# Adaptive Differentially Private Structural Entropy Minimization for
Unsupervised Social Event Detection

Zhiwei Yang  Institute of Information Engineering, Chinese Academy of
Sciences, & School of Cyber Security, University of Chinese Academy of
Sciences,Beijing, China<EMAIL_ADDRESS>,  Yuecen Wei  Beihang
UniversityBeijing, China<EMAIL_ADDRESS>,  Haoran Li  Hong Kong University
of Science and Technology,Hongkong, China<EMAIL_ADDRESS>,  Qian Li
Beijing University of Posts and Telecommunications,Beijing, China
<EMAIL_ADDRESS>,  Lei Jiang  Institute of Information Engineering,
Chinese Academy of Sciences,Beijing, China<EMAIL_ADDRESS>,  Li Sun  North
China Electric Power UniversityBeijing, China<EMAIL_ADDRESS>,  Xiaoyan
Yu  Beijing Institute of TechnologyBeijing, China<EMAIL_ADDRESS>,
Chunming Hu  Beihang UniversityBeijing, China<EMAIL_ADDRESS>and  Hao Peng
Beihang University,Beijing, China<EMAIL_ADDRESS>
(2024)

###### Abstract.

Social event detection refers to extracting relevant message clusters from
social media data streams to represent specific events in the real world.
Social event detection is important in numerous areas, such as opinion
analysis, social safety, and decision-making. Most current methods are
supervised and require access to large amounts of data. These methods need
prior knowledge of the events and carry a high risk of leaking sensitive
information in the messages, making them less applicable in open-world
settings. Therefore, conducting unsupervised detection while fully utilizing
the rich information in the messages and protecting data privacy remains a
significant challenge. To this end, we propose a novel social event detection
framework, ADP-SEMEvent, an unsupervised social event detection method that
prioritizes privacy. Specifically, ADP-SEMEvent is divided into two stages,
i.e., the construction stage of the private message graph and the clustering
stage of the private message graph. In the first stage, an adaptive
differential privacy approach is used to construct a private message graph. In
this process, our method can adaptively apply differential privacy based on
the events occurring each day in an open environment to maximize the use of
the privacy budget. In the second stage, to address the reduction in data
utility caused by noise, a novel 2-dimensional structural entropy minimization
algorithm based on optimal subgraphs is used to detect events in the message
graph. The highlight of this process is unsupervised and does not compromise
differential privacy. Extensive experiments on two public datasets demonstrate
that ADP-SEMEvent can achieve detection performance comparable to state-of-
the-art methods while maintaining reasonable privacy budget parameters.

Social Event Detection, Adaptive Differentially Private, Structural Entropy,
Mixed Sensitivity.

††journalyear: 2024††copyright: rightsretained††conference: Proceedings of the
33rd ACM International Conference on Information and Knowledge Management;
October 21–25, 2024; Boise, ID, USA.††booktitle: Proceedings of the 33rd ACM
International Conference on Information and Knowledge Management (CIKM ’24),
October 21–25, 2024, Boise, ID, USA††isbn: 979-8-4007-0436-9/24/10††doi:
10.1145/3627673.3679537††ccs: Security and privacy Social network security and
privacy

##  1\. Introduction

Figure 1. A toy example of an attack demonstrating the potential leakage of
sensitive information such as persons or geopolitical entities.

toy example.

Social event detection refers to extracting relevant clusters of messages from
social media message corpora or social message streams to represent specific
events in the real world. Privacy leakage is a crucial concern today and also
exists in social event detection tasks. Social events occur daily, often
containing rich information regarding collective social behaviors that
widespread public attention. Therefore, the task of social event detection
holds significant implications for sentiment analysis (Beck et al., 2021),
product recommendations, and decision-making (Liu et al., 2017a),
distinguishing between real and fake news (Mehta et al., 2022), as well as
monitoring and managing social crises (Pohl et al., 2015).

Most early methods for social event detection (Zhao et al., 2011; Amiri and
Daume III, 2016; Feng et al., 2015; Morabia et al., 2019; Gao et al., 2016;
Liu et al., 2017b; Wang and Zhang, 2017; Yan et al., 2015) primarily relied on
extracting attributes from text content and subsequently applying traditional
statistical or machine learning methods for detection. However, these
detection methods exhibited low accuracy and were not widely adopted due to
limitations such as insufficient richness in extracted text features and
inadequate interaction between different attributes. Later, with the
development of Graph Neural Networks (GNNs), GNN-based methods for social
event detection gradually gained popularity. GNNs can effectively facilitate
direct interactions between nodes and neighbors in a graph, leading many
researchers to improve performance by leveraging GNN models (Peng et al.,
2019, 2021; Cao et al., 2021; Peng et al., 2022). However, most methods have
been tested under the supervised learning assumption that social event
categories are predefined. The supervised assumption conflicts with the
acquisition of social events in the real world, i.e., the total number of
event categories in the real world frequently varies, making it impossible to
predefine the total number of social event categories. Therefore, supervised
GNN-based methods may not perform well in open environments. Recently, QSGNN
(Ren et al., 2022a) and HISEvent (Cao et al., 2024) proposed to enhance social
event detection in open environments. QSGNN successfully implemented social
event detection in open environments using a pseudo-labeling method. HISEvent
achieved state-of-the-art (SOTA) unsupervised social event detection using
Structural Entropy (Li and Pan, 2016) (SE) combined with hierarchical
clustering. Both of them can be applied effectively in the real world, but
they still suffer from a serious problem, which is the potential leakage of
sensitive private data due to their lack of protection for sensitive
attributes within the data. Graph-based social event detection methods
typically use various additional information to enrich the graph’s
connectivity (Peng et al., 2019; Ren et al., 2022a; Cao et al., 2024). Though
enriching edges can improve detection performance, it can also lead to
unintentional privacy leakage of user privacy data through member inference
attacks, attribute inference attacks, and other methods, Figure 1 shows a toy
example of the attack. To the best of our knowledge, among all the methods
mentioned above, there is no social event detection model that can handle both
unsupervised learning and privacy protection. We believe that both aspects are
necessary.

In this work, we propose a novel framework, Adaptive Differentially Private
Structural Entropy Minimization for Unsupervised Social Event Detection (ADP-
SEMEvent), which simultaneously achieves outstanding social event detection
performance with privacy protection on unsupervised learning. To respect
privacy, inspired by research on differential privacy (Dwork, 2006) (DP), we
implement an adaptive differentially private strategy in ADP-SEMEvent. We use
a mixed sensitivity strategy to achieve adaptive DP, enabling our model to
automatically select the most noise level based on different message sets on
different days. This adaptive nature effectively balances privacy protection
and detection accuracy. Specifically, we calculate the similarity between
nodes using the adaptive differentially private strategy. Then we explore the
edges connectivity of the graph using 1-dimensional Structural Entropy (1D SE)
and relevant attributes, ultimately obtaining a private message graph. For
unsupervised learning, we employ SE from information theory as a crucial
criterion during the clustering process, an indicator for evaluating the
amount of information contained within a graph. Subsequently, to overcome the
current 2-dimensional Structural Entropy (2D SE) minimization algorithm’s
performance and efficiency limitations, we designed a 2D SE minimization
method based on optimal subgraphs to cluster the message graph for social
event detection. The source code is available on
GitHub111https://github.com/SELGroup/ADP-SEMEvent.

Our experiments on two public Twitter datasets demonstrate that ADP-SEMEvent
outperforms baselines for both open and closed set settings and remains
competitive with state-of-the-art approaches within a certain range of privacy
budget settings. To the best of our knowledge, we are the first study to
implement unsupervised social event detection with differentially private
guarantees. Our contributions are as follows: 1) We incorporate an adaptive
differentially private strategy into the process of social event detection,
striking a balance between privacy protection and detection accuracy. 2) We
propose a novel 2D SE minimization algorithm based on optimal subgraphs to
partition the graph and use it for clustering the message graph to conduct
unsupervised social event detection. 3) Extensive experiments conducted on two
public Twitter datasets demonstrate that ADP-SEMEvent, under a reasonable
privacy budget, can achieve performance comparable to current state-of-the-art
models while preserving privacy.

##  2\. Preliminary

Figure 2. The proposed ADP-SEMEvent framework. ADP-SEMEvent consists of two
stages: the private message graph construction stage (stage 1) and the private
message graph clustering stage (stage 2). Messages with the same color
represent the same cluster according to the ground truth labels; orange edges
$E_{s}$ are derived from 1-dimensional structural entropy, and purple edges
$E_{a}$ are derived from relevant attributes; arrows of specific colors
indicate specific operations.

###  2.1. Differentially Private Message Graph

The release of messages graph refers to the process of transforming a series
of social messages set $D=\\{m_{1},\ldots,m_{N}\\}$ (generally, using Pre-
trained Language Models (PLMs) to convert the message into embeddings) into a
message graph $G=(V,E)$ using an algorithm $R$. $G$ is utilized for social
event detection. Under the background of protecting privacy, we present the
following definitions:

###### Definition 2.1.

(Differential Privacy (Dwork, 2006)). For algorithm $R$, given social message
sets $D$ and $D^{\prime}$ ($D$ and $D^{\prime}$ are neighboring datasets that
differ by only one message), and any output
$\mathcal{G}\subseteq\text{range}(D)$. For any $\epsilon>0$, if the algorithm
$R$ satisfies:

(1) $\Pr[R(D)\subseteq\mathcal{G}]\leq
e^{\epsilon}\Pr[R(D^{\prime})\subseteq\mathcal{G}]+\delta,$

we say that the algorithm $R$ satisfies $(\epsilon,\delta)$-DP, where
$\epsilon$ is the privacy budget. Particularly, if $\delta=0$, the algorithm
$R$ satisfies $\epsilon$-DP. $R$ can protect sensitive information contained
in the messages during the release of the message graph.

###### Definition 2.2.

(Sensitivity). Consider a function $R$ whose input is a message set and whose
output is in $\mathbb{R}^{k}$. The sensitivity $S_{R}$ of function $R$ is
defined as:

(2) $S_{R}=\max\|R(D)-R(D^{\prime})\|_{1}.$

Sensitivity is used to determine the magnitude of random perturbation noise to
protect private information, and the sensitivity includes global sensitivity
(Dwork et al., 2006b) ($S_{global}$) and local sensitivity (Nissim et al.,
2007) ($S_{local}$).

###### Definition 2.3.

(Laplace Mechanism for Message Similarity). We measure the similarity between
any two messages using the cosine similarity, then the Laplace mechanism
(Dwork et al., 2006b) is applied to generate the noise and perturb the
similarity as:

(3)
$\text{Cos}_{Lap}(m_{i},m_{j})=\text{Cos}(m_{i},m_{j})+\text{Lap}(\dfrac{S}{\epsilon}),$

where $S$ is typically calculated using global sensitivity, but we designed a
novel mixed sensitivity approach for $S$. (described in Section 3.3).
$\text{Lap}({S}/{\epsilon})$ denotes sampling from a Laplace distribution with
mean 0 and scale factor $S$.

