# Data Diversity Matters for Robust Instruction Tuning

Alexander Bukharin  Georgia Tech  Tuo Zhao  Georgia Tech

###### Abstract

Instruction tuning has emerged as a key step in aligning large language
models. One of the central challenges of instruction tuning is dataset
selection, as the composition of the instruction tuning dataset can
significantly impact downstream performance. In particular, researchers have
hypothesized that dataset diversity and dataset quality are important
indicators of downstream performance. However, it is not clear how to
automatically select high quality and diverse data or how exactly quality and
diversity affect instruction following ability. To resolve these issues, we
propose a new algorithm, Quality-Diversity Instruction Tuning (QDIT). QDIT
provides a principled algorithm to control dataset diversity and quality,
allowing us to conduct an in depth study on the effect of diversity and
quality on instruction tuning performance. From this study we draw two key
insights (1) there is a natural tradeoff between dataset diversity and quality
and (2) increasing dataset diversity significantly improves the worst case
instruction following performace, therefore improving robustness. We validate
the performance of QDIT on several large scale instruction tuning datasets,
where we find it can improve worst case performance by 18% while maintaining
or improving average performance compared to quality driven baselines.

##  1 Introduction

Large pre-trained language models have demonstrated a remarkable ability to
perform a wide range of natural language processing tasks (Vaswani et al.,
2017; Devlin et al., 2018; Radford et al., 2019; He et al., 2020; Brown et
al., 2020). Although these models are powerful, pre-trained models such as
GPT-3 can be quite difficult to work with and often do not follow user
instructions (Brown et al., 2020). To make language models more user-friendly
and to unlock instruction-following capabilities, researchers have turned to
instruction tuning, in which language models are trained to follow
instructions on a small set of example instruction-response pairs (Mishra et
al., 2021; Wei et al., 2021; Sanh et al., 2021; Wang et al., 2022a).
Instruction tuning has become extremely popular, allowing researchers to train
powerful and aligned language models with a relatively small budget (Taori et
al., 2023; Chiang et al., 2023; Xu et al., 2023).

One of the main challenges of instruction tuning is selecting the instruction
data. Recent works have found that the composition of the instruction-tuning
dataset significantly impacts the instruction following ability of the trained
model, with most works identifying two qualities that an instruction tuning
dataset should have: it should be high quality (Chen et al., 2023; Peng et
al., 2023) and it should cover a wide range of tasks (i.e. be diverse) (Wei et
al., 2021; Zhou et al., 2023; Gudibande et al., 2023). However, as of yet
there has been little study of how these two factors (quality and diversity)
affect on one another and how researchers can automatically select high
quality and diverse datasets. Instead, current work relies on manual data
curation (Zhou et al., 2023) or optimize one quantity only (Chen et al., 2023;
Peng et al., 2023; Wang et al., 2022b).

To bridge this gap we introduce a new algorithm, Quality-Diversity Instruction
Tuning (QDIT), to select diverse and high quality instruction tuning datasets.
QDIT is inspired by submodular optimization (Fujishige, 2005) and works by
iteratively selecting the datapoint that will improve the joint quality-
diversity score the most. QDIT provides an effective way to control the
diversity and quality of the instruction tuning dataset, allowing us to
conduct an in-depth study of diversity and quality in instruction tuning. From
this study we identify two key findings: (1) there is an inherent tradeoff
between dataset diversity and dataset quality and (2) improving dataset
diversity primarily improves the worst and average case instruction following
ability, while not affecting best case instruction following ability much.
Based on these finding, we are able to use QDIT to optimize the quality-
diversity tradeoff, improving worst case performance while maintaining or
improving best case and average performance for robust instruction following.
We extensively validate the QDIT algorithm with experiments on four large
scale instruction tuning datasets.

The rest of this article is organized as follows: Section 2 covers related
literature, Section 3 presents the QDIT algorithm, Section 4 contains our
experimental results, and Section 5 concludes our work.

##  2 Related Work

Given the widespread availability of large instruction tuning datasets (Ding
et al., 2023; Zheng et al., 2023a), a new question has emerged: How should we
select a training dataset from existing ones? Here data selection is necessary
as training on large scale instruction tuning datasets will have a large cost
and result in deterioration of model capabilities (Ouyang et al., 2022; Zhou
et al., 2023; Chen et al., 2023). Several strategies have been proposed to
answer this question, mainly focusing on improving dataset quality. Li et al.
(2023) propose to select instructions based on difficulty. Chen et al. (2023)
propose Alpagasus, in which a large language model such as ChatGPT is employed
to select the highest quality (instruction, response) pairs from a large
dataset. Peng et al. (2023) also seek to improve data quality, and do so by
rewriting answers to instructions with GPT-4. We seek to improve upon these
algorithms by including both diversity and quality in the selection criteria.

