MengluLi1 Xiao-PingZhang2,∗

# Interpretable Temporal Class Activation Representation for Audio Spoofing
Detection

###### Abstract

Explaining the decisions made by audio spoofing detection models is crucial
for fostering trust in detection outcomes. However, current research on the
interpretability of detection models is limited to applying XAI tools to post-
trained models. In this paper, we utilize the wav2vec 2.0 model and attentive
utterance-level features to integrate interpretability directly into the
model's architecture, thereby enhancing transparency of the decision-making
process. Specifically, we propose a class activation representation to
localize the discriminative frames contributing to detection. Furthermore, we
demonstrate that multi-label training based on spoofing types, rather than
binary labels as bonafide and spoofed, enables the model to learn distinct
characteristics of different attacks, significantly improving detection
performance. Our model achieves state-of-the-art results, with an EER of 0.51%
and a min t-DCF of 0.0165 on the ASVspoof2019-LA set.

###### keywords:

Deepfake audio, XAI, Speech Anti-spoofing, Deepfake detection,
Interpretability

##  1 Introduction

††∗X.-P. Zhang is the corresponding author.

Audio spoofing detection techniques have gained more attention recently due to
the threat brought to automatic speaker verification (ASV) systems and their
harmful impact of spoofed audio on societies. Spoofing countermeasures attempt
to improve detection performance by developing both classifier structure and
feature extraction methods. The detection algorithms usually operate upon the
hand-crafted acoustic feature [1, 2, 3] until the emergence of advanced
learnable front-ends, involving convolution-based networks [4, 5] or self-
supervised learning (SSL)-based architectures [6, 7], which outperform the
traditional feature extraction. Regarding architectures of the classifier,
Deep learning (DL)-based countermeasures have demonstrated their advance
compared to traditional machine learning models, such as GMM [8]. Notably,
light CNN [9, 10], ResNet [11, 12], and DARTS [13, 14] algorithms have made
great achievements in increasing the detection accuracy on both known and
unknown spoofing attacks generated by Text-to-Speech (TTS) and Voice
Conversion (VC) techniques.

While current state-of-the-art models achieve high accuracy on publicly
available datasets, they fail to provide explanations for their detection
outcomes or the decision-making process due to the black-box nature of DL-
based techniques. Designing a detection algorithm with higher
interpretability, especially through visualization, is crucial. It not only
clarifies how and why a detection model makes decisions, fostering trust in
the algorithm and its outputs, but also allows for understanding the
architecture of the algorithm and potential improvements by adjusting specific
segments. To address this problem, we propose an interpretable audio spoofing
detection model with a guaranteed high detection rate. We utilize a pre-
trained SSL model to extract frame-level features, which are then combined
with utterance-level information to form a high-level embedding of the audio
input. Our proposed detection pipeline incorporates an attention mechanism
along feature channels to generate a temporal class activation representation.
This representation effectively localizes the most discriminative frames
contributing to different labels in the detection process, while also making
them visually accessible.

The new contributions in this work are (1) We propose a novel audio spoofing
detection model that leverages an effective feature set comprising SSL-based
frame-level features and attentive utterance-level features. (2) The proposed
model provides a class activation map as a visualizable interpretation of
detection results, revealing the underlying temporal dynamics. (3) We
demonstrate the effectiveness of employing multi-label classification
training, rather than binary labels, to learn distinct characteristics of TTS
and VC-based artifacts.

##  2 Related Work

A group of existing works have utilized explainable artificial intelligence
(XAI) tools to uncover the behaviour of deep neural network algorithms in
detecting spoofed audio [15]. Ge et al. [16] utilize SHapley Additive
exPlanations (SHAP) [17] to identify the characteristics of artifacts relied
on various spoofing attacks. Lim et al. [18] apply both Deep Taylor [19] and
layer-wise relevance propagation (LRP) [20] to learn the attribution score of
audio formats in spectrograms. The Gradient-weighted Class Activation Mapping
(Grad-CAM) [21] is used in [22] to identify the significant regions in the
spectrogram. Motivated by Grad-CAM, we construct a learnable class activation
mechanism to localize the discriminative regions. However, unlike the existing
approaches that apply XAI tools externally, our method provides internal
justification for the decision-making process by considering both detection
capability and outcome interpretability simultaneously. We utilize the class
activation representation within our proposed detection model to identify and
visualize the crucial frames that determine detection outcomes.

##  3 Proposed Model

In this section, we elaborate the feature extraction module and the detection
pipeline of our model. The feature extraction module consists of both frame-
level and utterance-level representation. The detection pipeline includes
channel attention mechanism conditioning on the temporal features. The
architecture of the proposed detection model is illustrated in Figure 1.

Figure 1: Overall architecture of proposed audio spoofing detection model.

###  3.1 The SSL-based feature at the frame level

The SSL models [23, 24] have demonstrated its ability to generate latent
representations of raw audio waveform. Our proposed model utilizes a pre-
trained wav2vec 2.0 XLS-R model [25] as the front-end feature extractor to
obtain temporal representations for the raw audio inputs. The wav2vec 2.0
model consists of a CNN-based encoder module, a context network with the
Transformer architecture, and a quantization module, to produce a quantized
latent speech representation that captures the dependent information from the
entire audio sequence [23]. The selected wav2vec 2.0 XLS-R model, with 300
million parameters, is pre-trained on unlabelled speech data from various
sources with multiple languages. During the training phase, we fine-tune the
all parameters in this pre-trained model with our downstream classifier using
labelled training data, which makes this SSL-based front-end feature extractor
learn the deep embedding that more adapt to the spoofing detection task. Given
an input audio $x$, the corresponding frame-level feature representation
$\bm{S^{f}}\in\mathbb{R}^{T\times C}$ is extracted, where $T$ and $C$ refer to
the number of time frames and channels, respectively. The feature
representation is then fed to two stacks consisting of a fully connected (FC)
layer, batch normalization (BN) with ReLU activation, and a dropout layer for
data downsampling.

###  3.2 Attentive statistical feature at the utterance level

The deep embedding extracted from the wav2vec 2.0 model represent the speech
information at frame levels. Additionally, the utterance-level information is
also crucial to spoofing detection. Therefore, we implement attentive
statistical pooling [26] on the frame-level embedding to obtain an utterance-
level feature. Given the frame-level embedding, $\bm{S^{f}}$, we first
calculate its frame-level attentive score $e_{t}$ for each frame $t$ by:

$\displaystyle e_{t}$ $\displaystyle=f(W\bm{S}^{f}_{t}+b),$ (1)

where $f(\cdot)$ is tanh function, and the parameters $W$ and $b$ are shared
across all $C$ channels to avoid overfitting. Then, the score $e_{t}$ is
normalized over all frames using a softmax function:

$\displaystyle\alpha_{t}$
$\displaystyle=\frac{\exp(e_{t})}{\sum_{\tau}^{T}\exp(e_{\tau})}.$ (2)

The normalized attentive score represents the importance of each frame $t$,
and then it works as the weights to be applied on the embedding, $\bm{S^{f}}$,
to calculate the mean and standard deviation respectively, as follows:

$\displaystyle\tilde{\mu}$
$\displaystyle=\sum_{t}^{T}\alpha_{t}\bm{S}^{f}_{t},$ (3)
$\displaystyle\tilde{\sigma}$
$\displaystyle=\sqrt{\sum_{t}^{T}\alpha_{t}\bm{S}^{f}_{t}\odot\bm{S}^{f}_{t}-\tilde{\mu}\odot\tilde{\mu}}.$
(4)

The weighted mean $\tilde{\mu}$ and the weighted standard deviation
$\tilde{\sigma}$ are concatenated and projected into 1-D representation
$\bm{S^{u}}\in\mathbb{R}^{1\times C}$ as an utterance-level feature. This
weighted statistics describe the distribution of important frames across the
entire utterance. In this way, the utterance-level feature provides a higher-
level perspective by focusing on specific frames to emphasize discriminative
factors in the spoofing process.

The utterance-level feature $\bm{S^{u}}$ and the frame-level feature
$\bm{S^{f}}$ are concatenated along the time dimension to form the final
feature representation $\bm{S}\in\mathbb{R}^{T^{\prime}\times C}$, where
$T^{\prime}=T+1$.

###  3.3 The detection pipeline using temporal class activation

Our downstream detection pipeline receives the extracted audio features to
determine the type of spoofing while also learning to identify when the
spoofing sounds occur, as further detailed below.

####  3.3.1 Extracting channel attention vector (CAV)

Given the feature embedding $\bm{S}$, we extract the CAV to indicate the
importance of different feature channels contributing each class type $k$ as
formulated in Equation 5:

$\displaystyle\bm{A}_{k}$ $\displaystyle=W_{k}^{\top}\bm{S},$ (5)

where $W_{k}\in\mathbb{R}^{T^{\prime}}$ is the weight corresponding to $k$-th
class across all time frames, and $\bm{A}_{k}\in\mathbb{R}^{C}$ is the CAV for
each class.

####  3.3.2 Classifier on CAV with WCE loss

We pass $\bm{A}_{k}$ through a FC layer to make the first label prediction,
resulting in a prediction logit vector $\bm{z}\in\mathbb{R}^{K}$, where $K$ is
the total number of classes. The $\bm{z}$ and the utterance-level label
$\bm{y}$ are compared to compute a weighted multi-label cross-entropy (WCE)
loss in the following formula:

$\displaystyle\mathcal{L}_{CAV}$
$\displaystyle=-\frac{1}{K}\sum_{k=1}^{K}\bm{W}_{CE}[k]\cdot\log\frac{\exp(\bm{z}[k])}{\sum_{k}\exp(\bm{z}[k])}\cdot\bm{y}[k],$
(6)

where $\bm{z}[k]$ and $\bm{y}[k]\in\\{0,1\\}$ denote the predict logit and
ground-truth label of $k$-th class. $\bm{W}_{CE}$ is the weight assigned to
each class $k$.

####  3.3.3 Extracting temporal class activation (TCA) feature

We implement a learnable gating mechanism onto the CAV, $\bm{A}_{k}$, which
effectively selects and emphasizes the discriminative feature channels. The
gating mechanism is an FC layer with a softmax function along the dimension of
the feature channel, which gives an attentive tensor denoted as
$\bm{M}\in\mathbb{R}^{C\times K}$. Then, we apply $\bm{M}$ on the feature
embedding $\bm{S}$ through the inner product. Additionally, the prediction
logit vector $\bm{z}$ is used as a class-specific mask, thereby generating an
TCA feature $\bm{S^{\prime}}\in\mathbb{R}^{T^{\prime}\times C}$, which
highlights the discriminative regions along the temporal domain for each
class. We obtain $\bm{S^{\prime}}$ as follows,

$\displaystyle\bm{M}_{c,k}$
$\displaystyle=\frac{\exp(w_{gate}\bm{A}_{c,k})}{\sum_{c}^{C}\exp(w_{gate}\bm{A}_{c,k})},$
(7) $\displaystyle\bm{S^{\prime}}$
$\displaystyle=\bm{z}\cdot(\bm{S}\odot\bm{M}).$ (8)

where $\bm{A_{c,k}}$ denotes the $c$-th item of the channel attention vector
$\bm{A_{k}}$, and $w_{gate}$ acts as a scalar weight for the gating mechanism.

####  3.3.4 Classifier on TCA feature with WCE loss

The second classifier operates based on the TCA feature $\bm{S^{\prime}}$. As
$\bm{S^{\prime}}$ contains both frame-level and utterance-level information,
instead of aggregating all feature elements in $\bm{S^{\prime}}$ using global
pooling, we separate
$\bm{S^{\prime}}=\\{s^{\prime}_{1},s^{\prime}_{2},...,s^{\prime}_{T},s^{\prime}_{T+1}\\}$
into two segments
$\bm{S^{\prime}_{f}}=\\{s^{\prime}_{1},s^{\prime}_{2},...,s^{\prime}_{T}\\}$
of the length of $T$, and $\bm{S^{\prime}_{u}}=\\{s^{\prime}_{T+1}\\}$ and
then apply average pooling to each feature segment. An FC layer is applied
after the pooling operation to obtain a new prediction logit vector
$\bm{z^{\prime}}\in\mathbb{R}^{K}$ for $\bm{S^{\prime}}$. $\bm{z^{\prime}}$ is
also used to compute a weighted CE loss using Equation 6 resulting a
$\mathcal{L}_{TCA}$. Both $\mathcal{L}_{TCA}$ and $\mathcal{L}_{CAV}$ utilize
the same weight $\bm{W}_{CE}$ across each class.

