# Qualitative Analysis of a Graph Transformer Approach to Addressing Hate
Speech: Adapting to Dynamically Changing Content

Liam Hebert, Hong Yi Chen, Robin Cohen, Lukasz Golab

###### Abstract

Our work advances an approach for predicting hate speech in social media,
drawing out the critical need to consider the discussions that follow a post
to successfully detect when hateful discourse may arise. Using graph
transformer networks, coupled with modelling attention and BERT-level natural
language processing, our approach can capture context and anticipate upcoming
anti-social behaviour. In this paper, we offer a detailed qualitative analysis
of this solution for hate speech detection in social networks, leading to
insights into where the method has the most impressive outcomes in comparison
with competitors and identifying scenarios where there are challenges to
achieving ideal performance. Included is an exploration of the kinds of posts
that permeate social media today, including the use of hateful images. This
suggests avenues for extending our model to be more comprehensive. A key
insight is that the focus on reasoning about the concept of context positions
us well to be able to support multi-modal analysis of online posts. We
conclude with a reflection on how the problem we are addressing relates
especially well to the theme of dynamic change, a critical concern for all AI
solutions for social impact. We also comment briefly on how mental health
well-being can be advanced with our work, through curated content attuned to
the extent of hate in posts.

## Introduction

Online social platforms have allowed vast amounts of communication between
individuals at an unprecedented scale. Platforms such as Facebook have over
2.9 billion monthly active users who share opinions and connect with other
users111https://www.statista.com/statistics/264810/number-of-monthly-active-
facebook-users-worldwide/. A central tenet of these platforms is the removal
of traditional editorial barriers to reach a wider audience. Opinions or
commentary do not need to be regulated by editors before they can be published
and shared. However, this open approach to free speech has also led to the
explosion of propaganda, violence, and abuse against users based on their
race, gender, and religion (Das et al. 2020). In addition, widespread
dissemination of hateful speech has resulted in traumatizing mental health
effects for the victims (Vedeler, Olsen, and Eriksen 2019) and has ignited
social tensions and polarization between groups (Waller and Anderson 2021). To
combat this trend, social platforms have created rigorous community guidelines
which describe the kinds of content that can be
shared222https://transparency.fb.com/policies/community-standards/. These
guidelines are then enforced by teams of human moderators who manually allow
or disallow content. While effective, this approach can be insufficient for
coping with the growing scale of these platforms.

In an effort to allow improvements, platforms have also turned to the use of
automated methods to detect hate speech (Das, Banerjee, and Mukherjee 2022;
Mathew et al. 2021), aiming to classify the text that comprises the comment as
either hate speech or non-hate speech. However, we argue that this comment-
only scope is becoming increasingly limited and ineffective, due to the
importance of capturing context when deciding whether speech is hateful or
not.

To this end, we have designed an approach that goes beyond current hate speech
labelling efforts in three distinct ways (Hebert, Golab, and Cohen 2022).
First, we analyze entire discussions following a post, to detect hate speech.
Second, we support predicting when hate speech will occur, rather than simply
reacting to hateful posts once they are detected. All of this is achieved
using graph transformer networks, coupled with modelling attention and BERT
natural language processing. In so doing, we can capture the discussion
context and anticipate upcoming anti-social behaviour. This allows us to
analyze the conversational dynamics of different communities, being sensitive
to cases where the usage of a slur can be re-appropriated to appear to be non-
abusive. For example, the usage of certain slurs has been largely re-
appropriated in African American culture as a normal part of their vernacular
(Thomas 2007).

In this paper, we offer a detailed qualitative analysis of this solution for
hate speech detection in social networks, leading to insights into where the
method has the most impressive outcomes in comparison with competitors and
identifying scenarios where there are challenges to achieving ideal
performance. We draw out the key observation that comments on social platforms
have evolved to include images and external articles. These additional
elements can provide essential context to properly understanding the content
that follows.

We will conclude with a discussion on how to extend our method to encompass
the processing of images, towards the holistic processing of online posts. We
will also return to the concern of mental health well-being and discuss how
our more comprehensive automated solution for hate speech prediction can be
the basis for some significant steps forward. The social impact that we
anticipate coming from the research presented in this paper will be on social
media environments and their users.

## Data and Methods

### Comment-Only Hate Speech Models

To evaluate recent work in hate speech detection, we selected MuRIL by Das,
Banerjee, and Mukherjee (2022) and Bert-HateXplain by Mathew et al. (2021).
Both of these systems are based on the BERT transformer architecture, which
can create rich embeddings of text toward classification tasks (Devlin et al.
2019). We refer to these methods as comment-only hate speech models.

The main difference between the two methods is the data that both systems were
trained on. For Bert-HateXplain, the authors collected a combined dataset of
20,148 hateful tweets and posts from social platforms Twitter and Gab. For
MuRIL, the authors combined the HateXplain dataset with Founta et al. (2018)
(85,775 tweets) and Davidson et al. (2017) (24,783 tweets). This combined
approach was found to outperform HateXplain to become state-of-the-art in hate
speech detection (Das, Banerjee, and Mukherjee 2022).

### Graph Hate Speech Models

To study the usage of Graph Networks for hate speech detection, we focus on
our Graphormer approach proposed in Hebert, Golab, and Cohen (2022). This
model was novel in its ability to advance the study of hate speech detection
in social media by a) predicting where hate may arise rather than simply
reacting to posts that have been labelled as hate and b) leveraging graph
transformers for capturing contextual attention between comments in discussion
graphs.

Figure 1: Graphormer Architecture

Core to this model is the Graphormer architecture (Figure 1), which was
originally created by Ying et al. (2021) to predict molecular properties.
Graphormer uses a transformer model to create embeddings of atoms by computing
self-attention relationships between each atom of the molecule in relation to
their structure. The key to this approach is that self-attention can be
computed between all nodes of a graph irrespective of their structural
distance. This is contrasted by previous graph neural networks, which are
constrained to computing relationships between immediate neighbours of a node
(Wu et al. 2020).

To adapt Graphormer to social media analysis, we proposed reformulating hate
speech detection as a graph prediction task. Under this approach, comments are
represented as nodes and edges are the reply-to relationships between them. We
first initiate this graph by creating BERT embeddings of each comment. Then,
we aggregate and process the embeddings in relation to the discussion
structure using Graphormer, creating hate predictions for each node in the
graph. To focus on proactive predictions, the label of each node is an ordinal
value (0-4) based on how prevalent and encouraged the hate speech that follows
that comment. This training objective requires the model to reason about the
degree of hate throughout the entire discussion rather than being isolated to
the comment itself.