Our work is also related to several works that use submodular optimization to
increase dataset diversity in natural language processing (Kumar et al., 2019;
Kirchhoff and Bilmes, 2014). However, to the best of our knowledge we are the
first to include both data quality and diversity in the selection procedure.

##  3 Method

Before presenting QDIT, we discuss how to quantify and optimize dataset
diversity.

###  3.1 Quantifying Dataset Diversity and Quality

Given a set $A\subseteq V$, a natural way to measure the diversity of the set
$A$ with respect to $V$ is by the facility location function (Cornuéjols et
al., 1983)

$\displaystyle d(A)=\sum_{v\in V}\textrm{max}_{a\in A}sim(a,v)$ (1)

Intuitively, we can see that a set A that has a high diversity score $d(A)$
will have an element $a\in A$ close to each element $v\in V$ and will
therefore be representative of the set $V$.

In order to optimize the facility location function, researchers typically use
the greedy data selection algorithm (Nemhauser et al., 1978), where the
datapoint that most increases the diversity score of the current subset is
selected at each iteration. The greedy algorithm is shown in Algorithm 1,
where $d(a|A)=d(a\cup A)-d(A)$. We remark that this algorithm achieves the
best possible approximation ratio (in the worst case) that a polynomial time
algorithm can achieve unless $P=NP$ (Nemhauser et al., 1978).

To measure dataset quality, we follow prior works and measure the quality of
each (instruction, response) pair using a large language model such as ChatGPT
(Chen et al., 2023) or measure the quality of each datapoint using a reward
model trained on large amounts of human preference data (Ouyang et al., 2022;
Bai et al., 2022). We refer to such a function with $q(\cdot)$, and measure a
dataset’s overall quality by averaging the quality score of each datapoint.

###  3.2 Quality-Diversity Instruction-Tuning

In order to select data based on both quality and diversity, we alter the
greedy algorithm so that instead of selecting based on diversity alone at each
iteration, we select data based on a linear combination of quality and
diversity $(1-\alpha)d(a)+\alpha q(a)$. Here $\alpha$ is a hyperparameter that
can be used to control the diversity tradeoff. We remark that when $\alpha=1$,
QDIT becomes the quality driven selection algorithm proposed in Chen et al.
(2023) and when $\alpha=0$ QDIT becomes the classical greedy algorithm
(Nemhauser et al., 1978). QDIT will select points that most increase the joint
diversity-quality score of the data set, therefore satisfying our goal of
building a diverse and high quality dataset. The QDIT algorithm can be found
in Algorithm 2. In QDIT, we use the cosine similarity of instruction
embeddings as the similarity function in (1), where the instruction embeddings
are computed with sentence transformers (Reimers and Gurevych, 2019). See
appendix F for more details.

Algorithm 1 Greedy Data Selection

1:$K$ $\triangleright$ Desired subset size

2:$A\leftarrow\emptyset$

3:$R\leftarrow V$

4:$N\leftarrow 0$

5:while $N<K$ do

6: $a\leftarrow\textrm{argmax}_{a\in R}d(a|A)$

7: $A\leftarrow A\cup\\{a\\}$

8: $R\leftarrow R\setminus a$

9: $N\leftarrow N+1$

10:end while

Algorithm 2 QDIT Data Selection

1:$K,\alpha$

2:$A\leftarrow\emptyset$

3:$R\leftarrow V$

4:$N\leftarrow 0$

5:while $N<K$ do

6: $a\leftarrow\textrm{argmax}_{a\in R}(1-\alpha)*d(a|A)+\alpha*q(a)$

7: $A\leftarrow A\cup\\{a\\}$

8: $R\leftarrow R\setminus a$

9: $N\leftarrow N+1$

10:end while

Computational Complexity of QDIT.  Finding $a\in A$ that maximizes the
quality-diversity score at each time step has complexity
$\mathcal{O}(|V|^{3})$, leading to a total complexity of
$\mathcal{O}(|V|^{3}K)$ for QDIT based data selection. This is problematic,
since for some datasets $|V|$ is greater than 1 million. We take several
approaches to reduce data selection cost: (1) parallelization on GPUs, (2)
employing the lazy greedy selection algorithm of Minoux (2005), and (3) sub-
sampling according to Mirzasoleiman et al. (2015). These methods can be
employed simultaneously, allowing us to select from millions of datapoints
within a few hours. We provide more details in Appendix D, where we find that
data selection has minimal cost when compared to training.