####  3.3.5 Overall objective function

The overall objective function for the detection model is

$\displaystyle\mathcal{L}$
$\displaystyle=\lambda_{1}\mathcal{L}_{CAV}+\lambda_{2}\mathcal{L}_{TCA}$ (9)

where $\lambda_{1}$ and $\lambda_{2}$ are different weight values to balance
between two individual losses. $\lambda_{1}$ and $\lambda_{2}$ are set to 0.3
and 0.7 respectively in our model.

##  4 Experiment and Evaluation

###  4.1 Dataset and evaluation metrics

We use the ASVspoof2019 logical access (LA) dataset [27] for the experiments.
The spoofed data in the training and development sets are generated by four
TTS methods and two VC methods, while the evaluation set consists of 13
different and unseen methods to evaluate the generalization ability of the
detector. We fix all audio samples to the same length of 4 seconds either by
truncating the longer audio clips or concatenating the shorter audio clips
repeatedly. We evaluate the detection performance with two metrics: minimum
normalized tandem detection cost function (min t-DCF) [28] and the Equal Error
Rate (EER). A detection result with a lower min t-DCF value or EER score is
regarded to be more accurate.

Table 1: Performance on the ASVspoof 2019 evaluation set in terms of min t-DCF and pooled EER for state-of-the-art single systems and our proposed system. System | Front-end | min t-DCF | EER(%)
---|---|---|---
Ours | wav2vec 2.0 | 0.0165 | 0.51
Ma et al. [29] | raw waveform | 0.0187 | 0.64
Jung et al. [4] | SincNet | 0.0275 | 0.83
Li et al. [30] | SincNet | 0.0317 | 0.93
Ma et al. [31] | LFCC | 0.0294 | 0.98
Tak et al. [5] | SincNet | 0.0335 | 1.06
Li et al. [2] | LFCC | 0.0345 | 1.06
Luo et al. [1] | LFCC | 0.0328 | 1.07
Wang et al. [13] | wav2vec 2.0 | - | 1.08
Wang et al. [6] | wav2vec 2.0 | - | 1.28
Yang et al. [32] | CQT | 0.0490 | 1.54
Hua et al. [11] | raw waveform | - | 1.64
Ge et al. [14] | raw waveform | 0.0517 | 1.77
Table 2: Breakdown of EER (%) performance for all 13 attacks in ASVspoof2019 LA evaluation set with attack types specified (TTS, VC). Our proposed model, the single state-of-the-art and ablation study results are reported. Pooled EER is shown in the last column. System | A07 | A08 | A09 | A10 | A11 | A12 | A13 | A14 | A15 | A16 | A17 | A18 | A19 | EER(%)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
TTS | TTS | TTS | TTS | TTS | TTS | VC | VC | VC | TTS | VC | VC | VC
Proposed | 0.02 | 0.24 | 0.04 | 0.29 | 0.06 | 0.61 | 0.01 | 0.10 | 1.24 | 0.01 | 0.61 | 0.30 | 0.06 | 0.51
Ma et al. [29] | 0.15 | 0.15 | 0.02 | 0.41 | 0.10 | 0.06 | 0.02 | 0.35 | 0.41 | 0.30 | 2.19 | 0.27 | 0.42 | 0.64
w/o utterance-level feature | 0.01 | 0.02 | 0.01 | 1.32 | 0.02 | 0.42 | 0.00 | 0.08 | 2.71 | 0.00 | 5.74 | 0.61 | 0.04 | 1.12
w/o classifier on CAV | 0.00 | 0.02 | 0.00 | 0.24 | 0.02 | 0.06 | 0.01 | 0.06 | 0.83 | 0.00 | 2.01 | 1.47 | 0.08 | 0.61
w/ binary label | 0.00 | 0.55 | 0.01 | 0.83 | 0.04 | 2.57 | 0.05 | 0.53 | 9.45 | 0.00 | 14.4 | 1.06 | 0.08 | 4.26
Figure 2: Visualizing the temporal class activation feature on the selected
samples, each spoofed audio sample is labeled with its attack type. Different
colors denote the class-relevant attention, with color intensity representing
the level of contribution to detection results.

###  4.2 Model implementation details with multi-label training

The model is implemented with the PyTorch framework. We adopt the pre-trained
XLS-R model 111https://huggingface.co/facebook/wav2vec2-xls-r-300m with 300
million parameters based on the wav2vec 2.0 base model. The XLS-R model is
pre-trained on 436k hours of unlabeled speech in 128 languages, which produces
the speech embedding with a dimension size of 1024 in each 20 millisecond. The
resulting embedding from the wav2vec 2.0 model is compressed to a size of 128
by two linear layers with 512 and 128 hidden units, with dropout layers set at
a rate of 0.2.

During training and validation, we consider spoofing detection as a multi-
label classification problem instead of a binary classification. Based on the
spoofing generating types, the data labelled as spoofed in the training and
validation subsets are categorized into two groups, TTS and VC. Therefore, the
ground truth includes three classes of labels in total, which are bonafide,
TTS spoofed and VC spoofed. We believe that multi-label training will
encourage the model to learn more distinct characteristics to identify TTS and
VC-generated speech, thereby potentially increasing the accuracy of detecting
spoofing speech.

To manage the data imbalance in the training set, we utilize the WCE loss,
where the weights assigned to bonafide, TTS spoofed and VC spoofed are 8, 1,
and 1 respectively. An Adam optimizer [33] with a weight decay of $10^{-4}$ is
used. The model was trained with 50 epochs with a mini-batch size of 10 and a
learning rate of $10^{-5}$. The model with the minimum validation loss for the
development set was selected as the best model for evaluation. All experiments
were performed on a single GeForce RTX 3090 GPU and the implementation code is
publicly available 222https://github.com/menglu-lml/Interpretable-Detection-
IS24.

###  4.3 Experiment result

The performance result of our proposed model is presented in Table 1. Table 1
also illustrates the performance comparison between our proposed model and the
state-of-the-art single systems. The comparisons highlight that our model
outperforms not only other single models utilizing the SSL-based features but
also End-to-End detection systems and other systems employing a variety of
feature types, including learnable feature embedding and hand-crafted acoustic
features.

The first two rows of Table 2 presents the breakdown performance of our
proposed model and the state-of-the-art for each different spoofing attack in
the evaluation set. The results show that our model effectively detects both
TTS and VC-based spoofing speech, and outperforms the state-of-the-art method
on 8 attacks. In particular, our model achieves a notably low EER score on the
A17 attack, which is labelled as the worst-case scenario among all attacks
[34]. This is significant because the A17 utilizes direct waveform
concatenation method on bonafide human voice, resulting in a greater challenge
for detection.

###  4.4 Ablation study

The last three rows of Table 2 illustrates the results of the ablation
experiments to demonstrate the merit of the design choices. Removing the
utterance-level part in the feature embedding leads to a performance
degradation of 54.5% in terms of EER. It shows that the attentive utterance-
level feature effectively emphasizes discriminative frames contributing to the
detection process across the entire utterance. We demonstrate the underlying
connection of CAV to the detection process by ablating the WCE loss upon CAV
in the objective function. This leads to a 16% degradation in EER, dropping to
0.51%. Notably, with the loss upon CAV, our model enhanceperformances in
detecting VC attacks involving the direct conversion of human voices
(A17-A19). The effectiveness of multi-label training is also presented. Using
binary labels in training results in a degradation to 4.26% in EER, with the
decline primarily attributed to the failure to detect VC attacks. It shows
that multi-label training allows the detection model to learn the
discriminative factors in TTS and VC-based spoofed audio separately, which
gains a deeper understanding of the different characteristics of each attack
type.

###  4.5 Evaluation of the visual interpretability

As Figure 2 shows, we visualize the temporal class activation feature on the
audio samples within the evaluation dataset. The visualization uses different
colors to denote the detected audio types for each frame, including TTS-based
spoofed, VC-based spoofed, or bonafide. The intensity of color represents the
detection confidence. Notably, bonafide and TTS-based spoofed (A07-A12, A16)
audio samples are correctly classified in Figure 2. However, some VC-based
spoofed samples (A13-A15) are mislabelled as TTS-based, as indicated by the
activation feature's color. It occurs because the audio in A13-A15 are
generated by the combined VC-TTS spoofing systems, where the TTS voice serves
as the source speaker for the VC model. In such cases, our model effectively
detects the TTS-based voice source in attacks A13-A15, demonstrating that it
has learned the distinct characteristics of spoofed audio generated by TTS and
VC. It is further supported by the correct classification of spoofed audio
generated by pure VC models that utilize human voice as the source (A17-A19).
Additionally, Figure 2 localizes the most discriminative frames in the
detection process, providing justification for the decision made by our
proposed model.

##  5 Conclusion

We are the first to incorporate interpretability directly into the
architecture of audio spoofing detection models, enhancing the transparency of
their decision-making processes while ensuring a high detection rate. The
proposed learnable class activation mechanism identifies and visualizes the
crucial frames that contribute detection outcomes. Our model achieves an EER
of 0.51% on ASVspoof2019 LA set by leveraging utterance-level features and
multi-label classification training. We aim to apply this interpretable model
to detect partially spoofed audio by localizing the spoofing segments as
future work.