In this study, we also evaluate a baseline approach that uses Graph Attention
Networks (GAT) (Veličković et al. 2017). GAT models have previously been
adapted for hate speech detection by Parmentier and Cohen (2019); Parmentier
et al. (2021). Like Graphormer, GAT models utilize attention to create node
embeddings in a graph structure. However, this attention is constrained to
direct node neighbours, a limitation overcome by Graphormer by utilizing
transformers. This can result in predictions that focus more on the immediate
discussion context rather than the larger global context. Other work has
examined graph-based approaches for hate speech detection (Mishra et al. 2019;
Tian, Zhang, and Lau 2022); we confine our attention here to comparisons with
the models already described above.

### Reddit

We focus on the social platform Reddit to capture examples for this study. On
Reddit, discussions take place in topic-oriented communities called
subreddits. Each discussion is organized in tree-like structures where users
can create branching threads by replying to any comment in the tree. Prior
work analyzing Reddit communities has found that these communities exhibit
significant differences in their social makeup and communal behaviours (Waller
and Anderson 2021). For example, communities such as r/conservatives
demonstrate a right-leaning bias and community whereas r/politics contains a
polarizing left-leaning bias.

We analyzed 32 discussions from 16 different communities centred around
contentious topics and unique communities. Each discussion was chosen based on
the amount of comments it had. We also draw from Reddit conversations sampled
in Kurrek, Saleem, and Ruths (2020) for examples of reclaimed speech. For this
paper, we select five interesting examples to display in the section that
follows.

## Analysis

In this study, we focus on capturing examples from two categories of
conversational hate speech. First, we start with samples of contextual hate
speech, harmful comments that directly refer to or respond to the prior
discussion. Examples of this kind of hate speech would be a harmful commentary
on contentious topics, such as responding negatively to gay rights. We
hypothesize that comment-only methods would fail to capture the contextual
nuances that underpin the hatefulness of the target text. This can result in
false positives or false negatives when comments are judged in isolation.

The second category of hateful speech we study is inciteful hate speech:
comments that at first glance appear to be neutral but are designed to prompt
harmful discourse by other users. These examples aim to evaluate the ability
of graph hate speech models to proactively predict the direction of
conversations toward hatefulness, rather than only detect the hate of
individual comments. As a result, this direction also evaluates the ability of
graph methods to capture the dynamic nature of social media discussions, where
conversations are not static but grow over time as users add replies to the
content posted by other users. Examples of this kind of speech would be
comments concerning US president Biden in right-leaning political subreddits,
which can prompt hateful comments and threats.

For each comment in the discussion, we predict labels between zero and four
using comment-only and graph hate speech models. To match the ordinal
predictions given by Graphormer, we follow Hebert, Golab, and Cohen (2022) and
map the zero to one prediction given by comment-only models to bins of width
0.20 ([0-0.20], [0.20 - 0.40]. [0.40 - 0.60], [0.60-0.80], [0.80-1]). To
capture the ability of graph methods to adapt to evolving conversations, we
initialize the discussion graph with the initial post and immediate replies
(depth 1). We then iteratively predict the labels of each comment by gradually
increasing the depth of the discussion tree provided to the graph models. As
such, graph methods are constrained to make predictions about the direction of
conversations without seeing future comments.

### Contextual Hate Speech

Table 1: Conversation on r/gay containing contextual speech leading to false positive predictions Depth |  Text | Graphormer | GAT | Bert-HateXplain | MuRIL
---|---|---|---|---|---
0 |  Anyone else loving Lil Nas X meming on biggots? [image] | 2 | 1 | 3 | 0
1 |  What was this in reference too? | 1 | 1 | 0 | 0
2 |  Biggots are mad about his most recent song/music video | 1 | 2 | 1 | 0
3 |  Ah ok! I’ll have to check it out! | 0 | 0 | 0 | 0
4 |  The song is called Montero (call me by your name). | 0 | 0 | 0 | 0
5 |  My favorite line is ”I wanna f*ck you cuz I envy you”, and that hit hard | 0 | 0 | 2 | 4
6 |  ’I want that jet leg from f*ckin’ and flyin’ call me in | 0 | 1 | 3 | 4

We start by analyzing a conversation that took place on the subreddit /r/gay,
a community centred around LGBTQ topics (Table 1). In this conversation, users
are discussing a tweet from openly gay pop artist Lil Nas X. Our analysis of
this discussion thread demonstrates how a lack of context can lead to false
predictions about hatefulness.

The conversation begins with the initial post ”Anyone else loving Lil Nas X
meming on biggots”, referring to his tweet stating ”i thought y’all didn’t
like political correctness. what happened?”. This comment resulted in a high
hatefulness prediction from comment-only Bert-HateXplain and a moderate
prediction from the Graph methods. However, the hate predictions from the
graph methods quickly neutralize as the conversation leads into a discussion
of his latest song, Montero (depths 5 and 6). Given this context, it is clear
to these methods that the conversation is not hateful but rather just
discussing the lyrics of the song. However, this context is not available to
comment-only methods. The inaccuracy introduced without this important context
can be seen by the comment-only methods, in which both systems predicted high
hateful scores for both comments. This illustrates the ability of graph
methods to maintain conversational context when predicting hate scores.

Table 2: Conversation on /r/MensRights containing contextual affiliation-directed hate speech Depth |  Text | Graphormer | GAT | Bert-HateXplain | MuRIL
---|---|---|---|---|---
0 |  How to get banned from r/Feminism [image] | 3 | 3 | 0 | 0
1a |  i guess they were on their period and they want their feelings to matter at that point and you got banned for it | 3 | 3 | 1 | 0
1b |  Feminism is cancerous anyways | 3 | 2 | 1 | 0
1c |  Wow, they are almost as fragile as the donald. I guess all special snow flakes need their own safe space. | 1 | 2 | 1 | 0
1d |  Absolutely horrible. | 1 | 0 | 1 | 1

Next, we turn to an example of a discussion where context is needed to detect
hate speech. For this, we focus on the subreddit /r/MensRights. This community
advocates for increased men’s rights by discussing social issues that
adversely impact them, which frequently devolves into harmful misogyny
(Schmitz and Kazyak 2016).

This pattern of abuse can be seen in the discussion presented in Table 2.
Here, the user posts an image of a brief exchange they had in the r/Feminism
subreddit. In this exchange, the user was banned from that community for
advocating against the right for women to feel safe but rather that women only
have the right to be safe, regardless of reassurances and comfort. By posting
this exchange on r/MensRights, the user aimed to frame the r/Feminism
community in a negative light by stating that they were banned in response to
sharing a valid point.

In this discussion, graph methods were able to accurately understand the
negative context of feminism towards the hateful comments that followed. Most
surprisingly, none of the comments that followed were labelled as hate speech
by the comment-only methods. Such examples include ”Feminism is cancerous
anyways”, which was appropriately labelled as hateful by graph methods,
receiving a prediction of 3 from Graphormer, but mislabeled as innocuous by
text methods. We assume that this large difference in prediction comes
uniquely from the graph structure of the discussion.