##  4 Experiments

We first analyze how the QDIT algorithm affects dataset quality and diversity
and then present our main results and analysis.

###  4.1 Preliminary Analysis

To verify that QDIT can indeed control the quality and diversity of the
selected dataset, we first investigate whether the facility location function
is aligned with our intuitive understanding of diversity. Next, we study how
the $\alpha$ parameter affects the dataset’s quality and diversity. For this
analysis, we apply the QDIT algorithm to the Alpaca dataset (Taori et al.,
2023).

To qualitatively verify that the facility location function is aligned with
dataset diversity, we use the Berkeley Neural Parser (Kitaev and Klein, 2018;
Kitaev et al., 2019) to extract the root verb and first direct noun from each
instruction in the Alpaca dataset. We then plot the distribution of verb-noun
pairs in Figure 1 for random, quality driven, and QDIT data selection. From
Figure 1, we observe that selecting based on the facility location function
indeed improves the dataset diversity compared to random selection, while
selecting based on quality alone decreases dataset diversity.

(a) Random Selection

(b) Quality Driven Selection

(c) QDIT ($\alpha=0$)

Figure 1: Distribution of root verbs and first nouns selected by different
algorithms. The dataset size is 3000.

Next we plot how $\alpha$ in QDIT affects dataset quality and diversity in
Figure 2. Similar figures for other datasets can be found in Appendix A. From
these figures, we can observe that there is a tradeoff between quality and
diversity, and that QDIT allows us to smoothly control this tradeoff.
Moreover, we observe that QDIT is able to improve diversity without
significantly decreasing data quality. Altogether, these results indicate that
QDIT is a practical way to control dataset diversity and quality.

(a) 3K Selected Points

(b) 9K Selected Points

Figure 2: Effect of $\alpha$ on dataset quality and diversity. The red line
represents a randomly selected dataset.

###  4.2 Experimental Setup

Now that we can control quality and diversity with QDIT, we seek to study how
dataset quality and diversity affect instruction following ability.

$\diamond$ Training Setup. We use QDIT on four instruction tuning datasets:
Alpaca 52K (Taori et al., 2023), Ultrachat 1.3M (Ding et al., 2023), LMSYS-
Chat 1M (Zheng et al., 2023b), and a combined dataset of Alpaca 52K, Dolly 15K
(Conover et al., 2023), and the OIG-small-chip2 dataset (210K). We refer to
this dataset as “Mixed 270K”. For each dataset we consider two settings: a
small data setting and a large data setting. For our main experiments we
follow the training procedure from Taori et al. (2023) and use LLaMA-1 7B as
our base model (Touvron et al., 2023). In all settings we use the same number
of training epochs, meaning that the training cost is proportional to the
number of training instructions. Complete hyperparameter details can be found
in Appendix B.

$\diamond$ Evaluation. We evaluate the trained models in two main ways:
through AlpacaFarm (Dubois et al., 2023) and by using a reward model. For
AlpacaFarm, we evaluate the trained model versus a variety of reference
models, employing Claude 2 as the judge on five evaluation sets: InstructEval
(Wang et al., 2022b), WizardLM (Xu et al., 2023), Vicuna (Chiang et al.,
2023), Koala (Geng et al., 2023), and a set of 200 examples manually curated
from the ShareGPT dataset. In order to mitigate the effects of the judge LLM’s
positional bias, we evaluate the responses in both orders (i.e. QDIT response
shown first and QDIT response shown second). We then follow Chen et al. (2023)
and measure performance according to winning score ($\frac{\textrm{\\#
Win}-\textrm{\\# Lose}}{\textrm{Total comparisons}}+1$). Detailed comparison
plots can be found in Appendix E. In addition to language model based
evaluation, we evaluate our models based on the reward score achieved on each
evaluation dataset. We use the reward model from Dong et al. (2023), which is
trained on the Anthropic Helpful Harmless dataset and achieves a test accuracy
of over 75%. We refer to this score as “HH Score”

$\diamond$ Evaluating Robustness. Beyond evaluating average performance on the
test dataset, we also evaluate the worst and best case performance of each
model. We can evaluate the worst case performance with the HH Score by
calculating the average score achieved on the worst 10% of instructions for
each model (note that the worst instructions can change depending on the
model). Similarly, we can evaluate the worst case performance with AlpacaFarm
by measuring the losing rate on the worst 40% (smaller percents reduce the
test set size by too much) of instruction categories in the WizardLM
evaluation set. Here we use losing rate as a metric because a high losing rate
will indicate low robustness. Best case performance is measured in a similar
manner with win rate. Measuring best and worst case performance provides more
detailed insights into how diversity and quality affect model robustness.