## References

  * [1] A. Luo, E. Li, Y. Liu, X. Kang, and Z. J. Wang, ``A capsule network based approach for detection of audio spoofing attacks,'' in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 6359–6363.
  * [2] C. Li, F. Yang, and J. Yang, ``The role of long-term dependency in synthetic speech detection,'' _IEEE Signal Processing Letters_ , vol. 29, pp. 1142–1146, 2022.
  * [3] M. Li and X.-P. Zhang, ``Robust audio anti-spoofing system based on low-frequency sub-band information,'' in _2023 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA)_. IEEE, 2023, pp. 1–5.
  * [4] J.-w. Jung, H.-S. Heo, H. Tak, H.-j. Shim, J. S. Chung, B.-J. Lee, H.-J. Yu, and N. Evans, ``Aasist: Audio anti-spoofing using integrated spectro-temporal graph attention networks,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 6367–6371.
  * [5] H. Tak, J. weon Jung, J. Patino, M. Kamble, M. Todisco, and N. Evans, ``End-to-end spectro-temporal graph attention networks for speaker verification anti-spoofing and speech deepfake detection,'' in _Proc. 2021 Edition of the Automatic Speaker Verification and Spoofing Countermeasures Challenge_ , 2021.
  * [6] X. Wang and J. Yamagishi, ``Investigating self-supervised front ends for speech spoofing countermeasures,'' in _Proc. The Speaker and Language Recognition Workshop (Odyssey 2022)_ , 2022, pp. 100–106.
  * [7] H. Tak, M. Todisco, X. Wang, J.-w. Jung, J. Yamagishi, and N. Evans, ``Automatic speaker verification spoofing and deepfake detection using wav2vec 2.0 and data augmentation,'' in _The Speaker and Language Recognition Workshop_ , 2022.
  * [8] M. Li, Y. Ahmadiadli, and X.-P. Zhang, ``A comparative study on physical and perceptual features for deepfake audio detection,'' in _Proceedings of the 1st International Workshop on Deepfake Detection for Audio Multimedia_ , 2022, pp. 35–41.
  * [9] X. Ma, T. Liang, S. Zhang, S. Huang, and L. He, ``Improved lightcnn with attention modules for asv spoofing detection,'' in _2021 IEEE International Conference on Multimedia and Expo (ICME)_. IEEE, 2021, pp. 1–6.
  * [10] A. Tomilov, A. Svishchev, M. Volkova, A. Chirkovskiy, A. Kondratev, and G. Lavrentyeva, ``Stc antispoofing systems for the asvspoof2021 challenge,'' in _Proc. ASVspoof 2021 Workshop_ , 2021, pp. 61–67.
  * [11] G. Hua, A. B. J. Teoh, and H. Zhang, ``Towards end-to-end synthetic speech detection,'' _IEEE Signal Processing Letters_ , vol. 28, pp. 1265–1269, 2021.
  * [12] X. Li, N. Li, C. Weng, X. Liu, D. Su, D. Yu, and H. Meng, ``Replay and synthetic speech detection with res2net architecture,'' in _ICASSP 2021-2021 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2021, pp. 6354–6358.
  * [13] C. Wang, J. Yi, J. Tao, H. Sun, X. Chen, Z. Tian, H. Ma, C. Fan, and R. Fu, ``Fully automated end-to-end fake audio detection,'' in _Proceedings of the 1st International Workshop on Deepfake Detection for Audio Multimedia_ , 2022, pp. 27–33.
  * [14] W. Ge, J. Patino, M. Todisco, and N. Evans, ``Raw Differentiable Architecture Search for Speech Deepfake and Spoofing Detection,'' in _Proc. 2021 Edition of the Automatic Speaker Verification and Spoofing Countermeasures Challenge_ , 2021, pp. 22–28.
  * [15] M. Li, Y. Ahmadiadli, and X.-P. Zhang, ``Audio anti-spoofing detection: A survey,'' _arXiv preprint arXiv:2404.13914_ , 2024.
  * [16] W. Ge, M. Todisco, and N. Evans, ``Explainable Deepfake and Spoofing Detection: An Attack Analysis Using SHapley Additive exPlanations,'' in _Proc. The Speaker and Language Recognition Workshop (Odyssey 2022)_ , 2022, pp. 70–76.
  * [17] S. M. Lundberg and S.-I. Lee, ``A unified approach to interpreting model predictions,'' _Advances in neural information processing systems_ , vol. 30, 2017.
  * [18] S.-Y. Lim, D.-K. Chae, and S.-C. Lee, ``Detecting deepfake voice using explainable deep learning techniques,'' _Applied Sciences_ , vol. 12, no. 8, p. 3926, 2022.
  * [19] G. Montavon, S. Lapuschkin, A. Binder, W. Samek, and K.-R. Müller, ``Explaining nonlinear classification decisions with deep taylor decomposition,'' _Pattern recognition_ , vol. 65, pp. 211–222, 2017.
  * [20] H. Bharadhwaj, ``Layer-wise relevance propagation for explainable deep learning based speech recognition,'' in _2018 IEEE International symposium on signal processing and information technology (ISSPIT)_. IEEE, 2018, pp. 168–174.
  * [21] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, ``Grad-cam: Visual explanations from deep networks via gradient-based localization,'' in _Proceedings of the IEEE international conference on computer vision_ , 2017, pp. 618–626.
  * [22] B. Halpern, F. Kelly, R. van Son, and A. Alexander, ``Residual Networks for Resisting Noise: Analysis of an Embeddings-based Spoofing Countermeasure,'' in _Proc. The Speaker and Language Recognition Workshop (Odyssey 2020)_ , 2020, pp. 326–332.
  * [23] A. Baevski, Y. Zhou, A. Mohamed, and M. Auli, ``wav2vec 2.0: A framework for self-supervised learning of speech representations,'' _Advances in neural information processing systems_ , vol. 33, pp. 12 449–12 460, 2020.
  * [24] W.-N. Hsu, B. Bolte, Y.-H. H. Tsai, K. Lakhotia, R. Salakhutdinov, and A. Mohamed, ``Hubert: Self-supervised speech representation learning by masked prediction of hidden units,'' _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 29, pp. 3451–3460, 2021.
  * [25] A. Babu, C. Wang, A. Tjandra, K. Lakhotia, Q. Xu, N. Goyal, K. Singh, P. von Platen, Y. Saraf, J. Pino _et al._ , ``Xls-r: Self-supervised cross-lingual speech representation learning at scale,'' _arXiv preprint arXiv:2111.09296_ , 2021.
  * [26] Q. Wang, K. Okabe, K. A. Lee, H. Yamamoto, and T. Koshinaka, ``Attention mechanism in speaker recognition: What does it learn in deep speaker embedding?'' in _2018 IEEE Spoken Language Technology Workshop (SLT)_. IEEE, 2018, pp. 1052–1059.
  * [27] X. Wang, J. Yamagishi, M. Todisco, H. Delgado, A. Nautsch, N. Evans, M. Sahidullah, V. Vestman, T. Kinnunen, K. A. Lee _et al._ , ``Asvspoof 2019: A large-scale public database of synthesized, converted and replayed speech,'' _Computer Speech & Language_, vol. 64, p. 101114, 2020.
  * [28] T. Kinnunen, K. A. Lee, H. Delgado, N. Evans, M. Todisco, M. Sahidullah, J. Yamagishi, and D. A. Reynolds, ``t-dcf: a detection cost function for the tandem assessment of spoofing countermeasures and automatic speaker verification,'' in _Speaker Odyssey 2018 The Speaker and Language Recognition Workshop_ , 2018.
  * [29] Q. Ma, J. Zhong, Y. Yang, W. Liu, Y. Gao, and W. W. Ng, ``Convnext based neural network for anti-spoofing,'' _arXiv preprint arXiv:2209.06434_ , 2022.
  * [30] C. Li, F. Yang, and J. Yang, ``Multi-scale information aggregation for spoofing detection,'' _Available at SSRN 4251042_.
  * [31] X. Ma, S. Zhang, S. Huang, J. Gao, Y. Hu, and L. He, ``How to boost anti-spoofing with x-vectors,'' in _2022 IEEE Spoken Language Technology Workshop (SLT)_. IEEE, 2023, pp. 593–598.
  * [32] M. Yang, K. Zheng, X. Wang, Y. Sun, and Z. Chen, ``Comparative analysis of asv spoofing countermeasures: Evaluating res2net-based approaches,'' _IEEE Signal Processing Letters_ , 2023.
  * [33] D. P. Kingma and J. Ba, ``Adam: A method for stochastic optimization,'' _arXiv preprint arXiv:1412.6980_ , 2014.
  * [34] M. Todisco, X. Wang, V. Vestman, M. Sahidullah, H. Delgado, A. Nautsch, J. Yamagishi, N. W. D. Evans, T. H. Kinnunen, and K. A. LEE, ``Asvspoof 2019: Future horizons in spoofed and fake audio detection,'' in _Interspeech_ , 2019.


# A note on stability properties of powers of polymatroidal ideals

Amir Mafi and Dler Naderi  Amir Mafi, Department of Mathematics, University Of
Kurdistan, P.O. Box: 416, Sanandaj, Iran<EMAIL_ADDRESS>Dler Naderi,
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj,
Iran<EMAIL_ADDRESS>
###### Abstract.

Let $I$ be a matroidal ideal of degree $d$ of a polynomial ring
$R=K[x_{1},...,x_{n}]$, where $K$ is a field. Let $\operatorname{astab}(I)$
and $\operatorname{dstab}(I)$ be the smallest integers $m$ and $n$, for which
$\operatorname{Ass}(I^{m})$ and $\operatorname{depth}(I^{n})$ stabilize,
respectively. In this paper, we show that $\operatorname{astab}(I)=1$ if and
only if $\operatorname{dstab}(I)=1$. Moreover, we prove that if $d=3$, then
$\operatorname{astab}(I)=\operatorname{dstab}(I)$. Furthermore, we show that
if $I$ is an almost square-free Veronese type ideal of degree $d$, then
$\operatorname{astab}(I)=\operatorname{dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.

###### Key words and phrases:

Polymatroidal ideal, depth and associated primes stability number

###### 2010 Mathematics Subject Classification:

Primary 13A15; Secondary 13A30, 13C15

## Introduction

Throughout this paper, we assume that $R=K[x_{1},...,x_{n}]$ is the polynomial
ring in $n$ variables over a field $K$ with the maximal ideal
$\mathfrak{m}=(x_{1},...,x_{n})$, $I$ a monomial ideal of $R$ and $G(I)$ the
unique minimal monomial generators set of $I$. Let $\operatorname{Ass}(I)$ be
the set of associated prime ideals of $R/I$. Brodmann [2] showed that there
exists an integer $t_{0}$ such that
$\operatorname{Ass}(I^{t})=\operatorname{Ass}(I^{t_{0}})$ for all $t\geq
t_{0}$. The smallest such integer $t_{0}$ is called the index of Ass-stability
of $I$, and denoted by $\operatorname{astab}(I)$. Moreover,
$\operatorname{Ass}(I^{t_{0}})$ is called the stable set of associated prime
ideals of $I$. It is denoted by $\operatorname{Ass}^{\infty}(I)$. Brodmann [3]
also showed that there exists an integer $t_{0}$ such that
$\operatorname{depth}R/I^{t}=\operatorname{depth}R/I^{t_{0}}$ for all $t\geq
t_{0}$. The smallest such integer $t_{0}$ is called the index of depth
stability of $I$ and denoted by $\operatorname{dstab}(I)$. The first author
and Herzog [10] proved that if $n=3$ then, for any graded ideal $I$ of $R$,
$\operatorname{astab}(I)=\operatorname{dstab}(I)$. Also, they showed that for
$n=4$ the indices $\operatorname{astab}(I)$ and $\operatorname{dstab}(I)$ are
unrelated. Herzog, Rauf and Vladoiu [12] showed that for every polymatroidal
ideal of Veronese type $\operatorname{astab}(I)=\operatorname{dstab}(I)$ and
for every transversal polymatroidal ideal
$\operatorname{astab}(I)=1=\operatorname{dstab}(I)$. Herzog and Qureshi [11]
proved that if $I$ is a polymatroidal ideal of $R$, then
$\operatorname{astab}(I),\operatorname{dstab}(I)<\ell(I)$, where $\ell(I)$ is
the analytic spread of $I$, that is, the dimension of
$\mathcal{R}(I)/{{\mathfrak{m}}\mathcal{R}(I)}$, where $\mathcal{R}(I)$
denotes the Rees ring of $I$. Moreover, they conjectured that
$\operatorname{astab}(I)=\operatorname{dstab}(I)$ for all polymatroidal ideals
$I$. This conjecture does not have a positive answer in general, see [16] for
counterexamples. The ideals which provide these counterexamples are
polmatroidal ideals, but neither
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)$ nor $I$ is matroidal. Thus it
is an open question whether all matroidal ideals and all polymatroidal ideals
with $\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$ satisfy the equality
$\operatorname{astab}(I)=\operatorname{dstab}(I)$.

In this paper, we show that $\operatorname{astab}(I)=1$ if and only if
$\operatorname{dstab}(I)=1$. Also, we prove that if $I$ is a matroidal ideal
of degree $3$, then $\operatorname{astab}(I)=\operatorname{dstab}(I)$.
Furthermore, if $I$ is a polymatroidal ideal of degree $3$ and
$\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$, then
$\operatorname{astab}(I)=\operatorname{dstab}(I)$. In the end, we show that if
$I$ is an almost square-free Veronese type ideal of degree $d$, then
$\operatorname{astab}(I)=\operatorname{dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.

For any unexplained notion or terminology, we refer the reader to [9] or [18].
Several explicit examples were performed with the help of the computer algebra
system Macaulay2 [6].

##  1\. Preliminaries

In this section, we recall some definitions and known results which are used
in this paper. Let, as before, $K$ be a field and $R=K[x_{1},\ldots,x_{n}]$ be
the polynomial ring in $n$ variables over $K$ with each $\deg x_{i}=1$. For a
monomial ideal $I$ of $R$ and $G(I)=\\{u_{1},\ldots,u_{t}\\}$, we set
$\operatorname{supp}(I)=\cup_{i=1}^{t}\operatorname{supp}(u_{i})$, where
$\operatorname{supp}(u)=\\{x_{i}|u=x_{1}^{a_{1}}\ldots x_{n}^{a_{n}},a_{i}\neq
0\\}$ and we set $\gcd(I)=\gcd(u_{1},\ldots,u_{m})$. The linear relation graph
$\Gamma_{I}$ associated to a monomial ideal is the graph whose vertex set
$V(\Gamma_{I})$ is a subset of $\\{x_{1},\ldots,x_{n}\\}$ for which
$\\{x_{i},x_{j}\\}\in E(\Gamma_{I})$ if and only if there exist
$u_{k},u_{l}\in G(I)$ such that $x_{i}u_{k}=x_{j}u_{l}$ (see [11, Definition
3.1]). We say that the monomial ideal $I$ is full-supported if
$\operatorname{supp}(I)=\\{x_{1},\ldots,x_{n}\\}$. The monomial localization
of a monomial ideal $I$ with respect to a monomial prime ideal $\mathfrak{p}$
is the monomial ideal $I(\mathfrak{p})$ which is obtained from $I$ by
substituting the variables $x_{i}\notin\mathfrak{p}$ by $1$. The monomial
localization $I(\mathfrak{p})$ can also be described as the saturation
$I:(\prod_{x_{i}\notin{\mathfrak{p}}}x_{i})^{\infty}$ and when $I$ is a
square-free monomial ideal we see that
$I(\mathfrak{p})=I:(\prod_{x_{i}\notin{\mathfrak{p}}}x_{i})$. Let
$\mathfrak{p}$ be a monomial prime ideal of $R$. Then
$\mathfrak{p}=\mathfrak{p}_{A}$ for some subset $A\subseteq\\{1,\ldots,n\\}$,
where $\mathfrak{p}_{A}=\\{x_{i}|i\notin A\\}$.

A monomial ideal $I$ is called a polymatroidal ideal, if it is generated in a
single degree with the exchange property that for each two elements $u,v\in
G(I)$ such that $\deg_{x_{i}}(u)>\deg_{x_{i}}(v)$ for some $i$, there exists
an integer $j$ such that $\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$ and
$x_{j}(u/x_{i})\in I$. The polymatroidal ideal $I$ is called matroidal if $I$
is generated by square-free monomials (see [7]). For a polymatroidal ideal $I$
one can compute the analytic spread as $\ell(I)=r-s+1$, where
$r=|V(\Gamma_{I})|$ and $s$ is the number of connected components of
$\Gamma_{I}$ (see [11, Lemma 4.2]).