Table 3: Conversation on r/rupaulsdragrace containing reclaimed language and multi-modal context Depth |  Text | Graphormer | GAT | Bert-HateXplain | MuRIL
---|---|---|---|---|---
0 |  *SPOILERS* Always and Forever, paparazzi who? [image - Figure 2] | 1 | 1 | 1 | 0
1a |  Am I the only f*ggot that LIVED for this look? | 1 | 2 | 4 | 3
2a |  I honestly truly thought it was an amazing concept and I love the final result | 0 | 0 | 1 | 0
2b |  Not at all. F*ck fashion. I want fashion I’ll by the Vogue fall guide. Give me something creative I haven’t seen before. | 0 | 0 | 2 | 4
1b |  Say what you will about the look, but can we appreciate the fact that this b*tch got over a dozen Canon DSLRs plus the lenses on this dress? That shit ain’t cheap. | 1 | 1 | 3 | 4
1c |  This look is ridiculous. I love it, but my God, who thinks of this shit. | 0 | 0 | 1 | 4
Figure 2: Photo of the Drag Queen discussed in Table 3

Finally, we examine a conversation that contains reclaimed language that was
previously perceived as harmful to a given community. For this, we sample a
discussion from /r/rupaulsdragrace, a community dedicated to the LGBTQ drag
competition Ru Paul’s Drag Race (Table 3). In this conversation, users are
commenting on one competitor’s manufactured drag outfits. In this community,
slurs such as ”f*ggot” and ”b*tch” are reclaimed as positive terms to refer to
LGBTQ members and competitors333A common slogan on the show is ”Yass b*tch”,
which is used to cheer on competitors. However, these slurs are more often
used in a hateful context, proving a challenge for methods that do not
consider specific contexts.

Analyzing the predictions of each method, we see that comment-only methods
assign a high hate score for comments at depths 1a and 2b. However, both
comments are in fact positive and supportive of the competitor mentioned in
the initial post. Inspecting the content of these comments, we can infer that
the false positive prediction can likely be attributed to the usage of
reclaimed slurs without contextualizing them to the community and prior
discussion. This behaviour is contrasted with graph methods, which predicted
more accurate scores, likely due to the context provided by other comments
concerning the fashion of the dress.

It is important to note that both graph and comment-only methods can only
infer context by inspecting the text of comments. In this conversation, graph
methods were likely only able to infer context due to the other comments,
which discuss the fashion of the dress (depths 1b and 1c). However, upon
looking at the initial post, we see that the user accompanied their post with
an image of a drag queen (Figure 2). This picture provides immense context to
the discussion, such as the focus on fashion and the LGBTQ community. As such,
we hypothesize that future work that would include multi-modal posts could
provide important context to the comments that follow.

### Inciteful Hate Speech

Table 4: Conversation from r/conservatives containing inciteful speech
regarding Black Lives Matter
Depth |  Text | Graphormer | GAT | Bert-HateXplain | MuRIL
---|---|---|---|---|---
0 |  Black Guns Matter, as does election integrity [image] | 3 | 1 | 0 | 1
1 |  That’s Hilarious. I think its interesting how the left keeps trying to rub ”BLACK PEOPLE ARE BUYING GUNS” in the faces of conservatives, like we would somehow be opposed to that. […] glad that gun ownership is expanding among all demographics | 4 | 3 | 1 | 2
2a |  It’s on every conservative news and media platform as a big positive, but the left doesn’t pay attention to that literally all they think is conservatives racist, therefore black guns bad for them. | 3 | 2 | 1 | 2
2b |  They want us to be divided by race like they are. They are so racist they can’t imagine us being united by our love of fundamental rights | 4 | 3 | 0 | 0
|  […] |  |  |  |
4 |  I agree. The black panthers had the right idea. BLM members should arm too. | 2 | 0 | 0 | 0
5 |  They have been…. smh | 4 | 1 | 0 | 0

For our next set of examples, we investigate the ability of graph networks to
predict the direction of conversations. We start by analyzing a conversation
that took place on /r/conservatives, a community for discussing right-wing
policies (Table 4). These policies often include advocacy for gun rights and
strong support for election denial (Block 2021). Discussions in these
political communities have become increasingly polarized and hateful towards
members of the opposite party (Waller and Anderson 2021).

In this discussion, users are referring to a tweet from a black user
concerning the difficulty for people to purchase weapons due to a lack of
governmental IDs. The conversation is centred around confounding the Black
Lives Matter movement with gun advocacy and election denial (”Black Guns
Matter, as does election integrity”). The discussion then devolves into
affiliation-based hate as users claim left-leaning users are racists and
associate black activist groups (”the black panthers” and ”BLM”) with armed
violence.

Looking at the predictions of both groups of methods, we see that both graph
methods predict very high hatefulness scores for many of the comments within
the discussion. This can especially be seen in the comments at depth 2, in
which users delve into race accusations. However, it is important to note that
these comments are more akin to debate rather than explicit hate speech. This
is reflected in the predictions from the comment-only methods, which
consistently predict low hatefulness scores for these comments. As such, it
can be inferred that these high predictions originate from a belief that the
conversation will head in a hateful direction given the context thus far.
Indeed, this is the case as later comments (depth 5) intensify the discussion
and accuse members of the Black Lives Matter movement of armed violence.
However, it is also important to note that the earlier predictions appear to
conflate polarizing political discourse with hate (depth 1) with minimal
discussion context.

Table 5: Conversation from r/politics requiring long range forecasting from community cues Depth |  Text | Graphormer | GAT | Bert-HateXplain | MuRIL
---|---|---|---|---|---
0 |  Trump, who was impeached for withholding nearly $400 million in military aid from Ukraine, said ’this deadly Ukraine situation would never have happened’ if he were in office [article] | 2 | 2 | 0 | 0
1 |  This happened under Biden’s watch. That is a fact | 3 | 3 | 0 | 0
2a |  Russia has been threatening Ukraine for the last 8 years | 2 | 0 | 0 | 0
3 |  Whatever, still happened under Biden’s watch. Not Trump’s. | 3 | 2 | 0 | 0
4 |  Right, I’m sure the situation would’ve been so much better under the leadership of a failed jackass lapdog for Putin. | 4 | 3 | 3 | 0
2b |  You might want to do a little reading about Ukraine. Your comment is completely ludicrous | 3 | 3 | 0 | 0

To further examine the ability of Graphormer to predict the direction of
conversations, we investigate a conversation from /r/politics. This community
is known to have strong left-leaning political views and to have a distinct
and polarized user base from /r/conservatives (Waller and Anderson 2021).
Inside the sampled conversation (Table 5), users are discussing comments made
by former president Trump in relation to the Ukraine war. The conversation
begins benign but becomes combative, with one user shifting the blame for the
Ukrainian war onto the current left-leaning president. This trend culminates
into a climax at depth 4, where the user escalates to using hateful language
against Trump.