###  2.2. Structural Entropy

Social event detection can be conducted by constructing classification or
clustering models on the messages graph. Consistent with Cao et al.’s (Cao et
al., 2024) approach, we achieve unsupervised graph clustering using SE
minimization. We provide the relevant definitions below:

###### Definition 2.4.

(Encoding Tree of Messages Graph (Li and Pan, 2016; Li et al., 2015)). The
coding tree $T$ of the messages graph $G=(V,E)$ is a hierarchical clustering
partition of $G$ (illustrated as the blue rounded dashed line box in Figure
2). The coding tree $T$ includes all message nodes as leaf nodes. Each node
$\alpha$ in $T$ corresponds to a partitioning of message nodes, with the set
$T_{\alpha}={v_{a}^{1},\ldots,v_{a}^{j}}$, representing the successor nodes of
$\alpha$. The root node $\lambda$ of $T$ has the set $T_{\lambda}=V$,
indicating no partitioning. For each node $\alpha$ in $T$ (excluding
$\lambda$), the height $h(\alpha)$ is one less than that of its parent node.
The root node $\lambda$ has a height of 0. The height of $T$ is the maximum
height among all nodes in $T$.

###### Definition 2.5.

(1D SE). For a messages graph $G=(V,E)$ with $n$ vertices, its 1D SE is
defined as:

(4)
${H}^{(1)}(G)=-\sum_{i=1}^{n}\dfrac{d_{i}}{2m}\text{log}\dfrac{d_{i}}{2m},$

where $d_{i}$ represents the weighted degree of node $i$, and $m$ denotes the
sum of the weighted degrees of all nodes.

###### Definition 2.6.

(2D SE). For a messages graph $G=(V,E)$ with $n$ vertices,
$P=\\{p_{1},\ldots,p_{L}\\}$ is a partition of $V$, $p_{j}$ represents an
event cluster in the message graph. The 2D SE is defined as:

(5)
${H}^{(2)}(G)=-\sum_{j=1}^{L}\dfrac{V_{j}}{m}\sum_{i=1}^{n_{j}}\dfrac{d_{i}^{j}}{V_{j}}\text{log}_{2}{\dfrac{d_{i}^{j}}{V_{j}}}-\sum_{j=1}^{L}\dfrac{P_{cut_{j}}}{m}\text{log}_{2}\dfrac{V_{j}}{m},$

where $n_{j}$ is the number of nodes in partition $p_{j}$, $d_{i}^{j}$ is the
weighted degree of the $i$th node in $p_{j}$, $V_{j}$ is the sum of the
weighted degrees of all nodes in partition $p_{j}$, and $P_{\text{cut}_{j}}$
is the sum of the weights of the cut edges in $p_{j}$.

##  3\. Methodology

Input: social message embedding set $M=\\{m_{1},m_{2},m_{3},...,m_{N}\\}$;
maximum number of neighbors $k_{max}$.

Output: a message graph $G=(V,E)$.

1 Set $V=\\{1,2,3,...,N\\}$;

2 Get similarity $\leftarrow$ Eq. (14);

3 Set $k=1$, $E_{s}=\emptyset$, $E_{s}^{{}^{\prime}}=\emptyset$,
$\text{SE}=+\infty$;

4 while  _True_ do

5 initialize: $G=(V,E_{s})$, $E_{s}^{{}^{\prime}}=\emptyset$;

6 for  _each $v_{i}\in V$_ do

7 $E_{s}^{{}^{\prime}}\overset{\text{add to}}{\leftarrow}\text{Let $v_{i}$
connect the $k$ largest similarity nodes}$;

8 ${H}^{(1)}(G)$ $\leftarrow$ Eq. (4);

9 if  _${H}^{(1)}(G) <\text{SE}$ and $k<k_{max}$_ then

10 $\text{SE}={H}^{(1)}(G)$;

11 $E_{s}=E_{s}^{{}^{\prime}}$;

12 $k++$;

13

14 else

15 break;

16

17Set $E_{a}=\emptyset$;

18 for  _each $i\in[1,\text{size}(V)]$_ do

19 for  _each $j\in[i+1,\text{size}(V)]$_ do

20 if  _$v_{i}$ and $v_{j}$ have shared attributes_ then

21 $E_{a}\overset{\text{add to}}{\leftarrow}\text{similarity($v_{i}$,
$v_{j}$)}$;

22

23

24$E=E_{s}+E_{a}$;

25 $G=(V,E)$;

return $G$.

Algorithm 1 Message Graph Synthesis under Adaptive Differentially Private
Strategy

In this section, we provide a detailed description of ADP-SEMEvent. The entire
framework is illustrated in Figure 2. The private message graph construction
stage primarily utilizes our proposed adaptive differentially private strategy
(introduced in Section 3.2), combined with 1D SE and relevant attributes for
constructing the private message graph (discussed in Section 3.3). The private
message graph clustering stage involves a 2D SE minimization algorithm based
on optimal subgraphs (explained in Section 3.4).

###  3.1. Problem Formalization

Given a series of social messages $m_{1},\ldots,m_{N}$ forming a message graph
$G=(V,E)$, where the node set $V=\\{m_{1},\ldots,m_{N}\\}$ and the edge set
$E$ represents the relevance between messages. The task of social event
detection requires partitioning the message graph $G$ into several partitions,
denoted as $\\{e_{1},\ldots,e_{M}\\}$, where $e_{i}\subset V$, $e_{i}\cap
e_{j}=\emptyset$, $e_{1}\cup\ldots\cup e_{M}=V$. The partitions represent $M$
detected message clusters. Specifically, the message graph $G$ contains
private information and poses a risk of being attacked.

###  3.2. Adaptive Differentially Private Strategy

This section will introduce our mixed sensitivity differentially private
strategy and provide proof that this strategy satisfies $(\epsilon,\delta)$-DP
in the worst-case scenario.

####  3.2.1. Mixed Sensitivity Strategy

To prevent the waste of privacy budget $\epsilon$ caused by the global
sensitivity being much greater than the local sensitivity in a specific data
set, we design a mixed sensitivity strategy to prevent the waste of privacy
budget and the addition of unnecessary noise.

In Equation 2, $S_{global}$ refers to the maximum difference between the
results of a query for any two different data records in any data set.
$S_{global}$ is independent of the data set and is related only to the
calculation method itself. $S_{local}$ removes the restriction of any data set
from the definition of global sensitivity and only considers the current data
set. $S_{local}$ can be obtained through a limited number of calculations. In
general, we can not directly use $S_{local}$ in the Laplace mechanism because
if a malicious attacker knows the local sensitivity of a function $f$ on a
specific dataset, they can still infer some information about the dataset
(Nissim et al., 2007). Therefore, $S_{local}$ must be smoothed to prevent the
malicious attacker from deducing the local sensitivity. This is achieved using
smooth sensitivity (Nissim et al., 2007), denoted as $S_{smooth}$), which is
calculated as:

(6)
$S_{smooth}=2\text{exp}{(-\dfrac{\epsilon}{2}\cdot\log(\dfrac{2}{\delta}))}\cdot
S_{local},$

where $\delta$ is typically set to ${1}/{|D|^{2}}$($|D|$ is the size of $D$).

Then, we design the mixed sensitivity ($S_{mixed}$) strategy. $S_{mixed}$ is
calculated as follows:

(7) $S_{mixed}=\min\\{S_{global},S_{smooth}\\}.$

With the privacy budget $\epsilon$ unchanged, our strategy can automatically
determine fitness sensitivity based on daily events to achieve adaptive noise
perturbation.

####  3.2.2. Differential Privacy Proof

We prove that our strategy worst-case satisfies $(\epsilon,\delta)$-DP. This
entails demonstrating that global sensitivity satisfies $\epsilon$-DP and that
smooth sensitivity satisfies $(\epsilon,\delta)$-DP.

For $S_{smooth}$, Nissim et al. (Nissim et al., 2007) have provided a rigorous
proof that using $S_{smooth}$ for the Laplace mechanism always satisfies
$(\epsilon,\delta)$-DP, regardless of whether global sensitivity can be found.
Where $\delta$ represents the probability of failure, and in our strategy,
$\delta={1}/{|D|^{2}}$, is a small probability. For $S_{global}$, we provide
the following proof:

###### Proof.

For $D$ and $D^{\prime}$, $P_{D}$ and $P_{D^{\prime}}$ represent the
probability distributions of message similarities calculated on their
respective datasets. For any two messages $m_{i}$ and $m_{j}$ in the dataset,
the similarity is represented as $z=\text{Cos}(m_{i},m_{j})+Y$, where $Y$
follows Laplace distribution with scale ${S_{mixed}}/{\epsilon}$ and mean 0.
Therefore, we have:

(8)
$P_{D}(z)=\frac{\epsilon}{2S_{mixed}}\exp(-\frac{\epsilon|\text{Cos}_{D}(m_{i},m_{j})-z|}{S_{mixed}})$
(9)
$P_{D^{\prime}}(z)=\frac{\epsilon}{2S_{mixed}}\exp(-\frac{\epsilon|\text{Cos}_{D^{\prime}}(m_{i},m_{j})-z|}{S_{mixed}}).$

Next, verify whether it complies with differential privacy through
distribution:

(10) $\displaystyle\frac{P_{D}(z)}{P_{D^{\prime}}(z)}$
$\displaystyle=\frac{\exp\left(-\frac{\epsilon|\text{Cos}_{D}(m_{i},m_{j})-z|}{S_{mixed}}\right)}{\exp\left(-\frac{\epsilon|\text{Cos}_{D^{\prime}}(m_{i},m_{j})-z|}{S_{mixed}}\right)}$
(11)
$\displaystyle=\exp\left(\frac{\epsilon(|\text{Cos}_{D^{\prime}}(m_{i},m_{j})-z|-|\text{Cos}_{D}(m_{i},m_{j})-z|)}{S_{mixed}}\right)$
(12)
$\displaystyle\leq\exp\left(\frac{\epsilon(|\text{Cos}_{D^{\prime}}(m_{i},m_{j})-\text{Cos}_{D}(m_{i},m_{j})|)}{S_{mixed}}\right)$
(13) $\displaystyle\leq\exp\left(\epsilon\right).$

∎

In the process of proving, the last two steps follow from the triangle
inequality. According to Equation 1, our strategy satisfies $\epsilon$-DP. In
summary, our strategy worst-case satisfies $(\epsilon,\delta)$-DP.

Input: message graph $G=(V,E)$; Subgraph size $q$; initial partition
$P=\\{p_{1},\dots,p_{m}\\}$.

Output: Clustering results $C=\\{c_{1},\dots,c_{n}\\}$.