The product of polymatroidal ideals is again polymatroidal (see [5, Theorem
5.3]). In particular each power of a polymatroidal ideal is polymatroidal.
Also, $I$ is a polymatroidal ideal if and only if $(I:u)$ is a polymatroidal
ideal for all monomials $u$ (see [1, Theorem 1.1]). Furthermore, localizations
of polymatroidal ideals at monomial prime ideals are again polymatroidal [12,
Corollary 3.2]. According to [11] and [12], every polymatroidal ideal
satisfying the persistence property and non-increasing depth functions, that
is, if $I$ is a polymatroidal ideal then, for all $k$, there is the following
sequences: $\operatorname{Ass}(I^{k})\subseteq\operatorname{Ass}(I^{k+1})$ and
$\operatorname{depth}(R/I^{k+1})\leq\operatorname{depth}(R/I^{k}).$

In addition, every polymatroidal ideal is a normal ideal (see [12, Theorem
3.4]). One of the most distinguished polymatroidal ideals is the ideal of
Veronese type. Consider the fixed positive integers $d$ and $1\leq
a_{1}\leq...\leq a_{n}\leq d$. The ideal of Veronese type of $R$ indexed by
$d$ and $(a_{1},\ldots,a_{n})$ is the ideal $I_{(d;a_{1},\ldots,a_{n})}$ which
is generated by those monomials $u=x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}$ of $R$
of degree $d$ with $i_{j}\leq a_{j}$ for each $1\leq j\leq n$. Note that if
$a_{i}=1$ for all $i$, then we use $I_{d;n}$ instead of $I_{(d;1,\ldots,1)}$.
We say that $I$ is an almost square-free Veronese ideal of degree $d$ when
$I\neq 0$, $G(I)\subseteq G(I_{d;n})$ and
$\mid{G(I)}\mid\geq\mid{G(I_{d;n})}\mid-1$ (see [15]).

Herzog and Vladoiu [14] proved the following interesting results about
matroidal ideals.

######  Theorem 1.1.

Let $I$ be a matroidal ideal of $R$ generated in degree $d$, and denote as
before by $s$ the number of connected components of $\Gamma_{I}$. Let $I$ be
full-supported and $\gcd(I)=1$. The following statements hold:

  * (i)

$s\leq d$. In addition, $V(\Gamma_{I})=\\{x_{1},...,x_{n}\\}$ and $s=d$ if and
only if $\operatorname{dstab}(I)=1$;

  * (ii)

$I\subseteq{\mathfrak{p}_{1}}\cap\ldots\cap{\mathfrak{p}_{s}},$ where
$\mathfrak{p}_{1},\ldots,{\mathfrak{p}_{s}}$ are the monomial prime ideals
generated by the sets of vertices of the connected components
$\Gamma_{1},\ldots,\Gamma_{s}$ of $\Gamma_{I}$;

  * (iii)

$\operatorname{dstab}(I)=1$ if and only if
$I={\mathfrak{p}_{1}}\ldots{\mathfrak{p}_{d}}$, where
${\mathfrak{p}_{1}},\ldots,{\mathfrak{p}_{d}}$ are monomial prime ideals in
pairwise disjoint sets of variables.

From Theorem 1.1 (iii) one can conclude that for all full-supported matroidal
ideals with $\gcd(I)=1$ if $\operatorname{dstab}(I)=1$, then
$\operatorname{astab}(I)=1$ (see [14, Theorem 2.5] and [12, Corollary 4.6]).

##  2\. The results

Throughout this section, we assume that $I$ is a full-supported monomial ideal
and $gcd(I)=1$.

######  Lemma 2.1.

Let $I$ be a polymatroidal ideal of degree $d\geq 3$ and for all $i$,
$I(\mathfrak{p}_{\\{i\\}})=\mathfrak{p}_{i_{1}}\cap\mathfrak{p}_{i_{2}}\cap\ldots\cap\mathfrak{p}_{i_{d-1}}$,
where $G(\mathfrak{p}_{i_{j}})\cap G(\mathfrak{p}_{i_{k}})=\emptyset$ for all
$1\leq j\neq k\leq d-1$. Then
$I=\mathfrak{p}_{1}\cap\mathfrak{p}_{2}\cap\ldots\cap\mathfrak{p}_{d}$, where
$G(\mathfrak{p}_{r})\cap G(\mathfrak{p}_{s})=\emptyset$ for all $1\leq r\neq
s\leq d$.

###### Proof.

Suppose that
$I(\mathfrak{p}_{\\{i\\}})=\mathfrak{p}_{i_{1}}\cap\mathfrak{p}_{i_{2}}\cap\ldots\cap\mathfrak{p}_{i_{d-1}}$,
where $G(\mathfrak{p}_{i_{j}})\cap G(\mathfrak{p}_{i_{k}})=\emptyset$ for all
$1\leq j\neq k\leq d-1$. We may assume that
$I(\mathfrak{p}_{\\{1\\}})=I(\mathfrak{p}_{\\{1,\ldots,k_{1}\\}})$ for some
$k_{1}\geq 1$. Since $G(\mathfrak{p}_{i_{j}})\cap
G(\mathfrak{p}_{i_{k}})=\emptyset$ for all $1\leq j\neq k\leq d-1$, it follows
that $x_{1},\ldots,x_{k_{1}}\notin G(\mathfrak{p}_{i_{j}})$ for all $1\leq
j\leq d-1$. Without lose of generality of the proof by new labeling we may
assume that
$I(\mathfrak{p}_{\\{1\\}})=I(\mathfrak{p}_{\\{1,\ldots,k_{1}\\}})=\mathfrak{p}_{1}\cap\mathfrak{p}_{2}\cap\ldots\cap\mathfrak{p}_{d-1}$
such that $G(\mathfrak{p}_{j})\cap G(\mathfrak{p}_{k})=\emptyset$ for all
$1\leq j\neq k\leq d-1$ where
$\mathfrak{p}_{t}=(x_{{k_{t}}+1},\ldots,x_{k_{t+1}})$ for all $1\leq t\leq
d-1$ such that $x_{k_{d}}=x_{n}$. We claim that
$I=\mathfrak{p}_{1}\cap\mathfrak{p}_{2}\cap\ldots\cap\mathfrak{p}_{d}$, where
$G(\mathfrak{p}_{i})\cap G(\mathfrak{p}_{j})=\emptyset$ for all $1\leq i\neq
j\leq d$ and $\mathfrak{p}_{d}=(x_{1},\ldots,x_{k_{1}})$. We assume that the
claim is not true. Therefore
$I=\mathfrak{p}_{1}\cap\mathfrak{p}_{2}\cap\ldots\cap\mathfrak{p}_{d}\cap\mathfrak{p}_{d+1}\cap\ldots\cap\mathfrak{p}_{d+r}$
such that $r\geq 1$, $G(\mathfrak{p}_{j})\cap G(\mathfrak{p}_{k})=\emptyset$
for all $1\leq j\neq k\leq d-1$ and
$x_{s}\in\mathfrak{p}_{d}\cap\mathfrak{p}_{d+1}\cap\ldots\cap\mathfrak{p}_{d+r}$
for all $s\in\\{1,\ldots,k_{1}\\}$. Indeed, if
$x_{s}\notin\mathfrak{p}_{d}\cap\mathfrak{p}_{d+1}\cap\ldots\cap\mathfrak{p}_{d+r}$
for some $s\in\\{1,\ldots,k_{1}\\}$, then $x_{s}\notin\mathfrak{p}_{j}$ for
some $j\in\\{d,\ldots,d+r\\}$. Thus
$I(\mathfrak{p}_{\\{s\\}})\subseteq\mathfrak{p}_{1}\cap\mathfrak{p}_{2}\cap\ldots\cap\mathfrak{p}_{d-1}\cap\mathfrak{p}_{j}$
and this is a contradiction. Now, by our assumption there is
$m\in\\{0,1,\ldots,r\\}$ such that
$I(\mathfrak{p}_{\\{k_{1}+1\\}})=\mathfrak{p}_{2}\cap\mathfrak{p}_{3}\cap\ldots\cap\mathfrak{p}_{d-1}\cap\mathfrak{p}_{d+m}$,
where $G(\mathfrak{p}_{j})\cap G(\mathfrak{p}_{k})=\emptyset$ and
$G(\mathfrak{p}_{j})\cap G(\mathfrak{p}_{d+m})=\emptyset$ for all $2\leq j\neq
k\leq d-1$. Suppose that $m=0$ and so in this case
$I(\mathfrak{p}_{\\{k_{1}+1\\}})=\mathfrak{p}_{2}\cap\mathfrak{p}_{3}\cap\ldots\cap\mathfrak{p}_{d-1}\cap\mathfrak{p}_{d}$.
Hence $x_{k_{1}+1}\in\mathfrak{p}_{d+1}\cap\ldots\cap\mathfrak{p}_{d+r}$ and
there exists $i\in\\{2,\ldots,k_{2}-k_{1}\\}$ such that
$x_{k_{1}+i}\in\mathfrak{p}_{d}$, since otherwise we have
$\mathfrak{p}_{d}=(x_{1},\ldots,x_{k_{1}})$ and this is our claim. Also, there
exists $i\in\\{1,2,\ldots,k_{3}-k_{2}\\}$ such that
$I(\mathfrak{p}_{\\{k_{2}+i\\}})=\mathfrak{p}_{1}\cap\mathfrak{p}_{3}\cap\ldots\cap\mathfrak{p}_{d-1}\cap\mathfrak{p}_{d}$,
since otherwise
$\mathfrak{p}_{d}=(x_{1},\ldots,x_{k_{1}},x_{k_{2}+1},\ldots,x_{k_{3}})$ and
this is impossible because in this case
$\mathfrak{p}_{2}\subseteq\mathfrak{p}_{d}$. Suppose that $i=1$ and so
$I(\mathfrak{p}_{\\{k_{2}+1\\}})=\mathfrak{p}_{1}\cap\mathfrak{p}_{3}\cap\ldots\cap\mathfrak{p}_{d-1}\cap\mathfrak{p}_{d}$.
Thus there exists $j\in\\{2,\ldots,k_{3}-k_{2}\\}$ such that
$x_{k_{2}+j}\in\mathfrak{p}_{d}$, since otherwise
$\mathfrak{p}_{d}=(x_{1},\ldots,x_{k_{1}})$ and this is impossible. Suppose
that $j=2$ and so $x_{k_{2}+2}\in\mathfrak{p}_{d}$. Therefore
$I(\mathfrak{p}_{\\{k_{1}+1\\}})=\mathfrak{p}_{2}\cap\mathfrak{p}_{3}\cap\ldots\cap\mathfrak{p}_{d-1}\cap\mathfrak{p}_{d}$
such that $x_{k_{2}+2}\in\mathfrak{p}_{2}\cap\mathfrak{p}_{d}$ and this is
impossible. Hence the claim is true and this completes the proof.

The following example says that if $d=2$, then the above lemma does not hold.

######  Example 2.2.

Let
$I=(xz,xu,xv,xw,yz,yu,yv,yw,zv,zw,uv,uw)=(x,y,z,u)\cap(z,u,v,w)\cap(x,y,v,w)$
be a matroidal ideal of degree $2$ such that
$I(\mathfrak{p}_{\\{x\\}})=I(\mathfrak{p}_{\\{y\\}})=(z,u,v,w),I(\mathfrak{p}_{\\{z\\}})=I(\mathfrak{p}_{\\{u\\}})=(x,y,v,w)$
and $I(\mathfrak{p}_{\\{v\\}})=I(\mathfrak{p}_{\\{w\\}})=(x,y,z,u)$. But there
are no prime ideals $\mathfrak{p}_{1},\mathfrak{p}_{2}$ such that
$I=\mathfrak{p}_{1}\cap\mathfrak{p}_{2}$ and $G(\mathfrak{p}_{1})\cap
G(\mathfrak{p}_{2})=\emptyset$.

######  Theorem 2.3.

Let $I$ be a matroidal ideal of degree $d$. Then $\operatorname{astab}(I)=1$
if and only if $\operatorname{dstab}(I)=1$. In particular,
$I=\mathfrak{p}_{1}\mathfrak{p_{2}}\ldots\mathfrak{p}_{d}$ where
$G(\mathfrak{p}_{i})\cap G(\mathfrak{p}_{j})=\emptyset$ for all $1\leq i\neq
j\leq d$.