Investigating the predictions, we see a similar trend from the previous
example where each of the predictions from the comment-only methods remains
mostly neutral apart from the hateful comment at depth 4. However, the
contentiousness of the conversation given the previous instigating comments is
captured by graph methods, resulting in high predictions by both methods. This
can especially be seen with the prediction at depth 1, which seemingly
captured the contentious relationship between comments concerning Biden and
Trump. Indeed, we can see that the conversation did turn hateful later in the
conversation, validating this prediction.

## Discussion

In our analysis, we focused on analyzing two types of difficult hate speech:
contextual and inciteful. Contextual hate speech requires conversational
context to understand, and inciteful hate speech is not inherently hateful but
is designed to incite further hateful comments. Both types of speech present
difficulties for current comment-only approaches due to the heavy reliance on
the context in order to make correct predictions.

Starting with contextual hate speech, we analyzed three different
conversations originating from /r/gay, /r/MensRights, and /r/rupaulsdragrace.
Using comment-only methods, we found many predictions that were false
positives or false negatives depending on the text of the comment in
isolation. For false positives, we found that comment-only methods tended to
predict high hate scores for comments that contained slurs (Table 1 and Table
3). However, upon reading the rest of the conversation, it becomes clear that
many of these slurs are utilized in a non-derogatory context. The same can be
said for false negatives, where antagonistic replies can lose their hateful
context when considered in isolation (Table 2). However, in each of these
conversations, we found that graph methods perform well at capturing the vital
discussion context that is required to appropriately understand these
comments. Between the two graph methods we evaluated, both GAT and Graphormer
performed similarly well in the examples we explored.

To examine the ability of graph and comment-only methods to capture inciteful
speech, we analyzed two discussions from /r/conservatives and /r/politics,
communities with polarizing user-bases (Waller and Anderson 2021). We found
that graph methods are sensitive to counter-speech as evidence of inciting
hateful discourse. This can be especially seen in Table 5, where users
disagreeing on the cause of the Ukraine war lead to a high hatefulness
prediction from the two graph methods. While there may be some validity to
these predictions regarding their contentiousness, it does raise a concern
about how to moderate these heated debates. However, in the case of the
example in Table 4, the most inciteful comments (depth 2a, 2b, and 5) are
appropriately labelled as such. In each of these cases, comment-only methods
predicted each comment as neutral, even if the comments were hateful (depth
2b). We found that Graphormer was more sensitive to higher predictions than
GAT when faced with these types of comments.

In each of the examples, we also evaluated the ability of graph networks to
adapt to evolving social media conversations. Coping with this dynamic change
is essential for the successful real-life implementation of AI for social
impact. We evaluate this behaviour by iteratively predicting comments in the
discussion graph in a depth-wise fashion, differing from Hebert, Golab, and
Cohen (2022). As a result, we constrain graph models to predict labels from
only the context provided by previous comments, mirroring how the system would
be deployed in real situations. Despite this constraint, we still see that
graph systems are able to make accurate predictions. This can best be seen in
Table 1, where the graph models adapted their predictions to be less hateful
once the conversation developed.

We also found that many examples we retrieved were entered around multi-modal
posts. Such examples include the discussion in Table 1, involving an image of
a tweet, and the discussion in Table 5, involving an article concerning Trump
and the Ukraine war, among others. When investigating contextual hate speech,
Table 3 presents an example where the image (Figure 2) provides important
context to the comments that followed.

Figure 3: Examples of Multi-Modal Hate Speech

By analyzing this picture, it would be possible to understand that the
discussion concerns an LGBTQ drag queen competing in an elaborate dress.
However, without this context, we found that comment-only methods
misclassified supportive speech using reclaimed LGBTQ vernacular as hateful.
This is especially concerning given that these predictions could serve to
suppress communities that are vitally important to the mental health of
minority populations (Lucero 2017; Fish et al. 2020). Furthermore, memes sent
on online platforms are often only hateful if one considers both the image and
the text caption together, as seen in Figure 3 (Kiela et al. 2020). By taking
a holistic view of conversations by encoding images, text, and discussion
structure together, we hypothesize those hate speech detection methods would
be able to avoid many false predictions, such as the ones incurred in Table 3.
Furthermore, following Tian, Zhang, and Lau (2022), it would be possible to
include user-level information into this graph representation.

Finally, it is also important to analyze the mental health impact given by a
graph approach to hate speech. By reformulating hate speech as a graph
prediction task, we are able to train systems that can leverage discussion
context toward predicting the direction of conversations. This can allow
moderators on social platforms to be alerted of potentially harmful comments
and deploy mitigation strategies to shield users who are susceptible to mental
health effects. We see an example of such a discussion in Table 4, where users
that are susceptible to trauma from guns and race can be warned ahead of time
by utilizing the proactive graph predictions. Furthermore, by utilizing an
increasing ordinal scale (from zero to four) for predicting hate, users can
select their level of comfort by choosing the intensity of contentious
comments they are comfortable viewing. As the conversation develops, these
predictions can then be updated with further context and revised accordingly.
An example of where this would be useful is the discussion in Table 3, where
further comments add credence to the innocence of previous comments. By
providing these scores, platform owners can allow users to have control over
the content they see through self-moderation. Another valued opportunity for
deployment of our methods shown by qualitative analysis is in assiting
platforms to curtail hate speech proliferation: greater prediction of
impending escalating harm and caution in imposing penalites when discussion
isn’t hate can both be addressed.

## Conclusion

In this work, we explored the impact of Graph Transformer Networks on hate
speech detection (Hebert, Golab, and Cohen 2022). To do this, we performed an
extensive qualitative analysis of graph and comment-only methods on
conversations sampled from different communities on Reddit. When examining
contextual hate speech, we found that Graph Transformer Networks can prevent
both false positives and false negatives incurred by comment-only methods. In
these cases, context played a key role in understanding the nature of analyzed
comments. We also found similar gains in performance when analyzing
discussions that concerned inciteful speech. However, we also found that
debates were prone to high hate predictions despite being mostly civil.

Guided by this study, one promising direction for future work is to include
more modalities to better contextualize comments. Among the examples we
retrieved, many were centred around an image or article. We hypothesize that
utilizing a holistic view of conversations by including all modalities can
help prevent false positives. Most importantly, this approach could help catch
the most pervasive hate speech of all - discourse.

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# The subordinated processes controlled by a family of subordinators and
corresponding Fokker-Planck type equations

Long Shi1,2, Zu-Guo Yu1,3, Hai-Lan Huang1, Zhi Mao1 and Ai-Guo Xiao1
1 Hunan Key Laboratory for Computation and Simulation in Science and
Engineering and
Key Laboratory of Intelligent Computing and Information Processing of Ministry
of Education,
Xiangtan University, Xiangtan, Hunan 411105, China.
2 School of Science, Central South University of Forest and Technology,
Changsha, Hunan 410004, China.
3School of Mathematical Sciences, Queensland University of Technology,
GPO Box 2434, Brisbane, Q4001, Australia. Corresponding to<EMAIL_ADDRESS>
###### Abstract

In this work, we consider subordinated processes controlled by a family of
subordinators which consist of a power function of time variable and a
negative power function of $\alpha-$stable random variable. The effect of
parameters in the subordinators on the subordinated process is discussed. By
suitable variable substitutions and Laplace transform technique, the
corresponding fractional Fokker-Planck-type equations are derived. We also
compute their mean square displacements in a free force field. By choosing
suitable ranges of parameters, the resulting subordinated processes may be
subdiffusive, normal diffusive or superdiffusive.

