dielectric
tuning of exciton BE and QBG in 2D TMDCs. This integrated methodology provides
a streamlined pathway for determining exciton parameters in 2D TMDCs. Our
approach not only reveals the excitonic landscape in these materials but also
sheds light on the tunable nature of the QBG, contingent on the dielectric
environment of 2D TMDCs. This method can be a universal tool for exploring
excitonic properties in diverse 2D materials. Additionally, it uniquely allows
for determining the overall band type in atomically thin horizontal van der
Waals heterostructures, enhancing our understanding of their electronic
structure.

Figure 1: (a) Illustration of the electronic bandgap and the exciton Rydberg
series. Both the trion and exciton states are shown. Along with this, the key
fundamental parameters, such as trion BE, optical bandgap (OBG), exciton BE,
and the QBG, are represented. (b) Simplified schematic of (a). (c) Diagram of
the various substrate structures studied in this work. (d) An illustration for
the excitation of WS2 monolayer by 2.330 eV (532 nm) photon and the excitonic
emission at 2.025 eV (612 nm). (e) The PL spectrum for 1L-WS2, the trion
emission at 1.999 eV, is shown in green.

An illustration of the electronic bandgap and the exciton Rydberg states are
shown in Figure 1 (a) and (b). Due to quantum confinement and reduced
dielectric screening, the Coulomb attraction between the charge carriers is
strongly enhanced at the nanoscale regime. This creates a neutral two-body
quasiparticle with energy lower than the quasiparticle (single particle)
bandgap. The quantum states of the exciton are labeled by ‘n’ throughout the
manuscript. The term BE will correspond to the ground state (n=1) exciton
binding energy throughout the paper.

##  2 Results and discussion

In this work, we use the FDA method to investigate the dielectric tuning of
excitonic parameters in monolayer (1L) WS2. We explore three distinct
dielectric environments: (1) monolayer on a polydimethylsiloxane (PDMS)
substrate, (2) monolayers directly on a silicon substrate, and (3) top hBN
(13.28 nm) encapsulated on a silicon substrate, for (1) and (2) air is the top
dielectric medium. A schematic representation of the various substrates used
in this study is shown in Figure 1 (c). To quantify the dielectric
environment, we define the average dielectric constant of the environment
surrounding the monolayer as $\kappa$ = $\epsilon_{top}+\epsilon_{bottom}$/2.
For PDMS, Si and hBN-Si substrate, we have $\epsilon_{PDMS}$ = 2.04 52,
$\epsilon_{Si}$ = 11.7 53, $\epsilon_{hBN}$ = 4.5 54, which gives $\kappa$ as
1.52, 6.35 and 8.1 respectively. These configurations allow us to
systematically assess the influence of different dielectric conditions on
excitonic properties in 2D TMDCs. The photoluminescence (PL) spectrum for 1L
WS2 at room temperature is shown in Figure 1 (e), which was obtained by
pumping at 532 nm. A schematic representation of the excitation of the 1L WS2
at 2.330 eV (green) and its excitonic emission at 2.025 eV (red) is shown in
Figure 1 (d). In the collected PL spectrum, we found the presence of trion at
1.999 eV, represented by the green peak at the lower energy in Figure 1 (e).
From the difference between the exciton and trion emission peaks, we
calculated the trion binding energy of 26 meV.