1 $C=P$;

2 while  _True_ do

// Synthetic Super-graph based on partitions $C$

3 $V^{\prime}$ $\leftarrow$ The nodes in the same partition $c_{n}$ are
combined into a super-node;

4 $E^{\prime}$ $\leftarrow$ The edge weights between $V^{\prime}$ are summed
by the original nodes;

5 $G^{\prime}=(V^{\prime},E^{\prime})$ $\leftarrow$ $V^{\prime}$,
$E^{\prime}$;

// Get the optimal subgraphs

6 Calculate the number of subgraphs $k_{max}$ =
$\left\lceil\frac{|C|}{q}\right\rceil$;

7 for  _each $k$ $\in$ $[1,k_{max}]$_ do

8 Set $g_{sub_{k}}=(V_{sub_{k}},E_{sub_{k}})$, $V_{sub_{k}}=\emptyset$,
$E_{sub_{k}}=\emptyset$;

9 $g_{sub_{k}}$ $\overset{\text{add to}}{\leftarrow}$ The endpoint of the
largest edge in $G^{\prime}$;

10 while  _$|V_{sub_{k}}| <q$_ do

11 $g_{sub_{k}}\overset{\text{add to}}{\leftarrow}$ The node with the highest
cut weight to current $g_{sub_{k}}$;

12 $G^{\prime}=G^{\prime}-g_{sub_{k}}$;

// 2D SE minimization of optimal subgraphs

13 set $C=\emptyset$, $C_{last}=C$;

14 for  _each $g_{sub_{k}}$ $\in$ $\\{g_{sub_{1}},\dots,g_{sub_{k}}\\}$_ do

15 $T$ $\leftarrow$ Construct an initial coding tree according to definition
2.4;

16 $T_{best}$ $\leftarrow$ Run vanilla 2D SE minimization to minimize 2D SE;

17 $C\overset{\text{add to}}{\leftarrow}$ The partition corresponding to
$T_{best}$;

18 if  _$C=C_{last}$_ then

19 $q=2q$;

20 else if  _$k_{max}=1$_ then

21 break;

22 else $C_{last}=C$ ;

23

return $C$.

Algorithm 2 2D SE Minimization Based on Optimal Subgraphs

###  3.3. Private Message Graph Synthesis

To ensure the generation of rich and private message graphs, we propose a
method of message graph synthesis with edges $E_{s}$ based on SE and edges
$E_{a}$ based on attribute. We employ our proposed adaptive differentially
private strategy to protect privacy while computing edge weights.

We summarize our method into Algorithm 1, termed Message Graph Synthesis under
Adaptive Differentially Private Strategy. For $E_{s}$ (lines 3–14), we link
each message to its $k$ nearest neighbors, where the distance utilizes cosine
similarity under the adaptive differentially private strategy:

(14) $\text{Cos}_{Lap}(m_{i},m_{j})=\frac{m_{i}\cdot
m_{j}}{|m_{i}|\cdot|m_{j}|}+\text{Lap}(\frac{S_{mixed}}{\epsilon}),$

where $m_{k}$ represents the semantic vector corresponding to the message
node, obtained through PLMs (SBERT (Reimers and Gurevych, 2019)). It is worth
noting that, due to the cosine similarity’s range being [-1, 1],
$S_{global}=2$. We gradually increase the number of neighbors from $k=1$, then
compute the 1D SE under different $k$ neighbors, selecting the linkage method
with the smallest 1D SE as the final $E_{s}$. Specifically, to reduce
algorithmic overhead, we greedily select the first occurrence of local minimum
1D SE as our target rather than using the globally optimal target. For $E_{a}$
(lines 15–19), we draw inspiration from the majority of GNN-based social event
detection methods (Peng et al., 2019; Cao et al., 2021; Ren et al., 2022a;
Peng et al., 2022), extracting common attributes of social events such as
entities, user mentions, and user IDs. We set the edge weight between nodes
$v_{i}$ and $v_{j}$, sharing the same attributes to the cosine similarity
$\text{Cos}_{Lap}(m_{i},m_{j})$ corresponding to the nodes. Finally, we
combine $E_{a}$ and $E_{s}$ as all edges of the final private graph.

###  3.4. Event Detection via 2D SE Minimization

Figure 3. The impact of different ways of initializing partitions on the
results.

To efficiently cluster on the noisy message graph $G$, we design an algorithm
named 2D SE minimization based on optimal subgraphs. Existing approaches used
a greedy approach (vanilla 2D SE minimization (Li and Pan, 2016)) to
repeatedly merge any two nodes in the encoding tree $T$ for 2D SE
minimization, which is time-consuming on large graphs. Later, an incremental
method reduces runtime overhead by sequentially dividing large message graphs
into multiple subgraphs (Cao et al., 2024). Nevertheless, such methods
addressed the problem of high time complexity, but simply dividing the
subgraphs is unwise. The initial partitioning is crucial because it may force
strongly related message nodes to be divided into different parts, ultimately
preventing them from merging into the same cluster, as shown in Figure 3 (a).
So, we prevent strongly correlated message nodes from being forcibly separated
by constructing optimal subgraphs, as illustrated in Figure 3 (b).

Specifically, as shown in Algorithm 2, assuming a graph $G=(V,E)$, partition
$P=\\{p_{1},\ldots,p_{m}\\}$ (where $p_{m}\subset V,p_{i}\cap
p_{j}=\emptyset$), and subgraph size $q$ as input. In each iteration, we first
construct a super-graph based on the original graph and the initial partition
(lines 3–5), treating nodes in the same partition as a single new node and
using the sum of the cut edges between partitions as the edge weight between
new nodes. Then, using a greedy approach, we obtain the optimal subgraph
(lines 6–12), attempting to include edges with higher weights in the same
subgraph as much as possible. Finally, we perform 2D SE minimization on each
subgraph (lines 13–17), achieving the final clustering result based on
optimizing each encoding tree.

Table 1. Open-set results on Event2012.

Methods | $M_{1}$ | $M_{2}$ | $M_{3}$ | $M_{4}$ | $M_{5}$ | $M_{6}$ | $M_{7}$
---|---|---|---|---|---|---|---
AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI
BERT | 0.35 | 0.03 | 0.76 | 0.65 | 0.72 | 0.45 | 0.58 | 0.19 | 0.67 | 0.36 | 0.75 | 0.45 | 0.50 | 0.07
SBERT | 0.38 | 0.03 | 0.85 | 0.73 | 0.87 | 0.68 | 0.80 | 0.36 | 0.85 | 0.61 | 0.83 | 0.53 | 0.61 | 0.09
EventX | 0.06 | 0.01 | 0.29 | 0.45 | 0.18 | 0.09 | 0.19 | 0.07 | 0.14 | 0.04 | 0.27 | 0.14 | 0.13 | 0.02
KPGNN | 0.37 | 0.07 | 0.87 | 0.76 | 0.74 | 0.58 | 0.64 | 0.29 | 0.71 | 0.47 | 0.79 | 0.72 | 0.51 | 0.12
QSGNN | 0.41 | 0.07 | 0.80 | 0.77 | 0.76 | 0.59 | 0.68 | 0.29 | 0.73 | 0.48 | 0.80 | 0.73 | 0.54 | 0.12
HISEvent | 0.44 | 0.08 | 0.88 | 0.79 | 0.94 | 0.95 | 0.84 | 0.50 | 0.85 | 0.62 | 0.90 | 0.86 | 0.68 | 0.27
DPG-SEMEvent ($\epsilon$=10) | 0.18 | 0.05 | 0.72 | 0.72 | 0.64 | 0.69 | 0.37 | 0.21 | 0.61 | 0.58 | 0.66 | 0.67 | 0.34 | 0.20
DPG-SEMEvent ($\epsilon$=15) | 0.33 | 0.07 | 0.81 | 0.84 | 0.81 | 0.85 | 0.40 | 0.50 | 0.79 | 0.78 | 0.80 | 0.80 | 0.54 | 0.23
ADP-Spectral Clustering ($\epsilon$=10) | 0.05 | 0.07 | 0.01 | 0.02 | 0.05 | 0.08 | 0.02 | 0.10 | -0.01 | 0.01 | 0.02 | 0.11 | 0.12 | 0.07
ADP-Spectral Clustering ($\epsilon$=15) | 0.08 | 0.05 | 0.02 | 0.03 | 0.10 | 0.11 | 0.02 | 0.03 | 0.05 | 0.02 | 0.03 | 0.02 | 0.13 | 0.09
ADP-SEMEvent ($\epsilon$=10) | 0.22 | 0.08 | 0.77 | 0.80 | 0.73 | 0.73 | 0.51 | 0.63 | 0.62 | 0.59 | 0.80 | 0.83 | 0.35 | 0.25
ADP-SEMEvent ($\epsilon$=15) | 0.35 | 0.09 | 0.87 | 0.87 | 0.89 | 0.91 | 0.73 | 0.49 | 0.85 | 0.85 | 0.89 | 0.87 | 0.57 | 0.28
ADP-SEMEvent ($\epsilon$=None) | 0.46 | 0.09 | 0.87 | 0.87 | 0.92 | 0.91 | 0.85 | 0.55 | 0.85 | 0.62 | 0.90 | 0.93 | 0.65 | 0.35
Methods | $M_{8}$ | $M_{9}$ | $M_{10}$ | $M_{11}$ | $M_{12}$ | $M_{13}$ | $M_{14}$
AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI
BERT | 0.74 | 0.51 | 0.71 | 0.34 | 0.78 | 0.55 | 0.62 | 0.26 | 0.56 | 0.31 | 0.57 | 0.13 | 0.55 | 0.24
SBERT | 0.86 | 0.65 | 0.83 | 0.47 | 0.85 | 0.62 | 0.82 | 0.49 | 0.85 | 0.63 | 0.70 | 0.24 | 0.77 | 0.40
EventX | 0.21 | 0.09 | 0.19 | 0.07 | 0.24 | 0.13 | 0.24 | 0.16 | 0.16 | 0.07 | 0.16 | 0.04 | 0.14 | 0.10
KPGNN | 0.76 | 0.60 | 0.71 | 0.46 | 0.78 | 0.70 | 0.71 | 0.49 | 0.66 | 0.48 | 0.67 | 0.29 | 0.65 | 0.42
QSGNN | 0.75 | 0.59 | 0.75 | 0.47 | 0.80 | 0.71 | 0.72 | 0.49 | 0.68 | 0.49 | 0.66 | 0.29 | 0.66 | 0.41
HISEvent | 0.89 | 0.74 | 0.88 | 0.65 | 0.90 | 0.87 | 0.82 | 0.62 | 0.90 | 0.82 | 0.78 | 0.46 | 0.88 | 0.85
DPG-SEMEvent ($\epsilon$=10) | 0.33 | 0.14 | 0.35 | 0.32 | 0.76 | 0.80 | 0.59 | 0.48 | 0.46 | 0.38 | 0.56 | 0.36 | 0.51 | 0.49
DPG-SEMEvent ($\epsilon$=15) | 0.78 | 0.61 | 0.60 | 0.45 | 0.80 | 0.80 | 0.77 | 0.0 | 0.64 | 0.57 | 0.65 | 0.39 | 0.65 | 0.50
ADP-Spectral Clustering ($\epsilon$=10) | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.05 | 0.10 | 0.03 | 0.09 | 0.00 | 0.02 | 0.04 | 0.13
ADP-Spectral Clustering ($\epsilon$=15) | 0.06 | 0.02 | 0.06 | 0.04 | 0.02 | 0.04 | 0.03 | 0.04 | 0.04 | 0.02 | 0.14 | 0.05 | 0.01 | 0.02
ADP-SEMEvent ($\epsilon$=10) | 0.69 | 0.60 | 0.57 | 0.47 | 0.77 | 0.80 | 0.69 | 0.56 | 0.56 | 0.42 | 0.55 | 0.35 | 0.51 | 0.44
ADP-SEMEvent ($\epsilon$=15) | 0.73 | 0.53 | 0.85 | 0.76 | 0.88 | 0.85 | 0.81 | 0.64 | 0.74 | 0.74 | 0.72 | 0.43 | 0.71 | 0.58
ADP-SEMEvent ($\epsilon$=None) | 0.89 | 0.75 | 0.89 | 0.79 | 0.90 | 0.87 | 0.82 | 0.65 | 0.90 | 0.88 | 0.79 | 0.51 | 0.83 | 0.71
Methods | $M_{15}$ | $M_{16}$ | $M_{17}$ | $M_{18}$ | $M_{19}$ | $M_{20}$ | $M_{21}$
AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI
BERT | 0.43 | 0.07 | 0.71 | 0.43 | 0.56 | 0.22 | 0.52 | 0.24 | 0.59 | 0.28 | 0.60 | 0.32 | 0.54 | 0.17
SBERT | 0.67 | 0.17 | 0.78 | 0.50 | 0.77 | 0.35 | 0.81 | 0.52 | 0.83 | 0.54 | 0.80 | 0.52 | 0.70 | 0.24
EventX | 0.07 | 0.01 | 0.19 | 0.08 | 0.18 | 0.12 | 0.16 | 0.08 | 0.16 | 0.07 | 0.18 | 0.11 | 0.10 | 0.10
KPGNN | 0.54 | 0.17 | 0.77 | 0.66 | 0.68 | 0.43 | 0.66 | 0.47 | 0.71 | 0.51 | 0.68 | 0.51 | 0.57 | 0.20
QSGNN | 0.55 | 0.17 | 0.76 | 0.65 | 0.69 | 0.44 | 0.68 | 0.48 | 0.70 | 0.50 | 0.69 | 0.51 | 0.58 | 0.21
HISEvent | 0.72 | 0.27 | 0.87 | 0.83 | 0.81 | 0.56 | 0.80 | 0.70 | 0.87 | 0.63 | 0.81 | 0.69 | 0.69 | 0.45
DPG-SEMEvent ($\epsilon$=10) | 0.42 | 0.32 | 0.73 | 0.75 | 0.55 | 0.56 | 0.48 | 0.40 | 0.60 | 0.48 | 0.51 | 0.41 | 0.57 | 0.33
DPG-SEMEvent ($\epsilon$=15) | 0.56 | 0.33 | 0.82 | 0.78 | 0.69 | 0.58 | 0.66 | 0.58 | 0.86 | 0.56 | 0.68 | 0.58 | 0.57 | 0.33
ADP-Spectral Clustering ($\epsilon$=10) | 0.10 | 0.20 | 0.02 | 0.03 | 0.04 | 0.05 | 0.03 | 0.10 | 0.09 | 0.01 | 0.01 | 0.02 | 0.03 | 0.03
ADP-Spectral Clustering ($\epsilon$=15) | 0.02 | 0.03 | 0.02 | 0.03 | 0.06 | 0.04 | 0.04 | 0.02 | 0.18 | 0.10 | 0.03 | 0.01 | 0.05 | 0.05
ADP-SEMEvent ($\epsilon$=10) | 0.63 | 0.30 | 0.83 | 0.80 | 0.56 | 0.57 | 0.54 | 0.43 | 0.72 | 0.63 | 0.66 | 0.64 | 0.45 | 0.40
ADP-SEMEvent ($\epsilon$=15) | 0.60 | 0.35 | 0.84 | 0.84 | 0.74 | 0.57 | 0.73 | 0.65 | 0.84 | 0.67 | 0.76 | 0.64 | 0.66 | 0.42
ADP-SEMEvent ($\epsilon$=None) | 0.75 | 0.30 | 0.88 | 0.84 | 0.79 | 0.58 | 0.81 | 0.70 | 0.86 | 0.68 | 0.82 | 0.69 | 0.71 | 0.46