Key words: subordinated process; subordinator; Fokker-Planck-type equation;
mean square displacement; variable substitution; Laplace transform

##  1 Introduction

Anomalous diffusion is one of the most universal phenomena in nature [1-3]. In
one dimension, it is characterized by a mean square displacement (MSD) of the
form

$\langle(\Delta x)^{2}\rangle\sim Dt^{\alpha}$ (1)

with $\alpha\neq 1$, deviating from the linear dependence on time found for
normal diffusion [1]. The coefficient $D$ is generalized diffusion constant.
It is called subdiffusion for $0<\alpha<1$ and superdiffusion for $\alpha>1$
[4].

One of the simplest models to describe anomalous diffusion is continuous time
random walk (CTRW), introduced by Montroll and Weiss [5]. Let $W(x,t)$ be the
probability density function (PDF) of finding a particle (with continuous
distribution of step length) at point $x$ at time $t$, one has [5]

$W(x,t)=\sum_{n}f(x,n)g_{n}(t),$ (2)

where $f(x,n)$ is the PDF to find the particle at point $x$ after $n$ steps
and $g_{n}(t)$ is the probability to make exactly $n$ steps up to time $t$.
Here, the random number of steps $n$ plays the role of operational time
governing the evolution of the system.

A continuous realization of Eq.(2) is

$W(x,t)=\int_{0}^{\infty}f(x,\tau)g(\tau,t)d\tau,$ (3)

which has been considered by Fogedby in [6] in terms of a coupled system of
Itô stochastic differential equations for position $x$ and time $t$

$\frac{dx(\tau)}{d\tau}=F(x)+\Gamma(\tau),\hskip
28.45274pt\frac{dt(\tau)}{d\tau}=\eta(\tau),$ (4)

where $F(x)$ is an external force, $\Gamma(\tau)$ and $\eta(\tau)$ are random
noise sources that are assumed to be independent. In this frame, the position
$x$ and the physical time $t$ are parameterized in terms of the continuous
variable $\tau$ which may be considered as operational time. The random walk
in physical time $Y(t)$ is then given by the combined process $Y(t)=x(s(t))$,
where $\tau=s(t)$ is the inverse process to $t(\tau)$ defined as

$s(t)=inf\\{\tau>0:t(\tau)>t\\},$ (5)

a continuous analogue of the random number of the steps $n(t)$.

In formula (3) $W(x,t)$, $f(x,\tau)$ and $g(\tau,t)$ are the PDFs of the
processes $Y(t)$, $x(\tau)$ and $s(t)$, respectively. From the opinion of
subordination, which is originated by Bochner [7], the combined process $Y(t)$
can be said to be subordinated to parent process $x(\tau)$ using the
operational time $\tau=s(t)$. The process $s(t)$ is also called subordinator.
It is worth to note that the continuous time random processes in phase
(position-velocity) space based on the subordination were discussed in Refs.
[8-10].

In system (4), the most popular choice on random noise sources
$\Gamma(\tau),\eta(\tau)$ is taken $\Gamma(\tau)$ as a Gaussian noise and
$\eta(\tau)$ as a fully skew $\alpha-$stable Lévy noise (i.e. $s(t)$ is an
inverse $\alpha-$stable subordinator). In recent years, the system of coupled
Langevin equations (4) with inverse $\alpha-$stable subordinator is becoming a
hot topic [11-25]. The inverse tempered $\alpha-$stable subordinator and the
infinitely divisible inverse subordinators were also considered [26-34].

However, one cannot obtain directly characteristics of subordinator $s(t)$
from the definition (5). In [35], Meerschaert and Scheffler showed that the
$\alpha-$self-similar process $E(t)$ has the same distribution as the process
$t^{\alpha}Z^{-\alpha}$, where $Z$ is an $\alpha-$stable random variable, i.e.
$\langle e^{-uZ}\rangle=e^{-u^{\alpha}}$. Motivated by this, we assume that
the subordinator $s(t)$ has the following three forms
$s(t)=t^{\alpha}Z^{-\alpha}$, $s(t)=t^{\beta}Z^{-\alpha}$ or
$s(t)=t^{\beta}Z^{-\beta}$ and then discuss the resulting subordinated
processes $x(s(t))$.

This paper is organized as follows. In section 2, we introduce the
subordinated process controlled by subordinator
$s(t)=t^{\alpha}Z^{-\alpha}(0<\alpha<1)$ and connect the subordinated process
$x(s(t))$ with a time-fractional Fokker-Planck equation (FPE). In section 3,
we discuss the subordinated process $x(s(t))$ with
$s(t)=t^{\beta}Z^{-\alpha}(0<\alpha<1,0<\beta<2)$. The subordinated process
$x(s(t))$ with $s(t)=t^{\beta}Z^{-\beta}(0<\beta<2)$ is considered in section
4. The conclusions are presented in section 5.

##  2 The subordinated process with subordinator $s(t)=t^{\alpha}Z^{-\alpha}$

Assumed that the parent process $x(\tau)$ satisfies the Langevin equation

$\frac{dx(\tau)}{d\tau}=F(x(\tau))+\Gamma(\tau),$ (6)

where $\Gamma(\tau)$ is a white Gaussian noise with
$\langle\Gamma(\tau)\rangle=0,\langle\Gamma(\tau)\Gamma(\tau^{\prime})\rangle=2\delta(\tau-\tau^{\prime})$.
Thus, the PDF $f(x,\tau)$ obeys the following ordinary FPE [36]:

$\frac{\partial}{\partial\tau}f(x,\tau)=(-\frac{\partial}{\partial
x}F(x)+\frac{\partial^{2}}{\partial x^{2}})f(x,\tau)\triangleq
L_{FP}f(x,\tau).$ (7)

Now we assume $s(t)=t^{\alpha}Z^{-\alpha}(0<\alpha<1)$, where $Z$ is an
$\alpha-$stable random variable with the PDF $\psi_{\alpha}(t)$, i.e. the
Laplace transform $\langle
e^{-uZ}\rangle=\int_{0}^{\infty}\psi_{\alpha}(t)e^{-ut}dt=e^{-u^{\alpha}}$.