$\diamond$ Baselines. We primarily compare the QDIT algorithm with two
baselines: random data selection and quality based selection. We remark that
for the Alpaca dataset instruction-response quality is measured according to
ChatGPT (using the same setup as Chen et al. (2023)) and for all other
datasets quality is measured with the previously mentioned reward model.

We also introduce two variants of the QDIT algorithm that attempt to improve
data quality and diversity. The first variant, QDIT:Threshold, first orders
the data point by quality, and then iteratively selects instructions that do
not have a similarity score greater than $\tau$ with any point included
selected subset. This algorithm is different from QDIT in that it prevents the
dataset from having similar datapoints, which QDIT allows. The second variant
we propose is QDIT-Cluster, in which we first cluster the instructions and
then select an equal amount of datapoints from each cluster in a quality-
driven manner. Comprehensive details can be found in Appendix C, where we also
provide an analysis of how these algorithms select data.

Table 1: Instruction Tuning Results. Random 50K refers to a model trained on a
randomly sampled set of 50K points on the corresponding dataset. The top and
bottom 40% win and lose rate is computed on the WizardLM evaluation set.

Alpaca 52K | Average Performance | Worst Case Performance | Best Case Performance
---|---|---|---
|  Winning Score vs. Alpaca 52K $\uparrow$ |  Winning Score vs. Random 50K $\uparrow$ |  HH Score Mean $\uparrow$ |  Bottom 10% HH Score $\uparrow$ |  Bottom 40% Lose Rate $\downarrow$ |  Top 10% HH Score$\uparrow$ |  Top 40% Win Rate $\uparrow$
Random 9K | 0.975 | - | 3.756 | 0.489 | 19.23 | 6.942 | 21.77
Quality 9K | 1.013 | - | 3.944 | 0.59 | 30.38 | 7.478 | 31.82
QDIT 9K | 1.013 | - | 3.922 | 0.573 | 29.55 | 7.437 | 25.71
Random 3K | 0.928 | - | 3.601 | 0.368 | 37.35 | 6.876 | 26.25
Quality 3K | 0.923 | - | 3.732 | 0.352 | 30.63 | 6.972 | 20.45
[1pt/1pt] QDIT:Cluster 3K | 0.995 | - | 3.874 | 0.601 | 21.98 | 7.025 | 29.63
QDIT:Threshold 3K | 0.986 | - | 3.745 | 0.452 | 25.99 | 6.943 | 29
QDIT 3K | 0.961 | - | 3.809 | 0.5 | 25 | 7.03 | 27.06
Mixed 270K | Average Performance | Worst Case Performance | Best Case Performance
|  Winning Score vs. Alpaca 52K $\uparrow$ |  Winning Score vs. Random 50K $\uparrow$ |  HH Score Mean $\uparrow$ |  Bottom 10% HH Score $\uparrow$ |  Bottom 40% Lose Rate $\downarrow$ |  Top 10% HH Score$\uparrow$ |  Top 40% Win Rate $\uparrow$
Random 50K | 0.927 | 1 | 3.6 | 0.396 | 29.06 | 7.002 | 25.89
Quality 50K | 0.973 | 1.065 | 4.295 | 1.186 | 34.78 | 7.746 | 28.7
QDIT 50K | 0.975 | 1.051 | 4.33 | 1.223 | 35.22 | 7.761 | 26.32
Random 10K | 0.896 | 0.985 | 3.609 | 0.314 | 32.69 | 6.863 | 24.1
Quality 10K | 0.956 | 1.042 | 4.342 | 1.117 | 33.33 | 7.952 | 30
[1pt/1pt] QDIT:Cluster 10K | 0.962 | 1.035 | 4.408 | 1.157 | 37.74 | 7.842 | 30.61
QDIT:Threshold 10K | 0.98 | 1.06 | 4.282 | 1.051 | 33.96 | 8.005 | 30.7
QDIT 10K | 0.982 | 1.063 | 4.429 | 1.3 | 32.65 | 8.006 | 26.74
Ultrachat 1.3M | Average Performance | Worst Case Performance | Best Case Performance
|  Winning Score vs. Alpaca 52K $\uparrow$ |  Winning Score vs. Random 50K $\uparrow$ |  HH Score Mean $\uparrow$ |  Bottom 10% HH Score $\uparrow$ |  Bottom 40% Lose Rate $\downarrow$ |  Top 10% HH Score$\uparrow$ |  Top 40% Win Rate $\uparrow$