##  4\. Experiments

In this section, we evaluate the event detection and privacy protection
capabilities of ADP-SEMEvent. Specifically, our goal is to answer the
following questions: Q1: How does ADP-SEMEvent perform compared to other
methods? Q2: What is the capability of ADP-SEMEvent in privacy protection? Q3:
What is the impact of privacy budget $\epsilon$ on ADP-SEMEvent’s results? Q4:
How is the new 2D SE minimization algorithm executed? Q5: How does the
adaptive differential privacy strategy work?

Table 2. Open-set results on Event2018.

Methods | $M_{1}$ | $M_{2}$ | $M_{3}$ | $M_{4}$ | $M_{5}$ | $M_{6}$ | $M_{7}$ | $M_{8}$
---|---|---|---|---|---|---|---|---
AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI
BERT | 0.42 | 0.16 | 0.44 | 0.21 | 0.22 | 0.22 | 0.41 | 0.17 | 0.56 | 0.31 | 0.49 | 0.23 | 0.49 | 0.23 | 0.50 | 0.24
SBERT | 0.60 | 0.20 | 0.61 | 0.29 | 0.63 | 0.34 | 0.60 | 0.23 | 0.76 | 0.47 | 0.73 | 0.41 | 0.65 | 0.29 | 0.75 | 0.50
EventX | 0.11 | 0.02 | 0.12 | 0.02 | 0.11 | 0.01 | 0.14 | 0.06 | 0.24 | 0.13 | 0.15 | 0.08 | 0.12 | 0.02 | 0.21 | 0.09
KPGNN | 0.54 | 0.17 | 0.55 | 0.18 | 0.55 | 0.15 | 0.55 | 0.17 | 0.57 | 0.21 | 0.57 | 0.21 | 0.61 | 0.30 | 0.57 | 0.20
QSGNN | 0.56 | 0.18 | 0.57 | 0.19 | 0.56 | 0.17 | 0.59 | 0.18 | 0.59 | 0.23 | 0.59 | 0.21 | 0.63 | 0.30 | 0.55 | 0.19
HISEvent | 0.78 | 0.55 | 0.78 | 0.60 | 0.75 | 0.50 | 0.73 | 0.48 | 0.79 | 0.57 | 0.81 | 0.56 | 0.85 | 0.67 | 0.89 | 0.80
DPG-SEMEvent ($\epsilon$=10) | 0.40 | 0.35 | 0.45 | 0.43 | 0.36 | 0.24 | 0.40 | 0.23 | 0.39 | 0.40 | 0.44 | 0.42 | 0.41 | 0.35 | 0.30 | 0.26
DPG-SEMEvent ($\epsilon$=15) | 0.52 | 0.47 | 0.55 | 0.53 | 0.51 | 0.38 | 0.52 | 0.38 | 0.55 | 0.47 | 0.57 | 0.63 | 0.60 | 0.53 | 0.52 | 0.44
ADP-Spectral Clustering ($\epsilon$=10) | 0.01 | 0.03 | 0.02 | 0.02 | 0.01 | 0.02 | 0.05 | 0.02 | 0.10 | 0.09 | 0.02 | 0.04 | 0.02 | 0.01 | 0.08 | 0.02
ADP-Spectral Clustering ($\epsilon$=15) | 0.02 | 0.02 | 0.05 | 0.03 | 0.02 | 0.02 | 0.06 | 0.07 | 0.03 | 0.04 | 0.03 | 0.04 | 0.09 | 0.08 | 0.05 | 0.04
ADP-SEMEvent ($\epsilon$=10) | 0.48 | 0.41 | 0.57 | 0.55 | 0.45 | 0.38 | 0.53 | 0.26 | 0.56 | 0.44 | 0.56 | 0.51 | 0.58 | 0.41 | 0.41 | 0.37
ADP-SEMEvent ($\epsilon$=15) | 0.56 | 0.49 | 0.68 | 0.65 | 0.61 | 0.45 | 0.62 | 0.43 | 0.73 | 0.71 | 0.67 | 0.71 | 0.66 | 0.55 | 0.66 | 0.59
ADP-SEMEvent ($\epsilon$=None) | 0.75 | 0.60 | 0.77 | 0.70 | 0.75 | 0.54 | 0.73 | 0.51 | 0.79 | 0.58 | 0.82 | 0.73 | 0.85 | 0.65 | 0.89 | 0.82
Methods | $M_{9}$ | $M_{10}$ | $M_{11}$ | $M_{12}$ | $M_{13}$ | $M_{14}$ | $M_{15}$ | $M_{16}$
AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI | AMI | ARI
BERT | 0.42 | 0.17 | 0.46 | 0.19 | 0.48 | 0.18 | 0.54 | 0.32 | 0.40 | 0.18 | 0.52 | 0.27 | 0.53 | 0.28 | 0.43 | 0.21
SBERT | 0.63 | 0.23 | 0.72 | 0.39 | 0.70 | 0.31 | 0.76 | 0.54 | 0.65 | 0.34 | 0.68 | 0.43 | 0.71 | 0.40 | 0.65 | 0.25
EventX | 0.16 | 0.07 | 0.19 | 0.07 | 0.18 | 0.06 | 0.20 | 0.09 | 0.15 | 0.06 | 0.22 | 0.11 | 0.22 | 0.11 | 0.10 | 0.01
KPGNN | 0.46 | 0.10 | 0.56 | 0.18 | 0.53 | 0.16 | 0.56 | 0.17 | 0.60 | 0.28 | 0.65 | 0.43 | 0.58 | 0.25 | 0.50 | 0.13
QSGNN | 0.46 | 0.13 | 0.58 | 0.19 | 0.59 | 0.20 | 0.59 | 0.20 | 0.58 | 0.27 | 0.67 | 0.44 | 0.61 | 0.27 | 0.50 | 0.13
HISEvent | 0.76 | 0.52 | 0.80 | 0.52 | 0.81 | 0.50 | 0.85 | 0.73 | 0.89 | 0.83 | 0.89 | 0.81 | 0.83 | 0.70 | 0.68 | 0.30
DPG-SEMEvent ($\epsilon$=10) | 0.22 | 0.12 | 0.28 | 0.22 | 0.35 | 0.19 | 0.45 | 0.36 | 0.44 | 0.40 | 0.45 | 0.37 | 0.50 | 0.42 | 0.26 | 0.19
DPG-SEMEvent ($\epsilon$=15) | 0.36 | 0.12 | 0.50 | 0.42 | 0.52 | 0.35 | 0.64 | 0.67 | 0.56 | 0.60 | 0.60 | 0.51 | 0.64 | 0.63 | 0.50 | 0.25
ADP-Spectral Clustering ($\epsilon$=10) | 0.03 | 0.02 | 0.02 | 0.03 | 0.03 | 0.02 | 0.02 | 0.02 | 0.02 | 0.01 | 0.07 | 0.02 | 0.04 | 0.01 | 0.06 | 0.03
ADP-Spectral Clustering ($\epsilon$=15) | 0.05 | 0.06 | 0.02 | 0.04 | 0.03 | 0.03 | 0.05 | 0.07 | 0.01 | 0.03 | 0.06 | 0.02 | 0.02 | 0.09 | 0.04 | 0.06
ADP-SEMEvent ($\epsilon$=10) | 0.36 | 0.22 | 0.43 | 0.26 | 0.38 | 0.22 | 0.47 | 0.37 | 0.46 | 0.43 | 0.47 | 0.40 | 0.58 | 0.58 | 0.39 | 0.23
ADP-SEMEvent ($\epsilon$=15) | 0.53 | 0.48 | 0.60 | 0.52 | 0.63 | 0.43 | 0.70 | 0.73 | 0.66 | 0.69 | 0.71 | 0.69 | 0.76 | 0.68 | 0.52 | 0.27
ADP-SEMEvent ($\epsilon$=None) | 0.79 | 0.53 | 0.75 | 0.50 | 0.83 | 0.55 | 0.85 | 0.73 | 0.89 | 0.83 | 0.90 | 0.82 | 0.85 | 0.72 | 0.67 | 0.32