###### Proof.

If $\operatorname{dstab}(I)=1$, then by Theorem 1.1 there is nothing to prove.
Conversely, we use induction on $d$. If $d=2$, then by [16, Theorem 2.12] the
result follows. Suppose that $d>2$ and that the result has been proved for
smaller values of $d$. Since $\operatorname{astab}(I)=1$, we have
$\operatorname{Ass}(I)=\operatorname{Ass}^{\infty}(I)$ and since $I$ is a
matroidal ideal, it follows that
$\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$. Hence by [16, Propositions
2.8, 2.9] and the inductive hypothesis,
$1=\operatorname{astab}(I)=\operatorname{astab}(I(\mathfrak{p}_{\\{i\\}}))=\operatorname{dstab}(I(\mathfrak{p}_{\\{i\\}}))$.
Since $\operatorname{dstab}(I(\mathfrak{p}_{\\{i\\}}))=1$, by Theorem 1.1 we
have
$I(\mathfrak{p}_{\\{i\\}})=\mathfrak{p}_{i_{1}}\mathfrak{p}_{i_{2}}\ldots\mathfrak{p}_{i_{d-1}}$
for all $i$ such that $G(\mathfrak{p}_{i_{j}})\cap
G(\mathfrak{p}_{i_{k}})=\emptyset$ for all $1\leq j\neq k\leq d-1$. Now by
Lemma 2.1, $I=\mathfrak{p}_{1}\mathfrak{p}_{2}\ldots\mathfrak{p}_{d}$ such
that $G(\mathfrak{p}_{i})\cap G(\mathfrak{p}_{j})=\emptyset$ for all $1\leq
i\neq j\leq d$. Again by using Theorem 1.1 the result follows.

In the following result, let $s(I)$ be the number of connected components of
$\Gamma_{I}$ and we denote $u[j_{i}]$ the monomial
$x_{j_{1}}\ldots\widehat{x_{j_{i}}}\ldots x_{j_{t}}$, where the term
$x_{j_{i}}$ of $u=x_{j_{1}}\ldots{x_{j_{i}}}\ldots x_{j_{t}}$ is omitted and
$t\leq n$.

######  Lemma 2.4.

Let $I$ be a matroidal ideal of degree $d$. Then
$s(I(\mathfrak{p}_{\\{k\\}})\geq s(I)$.

###### Proof.

It is enough to prove that every edge of graph
$\Gamma_{I(\mathfrak{p}_{\\{k\\}})}$ is an edge of graph $\Gamma_{I}$. Suppose
that $\\{x_{i},x_{j}\\}\in E(\Gamma_{I(\mathfrak{p}_{\\{k\\}})})$. Then there
exists $u[k],v[k]\in G(I(\mathfrak{p}_{\\{k\\}}))$ such that
$x_{i}u[k]=x_{j}v[k]$. Thus $x_{i}u[k]x_{k}=x_{j}v[k]x_{k}$ and so
$x_{i}u=x_{j}v$. Therefore $\\{x_{i},x_{j}\\}\in E(\Gamma_{I})$. This
completes the proof.

######  Proposition 2.5.

Let $I$ be a polymatroidal ideal of degree $d$ and also
$I(\mathfrak{p}_{\\{k\\}})$ be a polymatroidal of degree $d-1$ in
$k[x_{1},\ldots,\widehat{x_{k}},\ldots\ldots,x_{n}]$ for some $k$. If
$\mathfrak{p}_{\\{k\\}}\in\operatorname{Ass}^{\infty}(I_{\\{k\\}})$, then
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)$.

###### Proof.

Suppose that
$\mathfrak{p}_{\\{k\\}}\in\operatorname{Ass}^{\infty}(I({\mathfrak{p}_{\\{k\\}}}))$.
Then by using [11, Corollary 1.6 and Lemma 4.2] we have
$s(I(\mathfrak{p}_{\\{k\\}}))=1$ and so by Lemma 2.4 $s(I)=1$. Thus
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)$, as required.

######  Lemma 2.6.

Let $I$ be a matroidal ideal of degree $d>1$ such that $s(I)<d$. Then there
are monomials $u,v,w\in G(I)$ and variables $x_{i},x_{j}$ in $R$ such that
$x_{i}u=x_{j}v$ and $x_{i}x_{j}|w$.

###### Proof.

We use induction on $d$. If $d=2$, then there is nothing to prove. Suppose
that $d>2$ and that the result has been proved for $d-1$. Now we prove that
the result holds for $d$. Since $I$ is a matroidal ideal, it is clear that
there are variables $x_{i},x_{j}$ in $R$ and monomials $u,v\in G(I)$ such that
$x_{i}u=x_{j}v$. Suppose that, by contrary, if $x_{i}u=x_{j}v$, then
$x_{i}x_{j}\nmid w$ for all monomials $w\in G(I)$. Since $x_{i}x_{j}\nmid w$
for every monomial $w\in G(I)$, it follows that if $x_{i}f\in G(I)$ for some
monomial $f$ of degree $d-1$, then $x_{j}f\in G(I)$. Suppose that
$A=\\{x_{1},\ldots,x_{k}\\}$ is a set of variables such that $x_{i}u=x_{j}v$
for some $u,v\in G(I)$. In this case we can write $I=(x_{1},\ldots,x_{k})J+L$
such that $J$ is a matroidal ideal of degree $d-1$ and
$x_{i}\notin\operatorname{supp}(L)$ for all $x_{i}\in A$. We claim that $L=0$.
Let $s^{{}^{\prime}}\in G(L)$ and $s\in G(J)$. By exchange property over
$s^{{}^{\prime}}$ and $x_{1}s$, there exists
$x_{l}\in\operatorname{supp}(s^{{}^{\prime}})$ such that $x_{l}s\in G(I)$.
Since $x_{1}x_{l}\nmid w$ for all monomials $w\in G(I)$, it follows that
$x_{l}\in A$. Therefore $L=0$ and so $I=(x_{1},\ldots,x_{k})J$. Since $J$ is a
matroidal ideal of degree $d-1$, if $s(J)<d-1$ then by induction hypothesis
there are variables $x_{k},x_{l}$ and monomials
$u^{{}^{\prime}},v^{{}^{\prime}},w^{{}^{\prime}}\in G(J)$ such that
$x_{k}u^{{}^{\prime}}=x_{l}v^{{}^{\prime}}$ and $x_{k}x_{l}|w^{{}^{\prime}}$.
In this case $x_{1}x_{k}u^{{}^{\prime}}=x_{1}x_{l}v^{{}^{\prime}}$,
$x_{k}x_{l}|x_{1}w^{{}^{\prime}}$ and
$x_{1}u^{{}^{\prime}},x_{1}v^{{}^{\prime}},x_{1}w^{{}^{\prime}}\in G(I)$. This
is a contradiction and so $s(J)=d-1$. By Theorem 1.1 we have
$J=\mathfrak{p}_{2}\ldots\mathfrak{p}_{d}$ such that $G(\mathfrak{p}_{i})\cap
G(\mathfrak{p}_{j})=\emptyset$ for all $2\leq i\neq j\leq d$. Therefore
$I=\mathfrak{p}_{1}\mathfrak{p}_{2}\ldots\mathfrak{p}_{d}$, where
$\mathfrak{p}_{1}=(x_{1},\ldots,x_{k})$ and $G(\mathfrak{p}_{i})\cap
G(\mathfrak{p}_{j})=\emptyset$ for all $1\leq i\neq j\leq d$. In this case
$s(I)=d$ and this is a contradiction. Thus the induction process is completed,
as required.

Following [8], let $I=(u_{1},\ldots,u_{s})$ be a monomial ideal with linear
quotients with respect to the ordering $u_{1},\ldots,u_{s}$. We denote by
$q_{i}(I)$ the number of variables which are required to generate the colon
ideal $(u_{1},\ldots,u_{i-1}):u_{i}$. Let $q(I)=\max\\{q_{i}(I)\mid 2\leq
i\leq s\\}$. It is proved in [13, Corollary 1.6] that the length of the
minimal free resolution of $R/I$ over $R$ is equal to $q(I)+1$. Hence
$\operatorname{depth}R/I=n-q(I)-1$. Thus in particular the integer $q(I)$ is
independent of the particular choice of the ordering of the monomials which
gives linear quotients. Polymatroidal ideals have linear quotients with
respect to the reverse lexicographical order of the generators, see [5,
Theorem 5.2]. Chiang-Hsieh in [4, Theorem 2.5] proved that if $I\subset R$ is
a full-supported matroidal ideal of degree $d$, then
$\operatorname{depth}R/I=d-1$.

######  Proposition 2.7.

Let $I$ be a matroidal ideal of degree $d$ such that
$\operatorname{dstab}(I)>1$. Then
$\operatorname{depth}(R/I^{2})<\operatorname{depth}(R/I)$.

###### Proof.

Since $\operatorname{dstab}(I)>1$, by [14, Proposition 2.3] it follows that
$s(I)<d$. Thus, by Lemma 2.6, there are monomials $u,v,w\in G(I)$ and
variables $x_{i},x_{j}$ in $R$ such that $x_{i}u=x_{j}v$ and $x_{i}x_{j}|w$.
After relabeling the variables we may assume that $x_{n-d}u=x_{n-d+1}v$ and
$x_{n-d}x_{n-d+1}|w$. Thus there is a monomial $s$ of degree $d-1$ such that
$u=x_{n-d+1}s$ and $v=x_{n-d}s$. Also, there is a monomial $w_{1}$ of degree
$d-2$ such that $w=x_{n-d}x_{n-d+1}w_{1}$. Hence $m=x_{n-d}x_{n-d+1}s^{2}\in
G(I^{2})$. Suppose that $\operatorname{supp}(s)=\\{x_{n-d+2},\ldots,x_{n}\\}$.
Let $J$ denote the monomial ideal generated by those monomials $r\in G(I^{2})$
such that $r$ is bigger than $m$ with respect to the reverse lexicographic
order induced by the ordering $x_{1}>x_{2}>\ldots>x_{n}$. For each $1\leq
l\leq{n-d-1}$, there is a monomial belonging to $G(I^{2})$ which is divided by
$x_{l}$. Thus there is a variable $x_{k}$ with $n-d\leq k\leq n$ such that
$x_{l}(m/x_{k})\in G(I^{2})$. Since $x_{k}<x_{l}$, it follows that
$x_{l}(m/x_{k})\in J$ and so $x_{l}m\in J$. Therefore $x_{l}\in J:m$ for all
$1\leq l\leq n-d-1$. Consequently, one has $q(I^{2})\geq n-d-1$. Now, by
exchange properties over elements $m=x_{n-d}x_{n-d+1}s^{2}$ and
$x_{n-d}^{2}x_{n-d+1}sw_{1}$ of $G(I^{2})$ there is $n-d+2\leq k\leq n$ such
that $x_{n-d}(x_{n-d}x_{n-d+1}s^{2}/x_{k})\in G(I^{2})$. Since
$x_{n-d}>x_{k}$, we have $x_{n-d}(x_{n-d}x_{n-d+1}s^{2}/x_{k})\in J$. Thus
$x_{n-d}m\in J$ and so $x_{n-d}\in J:m$. By the above argument over elements
$m=x_{n-d}x_{n-d+1}s^{2}$ and $x_{n-d}x_{n-d+1}^{2}sw_{1}$ of $G(I^{2})$, we
have $x_{n-d+1}\in J:m$. Therefore $q(I^{2})\geq n-d+1$ and so
$\operatorname{depth}(R/I^{2})<\operatorname{depth}(R/I)$, as required.

######  Proposition 2.8.

Let $I$ be a matroidal ideal of degree $3$ and
$\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$. Then
$\operatorname{astab}(I)=\operatorname{dstab}(I)\leq 2$.