Random 50K | 1.189 | 1 | 4.866 | 0.811 | 25 | 8.594 | 50
Quality 50K | 1.249 | 1.068 | 5.91 | 2.116 | 21.65 | 9.562 | 56.25
QDIT 50K | 1.26 | 1.094 | 5.909 | 2.185 | 22.52 | 9.69 | 57.45
Random 10K | 1.137 | 0.968 | 4.891 | 0.931 | 27.47 | 8.756 | 47.78
Quality 10K | 1.228 | 1.011 | 5.844 | 1.846 | 25.47 | 9.74 | 48.15
[1pt/1pt] QDIT:Cluster 10K | 1.254 | 1.036 | 5.802 | 1.808 | 20.68 | 9.52 | 51.04
QDIT:Threshold 10K | 1.242 | 1.037 | 5.734 | 1.937 | 19.64 | 9.64 | 57.73
QDIT 10K | 1.243 | 1.037 | 5.849 | 2.044 | 23.89 | 9.619 | 51.19
LMSYS 1M | Average Performance | Worst Case Performance | Best Case Performance
|  Winning Score vs. Alpaca 52K $\uparrow$ |  Winning Score vs. Random 50K $\uparrow$ |  HH Score Mean $\uparrow$ |  Bottom 10% HH Score $\uparrow$ |  Bottom 40% Lose Rate $\downarrow$ |  Top 10% HH Score$\uparrow$ |  Top 40% Win Rate $\uparrow$
Random 50K | 1.093 | 1 | 4.791 | 0.958 | 32.35 | 8.629 | 44.21
Quality 50K | 1.22 | 1.182 | 5.963 | 2.133 | 30 | 9.361 | 53.06
QDIT 50K | 1.217 | 1.168 | 6.074 | 2.278 | 34.48 | 9.484 | 51.55
Random 10K | 1.108 | 1.002 | 4.906 | 0.981 | 26.21 | 8.67 | 44.66
Quality 10K | 1.191 | 1.135 | 6.139 | 2.335 | 31.58 | 9.69 | 48.28
[1pt/1pt] QDIT:Cluster 10K | 1.218 | 1.143 | 6.125 | 2.085 | 28.44 | 9.787 | 51.02
QDIT:Threshold 10K | 1.22 | 1.173 | 5.978 | 2.319 | 26.13 | 9.709 | 55.42
QDIT 10K | 1.217 | 1.142 | 6.078 | 2.465 | 25.53 | 9.712 | 52.5

###  4.3 Main Results

The main experimental results can be found in Table 1. We mainly focus our
analysis on the low data settings, as we find models trained with QDIT can
match or beat the models trained in the high data settings while significantly
reducing training cost. Nonetheless, we include the high data setting for
reference.

Effect of Quality. Similar to prior works, we find that selecting data based
on quality significantly improves average performance, improving the winning
score versus Alpaca 52K by 5.41% and the average HH score by 17.14% in low
data settings. However, we also find that selecting based on quality alone can
hurt worst case performance, increasing the bottom 40% losing rate in two out
of four settings compared to random data selection. We hypothesize that this
performance drop is due to the fact that quality based selection hurts dataset
diversity.

Effect of Diversity. On the other hand, we find that data selection with QDIT
is able to achieve both a high average HH score (QDIT improves upon quality
driven selection by 2.56% for winning score vs Alpaca 52K and improves average
HH score by 1.29% in low data settings) while achieving a much better worst
case performance than both random and quality-driven selection. In particular,
we find that QDIT improves worst case HH score by 18.68 % and decreases worst
case losing rate vs Alpaca 52K by 11.45% when compared to quality-driven data
selection. We suspect that QDIT’s more diverse dataset teaches the model to a
respond to a wide range of instructions, therefore decreasing the probability
that it fails to follow evaluation instructions. We find that QDIT can also
maintain best case performance when compared to quality-diven selection, as
the best case HH score is improved by 0.12%. From these experiments we
conclude that by increasing data diversity while maintaining data quality,
QDIT can improve robust instruction following capability.