Table 3. Close-set results on Event2012 and Event2018. Methods | Event2012 | Event2018
---|---|---
AMI | ARI | AMI | ARI
BERT | 0.43 | 0.12 | 0.34 | 0.05
SBERT | 0.73 | 0.17 | 0.62 | 0.11
EventX | 0.19 | 0.05 | 0.16 | 0.03
KPGNN | 0.52 | 0.22 | 0.44 | 0.15
QSGNN | 0.53 | 0.22 | 0.44 | 0.16
HISEvent | 0.81 | 0.50 | 0.47 | 0.31
DPG-SEMEvent ($\epsilon$=10) | 0.50 | 0.20 | 0.20 | 0.13
DPG-SEMEvent ($\epsilon$=15) | 0.60 | 0.28 | 0.22 | 0.13
ADP-Spectral Clustering ($\epsilon$=10) | 0.10 | 0.05 | 0.08 | 0.03
ADP-Spectral Clustering ($\epsilon$=15) | 0.09 | 0.04 | 0.08 | 0.02
ADP-SEMEvent ($\epsilon$=10) | 0.51 | 0.41 | 0.20 | 0.14
ADP-SEMEvent ($\epsilon$=15) | 0.65 | 0.49 | 0.25 | 0.15
ADP-SEMEvent ($\epsilon$=None) | 0.81 | 0.52 | 0.49 | 0.33

###  4.1. Experimental Setup

Datasets. We experiment on two large public Twitter datasets. Event2012
(McMinn et al., 2013) has 68,841 English messages, including 503 events;
Event2018 (Mazoyer et al., 2020) has 64,516 French messages, including 257
events. We employ the time-splitting method proposed by Cao et al. (Cao et
al., 2024) and Ren et al. (Ren et al., 2022a) to partition the dataset.
Specifically, Event2012 is divided into $M_{0}$ , $\dots$ , $M_{21}$, a total
of 22 blocks; Event2018 is divided into $M_{0}$ , $\dots$ , $M_{16}$, a total
of 17 blocks. Our model is unsupervised and does not require a training set,
but some baselines are supervised, so the first day is the training set.

Baselines. We select 8 baselines. BERT (Devlin et al., 2019), and SBERT
(Reimers and Gurevych, 2019), which transform messages into message embeddings
and then employ K-means clustering for social event detection, requiring the
total number of events to be specified in advance; EventX (Liu et al., 2020),
which utilizes community detection methods for unsupervised social event
detection; KPGNN (Cao et al., 2021), a supervised GNN detection method; QSGNN
(Ren et al., 2022a), a supervised method based on pseudo labels, which
performs well in open environments; HISEvent (Cao et al., 2024), which
utilizes structural entropy for unsupervised clustering to achieve social
event detection, is the current SOTA. Additionally, DPG-SEMEvent, a variant of
our method, utilizes the Gaussian mechanism (Liu, 2018) for noise perturbation
without employing a mixed sensitivity strategy; ADP-Spectral Clustering,
running spectral clustering (Von Luxburg, 2007) directly on our published
privacy graph. Our results are the average of 5 runs.

Hyperparameters. For ADP-SEMEvent, we set the $\epsilon$ to three values: 10,
15, and None (indicating no privacy protection strategy). In open
environments, the parameter $q$ is set to 400, while in closed environments,
it is set to 300, and the same applies to the HISEvent, DPG-SEMEvent, and DPG-
SEMEvent. For KPGNN, QSGNN, and EventX, we adopt the settings reported in the
original papers.

Evaluation Metrics. We measure adjusted mutual information (AMI) and adjusted
rand index (ARI). Both metrics range from -1 to 1, where a value closer to 1
indicates better agreement between the predicted and true clustering.

Table 4. The results of attribute inference attack experiment.

Methods | PERSON (SVM) | GPE (SVM) | PERSON (LR) | GPE (LR) | PERSON (MLP) | GPE (MLP)
---|---|---|---|---|---|---
F1 | Precision | Accuracy | AUC | F1 | Precision | Accuracy | AUC | F1 | Precision | Accuracy | AUC
BERT | 0.72 | 0.68 | 0.86 | 0.98 | 0.75 | 0.68 | 0.87 | 0.98 | 0.74 | 0.76 | 0.89 | 0.98
SBERT | 0.71 | 0.78 | 0.76 | 0.96 | 0.68 | 0.70 | 0.83 | 0.97 | 0.73 | 0.69 | 0.81 | 0.97
HISEvent | 0.42 | 0.40 | 0.34 | 0.36 | 0.41 | 0.40 | 0.32 | 0.58 | 0.29 | 0.56 | 0.31 | 0.56
DPG-SEMEvent ($\epsilon$=10) | 0.33 | 0.34 | 0.33 | 0.33 | 0.32 | 0.35 | 0.31 | 0.50 | 0.25 | 0.45 | 0.21 | 0.46
DPG-SEMEvent ($\epsilon$=15) | 0.39 | 0.39 | 0.30 | 0.31 | 0.38 | 0.40 | 0.33 | 0.52 | 0.21 | 0.51 | 0.22 | 0.51
ADP-SEMEvent ($\epsilon$=10) | 0.33 | 0.35 | 0.32 | 0.34 | 0.35 | 0.35 | 0.31 | 0.49 | 0.21 | 0.46 | 0.21 | 0.45
ADP-SEMEvent ($\epsilon$=15) | 0.40 | 0.38 | 0.29 | 0.32 | 0.38 | 0.40 | 0.32 | 0.52 | 0.22 | 0.51 | 0.22 | 0.51

###  4.2. Overall Performance (Q1)

Tables 1 and 2 present the evaluation results of Event2012 and Event2018 in an
open environment, demonstrating the superior performance of ADP-SEMEvent. For
instance, the results from Event2012 show that in most cases, when
$\epsilon=\text{None}$, ADP-SEMEvent outperforms the current SOTA model
HISEvent. Optimally, the ARI in $M_{9}$ increases from 0.65 to 0.79,
representing a 21.5$\%$ improvement. When $\epsilon=10$ or $15$, compared to
the privacy-preserving models DPG-SEMEvent and ADP-Spectral Clustering, ADP-
SEMEvent surpasses the baselines on all metrics. The comparison with DPG-
SEMEvent indicates that our adaptive differential privacy strategy effectively
controls noise, and the comparison with ADP-Spectral Clustering highlights
that our Algorithm 2 can cluster private graphs more effectively, making it a
more efficient unsupervised method. In summary, ADP-SEMEvent has three
characteristics: 1) it does not require the specification of the total number
of events, making it unsupervised; 2) it emphasizes privacy protection; 3) it
exhibits excellent performance and robustness. No baseline simultaneously
satisfies all of these criteria.

Table 3 shows the evaluation results for Event2012 and Event2018 in a closed
environment. When $\epsilon$=None on large datasets, our ADP-SEMEvent achieves
SOTA performance. Performance decreases for $\epsilon$=10 and $\epsilon$=15,
but it still outperforms DPG-SEMEvent and ADP-Spectral Clustering. Test
results for Event2012 indicate that ADP-SEMEvent remains competitive; however,
the results for Event2018 show lower performance, possibly due to the closer
proximity of local sensitivity and global sensitivity in the Event2018
dataset.

###  4.3. Security Analysis (Q2)

We conduct a security analysis for all models. For Eventx, KPGNN, and QSGNN,
are supervised. Generally, machine learning models tend to exhibit
memorization, enabling malicious attackers to leverage membership inference
attacks to obtain data from the training set (Salem et al., 2018). For ADP-
SEMEvent and other baselines, unsupervised clustering methods are used,
eliminating the need for training data. Consequently, the risk of membership
inference attacks is greatly reduced. However, attackers can still design
attribute inference attacks (Gong and Liu, 2018; Zhao et al., 2021) to deduce
sensitive information about messages, such as relationships and locations.

To assess ADP-SEMEvent’s protection capabilities, we design attribute
inference attack experiments, selecting *Person* and *GPE* (Geopolitical
Entity) as the attack target. We manually select 535 pieces of data containing
sensitive attributes from Event2012. To comprehensively assess security, we
design binary and multi-class attack scenarios. Specifically, for *Person*, we
only consider whether the data contains the *Person* attribute, making it a
binary classification problem. We use F1 and precision as evaluation metrics.
For *GPE*, which includes 8 categories, it is a multi-classification problem.
We use accuracy and AUC as evaluation metrics. For BERT and SBERT, we directly
use embeddings as features of the messages; for other models, we use a simple
Node2Vec (Grover and Leskovec, 2016) model to obtain node embeddings from the
graph; 30% of the data is used as the training set. We choose support vector
product (SVM), logistic regression (LR), and multi-layer perceptron (MLP) as
attack models

Our test results are shown in Table 4. Since BERT and SBERT directly convert
messages into embeddings, sensitive information is directly exposed in the
embeddings, leading to the worst evaluation metrics. Although HISEvent’s
probability of a successful attack is lower than that of directly using
embeddings, its privacy protection capabilities remain limited due to a lack
of further processing of sensitive information. For ADP-SEMEvent and DPG-
SEMEvent, the attack model’s effectiveness on these models is relatively low,
indicating that the noise provided some degree of protection. However, ADP-
SEMEvent outperforms DPG-SEMEvent in event detection tasks, demonstrating its
higher application value.

Figure 4. Sensitivity analysis results of the model to privacy budget
$\epsilon$.

###  4.4. Hyperparameter Sensitivity (Q3)

To validate the sensitivity of ADP-SEMEvent to the privacy budget $\epsilon$
parameter, we conduct tests using Event2012 in an open environment. We
incrementally increase the value of the privacy budget $\epsilon$ from 1 to 10
with a step size of 1, then run the ADP-SEMEvent to observe its final
performance. This test is essential as it provides a straightforward
reflection of the relationship between the privacy budget $\epsilon$ and the
model performance, aiding algorithm users in balancing privacy protection
levels with detection performance in practical applications. Our test results
are illustrated in Figure 4. Analysis of the 4 cases indicates that the
privacy budget $\epsilon$ effectively controls the noise level. A smaller
$\epsilon$ adds more noise, resulting in lower data utility, while a larger
$\epsilon$ results in less noise and higher model performance.

Figure 5. Validity verification results of Optimized Hierarchical 2D SE
Minimization. ($\epsilon$ = None)

###  4.5. Ablation Study (Q4)

We compare the model’s performance before and after optimization to validate
the effectiveness of the 2D SE minimization algorithm based on optimal
subgraphs. As illustrated in Figure 5, in most cases, the algorithm’s
performance after optimization surpasses that of the unoptimized method. This
demonstrates that a well-initialized partition aids the subsequent iterative
process.

###  4.6. Noise Analysis (Q5)

To verify the noise level control by the adaptive sensitivity strategy, we use
sensitivity as an indicator of noise level, where a higher sensitivity
indicates greater perturbation to the original data. We conduct statistical
analyses of noise levels under different privacy budgets and across different
datasets. The results are shown in Figure 6. We find that when the privacy
budget $\epsilon\geq 5$, our mixed sensitivity begins to take effect, avoiding
noise waste. In a continuous 21-day simulation, the mixed sensitivity can
determine the most appropriate sensitivity based on the current day’s
situation, balancing performance and privacy.