In tungsten (W) based 2D TMDCs, the energy separation between the spin-orbit
split A and B exciton is large enough to observe the A exciton Rydberg series
55, 29. This enables the determination of the exciton binding energy and the
quasiparticle bandgap by fitting the experimentally found excitonic Rydberg
series to the fractional dimensional model. To experimentally determine the
excitonic Rydberg series, we carried out straightforward and effective
reflectance contrast ($\Delta R/R$) spectroscopy, eliminating the need for
complex methods like two-photon or magneto-optical characterization. The
reflectance contrast (RC) is the normalized difference between the reflectance
from the atomically thin sample and the substrate. $\Delta
R/R=(R_{sample}-R_{substrate})/(R_{substrate})$, in the limit of an atomically
thin sheet of materials, depends on the real part of the conductivity which is
proportional to the absorption 56. The RC spectrum for the three different
samples of monolayer WS2 is shown in Figure 2 (a)-(c). The three major peaks
in those spectra correspond to the primary exciton transition A, B, and C. The
A exciton is the fundamental excitation corresponding to n = 1 states,
occurring at the hexagonal Brillouin zone’s K and K′ valleys. The A exciton
transition was at 2.011 eV, 1.992 eV, and 1.988 eV for 1L WS2 on PDMS, Si, and
top hBN encapsulation on Si substrate, respectively. This corresponds to a
redshift of 23 meV for the A exciton peak when changing the dielectric
constant of the environment from that of a PDMS to hBN-Si. This effect of the
external dielectric environment on the monolayer is such that the reduction in
binding energy adjusts the QBG renormalization. Thus, the optical band gap (1s
state energy) remains relatively unchanged (23 meV shift). The B exciton peak
originates from the spin-orbit split valence band transition at the K and K′
valleys of the hexagonal Brillouin zone. The separation between A and B
exciton provides the value of spin-orbit splitting, which was found to be
around 400 meV. The origin of C exciton is due to the singularities in the
joint density of states (JDOS) known as van Hove singularities in the band
diagram, which leads to band nesting regions where a small section or region
of the band structure belonging to the conduction and valence band are
parallel to one another 57, 58, 59, 60.

Here, our primary focus is on the A exciton and its Rydberg series. To reveal
the minute spectral features of the higher energy A exciton Rydberg states, we
took the 1st derivative of the RC spectrum as shown in Figure 2 (a)-(c) bottom
panel. In these 1st derivative plots, at the position of A, B, and C exciton
peaks, there is a clear change in the curvature, and these inflection points
are indicated with dashed lines. Observing the regions between the A and B
exciton inflection points, there are additional points of change in curvature,
which correspond to higher energy states of the A exciton as highlighted in
Figure 2 (a)-(c) bottom panel. The excited state n = 2 was observed at 2.203
eV, 2.118 eV, and 2.093 eV on PDMS, Si, and top hBN encapsulation on the Si
substrate, respectively. The excitonic Rydberg states (2s and 3s) for the
three samples are shown in the inset of Figure 2(a)-(c) bottom panel, where
the peak position is shown by the dashed lines representing the inflection
point. In comparison to the energy shift of the 1s state, the 2s state has
redshifted by a much larger amount, 110 meV, because of the renormalization of
the QBG due to an increase of the dielectric constant of the surroundings from
PDMS to hBN-Si. This shift of the 100s of meV is a strong indication of the
effect of the dielectric environment on the BE and QBG.

Figure 2: Measured reflectance contrast ($\Delta R/R$) spectra and its first
derivative for monolayer WS2 on (a) PDMS, (b) Si and (c) top hBN encapsulation
on Si substrate. The top panel plots label the A, B, and C exciton peaks. The
position of the A, B, and C exciton, along with the A exciton higher energy
states 2s and 3s are marked by dashed lines in the bottom panel.

The BE is the difference between the QBG and the exciton Rydberg state energy
(RSE):

$E_{n}^{BE}=E_{QBG}-E_{n}^{RSE}$ (1)

The eigenenergy solution for the non-integer dimensional space $\alpha$ is
given by 41

$E_{n}^{BE}=-\frac{\mu
R_{y}}{m_{e}\epsilon_{eff}^{2}}\dfrac{1}{(n+\dfrac{\alpha-3}{2})^{2}}$ (2)

where $\mu$ the reduced mass of the exciton, Ry is the Rydberg energy (13.6
eV), me is the free electron mass, $\epsilon_{eff}$ is the effective
dielectric constant seen by the exciton. For integer dimensions where $\alpha$
= 3 and 2, we get the known $\dfrac{1}{n^{2}}$ and $\dfrac{1}{(n-1/2)^{2}}$
dependence, respectively 48. We calculated the BE and QBG for the three
samples by fitting the RSE for the A exciton, using Equations 1 and 2.