QDIT Variants. We also observe that the two variants of QDIT can improve
average performance compared to random data selection, but they do not
necessarily improve worst case performance compared to quality based
selection. Our experiments suggest that clustering based data selection is
highly sensitive to the performance of the clustering algorithm. For example,
clustering may be easy on datasets like Alpaca which are derived from a small
set of seed tasks, but difficult for larger and human generated instructions
like LMSYS. For threshold based selection, we hypothesize that constraining
intra-datapoint similarity is not as important as selecting instructions that
cover the entire training set.

###  4.4 Analysis

Effect of $\alpha$. We plot the effect of $\alpha$ on average and worst case
performance in Figure 3. From Figure 3, we can see that QDIT is relatively
robust to the value of $\alpha$, with values of $\alpha\in\\{0.5,0.7,0.9\\}$
typically having the highest worst case and average performance. However,
decreasing $\alpha$ by too much (i.e. $\alpha=0.1$) will result in a
significant drop in performance, as will increasing $\alpha$ by too much
($\alpha=1$). This highlights the need for a careful tradeoff between dataset
quality and diversity.

(a) Alpaca 3K

(b) Mixed 10K

(c) Ultrachat 10K

(d) LMSYS 10K

Figure 3: Effect of $\alpha$ on best case and worst case performance. The red
line represents a randomly selected dataset. Worst reward refers to the bottom
10 percent of HH scores.

Benchmark Performance. For a more comprehensive evaluation of QDIT, we follow
InstructEval (Chia et al., 2023) and evaluate our model on various benchmark
datasets including MMLU (Hendrycks et al., 2020), BBH (Suzgun et al., 2022),
DROP (Dua et al., 2019), and HumanEval (Chen et al., 2021). Although these
benchmarks are not fully aligned with instruction following performance, we
find that QDIT uniformly improves MMLU performance and performs best on BBH
and DROP in three out of the four scenarios when compared with quality driven
selection. These results further confirm the efficacy of QDIT.

Table 2: Evaluation on benchmark datasets. We bold the best result out of
quality based selection and QDIT based selection. Details can be found in
Appendix G.

| Alpaca 9K | Mixed 10K
---|---|---
| MMLU | BBH | DROP | HumanEval | MMLU | BBH | DROP | HumanEval
Random | 34.18 | 31.69 | 26.84 | 8.54 | 32.93 | 30.92 | 20.33 | 10.37
[1pt/1pt] Quality | 34.53 | 31.17 | 24.57 | 11.57 | 33.07 | 31.38 | 26.37 | 10.37
QDIT | 35.64 | 30.85 | 26.55 | 7.93 | 33.29 | 32.36 | 26.19 | 10.37
| Ultrachat 10K | LMSYS 10K
| MMLU | BBH | DROP | HumanEval | MMLU | BBH | DROP | HumanEval
Random | 32.12 | 33.19 | 26.24 | 13.41 | 33.05 | 32.57 | 25.06 | 15.85
[1pt/1pt] Quality | 35.44 | 32.06 | 17.01 | 12.2 | 34.74 | 32.32 | 25.95 | 10.36
QDIT | 35.78 | 32.66 | 26.73 | 13.41 | 35.9 | 32.76 | 26.22 | 11.58

Different Base Models. We evaluate the performance of QDIT with different base
models in Figure 4(a). From Figure 4(a) we find that QDIT can improve both
average performance and worst case performance for other base models,
supporting our main results. Interestingly we find that the Mistral model
performs best according to HH score, but worst according to AlpacaFarm
evaluation (Jiang et al., 2023). This highlights the importance of using
multiple evaluation strategies.

(a) Different Base Models

(b) Different Data Sizes (Thousands)

Figure 4: (a) Ablation with different base models. The train set is 10K
instructions selected from Ultrachat. (b) Ablation with different data sizes.
The worst reward is computed as the bottom 10 percent of HH scores, and the
winning score is vs. Alpaca 52K.

Data Size. We evaluate the performance of QDIT with different training data
sizes in Figure 4(b), where we find that QDIT results in a 3-4 times
improvement in data efficiency (and therefore training cost) as measured by
robust performance. We also find that increasing the dataset size beyond $50K$
only results in marginal gains of average performance, and can even decrease
worst case HH score. We hypothesize this phenomena is due to overfitting.