##  5\. Related Work

###  5.1. Differential Privacy

Differential privacy was first introduced by Dwork et al. (Dwork, 2006). The
most commonly used method of differentially private is the Laplace mechanism
(Dwork et al., 2006b), which adds appropriate noise to meet $\epsilon$-DP by
analyzing the global sensitivity of the query function. Subsequently, Dwork et
al. (Dwork et al., 2006a) proposed approximate differential privacy, which
relaxes the strict requirement of $\epsilon$-DP by using the Gaussian
mechanism to add noise, improving the applicability. To address the challenge
of computing global sensitivity, Dwork et al. (Dwork and Lei, 2009) proposed
the Propose-Test-Release framework, while Nissim et al. (Nissim et al., 2007)
introduced the smooth sensitivity and Sample and Aggregate frameworks.
Nowadays, due to the growing problem of privacy data leakage, many scholars
utilize differential privacy to protect data. For example, in private
clustering (Imola et al., 2023; Huang and Liu, 2018), private PageRank (Epasto
et al., 2022), private data release or synthesis (Yuan et al., 2023; Li et
al., 2023b; Ren et al., 2022b), and application in recommender system (Wei et
al., 2022), among others. However, existing DP methods do not effectively
adapt to the addition of noise based on each day’s events.

###  5.2. Social Event Detection

Existing research (Cao et al., 2024; Peng et al., 2019, 2022; Ren et al.,
2022b) typically formalizes the task of social event detection as clustering
relevant messages from social media sequences to represent events. We divide
social event detection methods into three categories based on the model input,
i.e., attribute-based, content-based, and Hybrid methods. 1) Attribute-based
methods (Hu et al., 2022; Singh and Kumari, 2021) require predefined rules or
patterns and then identify events by matching rules. Their drawback is the
inability to capture complex event features accurately, and updating rules for
newly emerged event types may be required, resulting in lower performance and
robustness. 2) Content-based methods (Wang and Zhang, 2017; Sihem Sahnoun and
Yahia, 2020) mostly rely on natural language processing to model and analyze
content. Their detection performance heavily depends on content modeling.
However, due to the complexity and variability of real-world content, methods
relying solely on content modeling have not shown good robustness. 3) Hybrid
methods (Peng et al., 2019; Cao et al., 2021; Ren et al., 2022a; Peng et al.,
2022; Ren et al., 2021; Li et al., 2023c, a) utilize GNNs for modeling,
effectively combining content and attributes into heterogeneous graphs to
achieve better performance. However, since GNNs contain rich node information,
they may inevitably include sensitive data that attackers should not be able
to access, making privacy protection crucial for the models.

##  6\. Conclusion

Figure 6. The results of the noise analysis. Above: The impact of privacy
budget on noise (Tested on M16, Event2012). Below: The impact of blocks on
noise (Privacy budget is set to 15).

In this work, we recognize the importance of privacy protection and have
explored how to conduct social event detection under the requirements of
privacy protection. We propose a novel unsupervised social event detection
framework, ADP-SEMEvent, implementing adaptive privacy protection. The
adaptive differentially private strategy we proposed maximizes the utilization
of the privacy budget and achieves a good balance between privacy and
accuracy. At the same time, the unsupervised approach eliminates the need to
determine the number of events in advance, providing better robustness in
real-world applications. Experiments show that ADP-SEMEvent can protect
privacy while achieving satisfactory results. Since PLMs have a significant
impact on both performance and privacy, we will explore efficient and private
PLMs in future work.

###### Acknowledgements.

This research is supported by the National Key R&D Program of China through
grant 2023YFC3303800, NSFC through grant 62322202, Shijiazhuang Science and
Technology Plan Project through grant 231130459A, and Guangdong Basic and
Applied Basic Research Foundation through grant 2023B1515120020.

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# Biexciton crystal in a two-dimensional semiconductor heteropentalayer.

Yi Huang (黄奕<EMAIL_ADDRESS>B. I. Shklovskii  School of Physics and
Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

(August 27, 2024; August 27, 2024)

###### Abstract

This paper is written for the Special Issue in Honor of Emmanuel Rashba. We
study the gas of indirect dipolar excitons created by an interband
illumination of a pentalayer WSe2/MoSe2/WSe2/MoSe2/WSe2. We show that two
colinear indirect excitons bind into a linear biexciton with twice larger
dipole moment. Two biexcitons with opposite dipole directions attract each
other at large distances and repel each other at short distances. Therefore,
biexcitons form a staggered crystal with anti-ferroelectric square lattice.
The electrostatic energy of this crystal per biexciton has a minimum at the
biexciton concentration $n=n_{c}=0.14d^{-2}$, where $d\sim 0.7$ nm is a single
layer thickness. At small illumination intensity, biexcitons condense into
sparse crystallites with $n=n_{c}$, where photoluminescence frequency is red
shifted and independent on the light intensity. We also study a capacitor made
of five identical semiconductor monolayers separated by hBN spacers where a
critical voltage applied between layers 1, 3, 5, and 2, 4 abruptly creates a
similar biexciton crystal. At this voltage, the differential capacitance
diverges.

##  I Introduction

Heterostructures of two-dimensional (2D) transition metal dichalcogenides
(TMDC) demonstrated wide spectrum of correlated states of excitons. For
example Rivera  _et al._ (2015); Calman  _et al._ (2020), in a bilayer
MoSe2/WSe2, the type II band alignment of MoSe2 and WSe2 monolayers allows the
formation of spatially indirect excitons, in which an electron in MoSe2 binds
to a hole in WSe2 (see Fig.1). Parallel dipolar excitons repel each other at
all distances between them or at all two dimensional exciton concentrations
$n$ tuned by the illumination intensity. Because of the weak overlap of the
electron and hole wave functions, these excitons decay slowly enough to form
the correlated ground state, which minimizes their repulsion. It was observed
Rivera  _et al._ (2015); Calman  _et al._ (2020) that in MoSe2/WSe2 devices,
the indirect exciton luminescence line is blue shifted due to the dipole-
dipole repulsion, and this effect becomes stronger as the illumination
intensity increases.

It was predicted recently Sammon and Shklovskii (2019) that in a trilayer
device WSe2/MoSe2/WSe2 (WMW) Baranowski  _et al._ (2017); Choi  _et al._
(2018) and similar devices with identical hexagonal boron nitride (hBN)
spacers between layers, repulsion of excitons at small distances is replaced
by their attraction at larger distances. This happens because in a WMW device,
the hole of an exciton originates with equal probability in either the upper
or lower WSe2 layers, while the electron is always at the middle layer (see
Fig. 2). At large distances, two oppositely oriented dipolar excitons attract
each other, while the repulsion takes over at short distances, because
electrons located in the middle layer approach each other. It was shown that
the interaction energy of directed up and down dipolar excitons has a minimum
near $n_{\mathrm{x}}^{-1/2}=2.8d$, where $n_{\mathrm{x}}$ is the two-
dimensional concentration of excitons and $d\sim 0.7$ nm Slobodkin  _et al._
(2020) is the distance between two layers. The ground state of these excitons
in a WMW device is a staggered square lattice of alternating up and down
dipoles (see Fig. 3). The interaction energy per exciton of such a crystal has
a minimum at the concentration $n_{{\mathrm{x}}c}=0.12d^{-2}$. Therefore, at
low illumination intensities, when $n_{\mathrm{x}}\ll n_{{\mathrm{x}}c}$, all
excitons condense into droplets of density $n_{{\mathrm{x}}c}$, which do not
interact with each other.

Ref. [Sammon and Shklovskii, 2019] ignored the tunneling of an exciton hole
between two possible locations in the lower and upper WSe2 layer. For a single
exciton this tunneling hybridizes two opposite orientations of a dipole moment
and creates a quadrupolar dark exciton Slobodkin  _et al._ (2020). However, in
a system of many interacting excitons this happens only at small enough
concentration $n$ when the energy splitting between symmetric and anti-
symmetric hole states is larger than the electrostatic energy required to
invert the orientation of an exciton dipole in the staggered classical
crystal. According to Ref. [Slobodkin  _et al._ , 2020] in WMW devices, the
quadrupolar exciton phase marginally loses to the staggered crystal at
$n_{\mathrm{x}}\sim n_{{\mathrm{x}}c}$.

Figure 1: Band alignment in MoSe2/WSe2 bilayer. Absorption of a light quantum
$\hbar\omega$ results in an electron in MoSe2 layer and a hole in WSe2 layer
bound by the Coulomb interaction in an spatially indirect exciton. Figure 2:
Schematic of a trilayer WSe2/MoSe2/WSe2. WSe2 (first and third) layers are red
and MoSe2 (second) layer is blue. When the device is illuminated at low
temperatures, the type II band alignment of neighboring WSe2/MoSe2
heterostructures allows the formation of indirect excitons consisting of an
electron in MoSe2 and a hole in WSe2. Excitons of opposite polarity attract
each other and form a staggered crystal with alternating exciton dipoles.

A recent experimental paper Bai  _et al._ (2022) used optical spectroscopy to
study exciton phases in a WMW device gated by two transparent graphene layers
separated by $\sim 10$ nm hBN spacers. The authors discovered the crystalline
phase of indirect excitons. Indeed, they observed sharper peaks in
photoluminescence due to stronger localization of excitons. They also measured
directly the diffusion coefficient of excitons and showed that it decreases in
the crystalline phase. The authors attributed the low density dark exciton
state to the predicted quadrupolar exciton Slobodkin  _et al._ (2020), and
showed how the transition from dark to bright exciton crystal happens with
increasing exciton concentration or with growing electric field.

Figure 3: The top view of an anti-ferroelectric square lattice of alternating
dipolar excitons or biexcitons. Black (white) circles correspond to dipoles
whose orientation points up (down). Straight lines are guides for eye.

In this paper we turn to TMDC pentalayers WSe2/MoSe2/WSe2/MoSe2WSe2 (WMWMW).
We claim that in a pentalayer, as shown in Fig. 4, spatially indirect excitons
of two consecutive bilayers attract each other and form linear biexcitons
111Similar exciton attraction was previously studied for two close bilayer
systems Cohen  _et al._ (2016); Hubert  _et al._ (2019, 2020); Choksy  _et
al._ (2021). Two biexcitons with opposite directions of their dipole moments
attract each other at large distances in the plane of the pentalayer. On the
other hand, at small distances they repel each other because they have in-
plane charges of the same sign in the three middle layers. This competition of
biexciton attraction and repulsion leads to an anti-ferroelectric biexciton
crystal in a WMWMW pentalayer. The top and side view of the biexciton crystal
are shown in Fig. 3 and 4.

Figure 4: Pentalayer WSe2/MoSe2/WSe2/MoSe2/WSe2 device. WSe2 layers are red
and MoSe2 layers are blue. When the device is illuminated at low temperatures,
linear biexcitons with alternating orientations are created. Biexcitons form a
staggered square lattice, and the top view of which is shown in Fig. 3.

