# DGoT: Dynamic Graph of Thoughts for Scientific Abstract Generation

###### Abstract

The method of training language models based on domain datasets has obtained
significant achievements in the task of generating scientific paper abstracts.
However, such models face problems of generalization and expensive training
costs. The use of large language models (LLMs) to solve the task of generating
paper abstracts saves the cost of model training. However, due to the
hallucination problem of LLM, it is often necessary to improve the reliability
of the results through multi-round query prompt approach such as Graph of
Thoughts (GoT), which also brings additional reasoning costs. In this paper,
we propose a Dynamic Graph of Thought (DGoT). It not only inherits the
advantages of the existing GoT prompt approach, but also dynamically adjust
the graph structure according to data characteristics while reducing model
reasoning cost. Experimental results show that our method’s cost-effectiveness
in abstract generation tasks is only 43.7% to 56.4% of other multi-round query
prompt approaches. Our code is available at https://github.com/JayceNing/DGoT.

Keywords: Abstract generation, large language models, prompt approaches

DGoT: Dynamic Graph of Thoughts for Scientific Abstract Generation

Xinyu Ning, Yutong Zhao, Yitong Liu††thanks:  Corresponding Author, Hongwen
Yang
---
School of Information and Communication Engineering
Beijing University of Posts and Telecommunications, China
{nxybupt, zhaoyutong, liuyitong<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

The abstract, as the essence of academic literature, aims to provide the core
ideas, methods, and results of research. The specific domain concepts and
professional terminology involved in scientific papers pose significant
challenges for researchers who wish to generate article abstracts through
automation. Nevertheless, the advancement of natural language processing (NLP)
technology has made it possible to accurately summarize articles and generate
abstracts through artificial intelligence (AI).

In pursuit of this target, previous methods usually involve collecting domain
data and training corresponding models to complete the task of text
summarization. The methods for generating scientific article abstracts can be
divided into two categories: not using citation information and using citation
information. For the former situation, Cohan et al. (2018) uses an RNN-based
encoder-decoder structural model to train abstract generation models
specifically tailored to two large-scale scientific paper datasets. Similarly,
Xiao and Carenini (2019) use LSTM-based models to fulfill abstract generation
tasks for scientific papers. However, some researchers contend that
incorporating citation information from articles can yield superior summary
outcomes. Yasunaga et al. (2019) integrated the original paper and its
citations, leveraging GCN and LSTM for crafting paper abstracts. The citation
graph-based model has also been adopted by other researchers An et al. (2021);
Luo et al. (2023), they respectively used LSTM and Transformer modules to form
an encoding and decoding model. Although their work has made significant
progress in the abstract generation of scientific papers, training models for
specific datasets often face generalization problems and also incur high
training costs.

In recent years, large language models (LLMs) have attracted the attention of
many researchers, such as GPT-3/4 Brown et al. (2020); Bubeck et al. (2023),
LLaMA Touvron et al. (2023), ChatGLM Du et al. (2021), or InternLM Team
(2023). The large language model exhibits strong generalization ability in
many natural language scenarios, which also brings new solutions to the task
of generating paper abstracts. Relying on an autoregressive token-based
mechanism by LLM, few-shot or zero-shot prompt engineering methods are used to
solve the target problem. Many prompt approaches have been proposed to
optimize the output results of LLM. Chain of Thought (CoT) Wei et al. (2022b)
improves the model’s ability to handle complex situations by adding problem-
solving reasoning processes to prompt words. Tree of Thoughts (ToT) Yao et al.
(2023) models the reasoning process of LLM using trees, enabling the model to
generate outcomes through multiple pathways and selecting the most favorable
ones via evaluators. Graph of Thoughts (GoT) Besta et al. (2023) integrates
the modeling of the above reasoning process using graphs, aggregating the
advantages of different paths on the basis of trees. However, existing GoT
approaches pre-define the number of edges, which means that the number of
conversations initiated with LLM is predetermined, possibly resulting in
needless resource consumption. To address this, the adaptive adjustment of the
graph structure is imperative to cater to specific task requirements.

In this paper, We propose a Dynamic Graph of Thought (DGoT) prompt method to
improve the quality of LLM-based generated literature abstracts while
minimizing the cost of using the model. Concretely, we divide the abstract
generation process based on LLM prompt approach into training process and
reasoning process. (See section 3 for details). Compared to the fixed graph
structure prompt method, our dynamic GoT has two main advantages: Firstly, our
approach inherits the advantages of GoT structure and improves the reasoning
performance of the model through effective prompt strategies. Secondly, our
approach evaluates the specified task during the training process and
dynamically determines the graph structure during the reasoning process,
improving the cost-effectiveness of the LLM model.

In addition, we propose different threshold settings, including simple mean
threshold and Gumbel threshold, to meet different requirements for the output
of GoT. Lower thresholds tend to degenerate the graph into a single path
during the reasoning process, while higher thresholds tend to maintain the
original structure of the graph.

We evaluated the proposed method on the PubMedCite Luo et al. (2023) dataset
for scientific literature abstract generation. Compared to other prompt
approaches based on multi-round query, our method achieved a two-fold
improvement in the cost-effectiveness metric. In addition, we verified the
consistency between the threshold setting and the resulting score, which can
provide a reference for users to choose the threshold when using DGoT-based
programs.

Our contributions can be summarized as follows:

  * •

We propose a Dynamic Graph of Thought (DGoT) method that improves the
performance of scientific literature abstract generation tasks while
minimizing the cost of large language models.

  * •

We defined a threshold setting method to guide the generation process of
dynamic graphs.

  * •

The experimental results on the PubMedCite dataset show that our method is
more cost-effective than other multi-round prompt approaches.

## 2\. Background & Notation

In this section, we introduce the impact of probability and specific prompt
approaches on the reasoning process of LLM. Additionally, we provide the
definition of symbols used in our discussion.

### 2.1. Probability in LLM

Following the established notation Yao et al. (2023), we use
$\textbf{X},\textbf{Y},\textbf{Z},...$ to denote input sentence. Each sentence
is consist of $n$ tokens, i.e. $\textbf{X}=[x_{1},x_{2},...,x_{n}]$. The
language model obtains the probability of the $i_{th}$ token through the
previous $i-1$ tokens, and samples according to the probability to obtain the
$i_{th}$ token:

$x_{i}\sim p_{\theta}(x_{i}|x_{1},x_{2},...,x_{i}-1)$ (1)

Where $\theta$ denotes the language model parameters, and $p_{\theta}$
represents the probability of the model predicting the $i_{th}$ token. The
process iterates until the model’s output corresponds to the stop token
Martins et al. (2020).

The probability distribution over target tokens $p_{\theta}$ is calculated by
softmax function using the output vector $v_{i}$ of the model Radford and
Narasimhan (2018), i.e. Transformer Vaswani et al. (2017). Using temperature
$T$ to adjust the probability distribution Holtzman et al. (2020):

$p_{\theta}(x_{i}|x_{1:i-1})=\mathrm{softmax}(v_{i})=\frac{exp(v_{i}/T)}{\sum_{i^{{}^{\prime}}}exp(v^{{}^{\prime}}_{i}/T)}$
(2)

Here, $T$ is confined to the interval $[0,1]$, and the larger $T$ is, the
stronger the randomness of the model output results, which lays the foundation
for the prompt approaches that obtain better results through multiple rounds
of questioning later.

### 2.2. Prompt approaches

We formalize the existing prompt methods as the baseline for comparison:

$\textbf{Y}\sim P_{\theta}(\textbf{Y}|f_{i}(\textbf{X},\mathrm{prompt_{i}}))$
(3)

Where $P_{\theta}$ denotes the language model, X is the basic information, and
Y is the model’s output. $\mathrm{prompt}$ signifies the text content of the
prompt word, which is combined with X using function $f$ to form the input
sequence.

#### 2.2.1. Input-Output (IO)

Perform one round of query directly to the LLM to obtain the output
$\textbf{Y}\sim
P_{\theta}(\textbf{Y}|f_{IO}(\textbf{X},\mathrm{prompt_{IO}}))$.

#### 2.2.2. Chain-of-Thought (CoT)

By refining the reasoning process, better output than IO can be obtained Wei
et al. (2022b). One implementation solution is to guide LLM in step-by-step
reasoning in prompt text of $\textbf{Y}\sim
P_{\theta}(\textbf{Y}|f_{CoT}(\textbf{X},\mathrm{prompt_{CoT}}))$.