Since

$P(s(t)\leq\tau)=P(Z\geq\frac{t}{\tau^{1/\alpha}})=\int_{\frac{t}{\tau^{1/\alpha}}}^{\infty}\psi_{\alpha}(t^{\prime})dt^{\prime},$
(8)

we have the PDF $g(\tau,t)$ of subordinator $s(t)$ as

$g(\tau,t)=\frac{1}{\alpha}\frac{t}{\tau^{1+1/\alpha}}\psi_{\alpha}(\frac{t}{\tau^{1/\alpha}}).$
(9)

Recalling the following Laplace transform formulas

$\int_{0}^{\infty}\varphi(at)e^{-ut}dt=\frac{1}{a}\int_{0}^{\infty}\varphi(t^{\prime})e^{-\frac{u}{a}t^{\prime}}dt^{\prime}=\frac{1}{a}\widetilde{\varphi}(\frac{u}{a}),\hskip
14.22636pta>0,$ (10)

and

$\int_{0}^{\infty}t\varphi(t)e^{-ut}dt=-\frac{d}{du}\int_{0}^{\infty}\varphi(t)e^{-ut}dt=-\frac{d}{du}\widetilde{\varphi}(u),$
(11)

we get the Laplace transform of $g(\tau,t)$ as

$\widetilde{g}(\tau,u)=u^{\alpha-1}\exp\\{-u^{\alpha}\tau\\}.$ (12)

Substituting $\widetilde{g}(\tau,u)$ into Eq.(3), we get the PDF $W(x,t)$ of
the subordinated process $Y(t)=x(s(t))$ in Laplace space:

$\begin{array}[]{lll}\widetilde{W}(x,u)&=&\int_{0}^{\infty}f(x,\tau)\widetilde{g}(\tau,u)d\tau\\\
\\\ &=&\int_{0}^{\infty}f(x,\tau)u^{\alpha-1}\exp\\{-u^{\alpha}\tau\\}d\tau\\\
\\\ &=&u^{\alpha-1}\widetilde{f}(x,u^{\alpha}).\end{array}$ (13)

Since $f(x,\tau)$ satisfies Eq.(7), one get in Laplace space

$u\widetilde{f}(x,u)-f(x,0)=L_{FP}\widetilde{f}(x,u).$ (14)

Assumed that $W(x,0)=f(x,0)=\delta(x)$, from Eq.(13) and Eq.(14), we have

$\begin{array}[]{lll}u\widetilde{W}(x,u)-W(x,0)&=&u^{\alpha}\widetilde{f}(x,u^{\alpha})-f(x,0)\\\
\\\ &=&L_{FP}\widetilde{f}(x,u^{\alpha})\\\ \\\
&=&u^{1-\alpha}L_{FP}\widetilde{W}(x,u).\end{array}$ (15)

Taking Laplace inverse transform for $u\rightarrow t$ in Eq.(15), we obtain
the following time-fractional FPE [4,37-39]

$\frac{\partial W(x,t)}{\partial t}=_{0}D_{t}^{1-\alpha}L_{FP}W(x,t),\hskip
14.22636pt0<\alpha<1,$ (16)

where ${}_{0}D_{t}^{1-\alpha}$ denotes the Riemann-Liouville fractional
derivative [40], defined by

$_{0}D_{t}^{1-\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\frac{\partial}{\partial
t}\int_{0}^{t}(t-s)^{\alpha-1}f(s)ds.$ (17)

Eq.(16) can be also derived from a generalised master equation [41], or from
continuous time random walks (CTRWs) [42].

We are also interested in the MSD of the subordinated process $Y(t)$ in free
force field. One can compute $\langle x^{2}\rangle(t)$ easily in terms of
Eq.(16) with natural boundary conditions:

$\langle
x^{2}\rangle(t)=\int_{-\infty}^{\infty}x^{2}W(x,t)dx=\frac{2}{\Gamma(\alpha+1)}t^{\alpha},\hskip
14.22636pt0<\alpha<1.$ (18)

From (18), we conclude that the proposed subordinated process is subdiffusive.

##  3 The subordinated process with subordinator $s(t)=t^{\beta}Z^{-\alpha}$

In this section, we assume that $x(\tau)$ satisfies Langevin equation (6) and
$s(t)$ has the form $s(t)=t^{\beta}Z^{-\alpha}(0<\alpha<1,0<\beta<2)$.

Since

$P(s(t)\leq\tau)=P(Z\geq\frac{t^{\beta/\alpha}}{\tau^{1/\alpha}})=\int_{\frac{t^{\beta/\alpha}}{\tau^{1/\alpha}}}^{\infty}\psi_{\alpha}(t^{\prime})dt^{\prime},$
(19)

we have

$g(\tau,t)=\frac{1}{\alpha}\frac{t^{\beta/\alpha}}{\tau^{1+1/\alpha}}\psi_{\alpha}(\frac{t^{\beta/\alpha}}{\tau^{1/\alpha}}).$
(20)

Setting

$h(\tau,t)=\frac{1}{\alpha}\frac{t}{\tau^{1+1/\alpha}}\psi_{\alpha}(\frac{t}{\tau^{1/\alpha}}),$
(21)

one can easily obtain

$g(\tau,t)=h(\tau,t_{1})$ (22)

with $t_{1}=t^{\beta/\alpha}$.

Since

$W(x,t)=\int_{0}^{\infty}f(x,\tau)g(\tau,t)d\tau,$ (23)

we have

$\begin{array}[]{lll}\frac{\partial}{\partial
t}W(x,t)&=&\int_{0}^{\infty}f(x,\tau)\frac{\partial}{\partial
t}g(\tau,t)d\tau\\\ \\\ &=&\int_{0}^{\infty}f(x,\tau)\frac{\partial
h(\tau,t_{1})}{\partial t_{1}}\frac{dt_{1}}{dt}d\tau\\\ \\\
&=&\frac{\partial}{\partial
t_{1}}\int_{0}^{\infty}f(x,\tau)h(\tau,t_{1})d\tau\frac{dt_{1}}{dt}.\end{array}$
(24)

Setting $W_{1}(x,t)=\int_{0}^{\infty}f(x,\tau)h(\tau,t)d\tau$, by comparing
Eq.(21) with Eq.(9), we obtain that $W_{1}(x,t)$ satisfies

$\frac{\partial W_{1}(x,t)}{\partial t}=_{0}D_{t}^{1-\alpha}L_{FP}W_{1}(x,t).$
(25)

Substituting Eq.(25) into Eq.(24), we have

$\begin{array}[]{lll}\frac{\partial}{\partial
t}W(x,t)&=&\frac{\partial}{\partial
t_{1}}W_{1}(x,t_{1})\cdot\frac{dt_{1}}{dt}\\\ \\\
&=&\frac{1}{\Gamma(\alpha)}\frac{\partial}{\partial
t_{1}}\int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1}L_{FP}W_{1}(x,s)ds\cdot\frac{dt_{1}}{dt}\\\
\\\ &=&\frac{1}{\Gamma(\alpha)}\frac{\partial}{\partial
t}\int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1}L_{FP}W_{1}(x,s)ds\\\ \\\
&=&\frac{1}{\Gamma(\alpha)}\frac{\partial}{\partial
t}\int_{0}^{t^{\beta/\alpha}}(t^{\beta/\alpha}-s)^{\alpha-1}L_{FP}W_{1}(x,s)ds.\end{array}$
(26)