Figure 3:  Experimentally measured exciton Rydberg state energies (RSE) for 1L
WS2 on the various substrates as a function of ‘n’. (a) The red, blue, and
black lines represent the FDA fit to the RSE data points for PDMS, silicon,
and top hBN encapsulation on a silicon substrate. (b) The horizontal red,
blue, and black line represents the QBG for PDMS, silicon, and top hBN
encapsulation on a silicon substrate, respectively, as calculated from the FDA
fit. (c) Schematic illustration of the effect of dielectric surroundings on
the exciton spatial size and the RSE. The modification of the strength of the
Coulomb interaction between electron-hole pair in two different dielectric
surroundings with $\epsilon_{1}<\epsilon_{2}$ leads to a decrease of the QBG
and BE.

By substituting Equation 1 in Equation 2, we get the final form of the
equation for the FDA fit. This final form contains four unknowns,
$\mu,\alpha,\epsilon_{eff}$, and the QBG. For the fit, we used the exciton
reduced mass of 0.16 me for 1L WS2 reported using magneto-optical spectroscopy
32. To ascertain the value of $\alpha$, we computed the following ratio
$\dfrac{E_{3s}-E_{1s}}{E_{2s}-E_{1s}}$, which only depends on the fractional
dimension $\alpha$ (see supplementary Section 2). Thus, the two unknown EQBG
and $\epsilon_{eff}$ were set as free parameters, and their values were
obtained through the fit. The FDA fit to the measured excitonic Rydberg series
for the three samples is shown in Figure 3 (a). For 1L WS2 on PDMS, Si and top
hBN encapsulation on Si substrate, the $\epsilon_{eff}$ is obtained from the
fit as 3.080 $\pm$0.75, 3.87 $\pm$ 1.13, 4.18 $\pm$ 1.06, respectively. The
calculated QBG from FDA and the exciton RSE series are also shown in Figure 3
(b). The details of the FDA for the three samples are summarized in Table 1.
To understand the prominent effect of the dielectric surrounding on the
exciton properties, we need to visualize the electric field distribution and
strength between an electron and the hole that constitutes an exciton. A
schematic diagram representing this is shown in Figure 3 (c) for two different
dielectric surroundings with $\epsilon_{1}<\epsilon_{2}$. For a higher
dielectric constant of the surrounding, there is a larger screening of the
Coulomb interaction between the electron-hole pairs, which reduces the
strength of the attractive potential and leads to an increase in the overall
size of the exciton 61. This leads to a decrease in the QBG and BE magnitude.

Substrate | $\alpha$ | QBG (eV) | BE (meV) | $\kappa$
---|---|---|---|---
1L WS2-PDMS | 2.86 | 2.272 $\pm$ 0.009 | 261 $\pm$ 9 | 1.52
1L WS2-Si | 2.82 | 2.165 $\pm$ 0.008 | 173 $\pm$ 8 | 6.35
hBN-1L WS2-Si | 2.85 | 2.131 $\pm$ 0.005 | 143 $\pm$ 5 | 8.1
Table 1: FDA results for the three samples used in this study. The fractional
dimension $\alpha$, QBG, B.E, and the average dielectric constant of the
environment $\kappa$ are summarized.

The bulk (3D) exciton BE (E3D) is linked to the BE as calculated through the
FDA by the following relationship42, 48:

$\frac{E^{FDA}}{E^{3D}}\approx\frac{108}{101}\dfrac{E_{2s}-E_{1s}}{E_{3D}}+\frac{20}{101}$
(3)

Using the experimentally found E3D for WS2 of 57 meV 62 and the E2s \- E1s
value as found in this work, we determined the value of EFDA as 216 meV, 146
meV, and 123 meV for 1L WS2 on PDMS, silicon, and hBN-Si substrates. Upon
inspection, it becomes clear that these values are in accordance with the
binding energy obtained from the FDA fit, as summarized in Table 1. We also
applied Equation 3 to find the EFDA for 1L WSe2. For which, we utilized the
established E3D value of 55 meV 62, coupled with the energy difference between
E2s \- E1s, which amounts to 130 meV 35 and 131 meV 37 from two distinct
research studies. Our calculations yielded EFDA values of 149 meV and 150 meV,
closely mirroring the experimentally observed values of 161 meV 35 and 168 meV
37 respectively, reported in the original studies.