#### 2.2.3. Tree-of-Thought (ToT)

ToT explicitly decomposes the reasoning process for specific problems and
models the reasoning process of the model as a tree structure Yao et al.
(2023). Each node in this tree encapsulates a state $s=[x,z_{1,...,i}]$, which
includes the input $x$ and output $z_{i}$ up to that point. It employs a
generator $G(p_{\theta},s,k)$ to initiate multiple rounds of questioning on
LLM to generate $k$ output nodes. The output results of each step of the model
are obtained by using CoT’s prompt sampling $z_{i}\sim
P_{\theta}(z_{i}|f_{CoT}([x,z_{1,...,i-1}],\mathrm{prompt_{CoT}}))$. These
output results are then evaluated using an evaluator
$\varepsilon(p_{\theta},S)$ and the tree structure is expanded through BFS or
DFS methods for better results.

#### 2.2.4. Graph-of-Thought (GoT)

Compared to tree structures, GoT further expands the universality of the
reasoning process by modeling it as a graph Besta et al. (2023). GoT is
characterized by a tuple $(G,\mathcal{T},\varepsilon,\mathcal{R})$ to
represent its core components. $G=(V,E)$ represents the reasoning process,
where $V$ corresponds to nodes, akin to the states $s$ in ToT, and $E$
signifies edges, denoting the relationships between different nodes.
$\mathcal{T}(G,p_{\theta})$ encompasses transformations, including
aggregation, refinement, and generation. $\varepsilon(v,G,p_{\theta})$ refers
to an evaluator, tasked with scoring a single thought $v$.
$\mathcal{R}(G,p_{\theta},h)$ is responsible for ranking the top $h$ thoughts.
Notably, this differs from ToT as GoT introduces the possibility of
aggregating the results from different paths, thereby integrating the
strengths of various thoughts. This enhancement boosts the efficiency of
prompt approaches.

GoT integrates the previous prompt approaches and can be degraded to other
methods under certain conditions. The existing GoT-based graph design methods
fix the graph structure before reasoning. For specific tasks, if the graph
structure is not complex enough, it may not achieve the expected effect, while
overly complex graphs may bring additional overhead. Therefore, this method
mainly aims to design a dynamic GoT for abstract generate tasks, which reduces
reasoning costs as much as possible while using graph structures to enhance
effectiveness.

## 3\. Method

Figure 1: The overall process of our method, including the training process
and reasoning process.

### 3.1. Procedure Overview

Human cognitive researchers believe that there are two modes of people’s
participation in decision-making — fast, automatic, unconscious mode (System
1), and slow, thoughtful, and conscious mode (System 2) Sloman (1996). This
happens to be the same way of thinking as the current approach based on large
language models, where unconscious text is output through LLM and thoughtful
results are obtained through prompt approaches.

Humans develop specific thinking patterns tailored to particular tasks to
enhance their problem-solving efficiency. The formation of human thinking
depends on their environment and the knowledge they have acquired. Therefore,
for the structure of the mind map in the prompt approaches, it is necessary to
train on a particular task to obtain the optimal map structure.

Fig. 1 shows the overall framework of our dynamic GoT, including the training
process and reasoning process. In the training stage, we fix the GoT
structure, that is, pre-define the number of nodes and edge connections in
$G(V,E)$. Utilizing the training data as input for the graph, we evaluate the
scores of each node in the process through $\varepsilon(p_{\theta},S)$. Based
on the different transformations $\mathcal{T}(G,p_{\theta})$ used, we divide
nodes into three categories: generating nodes, aggregating nodes, and
improving nodes, and record the scores of these three types of nodes
separately. Ultimately, we derive score distribution plots for the three
transformations throughout the training process. We then compute the
statistical characteristics of these distributions to establish a threshold.

During the inference stage, we initiate inquiries to LLM (transformations) in
sequence according to the graph structure. For three different types of
transformations, if the score of a query is greater than the threshold
obtained during the training process, the transformation is terminated.

### 3.2. Training Process

For three different types of transformations $\mathcal{T}$, we design the
process and text of the prompt separately. Consider the example of generation
transformations, which can be represented as:

$\mathcal{T}(G,p_{\theta})=\textbf{Y}\sim
P_{\theta}(\textbf{Y}|f_{Gen}(\textbf{X},\mathrm{prompt_{Gen}}))$ (4)

Here, $\textbf{X}=[x_{1},x_{2},...,x_{n}]$ represent basic information of the
article. Following SSN dataset format An et al. (2021), we define basic
information of original and reference articles, which is shown in Table 1.

Category |  Basic Information
---|---
Original Article |  Title
Abstract
Introduction
Other Section
Reference |  Title
Abstract
Table 1: Basic information of the article Figure 2: An example of a prompt
framework.

$\mathrm{prompt_{Gen}}$ is the prompt text of generation transformations, and
$f_{Gen}$ is the prompt framework. An example is shown in Fig. 2 (See Appendix
A.2 for other prompt frameworks).

$G=(V,E)$ contains the number of times $k$ that the LLM is queried for each
transformation process. The generation transformation, aggregation
transformation, and improving transformation are arranged in order to form the
basic structure of GoT.

For each node $V$ in $G$, it includes the result $v$ returned by LLM under one
transformation. Referring to Luo et al. (2023), we use ROUGE Lin and Hovy
(2003) as $\varepsilon(p_{\theta},S)$ to score generated abstract $v$. Due to
the fact that the true abstract should be unknown to the model during the
reasoning process, the ROUGE score is calculated using the generated abstract
and the introduction of the original article. Therefore, during the training
process, we also need to record the ROUGE scores in the generated summary and
original article introduction sections of the training data. We obtain the
score distribution plots of generation transformation, aggregation
transformation, and improving transformation, and calculate their statistical
characteristics as thresholds. For specific statistical parameters, we define
them as follows.

#### 3.2.1. Simple Mean Threshold

Simply use the mean of three transformation scores as the threshold.

$\mathrm{Thresh}_{\mathrm{Simple}}=\mu_{\mathrm{socre}}$ (5)

#### 3.2.2. Gumbel Threshold

The graph-based prompt method obtains the highest scoring result through
k-round questioning. According to generalized extreme value (GEV) distribution
Resnick (1987), When the number of samples is large enough, Gumbel
distribution can be used to model the distribution of maximum sampling values.
Its PDF is:

$p(x)=\frac{1}{\beta}e^{-(z+e^{-z})},\ \ where\ \ z=\frac{x-\mu}{\beta}$ (6)

During the training phase, the maximum value of the results obtained from each
transformation is recorded to estimate its distribution. We calculate its mean
$\mu_{Max}$ and variance $\sigma^{2}$, maximum likelihood estimation for $\mu$
and $\beta$ in the above equation.

$\beta^{2}=\frac{6\sigma^{2}}{\pi^{2}}$ (7)

$\mu=\mu_{max}-\gamma\beta$ (8)

Where $\gamma\approx 0.577215$ is Euler-Mascheroni constant.

The CDF of Gumbel distribution is:

$F(x;\mu,\beta)=e^{-e^{-(x-\mu)/\beta}}$ (9)

In the reasoning process, we define the Gumbel threshold as assuming that the
maximum value has been reached with a confidence level of
$p_{\mathrm{Thresh}}$. That is, $p_{\mathrm{Thresh}}=e^{-e^{-(x-\mu)/\beta}}$.
The corresponding score threshold can be solved as:

$\mathrm{Thresh}_{\mathrm{Gumbel}}=\mu-\beta\mathrm{ln}(-\mathrm{ln}p_{\mathrm{Thresh}})$
(10)

### 3.3. Reasoning Process

In the reasoning process, we initialize the graph structure first. We have
extended the functionality of the original GoT by adding Dynamic Generate and
Dynamic Aggregate modules.

Dynamic Generate module directly scores the results after each inquiry with
LLM, represented by $\mathcal{T}_{\mathrm{DG}}(G,p_{\theta},H)$, where
$H=[H_{\mathrm{Simple}},H_{\mathrm{Gumbel}}]$ represents the function used to
map the threshold. Corresponding to the two threshold setting methods in
section 3.2. $H_{\mathrm{Gumbel}}(p)$ receives the confidence probability
$p_{\mathrm{Thresh}}$ of the input parameter to calculate the gumbel
threshold. The generation transformation repeatedly queries the LLM before the
score reaches the set threshold, until the number of queries reaches $k$.