Case Study. We conduct a case study in Figure 5. We observe that quality based
selection does not select an instruction similar to the test instruction and
fails to respond correctly. On the other hand, QDIT does train on a similar
instruction due to its more diverse training set and correspondingly responds
to the instruction correctly. We remark that in practice the distance to the
closest training instruction does not always correlate to evaluation
performance, as we find embedding distance may not capture important aspects
of an instruction such as the format it is in or the desired response format.
In the future, we hope to investigate how different similarity metrics impact
QDIT and instruction tuning performance.

Figure 5: Spotify playlist case study.

##  5 Conclusion

In this paper we introduce QDIT, a method for automatically selecting high
quality and diverse instruction tuning datasets. QDIT allows us to flexibly
tradeoff quality and diversity, improving both average performance and model
robustness on a wide variety of datasets. As future lines of research, we
would like to investigate how different similarity metrics affect the
performance of QDIT and how QDIT performs with different model sizes.

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##  Appendix A Tradeoff between Quality and Diversity: Additional Results

We display additional results on the effect of $\alpha$ on dataset diversity
and quality in Figures 6, 7, 8. We again find that there is a tradeoff between
quality and diversity, and that $\alpha$ can be used to control this tradeoff.

(a) 1K Selected Points

(b) 5K Selected Points

(c) 10K Selected Points

Figure 6: Effect of $\alpha$ on dataset quality and diversity. The dataset is
Mixed 270K.

(a) 1K Selected Points

(b) 5K Selected Points

(c) 10K Selected Points

Figure 7: Effect of $\alpha$ on dataset quality and diversity. The dataset is
Ultrachat 1.3M.

(a) 1K Selected Points

(b) 5K Selected Points

(c) 10K Selected Points

Figure 8: Effect of $\alpha$ on dataset quality and diversity. The dataset is
LMSYS 1M.

##  Appendix B Training Details

We display the hyperparameter details in Table 3 and Table 4.

Table 3: General training details. The same hyperaparameter setting is used for every dataset and data selection strategy, following Chen et al. [2023]. Batch Size | Learning Rate | Epochs | Max Length | Weight Decay
---|---|---|---|---
128 | $2\times 10^{-5}$ | 3 | 512 | 0
Table 4: Selected values of $\alpha$. Dataset Size | Alpaca 52K | Mixed 270K | Ultrachat 1.3M | LMSYS 1M
---|---|---|---|---
3K | 0.7 | - | - | -
9K | 0.7 | - | - | -
10K | - | 0.7 | 0.7 | 0.7
50K | - | 0.9 | 0.9 | 0.7

##  Appendix C QDIT Variants

In this section we describe and analyze the variants of QDIT.

QDIT:Cluster. In this variant of QDIT, we first cluster all the instructions
based on their sentence-transformer embeddings using the $k$-means algorithm,
where $k=100$. We then select an equal number of points from each cluster, and
points are selected from each cluster based on quality.

QDIT:Threshold. In this variant of QDIT, we first sort the instructions based
on quality. We then iteratively remove instructions from the selected dataset
if they have a similarity with some other instruction in the dataset greater
than a threshold $\tau$. In our experiments we use the cosine similarity as
the similarity metric and set $\tau=0.5$.

Analysis. To further analyze why QDIT typically outperforms the two QDIT
variants, we plot the selection strategy of each variant on a sample problem
in Figure 9. From this figure we can observe the QDIT:cluster will sample
equally from across the sample space, leading to some poor quality datapoints
being sampled. We also observe that QDIT:Threshold does not necessarily
optimize for diversity and may not sample from large portions of the sample
space. On the other hand QDIT samples only high quality datapoints from each
portion of the sample space, therefore resulting in a high quality and diverse
dataset.

(a) QDIT:Cluster

(b) QDIT:Threshold

(c) QDIT

Figure 9: We display the selection strategy of the various QDIT algorithms on
a test dataset. Six datapoints are selected and the selected datapoints are
circled.

##  Appendix D Computational Cost of QDIT

We measure the wall time of QDIT as taking approximately 9.42 minutes to
select a dataset of size 10000. This experiment was conducted on a single A100
40G GPU. In contrast, training on Ultrachat 10K takes 48 minutes on 8 A100 40G
GPUs, meaning that data selection (given the embeddings and reward score) has
2% of the cost of training. Taking into account the inference and evaluation
process, our method has an even smaller relative cost. We remark that the
reward score and embedding generation can typically be done on a small GPU
very quickly, or it can even be done on a CPU for minimal cost.