###### Proof.

Since $\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$, by [16, Proposition
2.9 and Theorem 2.12] we have
$\operatorname{astab}(I)=\operatorname{astab}(I(\mathfrak{p}_{\\{i\\}}))=\operatorname{dstab}(I(\mathfrak{p}_{\\{i\\}}))$
for some $1\leq i\leq n$. We may assume $i=1$ and so
$\operatorname{astab}(I)=\operatorname{astab}(I(\mathfrak{p}_{\\{1\\}}))=\operatorname{dstab}(I(\mathfrak{p}_{\\{1\\}}))$.
We can consider two cases:
Case 1: Let $I(\mathfrak{p}_{\\{1\\}})$ be full-supported. Since
$deg(I(\mathfrak{p}_{\\{1\\}}))=2$, by the proof of [16, Corollary 2.13] we
have
$\operatorname{astab}(I(\mathfrak{p}_{\\{1\\}}))=\operatorname{dstab}(I(\mathfrak{p}_{\\{1\\}}))=1$
and so $\operatorname{astab}(I)=1$. Hence by Theorem 2.3 we have
$\operatorname{dstab}(I)=1=\operatorname{astab}(I)$.
Case 2: Suppose $I(\mathfrak{p}_{\\{1\\}})$ is not full-supported. In this
case we may assume
$I(\mathfrak{p}_{\\{1\\}})=I(\mathfrak{p}_{\\{1,\ldots,k\\}})$ and
$I(\mathfrak{p}_{\\{1,\ldots,k\\}})$ is full-supported in
$k[x_{k+1},\ldots,x_{n}]$. If
$\mathfrak{p}_{\\{1,\ldots,k\\}}\notin\operatorname{Ass}^{\infty}(I)$, then
$\mathfrak{p}_{\\{1,\ldots,k\\}}\notin\operatorname{Ass}^{\infty}(I(\mathfrak{p}_{\\{1,\ldots,k\\}}))$.
Hence
$\operatorname{astab}(I(\mathfrak{p}_{\\{1,\ldots,k\\}}))=\operatorname{dstab}(I(\mathfrak{p}_{\\{1,\ldots,k\\}}))=1$.
Thus $\operatorname{astab}(I)=1$ and so $\operatorname{dstab}(I)=1$. If
$\mathfrak{p}_{\\{1,\ldots,k\\}}\in\operatorname{Ass}^{\infty}(I)$, then
$\mathfrak{p}_{\\{1,\ldots,k\\}}\in\operatorname{Ass}^{\infty}(I(\mathfrak{p}_{\\{1,\ldots,k\\}}))$.
Thus by the proof of [16, Corollary 2.13] we have
$\operatorname{astab}(I)=\operatorname{astab}(I(\mathfrak{p}_{\\{1,\ldots,k\\}}))=\operatorname{dstab}(I(\mathfrak{p}_{\\{1,\ldots,k\\}}))=2$
and also by Theorem 2.3 we have $\operatorname{dstab}(I)>1$. By Proposition
2.7, we have $\operatorname{depth}(R/I^{2})\leq 1$. Since
$\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$, it follows
$\operatorname{depth}(R/I^{2})=\operatorname{depth}(R/I^{i})=1$ for all $i\geq
2$. Therefore $\operatorname{dstab}(I)=2$. This completes the proof.

######  Lemma 2.9.

Let $I$ be a polymatroidal ideal of degree $2$ and
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)$. Then
$\operatorname{astab}(I)=\operatorname{dstab}(I)\leq 2$

###### Proof.

By [16, Theorem 2.12], we have
$\operatorname{astab}(I)=\operatorname{dstab}(I)$. We have two cases:
Case 1: Let $G(I)$ have at least one pure power of a variable, say
$x_{i}^{2}\in G(I)$. Then $\operatorname{supp}(x_{i}^{2})=\\{x_{i}\\}$. Now by
using the same argument as used in the proof of Proposition 2.7 we conclude
that $q(I)\geq n-1$ and it therefore follows $\operatorname{depth}(R/I)=0$.
Thus $\operatorname{dstab}(I)=1$ and so
$\operatorname{astab}(I)=\operatorname{dstab}(I)\leq 2$.
Case 2: Let $G(I)$ do not have any pure power of the variables. Then $I$ is
square-free and so $I$ is a matroidal ideal. Thus, by [16, Corollary 2.13],
$\operatorname{astab}(I)=\operatorname{dstab}(I)\leq 2$. This completes the
proof.

######  Proposition 2.10.

Let $I$ be a polymatroidal ideal of degree $3$ and
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)\setminus\operatorname{Ass}(I)$.
Then $\operatorname{astab}(I)=\operatorname{dstab}(I)$.

###### Proof.

Suppose that $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$, where
$\mathfrak{p}\neq\mathfrak{m}$. Then there exists $t$ such that
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}(I^{t}(\mathfrak{p}))$.
Since $\deg(I(\mathfrak{p}))\leq 2$, by Lemma 2.9 we have
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}(I^{2}(\mathfrak{p}))$ and
so $\mathfrak{p}\in\operatorname{Ass}(I^{2})$. Hence
$\operatorname{Ass}^{\infty}(I)=\operatorname{Ass}(I^{2})\cup\\{\mathfrak{m}\\}$.
It therefore follows that $\operatorname{astab}(I)=\operatorname{dstab}(I)$.

Note that in Proposition 2.10, if $\mathfrak{m}\in\operatorname{Ass}(I)$ then
the result does not hold. The first author and Karimi [16, Example 2.21] have
given the following example:

######  Example 2.11.

Consider the polymatroidal ideal

$I=(x_{1}x_{2}x_{3},x_{2}^{2}x_{3},x_{2}x_{3}^{2},x_{1}x_{2}x_{4},x_{2}^{2}x_{4},x_{2}x_{4}^{2},x_{1}x_{3}x_{4},x_{3}^{2}x_{4},x_{3}x_{4}^{2},x_{2}x_{3}x_{4})$

of degree $3$. Then $\mathfrak{m}\in\operatorname{Ass}(I)$,
$\operatorname{dstab}(I)=1$ and $\operatorname{astab}(I)=2$.

The following corollary immediately follows from Proposition 2.10.

######  Corollary 2.12.

Let $I$ be a matroidal ideal of degree $3$ such that
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)$. Then
$\operatorname{astab}(I)=\operatorname{dstab}(I)$.

The following result easily follows from Corollary 2.12 and Proposition 2.8.

######  Corollary 2.13.

Let $I$ be a matroidal ideal of degree $3$. Then
$\operatorname{astab}(I)=\operatorname{dstab}(I)$.

######  Proposition 2.14.

Let $I$ be a polymatroidal ideal of degree $3$ such that
$\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$. Then
$\operatorname{astab}(I)=\operatorname{dstab}(I)$.

###### Proof.

Since $\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$, it follows that
$q(I)<n-1$ and so the polymatroidal ideal $I$ can not contain the pure power
of variables. If $x_{i}^{2}x_{j}\notin G(I)$ for all $i,j$, then $I$ is a
matroidal ideal and so, by Proposition 2.8,
$\operatorname{astab}(I)=\operatorname{dstab}(I)$. Now suppose
$x_{i}^{2}x_{j}\in G(I)$ for some $i,j$. In this case $q(I)\geq n-2$ and so
$\operatorname{depth}(R/I)\leq 1$. Again, since
$\mathfrak{m}\notin\operatorname{Ass}^{\infty}(I)$ it follows that
$1=\operatorname{depth}(R/I)=\operatorname{depth}(R/I^{i})$ for all $i$. Thus
$\operatorname{dstab}(I)=1$ and also
$\mathfrak{p}_{\\{i\\}}\notin\operatorname{Ass}^{\infty}(I(\mathfrak{p}_{\\{i\\}}))$.
Therefore, by [17, Remark 9] and [16, Remark 2.6, Theorem 2.12],
$\operatorname{Ass}(I^{t})=\bigcup_{i=1}^{n}\operatorname{Ass}((I(\mathfrak{p}_{\\{i\\}}))^{t})=\bigcup_{i=1}^{n}\operatorname{Ass}(I(\mathfrak{p}_{\\{i\\}}))=\operatorname{Ass}(I)$
for all $t$ and so $\operatorname{astab}(I)=1$. Therefore
$\operatorname{astab}(I)=\operatorname{dstab}(I)=1$, as required.

######  Proposition 2.15.

Let $J$ be an almost square-free Veronese type ideal of degree $d\geq 2$ and
$gcd(J)=1$. Then $\mathfrak{m}\in\operatorname{Ass}^{\infty}(J)$.

###### Proof.

If $J$ is the square-free Veronese type ideal, then by [12, Corollary 5.5]
there is nothing to prove. Now, suppose that $gcd(J)=1$ and we may assume that
$n\geq 4$. We use induction on $d$. Suppose $d=2$ and we may assume
$x_{n-1}x_{n}\notin G(J)$. In this case $x_{1}x_{2},x_{1}x_{3}\in G(J)$ and so
$\\{x_{2},x_{3}\\}\in E(\Gamma_{J})$. Also,
$x_{2}x_{3},x_{2}x_{4},\ldots,x_{2}x_{n}\in G(J)$. Hence
$\\{x_{1},x_{3}\\},\ldots,\\{x_{1},x_{n}\\}\in E(\Gamma_{J})$ and so
$\Gamma_{J}$ is connected. Therefore $\ell(J)=n$ and hence
$\mathfrak{m}\in\operatorname{Ass}^{\infty}(J)$. Suppose that $d>2$ and that
the result has been proved for $d-1$. Now we prove that the result holds for
$d$. We may assume that $x_{n-d+1}x_{n-d+2}\ldots x_{n}\notin G(J)$. Thus
$I_{d;n}=J+(x_{n-d+1}x_{n-d+2}\ldots x_{n})$ and so we have
$J=(x_{1},\ldots,x_{n-d})\cap I_{d;n}$. Thus $J(\mathfrak{p}_{\\{i\\}})$ is a
square-free Veronese type ideal for all $1\leq i\leq n-d$ and
$J(\mathfrak{p}_{\\{i\\}})$ is an almost square-free Veronese type for all
$n-d+1\leq i\leq n$. It therefore follows that
$s(J(\mathfrak{p}_{\\{i\\}}))=1$ for all $i$ and so by Lemma 2.4, $s(J)=1$.
Hence $\ell(J)=n$ and so $\mathfrak{m}\in\operatorname{Ass}^{\infty}(J)$, as
required.

Herzog, Rauf and Vladoiu [12] proved that If $I=I_{d;n}$ is a square-free
Veronese type ideal, then
$\operatorname{astab}(I)=\operatorname{dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.
The following theorem extends this result.

######  Theorem 2.16.

Let $J$ be an almost square-free Veronese type ideal of degree $d\geq 2$ and
$gcd(J)=1$. Then
$\operatorname{astab}(J)=\operatorname{dstab}(J)=\lceil\frac{n-1}{n-d}\rceil$.

###### Proof.

If $J$ is the square-free Veronese type ideal, then by [12, Corollary 5.7]
there is nothing to prove. Now, suppose that $gcd(J)=1$ and we may assume that
$d\leq n-2$. Assume that $k=\lceil\frac{n-1}{n-d}\rceil$. In this case
$\frac{n-1}{n-d}=1+\frac{d-1}{n-d}\leq d$ and so we may assume that $k\leq d$.
Let $I$ be the square-free Veronese type ideal of degree $d$ and we may assume
that $I=(J,u)$ such that $u=x_{n}x_{n-1}\ldots x_{n-d}x_{n-d+1}\notin G(J)$.
It is clear that $I^{k}:u^{k-1}x_{1}\ldots
x_{d-1}=\mathfrak{m}=J^{k}:u^{k-1}x_{1}\ldots x_{d-1}$. Indeed, if $1\leq
j\leq k-1$ or $d\leq j\leq n$ then $x_{j}u^{k-1}x_{1}\ldots
x_{d-1}=(x_{1}u)(x_{2}u)\ldots(x_{k-1}u)(x_{j}x_{k}\ldots
x_{d-1})=(\frac{x_{1}u}{x_{i_{1}}})(\frac{x_{2}u}{x_{i_{2}}})\ldots(\frac{x_{k-1}u}{x_{i_{k-1}}})(x_{i_{1}}\ldots
x_{i_{k-1}}x_{k}\ldots x_{d-1}x_{j})\in I^{k}$ and if $k\leq j\leq d-1$ then
$x_{j}u^{k-1}x_{1}\ldots
x_{d-1}=(x_{j}u)(x_{1}u)\ldots(x_{k-2}u)(x_{k-1}x_{k}\ldots x_{d-1})=\\\
(\frac{x_{j}u}{x_{i_{1}}})(\frac{x_{2}u}{x_{i_{2}}})\ldots(\frac{x_{k-2}u}{x_{i_{k-1}}})(x_{i_{1}}\ldots
x_{i_{k-1}}x_{k-1}x_{k}\ldots x_{d-1})\in I^{k}$, where
$x_{i_{1}},\ldots,x_{i_{k-1}}$ are distinctive elements of
$\\{x_{n-d+1},\ldots,x_{n}\\}$. Since
$\operatorname{dstab}(J)=\min\\{t|\mathfrak{m}\in\operatorname{Ass}(I^{t})\\}$,
it follows that
$\operatorname{dstab}(J)\leq\operatorname{dstab}(I)=\operatorname{astab}(I)\leq
d$ and $\operatorname{dstab}(J)\leq\operatorname{astab}(J)$. By Proposition
2.15 $\mathfrak{m}\in\operatorname{Ass}^{\infty}(J)$ and so we assume that $t$
is the smallest integer such that
$\operatorname{Ass}^{\infty}(J)=\operatorname{Ass}(J^{t})$. By using the above
argument we have $\mathfrak{m}\in\operatorname{Ass}^{\infty}(I)$. Thus
$\operatorname{astab}(I)=\operatorname{dstab}(I)\leq\operatorname{dstab}(J)$
and so
$\operatorname{dstab}(J)=\operatorname{dstab}(I)=\operatorname{astab}(I)=k$.
By induction on $d$ we will prove that
$\operatorname{astab}(J)=\operatorname{dstab}(J)=k$. If $d=2$, then by [16,
Proposition 2.12] the result follows. Let $d\geq 3$ and the result has been
proved for smaller values of $d$. By [16, Remark 2.6] we have
$\operatorname{Ass}(J^{k+j})\setminus\\{\mathfrak{m}\\}=\bigcup_{i=1}^{n}\operatorname{Ass}(J^{k+j}(\mathfrak{p}_{\\{i\\}}))$
for all $j$. By the induction hypothesis
$\operatorname{astab}(J(\mathfrak{p}_{\\{i\\}})=\operatorname{dstab}(J(\mathfrak{p}_{\\{i\\}})\leq
k$. This implies that
$\operatorname{Ass}(J^{k+j})\setminus\\{\mathfrak{m}\\}=\operatorname{Ass}(J^{k})\setminus\\{\mathfrak{m}\\}$
and so $\operatorname{Ass}(J^{k+j})=\operatorname{Ass}(J^{k})$ for all $j$.
Therefore $\operatorname{astab}(J)\leq k=\operatorname{dstab}(J)$ and so
$\operatorname{astab}(J)=\operatorname{dstab}(J)=\operatorname{dstab}(I)=\operatorname{astab}(I)=k$,
as required.