After replacing the variable $s$ by $t^{\prime\beta/\alpha}$ and using the
relation $W_{1}(x,t^{\beta/\alpha})=W(x,t)$ in Eq.(26), we obtain

$\begin{array}[]{lll}\frac{\partial}{\partial
t}W(x,t)&=&\frac{\beta}{\Gamma(\alpha+1)}\frac{\partial}{\partial
t}\int_{0}^{t}t^{\prime\beta/\alpha-1}(t^{\beta/\alpha}-t^{\prime\beta/\alpha})^{\alpha-1}L_{FP}W(x,t^{\prime})dt^{\prime}\\\
\\\ &=&\Phi_{t}L_{FP}W(x,t),\end{array}$ (27)

where

$\Phi_{t}f(t)=\frac{\beta}{\Gamma(\alpha+1)}\frac{\partial}{\partial
t}\int_{0}^{t}t^{\prime\beta/\alpha-1}(t^{\beta/\alpha}-t^{\prime\beta/\alpha})^{\alpha-1}f(t^{\prime})dt^{\prime}.$
(28)

When $\beta=\alpha$, Eq.(27) reduces into the time fractional FPE (16). It is
worthy to note that Mura et al. derived the integral form of Eq.(27) from a
non-Markovian forward drift equation and called it stretched time-fractional
diffusion equation [43].

The MSD $\langle x^{2}\rangle(t)$ of $Y(t)=x(t^{\beta}Z^{-\alpha})$ in a free
force field can be computed by

$\begin{array}[]{lll}\frac{d}{dt}\langle
x^{2}\rangle(t)&=&\int_{-\infty}^{\infty}x^{2}\frac{\partial W(x,t)}{\partial
t}dx\\\ \\\
&=&\frac{2\beta}{\Gamma(\alpha+1)}\frac{d}{dt}\int_{0}^{t}t^{\prime\beta/\alpha-1}(t^{\beta/\alpha}-t^{\prime\beta/\alpha})^{\alpha-1}dt^{\prime}\\\
\\\
&=&\frac{2\beta}{\Gamma(\alpha+1)}\frac{d}{dt}\int_{0}^{t^{\beta/\alpha}}\frac{\alpha}{\beta}(t^{\beta/\alpha}-s)^{\alpha-1}ds\\\
\\\ &=&\frac{2\beta}{\Gamma(\alpha+1)}t^{\beta-1}.\end{array}$ (29)

So, we have

$\langle x^{2}\rangle(t)=\frac{2}{\Gamma(\alpha+1)}t^{\beta},\hskip
14.22636pt0<\alpha<1,0<\beta<2.$ (30)

By comparing Eq.(30) with Eq.(18), we conclude that the diffusive process
$Y(t)$ is controlled by parameter $\beta$ in subordinator $s(t)$. It is easy
to obtain that the subordinated process $Y(t)$ is subdiffusive when
$0<\beta<1$, normal diffusive when $\beta=1$ and supdiffusive when
$1<\beta<2$.

##  4 The subordinated process with subordinator $s(t)=t^{\beta}Z^{-\beta}$

Here we assume that $x(\tau)$ satisfies Langevin equation (6) and $s(t)$ has
the form $s(t)=t^{\beta}Z^{-\beta}(0<\beta<2)$ where $Z$ is an $\alpha-$stable
random variable $(0<\alpha<1)$.

Since

$P(s(t)\leq\tau)=P(Z\geq\frac{t}{\tau^{1/\beta}})=\int_{\frac{t}{\tau^{1/\beta}}}^{\infty}\psi_{\alpha}(t^{\prime})dt^{\prime},$
(31)

we have

$g(\tau,t)=\frac{1}{\beta}\frac{t}{\tau^{1/\beta+1}}\psi_{\alpha}(\frac{t}{\tau^{1/\beta}}).$
(32)

After using the formulas (10) and (11), one can get the Laplace transform of
the PDF $g(\tau,t)$

$\widetilde{g}(\tau,u)=\frac{\alpha}{\beta}\tau^{\alpha/\beta-1}u^{\alpha-1}\exp\\{-u^{\alpha}\tau^{\alpha/\beta}\\}.$
(33)

It is worth to note that the subordinated process $Y(t)$ constructed in this
section can be connected to a CTRW with correlated waiting times (see
Ref.[21]) in terms of Eq.(33). Indeed, in Ref.[21], the authors modeled CTRW
with correlated waiting times by the following coupled Langevin equations

$\frac{dx(\tau)}{d\tau}=\Gamma(\tau),\hskip
28.45274pt\frac{dt(\tau)}{d\tau}=\int_{0}^{\tau}M(\tau-\tau^{\prime})\eta(\tau^{\prime})d\tau^{\prime},$
(34)

where $M(\tau)$ is a non-negative continuous memory function and $\eta(\tau)$
is a fully skew $\alpha-$stable Lévy noise. The Laplace transform of the PDF
$g(\tau,t)$ of the process $s(t)$ defined by Eq.(5) has the form

$\widetilde{g}(\tau,u)=u^{\alpha-1}\phi^{\prime}(\tau)\exp\\{-u^{\alpha}\phi(\tau)\\},$
(35)

where
$\phi(\tau)=\int_{0}^{\tau}d\tau^{\prime}(\int_{\tau^{\prime}}^{\tau}M(\tau^{\prime\prime}-\tau^{\prime})d\tau^{\prime\prime})^{\alpha}$.
By comparing Eq.(35) with Eq.(33), we have $\phi(\tau)=\tau^{\alpha/\beta}$,
which gives the connection between the subordinated process $Y(t)$ constructed
in this section and the CTRW with correlated waiting times introduced in Ref.
[21].

Some other kinds of correlated CTRW processes were given in Refs. [44-48]. The
CTRW with correlated jumps was studied in Ref.[44]. The CTRWs with correlated
waiting times defined as
$t(\tau)=\int_{0}^{\tau}|L_{\alpha}(\tau^{\prime})|d\tau^{\prime}$ in Refs.
[45-47] and with heterogeneous memorized waiting times defined as
$D_{0+}^{1-\mu}t_{x}(\tau)=|x|^{\theta/\gamma}|L_{\alpha}(\tau)|$ in Ref. [48]
are different from that introduced in Ref. [21].