Dynamic Aggregate module dynamically determines whether aggregation
transformation is needed and the number of times the transformation is
executed, denoted by $\mathcal{T}_{\mathrm{DA}}(G,p_{\theta},H)$. Due to
threshold settings, the previous step of this module may only retain one idea
$v$. In that case, this step is skipped directly. The improving transformation
is the same as the generation transformation, denoted by
$\mathcal{T}_{\mathrm{DI}}(G,p_{\theta},H)$, except for the prompt text used.

Ranking module is retained from the original modules of GoT, denoted by
$\mathcal{R}(G,p_{\theta},h)$, to preserve the best $h$ results after the
generating module.

After the initialization of the graph structure, reasoning modules will be
performed in order to obtain the final result.

## 4\. Experiments

### 4.1. Setup

#### 4.1.1. Datasets

To evaluate the effectiveness of our method, we conduct experiments on
PubMedCite datasets Luo et al. (2023). Due to the license of the dataset, we
did not obtain the original dataset of the author but downloaded the
corresponding paper through the official API of
PubMed111https://pubmed.ncbi.nlm.nih.gov/download/. We organized the data
according to the format in Table 1. Finally, 10,000 original literature and
reference training data pairs were obtained under the Inductive setting in
PubMedCite, as well as 5224 testing data pairs.

For the test dataset, the mean ROUGE scores of the abstract and introduction
parts of the source article are shown in the table 2. This ROUGE score can
serve as a reference for the highest achievable score when comparing the
generated results with the original paper introduction (In fact, the
summarization ability of LLM is higher than this score).

| R-1 | R-2 | R-L
---|---|---|---
Score | 0.332 | 0.132 | 0.164
Table 2: Comparison of ROUGE scores between the original abstract and
introduction sections of the test dataset.

#### 4.1.2. Baselines

We compare our method with other prompt methods, including IO, CoT, ToT, and
GoT. Based on the fairness principle, the tree and graph methods will initiate
the same number of LLM queries. Specifically, we set the branching factor
$k=3$ and the number of levels $L=3$ for ToT. The graph method arranges 3
different types of transformations — the generation transformation,
aggregation transformation, and improving transformation in sequence, each
with $E=3$ edges.

#### 4.1.3. Implementation Details

Our method is implemented by Python and Pytorch. Our baseline references the
design patterns of the corresponding prompt in
GoT222https://github.com/spcl/graph-of-thoughts. The specific design of DGoT
is extended based on the GoT code. We use ChatGLM2 Du et al. (2021) as LLM for
inference, and to our knowledge, ChatGLM does not use PubMedCite as training
data. We deployed the ChatGLM2-6B333https://github.com/THUDM/ChatGLM2-6B model
on three 24G 3090 GPUs and four 32G v100 GPUs respectively for local test.
During the training and reasoning process, we did not update the model
parameters but only deployed the model for inference, so the graphics memory
would become a bottleneck in the input length. Therefore, we limit the input
length to 20000 tokens, and any excess will be truncated. Both the top $p$ and
temperature $\mathrm{T}$ of the model are fixed at 0.7.

### 4.2. Training Data Distribution

Figure 3: Score distribution of training data. The dashed line represents the
mean of the corresponding transformation score. Figure 4: The distribution of
the maximum score for each transformation. The solid line represents the
kernel density estimation curve, while the dashed line represents the Gumbel
distribution curve estimated according to the method in 3.2.2. According to
Equ. 10, taking the generation transformation as an example, the corresponding
scores for confidence levels of 25%, 50%, and 75% are calculated.

As shown in Fig. 3, we plot the score distribution of training data under
three transformations. For the corresponding transformation, the solid line is
the kernel density estimation curve, while the dashed line is the mean score,
where the generation transformation is slightly larger than the other two
types. Fig. 4 shows the distribution of the maximum score for each
transformation. Since our graph structure has three edges for each
transformation, the maximum value is the highest score among the three
answers. According to the method in Section 3.2.2, the Gumbel distribution
calculated is represented by dashed lines. It can be observed that the Gumbel
distribution can fit well with the distribution of maximum values for each
transformation. In the reasoning process, the Gumbel threshold calculated from
the distribution is used as the hyper-parameter of GoT.

### 4.3. Main Results

Figure 5: Scores and costs of abstract generated by ChatGLM2-6B under different prompt approaches. Method |  R-1 |  R-2 |  R-L |  Prompt Tokens |  Response Tokens |  Cost |  Cost-effectiveness
---|---|---|---|---|---|---|---
IO |  0.303 |  0.081 |  0.166 |  10660.79 |  402.79 |  0.0167 |
CoT |  0.314 |  0.083 |  0.171 |  10644.81 |  358.77 |  0.0166 |
ToT |  0.356(0.042) |  0.098 |  0.190 |  82850.63 |  2606.48 |  0.1294 (0.1128) |  2.686
GoT |  0.354(0.040) |  0.099 |  0.190 |  99184.15 |  3219.40 |  0.1552 (0.1386) |  3.465
DGoT |  0.358(0.044) |  0.099 |  0.192 |  53414.97 |  1565.12 |  0.0833 (0.0667) |  1.516
Table 3: Main Results. R-1, R-2, and R-L represent the ROUGE scores of the
generated abstract and the actual abstract of the source articles
respectively. Prompt Tokens is the average number of tokens input to LLM
throughout the entire process of the method, while Response Tokens is the
average number of tokens returned by LLM. Cost is the cost corresponding to
the number of tokens. Here, we calculate the price of the local model based on
that setting of chatgpt-3.5 ($1.5/1M input tokens, $2/1M output tokens).

Table 3 compares the ROUGE scores and costs of our method with other prompt
approaches. The 5 approaches involved were validated on 5224 PubMedCite test
data we processed, among which the threshold setting method for DGoT is Simple
Mean Threshold. Since both IO and CoT only make one query to LLM, we compare
the results of ToT, GoT, and DGoT with the best of the two methods. It can be
seen that the multi-round query approach has significant performance
improvement compared to single-round query.

In Figure 5, it can be visually observed that DGoT has lower costs compared to
ToT or GoT. Additionally, DGoT demonstrates superior cost-effectiveness.

Cost-effectiveness is defined as the cost required to improve the performance
of a unit metric compared to a baseline method. Therefore, the smaller the
cost-effectiveness, the better. E.g. The values in the parentheses to the
right of cost in Table 3 (representing the additional cost required for multi-
round query approaches compared to the best single-round query method CoT)
divided by the values in the parentheses to the right of R-1 (representing the
improvement in R-1 score compared to CoT) yield the cost-effectiveness.
Compared to fixed tree or graph structures, our method has significant cost
advantages.

### 4.4. Effects of threshold function $H$ setting

Figure 6: The effect of different threshold settings on output score and cost.
Each associated with two box plots showing the ROUGE R-1 scores of the
generated abstract compared to the original introduction and the actual
abstract from left to right.

Table 4 compares the results under different threshold settings. This
experiment was validated on 100 papers in the test set. Since the scores
compared with the threshold are calculated in the generated abstract and
introduction sections of the original document, as the threshold increases,
the output result’s Intro.R-1 score continues to increase. However, the Abst.
ROUGE score compared with the actual abstract does not show a completely
linear growth trend.

Figure 6 shows that as the threshold increases, the required costs continue to
rise, but the rate of increase gradually decreases.

Setting |  Intro. R-1 |  Abst. R-1 |  Abst. R-2 |  Abst. R-L |  Cost
---|---|---|---|---|---
Simple |  0.303 |  0.313 |  0.075 |  0.176 |  0.0841
25% |  0.305 |  0.309 |  0.070 |  0.173 |  0.1075
50% |  0.314 |  0.309 |  0.075 |  0.170 |  0.1161
75% |  0.317 |  0.314 |  0.078 |  0.175 |  0.1186
Table 4: ROUGE scores and cost for different threshold settings. From top to
bottom, it represents simple threshold settings, as well as Gumbel threshold
settings with confidence levels of 25%, 50%, and 75%.

## 5\. Related Work

### 5.1. Large Language Models

In recent years, the scale of language models has been continuously increasing
Devlin et al. (2019); Radford and Narasimhan (2018); Brown et al. (2020);
OpenAI (2023), and they have also shown significant improvements in
performance in various downstream tasks, known as emergent capabilities Wei et
al. (2022a). The open-source large language model is becoming a new trend in
the field of artificial intelligence. Recent work includes LLaMA Touvron et
al. (2023), RWKV Peng et al. (2023), OpenFlamingo Awadalla et al. (2023),
GLM-130B Zeng et al. (2023) etc. Based on our method, the ceiling performance
of locally deployed open-source models on the task of paper abstracting can be
explored.