Acknowledgement: We would like to thank deeply grateful to the referee for the
careful reading of the manuscript and the helpful suggestions. The second
author has been supported financially by Vice-Chancellorship of Research and
Technology, University of Kurdistan under research Project No. 99/11/19299.

## References

  * [1] S. Bandari and J. Herzog, Monomial localizations and polymatroidal ideals, Eur. J. Comb., 34(2013), 752-763.
  * [2] M. Brodmann, Asymptotic stability of $\operatorname{Ass}(M/{I^{n}M})$, Proc. Am. Math. Soc., 74(1979), 16-18.
  * [3] M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc., 86(1979), 35-39.
  * [4] H. J. Chiang-Hsieh, Some arithmetic properties of matroidal ideals, Comm. Algebra, 38(2010), 944-952.
  * [5] A. Conca and J. Herzog, Castelnuovo-Mumford regularity of products of ideals, Collect. Math., 54(2003), 137-152.
  * [6] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
  * [7] J. Herzog and T. Hibi, Discrete polymatroids, J. Algebraic Combin., 16(2002), 239-268.
  * [8] J. Herzog and T. Hibi, Cohen-Macaulay polymatroidal ideals, Eur. J. Comb., 27(2006), 513-517.
  * [9] J. Herzog and T. Hibi, Monomial ideals, GTM., 260, Springer, Berlin, (2011).
  * [10] J. Herzog and A. Mafi, Stability properties of powers of ideals in regular local rings of small dimension, Pacific J. Math., 295(2018), 31-41.
  * [11] J. Herzog and A. A. Qureshi, Persistence and stability properties of powers of ideals, J. Pure and Appl. Algebra, 219(2015), 530-542.
  * [12] J. Herzog, A. Rauf and M. Vladoiu, The stable set of associated prime ideals of a polymatroidal ideal, J. Algebr Comb., 37(2013), 289-312.
  * [13] J. Herzog and Y. Takayama, Resolutions by mapping cones, Homology Homotopy Appl., 4(2002), 277-294.
  * [14] J. Herzog and M. Vladoiu, Squarefree monomial ideals with constant depth function, J. Pure and Appl. Algebra, 217(2013), 1764-1772.
  * [15] M. Jafari, A. Mafi and H. Saremi, Sequentially Cohen-Macaulay matroidal ideals, Filomat, 34 (2020), 4233-4244.
  * [16] Sh. Karimi and A. Mafi, On stability properties of powers of polymatroidal ideals, Collect. Math., 70(2019), 357-365.
  * [17] T. N. Trung, Stability of associated primes of integral closures of monomial ideals, J. Comb. Theory, Series A, 116(2009), 44-54.
  * [18] R. H. Villarreal, Monomial Algebras, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, (2015).


# Methods of Informational Trends Analytics and Fake News Detection on Twitter

Bohdan M. Pavlyshenko
Ivan Franko National University of Lviv, Ukraine
<EMAIL_ADDRESS>www.linkedin.com/in/bpavlyshenko/

###### Abstract

In the paper, different approaches for the analysis of news trends on Twitter
has been considered. For the analysis and case study, informational trends on
Twitter caused by Russian invasion of Ukraine in 2022 year have been studied.
A deep learning approach for fake news detection has been analyzed. The use of
the theory of frequent itemsets and association rules, graph theory for news
trends analytics have been considered.

Keywords: fake news detection, twitter, news trends, frequent itemsets,
transformers, deep learning, users’ communities.

##  1 Introduction

News have an essential impact in many areas of society, politics and business.
That is why one can see a lot of attempts to produce manipulative and fake
news to get a specified response in the society. One of horrible world events
is Russian invasion of Ukraine on February, 24 2022. It causes a large
informational news flow on social networks, including producing manipulative
and fake news to shape a specified explanation and justification of invasion.
One of the goals claimed by Russia was the ’denazification’ of Ukraine. One of
the allegations of Russian propaganda was that Ukraine was developing the
biological weapon in special laboratories.

Tweets, the messages of Twitter microblogs, have high density of semantically
important keywords. It makes it possible to get semantically important
information from tweets and generate the features of predictive models for the
decision-making support. Different studies of Twitter are considered in the
papers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In [14, 15, 16], we study
different approaches for the analysis of messages on Twitter, as well as the
use of tweet features for forecasting different kinds of events.

In this paper, we consider the methods for the analysis of Twitter trends and
for detecting fake news. As fake news, we will consider the news information
which is not true as well as the information which can contain real facts, but
with incorrectly specified accents, and the focuses that lead to distorted
conclusion and incorrect understanding of underlying processes. For our
analysis, we considered informational trends caused by Russian invasion of
Ukraine in 2022. In the study, we also consider the possible impact of
informational trends on different companies working in Russia during this
conflict.

##  2 News Trends on Twitter

For the analysis, we used a combination of keywords related to thematic areas
under consideration. The keywords related to the entity under consideration
can be treated as a thematic field. The use of semantic and thematic fields
for text analytics is considered in [17, 18, 19, 16]. To load tweets, we have
used Twitter API v2. For the ’ukraine nazi’ thematic field the following
Twitter API query "(ukraine OR ukrainian OR ukraine’s) (nazi OR nazism,
nazists OR neonazi OR neo-nazi)" has been used. For the ’ukraine biological
weapon’ thematic field, the query "ukraine biological (weapon OR weapons OR
warfare)" has been used. For the analysis, the following main python packages
were used: ’pandas’, ’matplotlib’, ’seaborn’, ’tweepy’. Figures 1-4 show the
time series for tweet counts for different queries. The lower case of query
keywords allows searching tweets with the keywords with both lower and upper
cases. As the results show, for the ’ukraine nazi’ thematic field, the
discussion of underlying problems rose dramatically after February 24, the
date of Russian invasion of Ukraine. The amount of tweets related to this
theme before that date was at the minimum level. That itself leads to the
conclusion that the problem with nazi in Ukraine was just a formal reason to
justify the invasion. Another claim of Russian propaganda was about biological
weapons that were allegedly being developed in Ukrainian laboratories (Figure
3). For instance, it was claimed that a special virus was being developed and
it was meant to be distributed through bats and migratory birds.

Figure 1: Time series of tweets for the query ’ukraine’ Figure 2: Time series
of tweets for the thematic field ’ukraine nazi’ Figure 3: Time series of
tweets for the thematic field ’ukraine biological weapon’  Figure 4: Time
series of tweets

##  3 Deep Learning Approach for Fake News Detection

Fake news can be detected and revealed by analyzing facts and comparing them
with reality and other news sources. Let us consider the methods which make it
possible to detect fake news automatically. It is not easy to develop an AI
system which can analyze the facts. But for manipulative news, it is typical
to amplify them artificially in different ways, e.g. by retweeting
manipulative tweets many times using different users’ accounts. Some accounts
can be bots which were artificially created, others can belong to real users.
It makes it possible to detect fake news using an approach which analyzes the
patterns of users’ behavior. Also, fake news have specific patterns in the
text of messages. Both users’ behavior and text patterns can be captured by
deep machine learning algorithms. As the features for a predictive model, we
used tweet texts and the list of users’ usernames who retweeted those tweets.
For model developing, evaluation and prediction, the Python framework
’pytorch’ was used. The ML model consists of several concatenated neural
subnetworks: subnetwork with DistilBERT transformer which ingests tokens of
tweet text,s subnetwork with the embedding layer with averaging which ingests
the mixture of encoded words of tweet texts and lists of usernames of
retweeters, as well as a subnetwork for the components of truncated singular
value decomposition of TF-IDF matrix for the list of usernames of retweeters.
Figure 5 shows the structure of the deep learning model for fake and
manipulative news detection. For our case study, the loaded tweets with the
thematic fields ’ukraine nazi’ and ’ukraine biological weapon’ were used. For
the model training and evaluation, the tweet datasets with a specified list of
users who retweeted those tweets were created. For the analysis, only the
tweets with a specified threshold for retweet counts were included. The
dataset was labeled using an appropriate tweet id, usernames, hashtags of
tweets which can be treated as fake or manipulative. Figure 6 shows model
evaluation results on the validation dataset.

Figure 5: Deep learning model structure Figure 6: Model evaluation results on
the validation dataset

The examples of tweets which were recognized by the deep learning model as
fake or manipulation news for the thematic fields ’ukraine nazi’ and ’ukraine
biological weapon’ are at [20].

##  4 Detecting Artificially Generated News

One of the ways to form manipulative trends is to produce artificially created
news tweet messages [21, 22, 23], e.g. by paraphrasing an initial text using
Seq2Seq neural network models. For this purpose, GPT-2, and BART transformers
can be used. Each pretrained transformer has its own patterns for generating
texts using an encoder-decoder approach. These patterns can be detected by
other transformers which are fine-tuned on a dataset with an artificialy
generated text. For the model fine-tuning, a TweepFake dataset of artificially
generated texts [22] was used. The framework ’pytorch’ was used for the model
developing and evaluation. We tried two models – the neural network with the
DistilBERT transformer layer and the neural network with the concatenation of
the DistilBERT transformer with embedding layers for the usernames of users
who post tweets. Figures 7, 8 show the model evaluation on the validation
dataset for these two models.

Figure 7: Evaluation results for the model with DisstilBERT transformer layer
Figure 8: Evaluation results for the model with the concatenation DisstilBERT
transformer layer and embedding layer for usernames

As the results show, the embedding layer for usernames can improve accuracy
scores, it means that usernames have predictive potential for detecting
artifitially created tweets. We applied the fine-tuned model to the datasets
of tweets of the semantic fields under consideration and have received the
following results: human - 80%, GPT-2-10%, RNN - 0.3%, Others - 3%.

##  5 The Impact of the Discussion About Company’s Behavior On Its Stock
Prices

Russian invasion of Ukraine has huge impact on users’ attitude to different
companies and shapes discussion trends on Twitter. Figure 9 shows time series
for tweet counts related to the consideration of different companies in Russia
during Russian invasion of Ukraine. For the case study, let us consider
McDonald’s. Figure 10 shows the tweet counts in the trend of considering this
company, a rolling mean of this time series, with a 7-day window and stock
price for the ’MCD’ ticker for the McDonald’s company.

Figure 9: Time series of tweets for trends related to different companies
Figure 10: Time series of tweets related to McDonald’s, rolling mean of the
time series, stock price time series for the ’MCD’ ticker

As a result of public discussion, it is known that McDonald’s has stopped its
activity in Russia. After this announcement, the stock price for the ’MCD’
ticker has returned to almost initial price value before the invasion period
(Figure 10). The stock price time series for the ’MCD’ ticker were loaded
using Python package ’yfinance’. It shows that public consideration of a
company’s behavior which is reflected on the social networks can have impact
on the stock market. It is supposed that consideration has some cumulative
effect, that is why the rolling mean of the time series of tweet counts
corresponds more precisely to company’s stock prices on the stock market. Let
us analyze the dependency of stock prices on the tweet counts rolling mean. It
is important to estimate the uncertainty of such dependency. For the modeling
of such a dependency, the Bayesian regression was applied. The probabilistic
model can be considered as follows:

$\begin{split}&y\sim Student_{t}(\nu,\mu,\sigma),\\\ &\mu=\alpha+\beta
x,\end{split}$ (1)

where $\nu$ is a distribution parameter, known as degrees of freedom. Target
variable $y$ is described by Student’s t-distribution which has fat tails that
makes it possible to take into account extreme events and values that enable
us to estimate uncertainties more accurately. For Bayesian inference
calculations, we used a Python package ’pystan’ for Stan platform for
statistical modeling [24]. For the analysis, as a feature independent
variable, we used z-scores for the rolling mean of tweet counts time series,
as a target variable, we used z-scores for stock price for ’MCD’ ticker. As a
result of sampling, the mean value as well as quantiles 0.01, 0.05, 0.95, 0.99
for the target variable were calculated. Figure 11 shows the results of
modeling. Figure 12 shows the probability density function for $\beta$
parameter. Quantile 0.05 for predicted target variable can be treated as the
value of risk (VaR) that is a quantitative characteristic for risk assessment.