The system’s dimensionality can be measured by adopting a physical parameter
dependent on the dimension as a valid criterion 45. Here, we have selected the
RSE’s binding energy ratio as such a criterion. The binding energy ratio for
the 1s:2s:3s states for the PDMS, Si, hBN-Si substrates are
$1:\dfrac{1}{3.78}:\dfrac{1}{13.73};1:\dfrac{1}{3.68}:\dfrac{1}{17.30}$; and
1:$\dfrac{1}{3.76}:\dfrac{1}{14.3}$, respectively. In contrast, when
considering the 2D and 3D systems with a Coulomb potential (1/r), the binding
energy ratios for the 1s:2s:3s states are $1:\dfrac{1}{9}:\dfrac{1}{25}$; and
1:$\dfrac{1}{4}:\dfrac{1}{9}$, respectively 63. Notably, the dimensionality of
monolayer WS2 within the three unique dielectric environments consistently
converges to a fractional dimension of approximately 2.8 and perfectly aligns
with the exciton Rydberg series. Upon comparison, it also becomes apparent
that the observed binding energy ratios for the three samples closely mirror
the ratios anticipated in a three-dimensional (3D) model. This robustly
corroborates our findings, reinforcing the dimensionality of approximately
2.8, as unveiled by our research.

Figure 4:  (a) The BE of 1L WS2 as a function of the number of graphene layers
as computed through the QEH model. The red horizontal line represents the BE
found on the silicon substrate from the FDA fit. (b) Experimentally measured
exciton RSE for monolayer MoS2 and MoSe2 measured by magneto-optical
spectroscopy techniques as a function of ‘n’. The red and blue line represents
the FDA fit to the data points taken from literature 63.

To analyze the values of BE and QBG found through the FDA method more
quantitatively, we utilize the newly developed quantum electrostatic
heterostructure (QEH) approach 64. In this model applying the spatially
dependent dielectric function, an electrostatic potential between the exciton
constituents is obtained. In this method, including the dielectric environment
is limited to van der Waals materials 64. Thus, we employ this approach for
two out of the three samples studied. Since 1L-graphene
($\epsilon_{1L-graphene}$ = 9.32 65) has a dielectric constant that is close
to that of the silicon ($\epsilon_{Si}$ = 11.7 53), we computed the BE for
1L-WS2 as a function of the number of graphene layers as the substrate. The
results of this computation are shown in Figure 4 (a). We notice that the BE
rapidly decreases when increasing the graphene layer numbers. Increasing the
substrate graphene layer number from 1 to 2 leads to a decrease of around 30
meV in the BE, whereas transitioning from 4 to 5 graphene layers leads to only
a few meV reduction. This points towards the importance of the spatial extent
in the out-of-plane direction that needs to be considered for understanding
the role of the screening caused by the dielectric surroundings. In Figure 4
(a), the exciton BE corresponding to the silicon substrate is represented as a
horizontal line, which intersects the BE vs graphene number of layers curve at
an intermediate value (1.62) between 1 and 2 layers of graphene. Thus, the
modification in BE and QBG caused by the silicon substrate is the same as that
resulting from $\approx$ 2 graphene layers. The BE of 173 meV found through
the FDA method on a silicon substrate is close to the reported BE of 214 meV
for 1L WS2 encapsulated by 2L graphene at 70 K 8. For the sample encapsulated
by $\approx$ 14 nm of hBN on Si substrate, the QEH model could now be directly
applied after replacing the Si substrate with 2 layers of graphene. For the
top hBN encapsulation of 13.28 nm, we used a configuration consisting of 40
layers of hBN. The BE calculated using the QEH approach for this configuration
of 40L hBN- 1L WS2 -2L graphene is 135 meV. This agrees with the BE of 143 meV
as calculated from the FDA method for the hBN encapsulated 1L WS2 on silicon
substrate.