### 5.2. Prompt Paradigms

In section 2.2, we introduced some mainstream prompt methods Wei et al.
(2022b); Yao et al. (2023); Besta et al. (2023). From the perspective of graph
theory, a path is a sequence of vertices in a graph, and a tree is an acyclic-
connected graph. Therefore, under certain conditions, GoT can degenerate into
ToT or CoT. Other prompt approaches include Self-consistent CoT Wang et al.
(2023), which samples a set of different reasoning paths and then selects the
most consistent answer.

Cumulative Reasoning Zhang et al. (2023) decomposes the reasoning process into
three components: proposer, verifier, and reporter, and uses an accumulative
and iterative approach to reasoning. Chain-of-Verification Dhuliawala et al.
(2023) drafts verification questions to promote model optimization of the
initial output results. Due to the scalability of the GoT system architecture,
the above methods can be incorporated into the system by adding functional
modules. In this paper, the final graph form may be a path or a tree by
dynamically adjusting the GoT graph structure, in order to minimize the
reasoning cost while completing scientific paper abstract generation.

### 5.3. LLM in Scientific Research

Traditional AI-assisted scientific paper tasks typically involve training
models on data of designated domains. The large language model can complete
downstream tasks with few or zero-shot approaches, so its application in
completing scientific research tasks has attracted increasing attention
Birhane et al. (2023); Boiko et al. (2023); Bran et al. (2023); Liang et al.
(2023). Large language models trained on scientific paper datasets include
Galactica Taylor et al. (2022), Darwin Xie et al. (2023) and Mozi Lan et al.
(2023) etc. This article has attempted to evaluate the performance of the
universal open-source large model in the task of paper abstracts, and related
evaluation tests can be conducted on the scientific article large model in the
future.

## 6\. Conclusion

In this work, we propose a Dynamic Graph of Thought prompt approach that can
adaptively adjust the graph structure during the reasoning process to reduce
the language model cost. We define a threshold-setting mechanism for the GoT
evaluation function to provide a reference for the trade-off between
performance and cost. Our experiments show that on the task of scientific
literature abstract generation, this method achieves the best cost-
effectiveness compared to other multi-round prompt approaches.

## Acknowledgements

This work was supported by National Demonstration Center for Experimental
Electronic Information Education (Beijing University of Posts and
Telecommunications) and computing resources are supported by the High-
performance Computing Platform of BUPT.

## Bibliographical References

##

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##  Appendix A Appendix

### A.1. Other Potential Influencing Factors on the Results

To investigate the effects of prompt length, Branching Factors, and different
models on experimental outcomes, we conducted validations on the initial 100
records from both the training and testing sets of PubMedCite. These
validations were conducted on a server equipped with a 24G 3090 graphics card.
For ChatGLM2-6B, we deployed it using the same API approach as described in
Section 4.1.3 of the main text. For
InternLM2-Chat-7B444https://github.com/InternLM/InternLM, we utilized
LMDeploy555https://github.com/InternLM/lmdeploy to accelerate its inference
process. We also provided the inference duration for each method under each
model for time comparison. The pertinent code is available in our GitHub
repository666https://github.com/JayceNing/DGoT, along with a readily
deployable Docker image.

#### A.1.1. Effect of Prompt Length

Figure 7: Scores and costs of abstract generated by ChatGLM2 using GoT prompt
approach under various prompt length settings. Figure 8: Scores and costs of
abstract generated by InternLM2-chat-7B using GoT prompt approach under
various prompt length settings.

In Section 4.1.3, longer input texts are truncated before inputting the model.
In order to explore the influence of prompt length (Includes prompt text
within the prompt framework, along with the corresponding filled-in
information) on the result, we adopt the same GoT setting as Section 4.1.2 to
validate the two models.

For prompt length’s effect on R-1 scores, Figure 7 highlights ChatGLM2’s peak
R-1 score at input length 2048, while Figure 8 shows InternLM2’s peak at
length 4096. Both models exhibit an initial increase followed by a decrease in
R-1 scores as prompt length increases.

Table 5 lists the same result metrics as in Section 4.3, in addition to the
proportion of truncated input text and the inference time for each prompt
length. Longer input leads to higher inference time and cost. The rate of
input truncation is not linearly correlated with the resulting scores.
Different models exhibit varied inference performance, with ChatGLM2 excelling
at retaining information from the introduction, while InternLM2 demonstrates
stronger abstract generation capabilities.

#### A.1.2. Effect of Branching Factors

In A.1.1, InternLM2 demonstrates its best abstract generation capability when
the prompt length is 4096. Under this condition, we test the impact of
different branching factors $k$ on the results.

Figure 10 demonstrates that as the branching factor $k$ increases, the R-1
scores between the output and the original introduction gradually improve.
This is because the scoring function is designed to select responses with the
highest R-1 scores compared to the introduction. However, there is no
improvement in R-1 scores between the output and the actual abstract.

Table 6 shows that the Introduction R-1 scores gradually increase as the
branching factor grows, but the rate of increase diminishes. This is reflected
in the rise of Cost-effectiveness, indicating that the cost required to
improve each unit score becomes increasingly higher. This to some extent
demonstrates that the number of Agents also adheres to Scaling Laws Kaplan et
al. (2020), but the performance gain it brings is not limitless.

Figure 9 provides a more intuitive presentation of this conclusion. Its
horizontal axis represents the

Model/ Method |  Prompt Length |  Cut Ratio |  Intro. R-1 |  Abst. R-1 |  Abst. R-2 |  Abst. R-L |  Prompt Tokens |  Resp. Tokens |  Infer. Time(s) |  Cost
---|---|---|---|---|---|---|---|---|---|---
Chat- |  256 |  0.993 |  0.339 |  0.367 |  0.091 |  0.189 |  2623.29 |  2062.74 |  76.61 |  0.008
GLM2/ |  512 |  0.973 |  0.456 |  0.400 |  0.111 |  0.192 |  5360.43 |  2613.06 |  96.05 |  0.013
GoT |  1024 |  0.852 |  0.517 |  0.431 |  0.144 |  0.216 |  12881.78 |  3545.09 |  126.69 |  0.026
|  2048 |  0.775 |  0.520 |  0.435 |  0.154 |  0.221 |  23189.33 |  3223.30 |  120.00 |  0.041
|  4096 |  0.693 |  0.510 |  0.391 |  0.132 |  0.203 |  40675.92 |  3979.97 |  154.09 |  0.068
|  8192 |  0.465 |  0.436 |  0.366 |  0.104 |  0.191 |  70249.50 |  5711.58 |  282.49 |  0.116
Intern- |  256 |  0.993 |  0.317 |  0.368 |  0.097 |  0.187 |  3368.73 |  1440.99 |  37.16 |  0.007
LM2/ |  512 |  0.973 |  0.450 |  0.418 |  0.125 |  0.200 |  5949.60 |  1892.68 |  47.75 |  0.012
GoT |  1024 |  0.830 |  0.418 |  0.456 |  0.164 |  0.235 |  13642.57 |  1894.43 |  48.89 |  0.024
|  2048 |  0.740 |  0.447 |  0.471 |  0.176 |  0.240 |  23849.84 |  1965.58 |  53.24 |  0.039
|  4096 |  0.670 |  0.447 |  0.482 |  0.190 |  0.259 |  39069.64 |  2139.46 |  60.76 |  0.062
|  8192 |  0.436 |  0.422 |  0.479 |  0.183 |  0.250 |  65586.77 |  2049.10 |  72.41 |  0.102
Table 5: Effect of Prompt Length. The performance of ChatGLM2 and InternLM2 models using GoT as prompt approach was tested under different input prompt lengths. If the length of the input exceeds the specified Prompt Length, the input will be truncated. The Cut Ratio indicates the proportion of the input to be truncated. Intro. R-1 represents the ROUGE scores of the generated abstract and the introduction of the source articles. Abst. R-1, Abst. R-2, and Abst. R-L represent the ROUGE scores of the generated abstract and the actual abstract of the source articles respectively. Prompt Tokens is the average number of tokens input to LLM throughout the entire process of the method, while Resp. Tokens is the average number of tokens returned by LLM. Infer. Time refers to the time required for a specific model to generate the abstract of a research paper using GoT method. Cost is the cost corresponding to the number of tokens. Here, we calculate the price of the local model based on that setting of chatgpt-3.5 ($1.5/1M input tokens, $2/1M output tokens). $k$ |  Introduction R-1 |  Abst. R-1 |  Abst. R-2 |  Abst. R-L |  Prompt Tokens |  Resp. Tokens |  Infer. Time(s) |  Cost |  C/E
---|---|---|---|---|---|---|---|---|---
3 |  0.448 |  0.481 |  0.191 |  0.264 |  39940.46 |  2088.17 |  60.86 |  0.064 |
6 |  0.492(0.044) |  0.480 |  0.187 |  0.250 |  80091.36 |  4360.62 |  122.64 |  0.128(0.064) |  1.454
9 |  0.507(0.059) |  0.481 |  0.188 |  0.255 |  120196.65 |  6761.61 |  187.29 |  0.193(0.129) |  2.186
12 |  0.524(0.076) |  0.475 |  0.190 |  0.255 |  160543.10 |  9139.17 |  251.58 |  0.259(0.195) |  2.565
15 |  0.530(0.082) |  0.478 |  0.187 |  0.251 |  200225.20 |  11821.69 |  322.74 |  0.323(0.259) |  3.158
Table 6: Effect of Branching Factors $k$. C/E represents Cost-effectiveness
(see section 4.3). Figure 9: The effect of different branching factors $k$
settings on output Intro. R-1 score.