Figure 11: Normalized time series of stock price for the ’MCD’ ticker (y),
prediction for stock price (pred), quantiles for prediction (0.01 - pred_q01,
0.05 - pred_q05, 0.95 - pred_q95, 0.99 - pred_q99 ) Figure 12: Probability
density function for $\beta$ model parameter

##  6 Analysis of Tweets Using Frequent Itemsets

The frequent itemsets and associative rules theory is often used in the
intellectual analysis [25, 26, 27, 28, 29, 30, 31, 32]. It can be used in a
text data analysis to identify and analyze certain sets of objects, which are
often found in large arrays and are characterized by certain features. Let us
consider the algorithms for detecting frequent sets and associative rules on
the example of processing microblog messages on tweets. Figure 13 shows
keyword frequencies for the specified thematic field ’ukraine nazi’. Using
these keywords, one can calculate frequent itemsets. Figure 14 shows the graph
of frequent itemsets which describes the semantic structure of entities for a
specified thematic field. Figure 14 shows the graph of the subset of
association rules, Figure 16 shows the association rules represented by a
grouped matrix. Figure 17 shows the association rules which contain the
keyword ’fake’. Figures 18–20 show the similar calculation for the thematic
field ’ukraine biological weapon’. Figures 21–23 show the subset of these
frequent itemsets and association rules which contain the keyword ’bird’. The
quantitative characteristics of frequent itemsets like value of support can be
used as a predictive feature in machine learning models.

Figure 13: Keyword frequencies related to the thematic field ’ukraine nazi’
Figure 14: Graph of semantic frequent itemsets Figure 15: Graph of association
rules Figure 16: Association rules represented by a grouped matrix Figure 17:
Association rules represented by a grouped matrix with the keyword ’fake’
Figure 18: Keyword frequencies related to the thematic field ’ukraine
biological weapon’ Figure 19: Graph of semantic frequent itemsets Figure 20:
Graph of association rules Figure 21: Graph of semantic frequent itemsets with
the keyword ’bird’ Figure 22: Graph of association rules with the keyword
’bird’ Figure 23: Association rules with the keyword ’bird’ represented by a
grouped matrix

##  7 Graph Structure of Tweets

The relationships among users can be considered as a graph, where vertices
denote users and edges denote their connections. Using graph mining
algorithms, one can detect user communities and find ordered lists of users by
various characteristics, such as Hub, Authority, PageRank, Betweenness. To
identify user communities, we used the Community Walktrap algorithm and to
visualize them we used Fruchterman-Reingold algorithm, which are implemented
in the package ’igraph’ [33] for the R programming language environment. The
Community Walktrap algorithm searches for related subgraphs, also called
communities, by random walk [34]. A graph which shows the relationships
between users can be represented by Fruchterman-Reingold algorithm [35]. The
qualitative structure of user’s connections can be used for aggregating
different quantitative time series and, in such a way, creating new features
for predictive models which can be used, for example, for predicting target
variables. Figure 24 shows users connections and revealed communities for the
subset of tweets which are related to the trends under consideration. The
results show that some communities marked by different colors are highly
isolated and have only few connections outside. This kind of communities can
be treated as suspicious, since artificially created communities for
amplifying manipulative news are also highly isolated and their activity is
often concentrated on amplifying by retweeting tweets from a limited set of
users. Therefore, the numerical characteristics of users communities can have
a predictive potential.

Figure 24: Graph of users’ connections

##  8 Conclusion

The obtained results show that an effective system for detecting fake and
manipulative news can be developed using combined neural network which consist
of three concatenated subnetworks: the subnetwork with DistilBERT transformer
for tweet texts, the subnetwork with embedding of tweet words and usernames of
users who retweeted tweets, and the subnetwork for the components of singular
value decomposition of TF-IDF matrix for lists of usernames of users who
retweeted tweets. Discussions on social networks about companies behavior has
impact on their business and their stock prices on the stock market. To
analyze such an impact and make risk assessment, Bayesian regression can be
used. Using the theory of frequent itemsets and association rules along with
thematic fields of keywords, makes it possible to reveal the semantic
structure for entities in news messages. The quantitative characteristics
association rules like support and confidence can be used as features for
machine learning predictive models. Using the graph theory, the hidden
communities of users can be revealed and their characteristics can be used in
a machine learning approach for fake and manipulative news detection.

## References

  * [1] A. Java, X. Song, T. Finin, and B. Tseng, ‘‘Why we twitter: understanding microblogging usage and communities,’’ in Proceedings of the 9th WebKDD and 1st SNA-KDD 2007 workshop on Web mining and social network analysis, pp. 56–65, 2007.
  * [2] H. Kwak, C. Lee, H. Park, and S. Moon, ‘‘What is Twitter, a social network or a news media?,’’ in Proceedings of the 19th international conference on World wide web, pp. 591–600, 2010.
  * [3] A. Pak and P. Paroubek, ‘‘Twitter as a corpus for sentiment analysis and opinion mining,’’ in LREc, vol. 10, pp. 1320–1326, 2010.
  * [4] M. Cha, H. Haddadi, F. Benevenuto, and K. Gummadi, ‘‘Measuring user influence in twitter: The million follower fallacy,’’ in Proceedings of the International AAAI Conference on Web and Social Media, vol. 4, 2010.
  * [5] F. Benevenuto, T. Rodrigues, M. Cha, and V. Almeida, ‘‘Characterizing user behavior in online social networks,’’ in Proceedings of the 9th ACM SIGCOMM conference on Internet measurement, pp. 49–62, 2009.
  * [6] J. Bollen, H. Mao, and X. Zeng, ‘‘Twitter mood predicts the stock market,’’ Journal of computational science, vol. 2, no. 1, pp. 1–8, 2011.
  * [7] S. Asur and B. A. Huberman, ‘‘Predicting the future with social media,’’ in 2010 IEEE/WIC/ACM international conference on web intelligence and intelligent agent technology, vol. 1, pp. 492–499, IEEE, 2010.
  * [8] D. Shamma, L. Kennedy, and E. Churchill, ‘‘Tweetgeist: Can the twitter timeline reveal the structure of broadcast events,’’ CSCW Horizons, pp. 589–593, 2010.
  * [9] M. Wang and G. Hu, ‘‘A novel method for twitter sentiment analysis based on attentional-graph neural network,’’ Information, vol. 11, no. 2, p. 92, 2020\.
  * [10] V. Balakrishnan, S. Khan, and H. R. Arabnia, ‘‘Improving cyberbullying detection using twitter users’ psychological features and machine learning,’’ Computers & Security, vol. 90, p. 101710, 2020.
  * [11] N. Grinberg, K. Joseph, L. Friedland, B. Swire-Thompson, and D. Lazer, ‘‘Fake news on twitter during the 2016 us presidential election,’’ Science, vol. 363, no. 6425, pp. 374–378, 2019.
  * [12] O. Ajao, D. Bhowmik, and S. Zargari, ‘‘Fake news identification on twitter with hybrid cnn and rnn models,’’ in Proceedings of the 9th international conference on social media and society, pp. 226–230, 2018.
  * [13] S. Helmstetter and H. Paulheim, ‘‘Weakly supervised learning for fake news detection on twitter,’’ in 2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), pp. 274–277, IEEE, 2018.
  * [14] B. M. Pavlyshenko, ‘‘Forecasting of Events by Tweets Data Mining,’’ Electronics and information technologies, no. 10, pp. 71–85, 2018.
  * [15] B. M. Pavlyshenko, ‘‘Can Twitter Predict Royal Baby’s Name ?,’’ Electronics and information technologies, no. 11, pp. 52–60, 2019.
  * [16] B. M. Pavlyshenko, ‘‘Forming predictive features of tweets for decision-making support,’’ in International Scientific Conference “Intellectual Systems of Decision Making and Problem of Computational Intelligence”, pp. 479–490, Springer, 2021.
  * [17] B. Pavlyshenko, ‘‘Classification analysis of authorship fiction texts in the space of semantic fields,’’ Journal of Quantitative Linguistics, vol. 20, no. 3, pp. 218–226, 2013.
  * [18] B. Pavlyshenko, ‘‘Clustering of authors’ texts of english fiction in the vector space of semantic fields,’’ Cybernetics and Information Technologies, vol. 14, no. 3, pp. 25–36, 2014.
  * [19] B. Pavlyshenko, ‘‘Genetic optimization of keyword subsets in the classification analysis of authorship of texts,’’ Journal of Quantitative Linguistics, vol. 21, no. 4, pp. 341–349, 2014.
  * [20] ‘‘Examples of Automatically Recognized Tweets As Fake or Manipulative.’’ URL: https://bit.ly/37u4viL.
  * [21] G. Jawahar, M. Abdul-Mageed, and L. V. Lakshmanan, ‘‘Automatic detection of machine generated text: A critical survey,’’ arXiv preprint arXiv:2011.01314, 2020.
  * [22] T. Fagni, F. Falchi, M. Gambini, A. Martella, and M. Tesconi, ‘‘Tweepfake: About detecting deepfake tweets,’’ Plos one, vol. 16, no. 5, p. e0251415, 2021.
  * [23] T. Murayama, ‘‘Dataset of fake news detection and fact verification: A survey,’’ arXiv preprint arXiv:2111.03299, 2021.
  * [24] B. Carpenter, A. Gelman, M. D. Hoffman, D. Lee, B. Goodrich, M. Betancourt, M. Brubaker, J. Guo, P. Li, and A. Riddell, ‘‘Stan: A probabilistic programming language,’’ Journal of statistical software, vol. 76, no. 1, 2017.
  * [25] R. Agrawal, R. Srikant, et al., ‘‘Fast algorithms for mining association rules,’’ in Proc. 20th int. conf. very large data bases, VLDB, vol. 1215, pp. 487–499, 1994.
  * [26] R. Agrawal, H. Mannila, R. Srikant, H. Toivonen, A. I. Verkamo, et al., ‘‘Fast discovery of association rules.,’’ Advances in knowledge discovery and data mining, vol. 12, no. 1, pp. 307–328, 1996.
  * [27] C.-K. Chui, B. Kao, and E. Hung, ‘‘Mining frequent itemsets from uncertain data,’’ in Pacific-Asia Conference on knowledge discovery and data mining, pp. 47–58, Springer, 2007.
  * [28] K. Gouda and M. J. Zaki, ‘‘Efficiently mining maximal frequent itemsets,’’ in Proceedings 2001 IEEE International Conference on Data Mining, pp. 163–170, IEEE, 2001.
  * [29] R. Srikant, Q. Vu, and R. Agrawal, ‘‘Mining association rules with item constraints.,’’ in Kdd, vol. 97, pp. 67–73, 1997.
  * [30] M. Klemettinen, H. Mannila, P. Ronkainen, H. Toivonen, and A. I. Verkamo, ‘‘Finding interesting rules from large sets of discovered association rules,’’ in Proceedings of the third international conference on Information and knowledge management, pp. 401–407, 1994.
  * [31] N. Pasquier, Y. Bastide, R. Taouil, and L. Lakhal, ‘‘Discovering frequent closed itemsets for association rules,’’ in International Conference on Database Theory, pp. 398–416, Springer, 1999.
  * [32] S. Brin, R. Motwani, and C. Silverstein, ‘‘Beyond market baskets: Generalizing association rules to correlations,’’ in Proceedings of the 1997 ACM SIGMOD international conference on Management of data, pp. 265–276, 1997.
  * [33] G. Csardi, T. Nepusz, et al., ‘‘The igraph software package for complex network research,’’ InterJournal, complex systems, vol. 1695, no. 5, pp. 1–9, 2006.
  * [34] P. Pons and M. Latapy, ‘‘Computing communities in large networks using random walks,’’ in International symposium on computer and information sciences, pp. 284–293, Springer, 2005.
  * [35] T. M. Fruchterman and E. M. Reingold, ‘‘Graph drawing by force-directed placement,’’ Software: Practice and experience, vol. 21, no. 11, pp. 1129–1164, 1991.