Let us turn to the subordinated process $Y(t)$. In Laplace space, the PDF
$W(x,t)$ can be represented by

$\widetilde{W}(x,u)=\int_{0}^{\infty}f(x,\tau)\widetilde{g}(\tau,u)d\tau.$
(36)

Inserting Eq.(33) into Eq.(36), we have

$\begin{array}[]{lll}\widetilde{W}(x,u)&=&\int_{0}^{\infty}f(x,\tau)\frac{\alpha}{\beta}\tau^{\alpha/\beta-1}u^{\alpha-1}\exp\\{-u^{\alpha}\tau^{\alpha/\beta}\\}d\tau\\\
\\\ &=&u^{\alpha-1}\int_{0}^{\infty}f(x,s^{\beta/\alpha})e^{-u^{\alpha}s}ds\\\
\\\ &=&u^{\alpha-1}\int_{0}^{\infty}f_{1}(x,s)e^{-u^{\alpha}s}ds\\\ \\\
&=&u^{\alpha-1}\widetilde{f}_{1}(x,u^{\alpha}),\end{array}$ (37)

where $f_{1}(x,\tau)=f(x,\tau^{\beta/\alpha})$. Obviously there exists the
relation

$\frac{\partial}{\partial\tau}f_{1}(x,\tau)=\frac{\partial
f(x,\tau_{1})}{\partial\tau_{1}}\cdot\frac{d\tau_{1}}{d\tau},$ (38)

where $\tau_{1}=\tau^{\beta/\alpha}$.

Since the PDF $f(x,\tau)$ satisfies Eq.(7), we have

$\begin{array}[]{lll}\frac{\partial}{\partial\tau}f_{1}(x,\tau)&=&\frac{\partial
f(x,\tau_{1})}{\partial\tau_{1}}\cdot\frac{d\tau_{1}}{d\tau}\\\ \\\
&=&L_{FP}f(x,\tau_{1})\cdot\frac{d\tau_{1}}{d\tau}\\\ \\\
&=&L_{FP}f_{1}(x,\tau)\frac{\beta}{\alpha}\tau^{\beta/\alpha-1}.\end{array}$
(39)

We Set $q(\tau)=\frac{\beta}{\alpha}\tau^{\beta/\alpha-1}$, whose Laplace
transform is
$\widetilde{q}(u)=\frac{\Gamma(\beta/\alpha+1)}{u^{\beta/\alpha}}$. Taking
Laplace transform for variable $\tau$ in Eq.(39), one gets

$\begin{array}[]{lll}u\widetilde{f}_{1}(x,u)-f_{1}(x,0)&=&\int_{0}^{\infty}L_{FP}f_{1}(x,\tau)q(\tau)e^{-u\tau}d\tau\\\
\\\ &=&\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}L_{FP}\widetilde{f}_{1}(x,s)\widetilde{q}(u-s)ds,\end{array}$
(40)

with $0<a<Re(u)$.

From Eq.(37), with the initial condition $W(x,0)=f_{1}(x,0)=\delta(x)$, we
have

$\begin{array}[]{lll}u\widetilde{W}(x,u)-W(x,0)&=&u^{\alpha}\widetilde{f}_{1}(x,u^{\alpha})-f_{1}(x,0)\\\
\\\ &=&\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}L_{FP}\widetilde{f}_{1}(x,s)\widetilde{q}(u^{\alpha}-s)ds\\\
\\\ &=&\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}L_{FP}\widetilde{W}(x,s^{1/\alpha})s^{\frac{1-\alpha}{\alpha}}\frac{\Gamma(\beta/\alpha+1)}{(u^{\alpha}-s)^{\beta/\alpha}}ds\\\
\\\ &=&\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}s^{\frac{1-\alpha}{\alpha}}\frac{\Gamma(\beta/\alpha+1)}{(u^{\alpha}-s)^{\beta/\alpha}}ds\int_{0}^{\infty}L_{FP}W(x,t^{\prime})e^{-s^{1/\alpha}t^{\prime}}dt^{\prime}.\end{array}$
(41)

Setting
$\widetilde{k}(u,s)=s^{\frac{1-\alpha}{\alpha}}\frac{\Gamma(\beta/\alpha+1)}{(u^{\alpha}-s)^{\beta/\alpha}}$
and taking inverse Laplace transform for $u\rightarrow t$ in Eq.(41), we
obtain

$\frac{\partial}{\partial t}W(x,t)=\Phi_{t}L_{FP}W(x,t),$ (42)

where

$\Phi_{t}h(t)=\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}k(t,s)ds\int_{0}^{\infty}h(t^{\prime})e^{-s^{1/\alpha}t^{\prime}}dt^{\prime}.$
(43)

Next we also compute the MSD $\langle x^{2}\rangle(t)$ of the subordinated
process $Y(t)=x(t^{\beta}Z^{-\beta})$ in a free force filed.

Since

$\langle x^{2}\rangle(t)=2\int_{0}^{\infty}\tau g(\tau,t)d\tau,$ (44)

we have

$\begin{array}[]{lll}\langle
x^{2}\rangle(u)&=&2\int_{0}^{\infty}\tau\widetilde{g}(\tau,u)d\tau\\\ \\\
&=&2\int_{0}^{\infty}\tau\frac{\alpha}{\beta}\tau^{\alpha/\beta-1}u^{\alpha-1}\exp\\{-u^{\alpha}\tau^{\alpha/\beta}\\}d\tau\\\
\\\ &=&2u^{\alpha-1}\int_{0}^{\infty}s^{\beta/\alpha}e^{-u^{\alpha}s}ds\\\ \\\
&=&\frac{2\Gamma(\frac{\beta+\alpha}{\alpha})}{u^{\beta+1}}.\end{array}$ (45)

So

$\langle
x^{2}\rangle(t)=\frac{2\Gamma(\beta/\alpha)}{\alpha\Gamma(\beta)}t^{\beta}.$
(46)

From Eq.(46), we conclude that the subordinated process is subdiffusive when
$0<\beta<1$, normal diffusive when $\beta=1$ and supdiffusive when
$1<\beta<2$.

##  5 Conclusions

In this work, we consider the subordinated process $Y(t)$ which is controlled
by a family of subordinators. For the case of $s(t)=t^{\alpha}Z^{-\alpha}$,
the PDF $W(x,t)$ of the subordinated process $Y(t)$ satisfies a time-
fractional Fokker-Planck equation. For the case of
$s(t)=t^{\beta}Z^{-\alpha}$, the corresponding diffusion equation is a
stretched time-fractional Fokker-Planck equation. For the case of
$s(t)=t^{\beta}Z^{-\beta}$, we obtain that the Laplace transform of the PDF
$g(\tau,t)$ is of the form
$\widetilde{g}(\tau,u)=\frac{\alpha}{\beta}\tau^{\alpha/\beta-1}u^{\alpha-1}\exp\\{-u^{\alpha}\tau^{\alpha/\beta}\\}$.
The resulting process can be connected with CTRW with correlated waiting time.

## Acknowledgments

This project was supported by the Natural Science Foundation of China (Grant
Nos. 11371016 and 11271311), the Chinese Program for Changjiang Scholars and
Innovative Research Team in University (PCSIRT) (Grant No. IRT1179), the Lotus
Scholars Program of Hunan province of China.

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