Intro. R-1 scores. The purple line shows the Gumbel distribution curve, and
the gray dashed line marks the mean position. As the branching factor rises,
the increase in average output score diminishes.

#### A.1.3. Results under Optimal Prompt Length

Model |  Meth- od |  Introduction R-1 |  Abst. R-1 |  Abst. R-2 |  Abst. R-L |  Prompt Tokens |  Resp. Tokens |  Infer. Time(s) |  Cost |  C/E
---|---|---|---|---|---|---|---|---|---|---
Chat- |  IO |  0.311 |  0.387 |  0.126 |  0.204 |  2274.14 |  233.51 |  10.53 |  0.003 |
GLM2 |  CoT |  0.305 |  0.401 |  0.129 |  0.213 |  2269.77 |  214.12 |  9.84 |  0.003 |
|  ToT |  0.476(0.171) |  0.390 |  0.130 |  0.199 |  20465.34 |  2376.15 |  96.11 |  0.035(0.032) |  0.187
|  GoT |  0.475(0.170) |  0.382 |  0.128 |  0.196 |  20409.60 |  2442.31 |  97.25 |  0.035(0.032) |  0.188
|  DGoT |  0.418(0.113) |  0.395 |  0.129 |  0.199 |  10602.23 |  1256.39 |  55.19 |  0.018(0.015) |  0.132
Intern- |  IO |  0.317 |  0.439 |  0.164 |  0.242 |  4420.49 |  239.16 |  8.14 |  0.007 |
LM2 |  CoT |  0.279 |  0.436 |  0.158 |  0.237 |  4417.75 |  195.71 |  7.19 |  0.007 |
|  ToT |  0.477(0.198) |  0.414 |  0.148 |  0.212 |  39812.07 |  2241.35 |  67.43 |  0.064(0.057) |  0.287
|  GoT |  0.456(0.177) |  0.419 |  0.156 |  0.220 |  39732.42 |  2225.52 |  67.26 |  0.064(0.057) |  0.322
|  DGoT |  0.399(0.120) |  0.422 |  0.152 |  0.222 |  19690.67 |  1016.34 |  33.78 |  0.031(0.024) |  0.200
Table 7: Results under Optimal Prompt Length. Figure 10: The effect of
different branching factors $k$ settings on output score and cost. Each $k$
setting corresponds to two box plots, representing the ROUGE R-1 scores of the
generated abstract compared to the original literature introduction and the
actual abstract from left to right.

Section A.1.1 Experimental results show that the optimal prompt length for the
ChatGLM2 model is 2048, while for InternLM2 it is 4096. Section A.1.2
demonstrates that the optimal branching factor $k$ is 3. Under this
experimental configuration, the performance of both models is tested using
different prompt approaches. The dataset used for testing consists of the
first 100 entries in the test set, with DGoT threshold set to Simple Mean
Threshold.

Figure 11 and 12 present the experimental results of the two models on a small
test set, which roughly align with the trends discussed in Section 4.3.
Additionally, here we provide the R-1 scores of the output results compared to
the original text introductions. It can be observed that, compared to single-
round query method, multi-round query approach generally shows improvement in
this score.

Figure 11: Scores and costs of abstract generated by ChatGLM2-6B under
different prompt approaches. Figure 12: Scores and costs of abstract generated
by InternLM2-Chat-7B under different prompt approaches.

Table 7 presents more detailed results. The highest ROUGE scores obtained from
different prompting approaches within the two major categories of single-round
query and multi-round query are highlighted in bold. For different models,
using IO or CoT for single-round query yields varying results. InternLM2
performs better with IO, while ChatGLM2 performs better with CoT. DGoT, as a
multi-round query approach, generally demonstrates good performance in ROUGE
scores for generating abstract compared to the original abstract, with
superior Cost-effectiveness.

Nevertheless, the performance of Intro. R-1 of ToT is better than that of GoT
in the current multi-round query approach, which indicates that aggregation
transformation under the current prompt word framework does not bring
performance improvement compared with generation transformation. In addition,
the increase in Intro. R-1 scores was accompanied by a decrease in Abst. R-1
scores. This indicates that there is a tradeoff between these two evaluation
indicators. Finding a suitable evaluation function to improve the output of
the prompt method is still a problem worth studying.

### A.2. Prompt Framework for Transformations

In the main text, Figure 2 shows the prompt framework for generation
transformation. Here, Figures 13 and 14 show the prompt frameworks for
aggregation transformation and improving transformation, respectively.

Figure 13: Prompt framework for Aggregation Transformation. Figure 14: Prompt
framework for Improving Transformation.


# Redshift Evolution of the H2/HI Mass Ratio In Galaxies

Laura Morselli1,2, A. Renzini2, A. Enia3,4, G. Rodighiero1,2

1 Dipartimento di Fisica e Astronomia, Università di Padova, vicolo
dell’Osservatorio 3, I-35122 Padova, Italy
2 INAF $-$ Osservatorio Astrofisico di Padova, vicolo dell’Osservatorio 5,
I-35122 Padova, Italy
3 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti
93/2, I-40129, Bologna, Italy
4 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, Via Gobetti 93/3,
I-40129, Bologna, Italy E-mail<EMAIL_ADDRESS>
(Accepted 2021 January 18. Received 2021 January 12; in original form 2020
December 11)

###### Abstract

In this paper we present an attempt to estimate the redshift evolution of the
molecular to neutral gas mass ratio within galaxies (at fixed stellar mass).
For a sample of five nearby grand design spirals located on the Main Sequence
(MS) of star forming galaxies, we exploit maps at 500 pc resolution of stellar
mass and star formation rate ($M_{\star}$ and SFR). For the same cells, we
also have estimates of the neutral ($M_{\rm HI}$) and molecular ($M_{\rm
H_{2}}$) gas masses. To compute the redshift evolution we exploit two
relations: i) one between the molecular-to-neutral mass ratio and the total
gas mass ($M_{\rm gas}$), whose scatter shows a strong dependence with the
distance from the spatially resolved MS, and ii) the one between
$\log(M_{\rm{H_{2}}}/M_{\star})$ and $\log(M_{\rm{HI}}/M_{\star})$. For both
methods, we find that $M_{\rm H_{2}}$/$M_{\rm HI}$ within the optical radius
slightly decreases with redshift, contrary to common expectations of galaxies
becoming progressively more dominated by molecular hydrogen at high redshifts.
We discuss possible implications of this trend on our understanding of the
internal working of high redshift galaxies.