The electrostatic energy of the staggered crystal per biexciton in a
pentalayer WMWMW is calculated below. The result is shown by the full black
curve as a function of $nd^{2}$ in Fig. 5, where $n$ is the 2D concentration
of biexcitons. This curve has a minimum at $n=n_{c}=0.14d^{-2}$. In the same
figure, by the blue dash curve we reproduce the electrostatic energy per
exciton of a staggered crystal in a WMW trilayer as a function of
$n_{\mathrm{x}}d^{2}$, where $n_{\mathrm{x}}$ is the concentration of excitons
Sammon and Shklovskii (2019). We see that although both energies have minimums
at close values of $nd^{2}$ or $n_{\mathrm{x}}d^{2}$, the minimum of cohesive
energy of biexcitons is approximately 3 times deeper. This allows to study the
biexciton crystal in a WMWMW device at higher temperatures. We also show below
that, in the pentalayer WMWMW, hybridization of biexcitons with opposite
directions of their dipole moments into dark quadrupolar boexcitons plays a
minor role, so that the staggered anti-ferroelectric crystal of biexcitons
dominates the phase diagram.

We therefore predict that at $n<n_{c}$ biexcitons condense into crystallites
with $n=n_{c}$. This crystallization can be observed spectroscopically, via
additional to isolated biexciton red shift of the exciton photoluminescence.
It originates due to interaction of recombining an electron-hole pair with
surrounding biexciton crystal, grows during condensation, and saturates at the
level independent on the illumination intensity, when the final density of the
biexciton crystal $n_{c}$ is reached.

The plan of the rest of the paper is as follows. In section II, we address the
issue of biexciton hybridization and show that in the pentalayer case it can
be neglected. In section III, we present our calculation of the interaction
energy of the biexciton crystal. In section IV, we calculate of the energy
cost to flip the orientation of a biexciton in a fixed staggered crystal. In
section V, we consider a device made of five layers of MoSe2 separated by
identical hBN spacers and show that a critical value of voltage applied to its
layers 1, 3, 5 with respect to layers 2, 4 abruptly induces a finite-
concentration biexciton crystal. This means that at this voltage, the
pentalayer 135-24 differential capacitance is infinite. In section VI, we
briefly comment on the four-layer contactless device.

Figure 5: The full black curve shows the dimensionless electrostatic energy
per biexciton $U/(e^{2}/\kappa d)$ in a staggered lattice (c.f. Fig. 3) versus
the dimensionless biexciton density $nd^{2}$ in a WMWMW device. The blue
dashed curve reproduces the electrostatic energy of a staggered lattice per
exciton versus the dimensionless density of excitons $n_{\mathrm{x}}d^{2}$ in
a WMW device Sammon and Shklovskii (2019).

##  II Competition between dipolar and quadrupolar biexcitons

Let us show that the staggered crystal of dipolar biexcitons in a WMWMW device
easily survives the competition of a crystal of quadrupolar dark biexcitons.
Indeed, there is a possibility that the quantum mechanical tunneling of holes
between bottom and top WSe2 layers hybridizes two biexcitons with opposite
orientations of their dipole moments into a symmetric quadrupolar dark
biexciton. Whether the quadrupolar biexcitons win depends on the relationship
between the quantum energy splitting of symmetric and antisymmetric biexciton
states and the classical electrostatic energy cost of inverting the direction
of a single biexciton in a fixed staggered lattice of its neighbors. We know
that near $n_{\mathrm{x}}=n_{{\mathrm{x}}c}$ in a WMW device, the staggered
biexciton crystal phase has smaller energy and marginally dominates the phase
diagram Slobodkin  _et al._ (2020). Now we should look at this competition in
a pentalayer device.

Below we calculate the Coulomb energy cost $\Delta U$ of inverting the
orientation of a dipolar biexciton in a WMWMW device. It is plotted as a
function of $nd^{2}$ in Fig. 6 together with the cost of inverting a dipolar
exciton for a WMW device as a function of $n_{\mathrm{x}}d^{2}$. We see that
the former is approximately twice larger than the latter at small $n$ and
$n_{\mathrm{x}}$. On the other hand, the quantum energy splitting in a WMWMW
device is much smaller than in a WMW device. Indeed, the tunneling of a hole
from the top WSe2 layer to the bottom one requires simultaneous hopping of two
holes. The hole from the middle WSe2 layer hops to the bottom WSe2 layer,
while another hole of the same biexciton hops from the top WSe2 layer to the
middle WSe2 layer. As a result in a WMWMW device, the quantum energy splitting
looses the competition by a large margin and our classical theory is
applicable practically at all interesting concentrations. Together with a more
stable staggered lattice due to a larger interaction energy per biexciton,
this makes the physics of a WMWMW device simpler than that of a WMW one.

Figure 6: The full black curve shows the dimensionless energy cost $\Delta
U/(e^{2}/\kappa d)$ of inverting of a biexciton dipole moment in a pentalayer
device (c.f. Fig. 4) by moving a hole between the top WSe2 layer and the
bottom WSe2 layer versus the dimensionless biexciton density $nd^{2}$. The
blue dashed curve shows the energy of inverting an exciton in a trilayer
device (c.f. Fig. 2) by moving a hole from the top WSe2 layer to the bottom
one versus the exciton density $n_{\mathrm{x}}d^{2}$.

##  III Calculation of the energy of staggered biexciton crystal

In the next two sections, we explain how we arrived at results shown in Fig. 5
and Fig. 6. Without loss of generality, we pick in Fig. 4 a biexciton with its
dipole moment pointing up as the reference point, and calculate its
interaction with all other charges of the lattice. For convenience we separate
$U$ into four terms $U=U_{h1}+U_{e1}+U_{h2}+U_{e2}$, where $h1,\,h2$
($e1,\,e2$) label the first and second holes (electrons) counting from the top
in the reference biexciton. In the calculation of the biexciton interaction
energy we ignore the interaction between electrons and holes of the same
biexciton which are responsible only for its self-energy.

Each of four terms of $U$ consists of contributions of two sublattices labeled
by $\bullet$ (up) and $\circ$ (down) in Fig. 3 ($r_{\alpha}$ is the 2D
coordinate of a biexciton $\alpha$):

$\displaystyle U_{h1}=\frac{e^{2}}{2\kappa}\sum\limits_{\alpha}^{\circ}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(4d)^{2}}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{r_{\alpha}^{2}+d^{2}}}-\frac{1}{\sqrt{r_{\alpha}^{2}+(3d)^{2}}}\right)$
$\displaystyle+\frac{e^{2}}{2\kappa}\sum\limits_{\alpha\neq 0}^{\bullet}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{r_{\alpha}^{2}+d^{2}}}-\frac{1}{\sqrt{r_{\alpha}^{2}+(3d)^{2}}}\right),$
(1) $\displaystyle U_{e1}=\frac{e^{2}}{2\kappa}\sum\limits_{\alpha}^{\circ}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{r_{\alpha}^{2}+d^{2}}}-\frac{1}{\sqrt{r_{\alpha}^{2}+(3d)^{2}}}\right)$
$\displaystyle+\frac{e^{2}}{2\kappa}\sum\limits_{\alpha\neq 0}^{\bullet}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}-\frac{2}{\sqrt{r_{\alpha}^{2}+d^{2}}}\right),$
(2) $\displaystyle U_{h2}=\frac{e^{2}}{2\kappa}\sum\limits_{\alpha\neq
0}\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}-\frac{2}{\sqrt{r_{\alpha}^{2}+d^{2}}}\right),$
(3) $\displaystyle U_{e2}=\frac{e^{2}}{2\kappa}\sum\limits_{\alpha}^{\circ}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}-\frac{2}{\sqrt{r_{\alpha}^{2}+d^{2}}}\right)$
$\displaystyle+\frac{e^{2}}{2\kappa}\sum\limits_{\alpha\neq 0}^{\bullet}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{r_{\alpha}^{2}+d^{2}}}-\frac{1}{\sqrt{r_{\alpha}^{2}+(3d)^{2}}}\right),$
(4)

where $\kappa$ is the dielectric constant of the system and $\alpha\neq 0$
means we exclude the coordinate origin (a dipole pointing up) in the
summation. As a result, we obtains the total interaction energy

$\displaystyle U=\frac{e^{2}}{\kappa d}f(nd^{2}),$ (5)

where the dimensionless function $f(x)$ is given by

$\displaystyle
f(x)=2\sqrt{x}\sum\limits_{i=1}^{\infty}\sum\limits_{j=0}^{\infty}\left(\frac{4}{\sqrt{i^{2}+j^{2}}}+\frac{4}{\sqrt{i^{2}+j^{2}+2^{2}x}}\right.$
$\displaystyle-\left.\frac{6}{\sqrt{i^{2}+j^{2}+x}}-\frac{2}{\sqrt{i^{2}+j^{2}+3^{2}x}}\right)$
$\displaystyle+2\sqrt{x}\sum\limits_{i=0}^{\infty}\sum\limits_{j=0}^{\infty}\quantity(2-\delta_{j0})\left(\frac{1}{\sqrt{(2i+1)^{2}+(2j)^{2}+4^{2}x}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{(2i+1)^{2}+(2j)^{2}}}\right).$ (6)

The result is shown by the full black curve as a function of $nd^{2}$ in Fig.
5, where $n$ is the 2D concentration of biexcitons.

Comparison with WMW trilayer Sammon and Shklovskii (2019) shows that minimum
of energy is 3 times deeper in a pentalayer. The reason for this is that
biexcitons in a pentalayer device have twice larger dipole moment than
excitons in a trilayer device. Thus, the absolute value of the attraction
energy is four times larger. Repulsion between two biexcitons at small
distances is substantially larger than the case of excitons as well. This
happens because for biexcitons it originates from same-sign charges repulsion
in three layers instead of one layer charge repulsion for excitons in the WMW
device.

##  IV Calculation of the energy cost of biexciton dipole moment inversion

Next we calculate the Coulomb energy cost to flip the orientation of a
biexciton in the staggered lattice of a WMWMW pentalayer (c.f. Fig. 4). By
moving the hole $h1$ of a reference biexciton in the top WSe2 layer to the
bottom WSe2 layer, i.e. to the position which we label as $h3$, we find its
electrostatic energy

$\displaystyle U_{h3}=\frac{e^{2}}{2\kappa}\sum\limits_{\alpha}^{\circ}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{r_{\alpha}^{2}+d^{2}}}-\frac{1}{\sqrt{r_{\alpha}^{2}+(3d)^{2}}}\right)$
$\displaystyle+\frac{e^{2}}{2\kappa}\sum\limits_{\alpha\neq 0}^{\bullet}$
$\displaystyle\left(\frac{1}{\sqrt{r_{\alpha}^{2}+(2d)^{2}}}+\frac{1}{\sqrt{r_{\alpha}^{2}+(4d)^{2}}}\right.$
$\displaystyle-\left.\frac{1}{\sqrt{r_{\alpha}^{2}+d^{2}}}-\frac{1}{\sqrt{r_{\alpha}^{2}+(3d)^{2}}}\right).$
(7)

The energy cost of inverting a biexciton orientation is then given by $\Delta
U=U_{h3}-U_{h1}$. This energy is plotted as the full black curve in Fig. 6,
where the energy cost to flip an exciton for a WMW trilayer is shown by the
dashed blue curve for comparison. At very small concentrations
$nd^{2},n_{\mathrm{x}}d^{2}\to 0$, we have $\Delta U\propto
n^{3/2},n_{\mathrm{x}}^{3/2}$ with the coefficient of a pentalayer device 4
times larger than that for a trilayer. At larger concentration
$n,n_{\mathrm{x}}\gtrsim n_{c}=0.14d^{-2}$, the energy cost to flip a
biexciton in a pentalayer is 2 times larger than to flip an exciton in a
trilayer.