Table 2: Summary of the fractional dimensional analysis for 1L MoS2 and 1L
MoSe2. The fractional dimension $\alpha$ and the average dielectric constant
of the environment $\kappa$ are shown. Additionally, the QBG and B.E values as
calculated from the FDA fit and found experimentally in the original
experiment 63 are summarized.

2D Material |  $\alpha$ |  FDA-QBG (eV) |  FDA-BE (meV) |  $\kappa$ |  Exp-QBG (eV) |  Exp-BE (meV)
---|---|---|---|---|---|---
hBN-1L MoS2-hBN |  2.96 |  2.164 $\pm$ 0.001 |  227 $\pm$ 1 |  4.45 |  2.160 |  221
hBN-1L MoSe2-hBN |  2.93 |  1.870 $\pm$ 0.003 |  252 $\pm$ 3 |  4.45 |  1.874 |  231

  * a

The $\epsilon_{eff}$ value as found from the FDA fit for 1L MoS2 and 1L MoSe2
encapslated in hBN is 4.12 and 4.47, respectively.

In order to firmly establish the importance of the FDA method for 2D TMDCs
other than WS2, we applied it to find the QBG and BE for 1L MoS2 and 1L MoSe2
for which we used the available experimental data of their Rydberg series at
zero magnetic field as found by magneto-optical spectroscopy in the literature
63. The FDA fit to the Rydberg states (1s-3s) for the monolayer of MoS2 and
MoSe2 is shown in Figure 4 (b). The $\mu$, exciton reduced mass used is 0.275
and 0.350 for MoS2 and MoSe2 as calculated by performing magneto-optical
spectroscopy at 91 T 63. The details of the FDA for these 2D TMDCs are
summarized in Table 2. The BE ratio for the 1s:2s:3s states for the MoS2 and
MoSe2 are $1:\dfrac{1}{3.98}:\dfrac{1}{9.86}$ and
$1:\dfrac{1}{3.87}:\dfrac{1}{10.95}$ respectively. These ratios are similar to
the BE ratios for 3D Coulomb potential which is 1:$\dfrac{1}{4}:\dfrac{1}{9}$.
Thus, the dimensionality of $\approx$ 3 as calculated through FDA matches well
with the dimension found from the Rydberg series ladder.

In summary, our study not only establishes the robust applicability of the FDA
to characterize Rydberg state energies in monolayer WS2 but also unveils a
profound influence of dielectric environments on excitonic properties. The
achieved 46% reduction (118 meV) in BE and a substantial 141 meV decrease in
QBG by varying $\kappa$ from 1.52 to 8.1 underscore the sensitivity of
excitons to their surroundings. Remarkably, our findings transcend the
specific dielectric conditions, showcasing the versatility of the FDA. The
fractional dimension, consistently approximating 2.8, signifies the resilience
of the monolayer WS2 excitonic system to changes in the average dielectric
environment, even when increased fivefold. Furthermore, our work paves the way
for potential extensions to trion states in 2D semiconductors, opening avenues
for future investigations in this promising direction. The demonstrated
efficacy of the FDA in unraveling excitonic behaviors in diverse scenarios
positions it as a valuable tool for the broader exploration of 2D materials.
In conclusion, our results not only contribute significantly to the
fundamental understanding of excitons in 2D semiconductors but also set the
stage for further breakthroughs in tailoring excitonic properties for diverse
applications. We anticipate that our work will inspire new avenues of research
and exploration in the exciting realm of 2D materials and their unique
electronic characteristics.

Details of the sample fabrication, optical spectroscopy, fractional dimension
calculation, optical spectroscopy results of 2D WS2, atomic force microscopy
of hBN-1LWS2-Si sample, and dielectric screening length calculation are
provided in the supporting information.

We acknowledge the financial support of the Academy of Finland Flagship
Programme (PREIN) (320165). C.R.F acknowledges the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 895369.

## Data availability

The authors declare that the data supporting the findings of this study are
available within the paper and in the supplementary material files.

##  3 Competing interests

The authors declare no competing interests.

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