###### keywords:

galaxies: evolution – galaxies: star formation – galaxies: spirals

††pubyear: 2021††pagerange: Redshift Evolution of the H2/HI Mass Ratio In
Galaxies–References

##  1 Introduction

Our understanding of galaxy formation and evolution is strictly connected to
the accretion of cold gas on galaxies across cosmic time: this gas coming from
the cosmic web cools down to form atomic hydrogen (HI) first, and then
molecular hydrogen (H2), that can eventually collapse under gravitational
instability to form new stars. Feedback from star formation also plays a
crucial role, as it is a necessary ingredient to ensure a low efficiency of
the star formation process itself: without feedback the gas in a galaxy would
be consumed almost completely over a free-fall time, turning most baryons into
stars, as opposed to the $\sim 10$ per cent of baryons being locked into stars
as actually observed in the local Universe (e.g. Bigiel et al., 2008; Krumholz
et al., 2012; Hayward & Hopkins, 2017). Feedback from star formation includes
photo-dissociation of H2 into HI due to the radiation emitted by young stars
(e.g. Allen et al., 2004; Sternberg et al., 2014). Therefore, HI is not only
an intermediate gas phase towards star formation, but also one of its
products, and it is key in establishing the self-regulating nature of the star
formation process. Unfortunately, till now our knowledge of the HI content in
individual galaxies is restricted to the low redshift Universe, where HI is
detected in emission via the 21cm line. Several surveys have targeted HI in
galaxies at $z<0.05$: HIPASS (Meyer et al., 2004), ALFALFA (Giovanelli et al.,
2005), xGASS (Catinella et al., 2018), HI-MaNGA (Masters et al., 2019). At
higher redshift, the HIGHz survey (Catinella & Cortese, 2015) targeted the HI
emission of massive galaxies at $z\sim 0.2$, while the CHILES survey pushed
the limit of individual detections up to $z\sim 0.4$ (Fernández et al., 2016).
At even higher redshift our knowledge of HI content is entirely obtained by
stacking analysis: Kanekar et al. (2016) at $z\sim 1.3$ and Chowdhury et al.
(2020, C20 hereafter) at $z\sim 1$. Damped Ly$\alpha$ or MgII absorption line
systems give us the chance to estimate the HI content at $z\gtrsim$1.5, with
the caveat that they trace HI located well outside the optical disk of
galaxies, hence revealing little about what is going on inside their star-
forming body.
Recently, in Morselli et al. (2020, M20 hereafter) we analyzed the HI and H2
content of five nearby, grand-design, massive main sequence (MS) galaxies on
scales of $\sim 500$pc and linked the availability of molecular and neutral
hydrogen to the star formation rate (SFR) of each region. We found that H2/HI
increases with gas surface density, and at fixed total gas surface density it
decreases (increases) for regions with a higher (lower) specific star
formation rate (sSFR). In this paper we exploit tight correlations to estimate
the evolution with redshift of the H2/HI mass ratio within galaxies. It is
generally assumed that this ratio increases with redshift, because galaxies
are more gas rich and as the gas surface density increases, recombination is
favored. However, galaxies at high redshift are also more star forming, and
higher levels of star formation favor photo-dissociation of the H2 molecule,
hence it is not a priori obvious which trend would dominate over the other.

##  2 Data: M∗, SFR, H2 and HI at 500 pc Resolution

The methodology to retrieve estimates of the stellar mass ($M_{\star}$), SFR,
HI mass ($M_{\rm HI}$) and H2 mass ($M_{\rm{H_{2}}}$) is detailed in Enia et
al. (2020) and M20. Briefly, starting from the DustPedia archive (Davies et
al., 2017; Clark et al., 2018) we built a sample of five nearby, face-on,
grand design spiral galaxies with stellar mass in the range
$10^{10.2-10.7}M_{\odot}$, that lie on the MS relation at $z=0$. These sources
have been observed in at least 18 bands from the far ultraviolet (FUV) to the
far infrared (FIR). We used the photometric data from FUV to FIR to run SED
fitting with MAGPHYS (da Cunha et al., 2008) on cells of 500pc$\times$500pc.
We obtained the SFR as the sum of the un-obscured (SFRUV) and obscured (SFRIR)
contributions. To this aim, SFRUV and SFRIR have been computed using the
scaling relations of Bell & Kennicutt (2001) and Kennicutt (1998),
respectively, where the UV and IR luminosities ($L_{\rm UV}$ and $L_{\rm IR}$)
are evaluated from the best-fit SED (see Enia et al., 2020). Finally, as these
sources are included in the HERACLES (Leroy et al., 2009) and THINGS (Walter
et al., 2008) surveys, they have been observed in CO(2-1) and HI at 21 cm.
Hereafter, we make use of the H2 estimated using $\alpha_{\rm CO}$ =
4.3$M_{\odot}$${\rm(K\cdot km\cdot s^{-1}pc^{2})^{-1}}$ (e.g. Bolatto et al.,
2013). Details on how the HI and H2 maps at 500pc resolution were obtained can
be found in M20, where the consistency of the results using a constant or
metallicity-dependent $\alpha_{\rm CO}$ is discussed.

##  3 the H2/HI mass ratio at high redshift

In this paper we exploit local correlations observed at 500pc resolution to
estimate the redshift evolution of the H2/HI ratio. An important caveat of
this procedure is the validity on galactic scales of correlations observed on
sub-galactic scales or, in other words, whether integrated quantities can be
estimated from spatially-resolved relations. In recent years several studies
have indeed revealed that the "main" correlations involved in the star
formation process, the MS of star forming galaxies and the molecular gas Main
Sequence (MGMS, e.g. Lin et al., 2019) have very similar slopes when analyzed
on sub-galactic or galactic scales (e.g. Hsieh et al., 2017; Lin et al., 2019;
Cano-Díaz et al., 2019; Enia et al., 2020).

Figure 1: Left panel: ${\rm log({H_{2}}/{HI})}$ \- ${\rm log}\Sigma_{\rm gas}$
plane, adapted from Figure 8 of M20. Each cell is color-coded according to the
average value of $\Delta_{\rm MS}$. The blue solid line is the best fit to the
cells having an average value of $\Delta_{\rm{MS}}$ in the range [-0.2,0.2];
the slope of this best fit is $m_{1}$. The gray shaded area includes the
values for which we compute $\Delta_{\rm MS}$ as a function of H2/HI ratio, as
shown in the right panel: the slope of the best fit (blue solid line) give us
$m_{2}$.

###  3.1 Method 1

To estimate the redshift evolution of $M_{\rm H_{2}}$/$M_{\rm HI}$ in MS
galaxies we proceed as follows. We define the variable $Y$ as the log of
$M_{\rm H_{2}}$/$M_{\rm HI}$ and express it as a function the the total gas
mass ($M_{\rm gas}=M_{\rm H_{2}}+M_{\rm HI}$) and SFR :

$Y={\rm log}\frac{M_{\rm H_{2}}}{M_{\rm HI}}=f(M_{\rm gas},{\rm SFR}).$ (1)

It follows that:

${dY\over d{\rm log}(1+z)}=m_{1}{d{\rm log}M_{\rm gas}\over d{\rm
log}(1+z)}+m_{2}{d{\rm log(SFR)}\over d{\rm log}(1+z)},$ (2)

where:

${\partial Y\over\partial{\rm log}M_{\rm gas}}\simeq m_{1}\quad{\rm
and}\quad{\partial Y\over\partial{\rm log(SFR)}}\simeq m_{2},$ (3)

with $m_{1}$ describing the conversion of HI into H2 and $m_{2}$ the opposite
conversion from H2 to HI due to photo-dissociation. From Tacconi et al. (2018)
we have that, at fixed stellar mass,

${d{\rm log}M_{\rm H_{2}}\over d{\rm log}(1+z)}=2.6$ (4)

which refers only to $M_{\rm{H_{2}}}$, not to $M_{\rm gas}$. For the redshift
evolution of the SFR (at fixed stellar mass) we adopt the scaling from Speagle
et al. (2014):

${d{\rm log}{\rm SFR}\over d{\rm log}(1+z)}=3.5.$ (5)

Therefore, Equation (2) becomes:

${dY\over d{\rm log}(1+z)}=m_{1}{d{\rm log}M_{\rm gas}\over d{\rm
log}(1+z)}+3.5m_{2}.$ (6)

As a next step, we need to derive ${d{\rm log}M_{\rm gas}\over d{\rm
log}(1+z)}$. Since we have:

${\rm log}M_{\rm HI}={\rm log}M_{\rm H_{2}}-Y,$ (7)

then:

$M_{\rm gas}=M_{\rm H_{2}}\times(1+10^{-Y}),$ (8)

and the derivative becomes:

$\begin{split}{d{\rm log}M_{\rm gas}\over d{\rm log}(1+z)}={d{\rm log}M_{\rm
H_{2}}\over d{\rm log}(1+z)}+{d{\rm log}(1+10^{-Y})\over d{\rm log}(1+z)}=\\\
2.6-\left(1+{M_{\rm H_{2}}\over M_{\rm HI}}\right)^{-1}{dY\over d{\rm
log}(1+z)}\end{split}$ (9)

where the first derivative is given by Equation (4). Therefore, using Equation
(9), Equation (6) becomes:

$\begin{split}{dY\over d{\rm log}(1+z)}=-m_{1}\left(1+{M_{\rm H_{2}}\over
M_{\rm HI}}\right)^{-1}{dY\over d{\rm log}(1+z)}+2.6m_{1}+3.5m_{2}\end{split}$
(10)

Now we integrate the left and right sides of Equation (10) between $z=0$ and
$z$:

$\begin{split}&\int_{0}^{z}\left(1+{m_{1}\over
1+10^{Y}}\right)\,{dY}=(2.6m_{1}+3.5m_{2})\int_{0}^{\log(1+z)}\,{d{\rm
log(}1+z)}\end{split}$ (11)

By solving the integrals of the left and right sides of Equation (11) we get:

$\begin{split}&(1+m_{1})(Y_{z}-Y_{0})+m_{1}(\log(1+10^{Y_{0}})-\log(1+10^{Y_{z}}))\\\
&=(2.6m_{1}+3.5m_{2})\log(1+z),\end{split}$ (12)

where the subscript 0 ($z$) refers to the values at redshift 0 ($z$). Thus,
this equation is meant to describe the redshift evolution of the H2/HI mass
ratio at fixed stellar mass. To proceed with the numerical solution of
Equation (12), we need the values of $m_{1}$ and $m_{2}$ that we obtain from
Figure 8 of M20, reported here in the left panel of Figure 1. This Figure
shows how the ratio of molecular to atomic hydrogen varies as a function of
the total gas surface density and distance from the spatially resolved MS
relation, ${\rm\Delta_{MS}}$, which is defined as the difference between
log(SFR) of a region and its MS value at the same stellar mass. Inside
galaxies, the H${}_{\rm{}_{2}}$/HI mass ratio is very strongly correlated with
the total gas surface density and anticorrelated with the local SFR, as
quantified by ${\rm\Delta_{MS}}$. In M20 we interpret this anticorrelation as
evidence that the UV radiation from recently formed, massive stars has the
effect of photo-dissociating molecular hydrogen, a manifestation of the self-
regulating nature of the star formation process.

We estimate $m_{1}$ by fitting the relation between ${\rm log({H_{2}}/{HI})}$
and ${\rm log}\Sigma_{\rm gas}$ along the MS ($\Delta_{\rm{MS}}\sim 0$): the
best fit returns a slope of 1.49 (blue solid line in the left panel of Figure
1). To estimate $m_{2}$, we calculate the slope of the ${\rm
log({H_{2}}/HI)}\\!-\\!\Delta_{\rm{MS}}$ relation at fixed ${\rm
log}\Sigma_{\rm gas}$, considering a narrow range of ${\rm log}\Sigma_{\rm
gas}$ values where data exist over the widest range of the ${\rm{H_{2}}/{HI}}$
mass ratio (the vertical grey region in the left panel), hence offering the
best possible estimate of this derivative. The best fit returns a slope of
$-1.55$ (right panel of Figure 1). We adopt these two derivatives as proxies
for $m_{1}$ and $m_{2}$ as defined by Equations (3), based on the
aforementioned similarity between the corresponding spatially resolved and
global relations. For simplicity, in the following we assume $m_{1}$=1.5 and
$m_{2}$=$-1.5$, values which are perfectly consistent with the best fit ones.
Under this assumption, Equation (10) becomes:

${dY\over d{\rm log}(1+z)}=-{1.35\over 1+1.5\left(1+{M_{\rm H_{2}}\over M_{\rm
HI}}\right)^{-1}}\quad.$ (13)

Equation (13) implies that the redshift derivative of $Y$ is always negative,
i.e., the phase equilibrium shifts in favour of HI in high redshift galaxies.
This comes from the SFR increasing with redshift faster than the molecular gas
mass, see the above Equations (4) and (5). Let us consider three limiting
cases. If $M_{\rm H_{2}}$ largely dominates over $M_{\rm HI}$, then the
denominator in Equation (13) is $\sim 1$ and the derivative is $-1.35$. If
$M_{\rm HI}$ largely dominates, the derivative becomes -0.54. Finally, if the
two phases are nearly equal in mass the denominator is $\sim$ 1.75 and the
derivative becomes -0.77. So, the derivative will always be between -0.54 and
$-1.35$.

However, an analytical solution of Equation 12 is also possible, and it is
shown in Figure 2 (solid lines) for $m_{1}=1.5$, $m_{2}=-1.5$, and for $M_{\rm
H_{2},0}$/$M_{\rm HI,0}$ = 1/3, 1 and 3, i.e., three typical values of the
H2/HI mass ratio within the optical radius of MS galaxies in the local
Universe (Casasola et al., 2020). Galaxies that at $z=0$ are HI dominated, or
in which the two phases are equal in mass, show just a mild evolution of
$M_{\rm H_{2}}$/$M_{\rm HI}$, implying that by $z\sim 2$ HI still holds the
majority share. However, galaxies that locally are H2 dominated will tend to
show a slightly steeper evolution, to reach $M_{\rm H_{2}}$/$M_{\rm HI}$
$\sim$ 1.2 at $z=2$. We notice that lower values than 3.5 in Equation (5) can
be found in the literature: they would imply a flatter evolution of $M_{\rm
H_{2}}$/$M_{\rm HI}$ compared to our results.

Figure 2: Redshift evolution at fixed stellar mass of the H2/HI mass ratio,
obtained applying Method 1 (solid lines) and Method 2 (dashed lines), for
three different values of ($M_{\rm H_{2}}/M_{\rm HI})_{z=0}$= 1/3 (turquoise),
1 (gray) and 3 (black). The values obtain from the HI detection of C20 at
z=1.04 are marked with the white-to-black colored bar, with the gradient
indicating variations of the fraction of HI inside the optical radius. The
values estimated from the correlations of Zhang et al. (2020) at z=0, 0.83 and
1.23 are indicated with the yellow-to-purple colored bar, with the gradient
indicating the variations in stellar mass.

###  3.2 Method 2

With the data for the five galaxies in the sample of M20 we analyze how
$M_{\rm HI}$ and $M_{\rm H_{2}}$ are linked on scales of 500 pc. We observe a
slightly super-linear correlation between log($M_{\rm HI}/M_{\star}$) and
log($M_{\rm H_{2}}/M_{\star}$), characterized by a slope of 1.13, a Spearman
coefficient of 0.62 and $p$-value $\sim$ 0:

${\rm log}\frac{M_{\rm HI}}{M_{\star}}\propto 1.13\ {\rm log}\frac{M_{\rm
H_{2}}}{M_{\star}}$ (14)

and the correlation is shown in Figure 3. We note that one of our five
galaxies, NGC5194 (M51), has a significantly flatter slope and smaller
Spearman coefficient, and interestingly is the only galaxy in the sample to be
experiencing an interaction (with M51b) as well as the only one to have T-type
= 4 (while the rest of the galaxies have T-type between 5.2 and 5.9). We
decided to keep NGC5194 in our sample for consistency with Method 1, but
noting that the slope for the remaining four galaxies is slightly steeper
(1.24). This correlation gives us the possibility to estimate the evolution of
$M_{\rm HI}/M_{\star}$ with $z$ just by considering the evolution of the
molecular gas (at fixed stellar mass), expressed in Equation (4). Hence,
Equation (14) becomes:

$\frac{M_{\rm HI}}{M_{\star}}\propto\left(\frac{M_{\rm
H_{2}}}{M_{\star}}\right)^{1.13}\propto(1+z)^{1.13\times 2.6}$ (15)

and thus:

$\frac{M_{\rm H_{2}}}{M_{\rm
HI}}\propto(1+z)^{2.6}\times(1+z)^{-2.94}\propto(1+z)^{-0.34}.$ (16)

The trend expressed by Equation (16) is shown in Figure 2 (dashed lines) for
the three values of $M_{\rm H_{2}}$/$M_{\rm HI}$ at $z$ = 0 used in Method 1:
1/3, 1 and 3. The two methods appear to give basically consistent results,
with only a modest evolution of $M_{\rm H_{2}}$/$M_{\rm HI}$ with redshift in
favor of HI, which is more pronounced in Method 1 (we note that a steeper
slope than the one expressed in Equation (15) would increase the consistency
between the two methods). This agreement may not be surprising, as the two
methods are in fact more similar than they appear. Indeed, in Method 1 the
effect of the SFR on $M_{\rm H_{2}}$/$M_{\rm HI}$ is treated explicitly,
whereas in Method 2 it is implicit in the $M_{\rm H_{2}}$-$M_{\rm HI}$
correlation.