Above we used the three-dimensional Coulomb interaction between charges,
because we assumed that all studied devices have relatively thick hBN layers
below and above them which make dielectric constant contrast relatively small.
These hBN layers are used for gating and are not shown in Figs. 2 and 4.

Our theory also assumes that the electrons (holes) are point-like particles
located in the middle of a layer. This is a good assumption if $n^{-1/2}\gg
a$, where $a$ is the localization length of electrons (holes). Using an
estimate $a=\hbar/\sqrt{2m^{\star}E_{b}}$, where the effective mass
$m^{\star}\sim 0.5m_{0}$ 222For electrons in MoSe2 the effective mass is
$m^{\star}_{e}=0.49\,m_{0}$ Liu  _et al._ (2020, 2021) and for holes in WSe2
is $m^{\star}_{h}=0.42\,m_{0}$ Nguyen  _et al._ (2019)., $m_{0}$ is the free
electron mass, and $E_{b}\sim 250$ meV Gillen and Maultzsch (2018); Kamban and
Pedersen (2020) is the binding energy of an exciton, we get $a\sim 0.6$ nm. On
the other hand, $n_{c}^{-1/2}\sim 2.7d\sim 1.9$ nm, so our classical theory is
correct for $n<n_{c}$ and needs a revision for larger $n>n_{c}$ where
substantial overlap of wave functions leads to decay of biexcitons into free
excitons and eventually to electron-hole plasmas Xu and Fogler (2021). This,
however, does not change our prediction that at $n<n_{c}$ biexcitons condense
into droplets with $n\sim n_{c}$, which produces red shifted photoluminescence
independent on the light intensity.

So far we dealt with zero temperature and ignored Coulomb impurities in the
system. Our predictions remain quantitatively correct if the temperature $T$
is much smaller than the energy of the crystal per exciton $0.054e^{2}/\kappa
d\sim 180\,\mathrm{K}$ and concentration of Coulomb impurities $n_{i}\ll
n_{c}\sim 3\times 10^{13}$ cm-2, where we use $d\sim 0.7$ nm and $\kappa\sim
7$ Laturia  _et al._ (2018).

##  V Infinite differential capacitance of voltage induced biexciton crystal

Above we dealt with biexcitons in optically excited WMWMW heterolayers. One
can study similar biexciton physics in a device made of five monolayers of
MoSe2 separated by identical hBN spacers. If layers 1, 3, and 5 are biased
with respect to layers 2 and 4 by a positive voltage $V$, one can study its
nontrivial 135-24 capacitance (c.f. Fig. 7). Theory of a similar capacitance
was the main subject of the previous study of a three-layer device Sammon and
Shklovskii (2019). In the case of a pentalayer, all the electrons and holes of
neighboring layers form excitons, which in turn form biexcitons with two-
dimensional concentration $n$ (c.f. Fig. 7). Thus, the voltage-biased
pentalayer has the same configuration of biexcitons as the contactless but
illuminated WMWMW device studied above. Therefore, the charge of the 135-24
capacitor per unit area is $2ne$ can be calculated using the energy of the
biexciton crystal of a pentalayer $U(n)$ shown in Fig. 5.

Figure 7: Cross section of a capacitor made of five MoSe2 monolayers shown in
blue (dark) color. hBN spacers of thickness $d_{0}$ are shown by yellow
(light) color. The distance between two charges of an exciton is equal to $d$.
A voltage $V$ applied between layers 1, 3, 5 and layers 2, 4, induces
biexcitons which form a staggered anti-ferroelectric square lattice with a
lattice constant $n^{-1/2}$.  Figure 8: Dimensionless biexciton density
$nd^{2}$ as a function of dimensionless voltage $(V-V_{2})/(e/\kappa d)$ for
the pentalayer capacitor shown in Fig. 7. The dashed curve shows the function
$n(V)$ obtained from Eq. (11), while the solid red curve shows the equilibrium
function $n(V)$ obtained using the Maxwell’s rule Eq. (13). We see that in
equilibrium, $n(V)$ jumps from zero to $n_{c}$ at the critical voltage
$V_{c}$.

One could expect that the first biexciton enters the capacitor at the voltage

$\displaystyle V_{2}=(2E_{g}-E_{b})/2e,$ (8)

where $E_{g}$ is the band gap of MoSe2 and $E_{b}>0$ is the ionization energy
of an biexciton (energy necessary to make of two free electrons and two free
holes from it). Remarkably, at low temperatures, the attraction between
biexcitons in the pentalayer device causes a first order phase transition as
the applied voltage $V$ grows (c.f. Fig. 8). While at small $V$, there are no
excitons or biexcitons in the pentalayer and the capacitor remains uncharged,
at some critical value $V=V_{c}<V_{2}$ the whole lattice of alternating
biexcitons with concentration $n_{c}$ emerges. This means that a macroscopic
charge $Q_{c}=2eSn_{c}$, where $n_{c}=0.14d^{-2}$ and $S$ is the device area,
enters this capacitor. Thus, the differential capacitance $C=dQ/dV$ has a
$\delta$-peak at $V=V_{c}$.

Figure 9: The non-singular capacitance $C_{u}/(4C_{g})$ as a function of
dimensionless voltage $(V-V_{c})/(e/\kappa d)$ for the pentalayer MOSe2 device
shown in Fig. 7. The total geometric capacitance for the pentalayer is equal
to $4C_{g}$, where $C_{g}=\kappa S/4\pi d$ is the geometric capacitance
between two MoSe2 layers.

The differential capacitance of such a device can be determined from the total
electrostatic energy $E$ of the system as

$\displaystyle
C^{-1}=\frac{d^{2}E}{dQ^{2}}=\frac{1}{(2eS)^{2}}\frac{d^{2}E}{dn^{2}}.$ (9)

The energy $E$ of this system of classical charges can be written as

$\displaystyle E=2enSV_{2}+nSU,$ (10)

where $V_{2}$ is the voltage necessary to create a single isolated biexciton
and is given by Eq. (8), while $U$ is the interaction energy of the anti-
ferroelectric biexciton crystal per biexciton. Using Eqs. (5) and (10), one
can calculate the voltage as

$\displaystyle V=\frac{dE}{dQ}=V_{2}+\frac{1}{2e}\frac{d(nU)}{dn}.$ (11)

Eq. (11) results in the dimensional density $n(V)d^{2}$ as a function of the
voltage $(V-V_{2})/(e/\kappa d)$ shown by the dashed curve in Fig. 8. Note
that in a range of voltages this curve has three branches: a lower branch
$n=0$, a middle branch, and an upper branch. Within the middle branch, the
differential capacitance defined by Eq. (9) is negative, so that this region
is thermodynamically unstable and inaccessible. Thus, in experiments, we do
not expect the density to change continuously along the dashed curve, but
instead it should follow the red full curve where the density of biexcitons
$n$ jumps from zero to a finite value $n_{c}=0.14d^{-2}$ at the critical
voltage

$V_{c}=V_{2}-0.027\frac{e}{\kappa d}.$ (12)

Here $V_{c}$ is determined by the Maxwell area rule Landau and Lifshitz (1980)

$\int_{0}^{n_{c}}n(V)dV=0,$ (13)

where the integral is taken along the dashed curve in Fig. 8. At $V=V_{c}$ the
two regions lying between the vertical red line and the dashed curve have
equal areas. It is worth noting that $n_{c}$ obtained from Maxwell’s area rule
is the same $n_{c}$ at which $U$ reaches its minimum value (c.f. Fig. 5). As
the density abruptly jumps, there is a $\delta$-peak in the capacitance at
$V=V_{c}$. For $V\geq V_{c}$ we can write the capacitance as

$C(V)=2eSn_{c}\delta(V-V_{c})+C_{u}(V),$ (14)

where the non-singular capacitance $C_{u}(V)=2eS(dn/dV)$ is obtained by
differentiating the upper branch of the $n(V)$ red curve shown in Fig. 8 with
respect to $V$, and the result is shown in Fig. 9. As $V$ approaches $V_{c}$
from above, $C_{u}(V)$ grows as $(V-V_{c}+0.01e/\kappa d)^{-1/2}$, and reaches
a very large maximum value $C_{u}(V_{c})\simeq 64C_{g}$, where $C_{g}=\kappa
S/4\pi d$ is the geometrical capacitance of the capacitor formed by two MoSe2
planes separated by distance $d$. (The origin of small number 0.01 and large
number 64 can be traced to small value $n_{c}d^{2}=0.14$).

Above we dealt with the strong effect of correlations between electrons and
holes, which makes the capacitance much larger than the geometrical one.
Historically, relatively small addition to the geometrical capacitance of a
plane capacitor due to strong electron correlations leading to the negative
compressibility of electron gas in one of the capacitor plates was predicted
long ago Bello  _et al._ (1981) and observed in Si MOSFET Kravchenko  _et al._
(1990) and GaAs two-dimensional gas Eisenstein  _et al._ (1992). Later it was
predicted that in a thin capacitor with a small charge, electron and holes
form compact dipoles weakly repelling each other at large distances. As a
result, its capacitance becomes much larger than the geometrical one Skinner
and Shklovskii (2010). Indeed, a substantial capacitance enhancement was
simultaneously observed Li  _et al._ (2011).

In this section, we predicted that in the clean pentalayer capacitor, the
135-24 differential capacitance diverges at some critical voltage at zero
temperature due to attraction of biexcitons. Such a divergence was previously
predicted for two other kinds of capacitors, where attraction between dipoles
dominates their repulsion Chen  _et al._ (2011); Sammon and Shklovskii (2019).
Of course, in the real world the maximum capacitance has a much larger than
geometrical but finite value, due to a finite disorder and a finite
temperature.

##  VI Four-layer devices

So far in this paper we dealt with pentalayer devices and their comparison
with trilayer ones. Let us briefly dwell on the contactless, but illuminated
by interband light, four layer devices. In a asymmetric four layer WMWM
device, all excited by light excitons at small density bind into identical
linear dipolar biexcitons, which repel each other and form a ferroelectric
(see Fig. 10).

Alternatively, one can imagine a symmetric four layer WMMW device, where
oppositely directed excitons emerge in the two bottom and the two upper
layers. They attract each other at large distances and repel at small ones.
The ground state of such four layer device is an anti-ferroelectric staggered
crystal similar to the one shown in Fig. 2 for WMW layer. However, for a given
two-dimensional concentration of excitons $n$, both attraction and repulsion
are weaker. We found that the staggered crystal cohesive energy per exciton in
a WMMW tetralayer is 8 times smaller than in a WMW trilayer. A single exciton
in the upper two layers can in principle hybridize with the one in the lower
two layers, resulting in a quadrupolar dark exciton. But the related matrix
element and the quantum energy splitting are probably much weaker even than
the Coulomb energy cost of inverting an exciton dipole moment in a staggered
crystal and therefore can be ignored.

Figure 10: Asymmetric four layer WSe2/MoSe2/WSe2/MoSe2 device. WSe2 layers are
red and MoSe2 layers are blue. When the four layer device is illuminated at
low temperatures, identical linear biexcitons which repel each other are
formed.

###### Acknowledgements.

We are grateful to L. Butov, M.M. Glazov, M. Sammon and X.-Y. Zhu for helpful
discussion. Y.H. gratefully acknowledges support from Larkin Fellowship at the
University of Minnesota.

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