##  4 Discussion

###  4.1 Comparison with other estimates of the HI content of high redshift
galaxies

We compare these trends with the recent detection of HI in emission in $z\sim
1$ galaxies, obtained by C20 via stacking analysis over 7,653 star forming
galaxies. They find that in their sample, with a mean stellar mass of
$9.4\cdot 10^{9}\,M_{\odot}$, the mean HI mass is $1.19\cdot
10^{10}\,M_{\odot}$. To compute the mean H2/HI mass ratio in the galaxies
observed by C20 we proceed as follows. We consider the mean molecular-to-
stellar mass ratio in the local Universe for galaxies with $M_{\star}\sim
10^{10}\,M_{\odot}$ to be $\sim$ 0.1 (e.g. Casasola et al., 2020; Hunt et al.,
2020). Thus, the mean molecular gas mass of galaxies having a mean stellar
mass of $9.4\cdot 10^{9}\,M_{\odot}$ is $\sim 9.4\cdot 10^{8}\,M_{\odot}$. By
applying the scaling from Tacconi et al. (2018), expressed by Equation (4),
the expected mean $M_{\rm H_{2}}$ in $z=1$ galaxies turns out to be $\sim
5.7\cdot 10^{9}\,M_{\odot}$, hence:

$\left({\frac{M_{\rm H_{2}}}{M_{\rm HI}}}\right)_{z=1}=\frac{5.7\cdot
10^{9}}{1.19\cdot 10^{10}}=0.48.$ (17)

This value is obtained assuming that the HI detected by C20 ($M_{\rm
HI_{tot}}$) lies completely within the optical radius ($R_{25}$) of the
galaxies in the sample (i.e., $f_{\rm R25}=\frac{M_{\rm HI_{R25}}}{M_{\rm
HI_{tot}}}=1$, with $M_{\rm HI_{R25}}$ the HI mass within $R_{25}$). The
average beam of the observations described in C20 is between 30 and 60 kpc,
thus it is likely that a certain fraction of the observed HI lies outside the
optical radius, resulting in an underestimation of the H2/HI mass ratio within
the optical radius. The white-to-black bar in Figure 2 represents the estimate
of $M_{\rm H_{2}}/M_{\rm HI}$ at $z$=1, assuming that C20 have sampled a
fraction of HI inside the optical radius, varying from 100 per cent (white) to
25 per cent (black). In particular, we find that $M_{\rm HI}>M_{\rm H_{2}}$ at
$z\sim$1 for $f_{\rm R25}>0.4$. For $f_{\rm R25}<0.4$, a value consistent with
the $z=0$ estimate of Hunt et al. (2020) of galaxies having on average 30$\%$
of their total HI inside the optical radius, we get $M_{\rm H_{2}}>M_{\rm HI}$
at $z\sim 1$, but even when this fraction is only 20%, $M_{\rm H_{2}}$ is only
a factor of 2 higher than $M_{\rm HI}$.
In Figure 2 we also include, with yellow-to-purple vertical bars, the recent
estimates of $M_{\rm HI}$ obtained by Zhang et al. (2020) from the local
correlations between log${\frac{M_{\rm HI}}{M_{\star}}}$ and the $(NUV-r)$
color, which is a proxy for the specific SFR. We report their results at three
different redshifts: $z$ = 0, 0.83 and 1.23. As above, we use the evolution of
$M_{\rm H_{2}}$ with redshift as given by Equation (4) to estimate $M_{\rm
H_{2}}$ at the three redshifts, while $M_{\rm HI}$ is obtained for M⋆ varying
between $10^{9}\,\hbox{$M_{\odot}$}$ (in yellow in Figure 2) and
$10^{10.5}\,\hbox{$M_{\odot}$}$ (in purple in Figure 2). It is worth noting
that the HI estimates used in Zhang et al. (2020) do not refer to values
within the optical radius; this is clear at $z=0$, where the estimates of
$M_{\rm H_{2}}$/$M_{\rm HI}$ are significantly smaller than those of Casasola
et al. (2020) computed within the optical radius.
While our two methods and the one of Zhang et al. (2020) yield similar results
(in that they suggest a non-vanishing HI contribution at high redshift), it is
worth recapping the underlying physical motivations of each of them. Method 1
is built on the observed scaling of the $M_{\rm H_{2}}/M_{\star}$ ratio with
redshift, Eq. (4), and attempts to include the effect of photo-dissociation of
the H2 molecules by young stars. Method 2 assumes that the local correlation
between $M_{\rm H_{2}}$ and $M_{\rm HI}$ holds at all redshifts, and the
rationale of it is that if galaxies have more H2 they must have also more HI,
which is the necessary step to form H2. The method of Zhang et al. (2020)
assumes that the local correlation between $M_{\rm HI}$ and the ultraviolet-
optical colour (a SFR proxy) holds also at all redshifts: as the SFR increases
with redshift, so has to do $M_{\rm HI}$ as well. We notice that only our two
methods use the observed increasing trend with redshift of the H2 to stellar
mass ratio.

Figure 3: Correlation between log($M_{\rm HI}$/$M_{\star}$) and log($M_{\rm
H_{2}}$/$M_{\star}$) at 500 pc resolution, for the 5 galaxies of M20. The best
fit correlation (solid orange line) has a slope of 1.13 and a Spearman
coefficient of 0.62.

###  4.2 Implications and conclusions

All the above results rely on extrapolations from local trends that may or may
not hold when applying them to high redshift, thus at this stage we consider
the results tentative. Yet, in all methods the H2/HI mass ratio is expected to
decrease with redshift, contrary to the notion that it would increase, with H2
dominating at high redshift. Thus, these results suggest that HI cannot be
neglected at high redshift and we discuss below some implications for our
understanding of high redshift galaxies.

The first one concerns star formation, namely the gas depletion time $M_{\rm
gas}$/SFR and the star formation efficiency (SFE). For lack of direct evidence
on the HI mass, the H2 mass has been generally used as a proxy for the total
gas mass. If our projections are correct, and if some (if not all) of the HI
observed within the optical disk of galaxies comes from H2 photo-dissociation,
then the total gas depletion time should be at least a factor of $\sim 2$
longer than previously estimated (e.g. Scoville et al., 2017; Tacconi et al.,
2018). In the end, the HI to H2 to stars conversion is not a one-way process
inside galaxies, but rather a cycle in which part of the H2 in converted back
to HI. Thus, the total gas depletion time is a more informative quantity
compared to the molecular gas depletion time.

The second implication concerns the contribution of HI to the total baryonic
mass inside the stellar disk of high redshift galaxies. Even when ignoring HI,
spatially-resolved dynamical studies have shown that $z\sim 2$ galaxies are
strongly baryon dominated inside their effective radius (Genzel et al., 2017,
2020). If the mass of HI is comparable to that of H2, as it is at $z\sim 0$,
then these galaxies may turn out even more baryon dominated than estimated
thus far. Similarly, a higher gas fraction due to the addition of the HI
component would lower the Toomre parameter, making disks more prone to clump
formation instabilities.

For a direct assessment of the HI content of star forming galaxies at high
redshifts we will have to wait for the planned surveys with the Square
Kilometer Array (SKA). Indeed, ultra-deep SKA1 surveys may probe massive
galaxies (with $M_{\rm HI}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}10^{10}\,\hbox{$M_{\odot}$}$) up to
$z\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1.7$ (Blyth et al., 2015),
or even beyond via stacking. Strawman HI surveys with SKA1 foresee two
medium/high redshift surveys (Blyth et al., 2015): a deep survey (150 deg2)
that will detect the mentioned amount of HI up to $z\sim 0.7$ and an ultra-
deep survey (2 deg2) that will reach $z\sim 1.7$. These observations should be
amply sufficient to check the extent to which our projections are correct.

## acknowledgments

We are grateful to the anonymous referee for a careful consideration of our
manuscript, to Leslie Hunt for useful comments on an early version, and to
Lucia Rodríguez-Muñoz, Arianna Renzini, Bhaskar Agarwal and Hannah Übler for
fruitful discussion and valuable inputs. LM acknowledges support from the BIRD
2018 research grant from the Universit$\grave{\rm a}$ degli Studi di Padova.
AE and GR acknowledge the support from grant PRIN MIUR 2017 - 20173ML3WW 001.

## Data Availability

The derived data underlying this article will be shared on reasonable request
to the corresponding author.

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