##  Appendix E Complete Win-Tie-Lose Results

In this appendix we show the win, ties, and losses achieved versus the
reference models as judged by Claude 2. The results can be seen in Figures 10,
11, 12, 13, 14, and 15.

(a) Random 10K vs. Alpaca 52K

(b) Quality 10K vs. Alpaca 52K

(c) QDIT 10K vs. Alpaca 52K

Figure 10: We display the results on Ultrachat as judged by Claude 2. The base
model here is LLaMA-2 7B.

(a) Random 10K vs. Alpaca 52K

(b) Quality 10K vs. Alpaca 52K

(c) QDIT 10K vs. Alpaca 52K

Figure 11: We display the results on Ultrachat as judged by Claude 2. The base
model here is LLaMA-1 7B.

(a) Random 10K vs. Random 50K

(b) Quality 10K vs. Random 50K

(c) QDIT 10K vs. Random
50K

Figure 12: We display the results on Ultrachat as judged by Claude 2. The base
model here is LLaMA-1 7B.

(a) Random 10K vs. Alpaca 52K

(b) Quality 10K vs. Alpaca 52K

(c) QDIT 10K vs. Alpaca 52K

Figure 13: We display the results on LMSYS as judged by Claude 2. The base
model here is LLaMA-1 7B.

(a) Random 10K vs. Random
50K

(b) Quality 10K vs. Random
50K

(c) QDIT 10K vs. Random
50K

Figure 14: We display the results on LMSYS as judged by Claude 2. The base
model here is LLaMA-1 7B.

(a) Quality 10K vs. Alpaca
52K

(b) QDIT 10K vs. Alpaca
52K

Figure 15: We display the results on Ultrachat as judged by Claude 2. The base
model here is Mistral 7B.

##  Appendix F Similarity Metrics

In QDIT, we use the cosine similarity of instruction embeddings as the
similarity function in (1), where the instruction embeddings are computed with
sentence transformers [Reimers and Gurevych, 2019]. More specifically, we use
the all-mpnet-base-v2 model available from Huggingface.

##  Appendix G Benchmark Environments

For evaluation on common NLP benchmarks, we follow Chia et al. [2023]. In
particular, we conduct five shot evaluation on MMLU, three shot on BBH, three
shot on DROP, and zero shot on HumanEval.


# Cross-link Interference Modeling in 6G Millimeter Wave and Terahertz LEO
Satellite Communications

Sergi Aliaga, Vitaly Petrov, Josep M. Jornet
Northeastern University, Boston, MA, USA
{aliaga.s, v.petrov<EMAIL_ADDRESS>
###### Abstract

One of the important questions when discussing next-generation near-Earth
mmWave and terahertz (THz) band satellite communications as an integral part
of the 5G-Advanced and 6G landscape is the potential interference-related
issues when deploying such systems. While the space-to-ground and ground-to-
space interference has been explored in multiple works already, the
interference at mmWave and THz cross-links, the links between the satellites
themselves, have not been extensively studied yet. However, severe cross-link
interference may both challenge the reliability of the data exchange within
the constellation, as well as compromise the efficient co-existence of
multiple satellite constellations (i.e., by different providers) covering the
same or neighboring areas. In this paper, both relevant mathematical models
and extensive simulation studies are presented for cross-link mmWave and THz
satellite communications. Our results indicate that the cross-link
interference in the considered setups is a non-negligible factor that must be
further explored and accounted for in the design and deployment of next-
general mmWave and THz satellite communication systems.

##  I Introduction

Seamless integration of high-rate satellite communications into the networking
landscape is one of the major innovations on the way from 5G through
5G-Advanced to 6G systems [1]. Particularly, large constellations of low-
earth-orbit (LEO) satellites are envisioned as key enablers for ubiquitous
global high-rate connectivity [2]. Complementing wireless local area networks
and cellular networks, these near-Earth systems thus complete the
_“connectivity triad”_ providing high-rate and low-latency Internet access for
the users on land, sea, and air [3].

To achieve the prospective 6G-grade performance levels, the use of highly-
directional transmissions is envisioned over wide bands in millimeter-wave
(mmWave, $\approx$$30$ GHz–$300$ GHz, including the Ka band [4]) and even
terahertz (THz, $300$ GHz–$3$ THz) frequencies [5, 6]. Such directional high-
rate links may be used for both satellite-to-ground links (Uplink and
Downlink) and for inter-satellite links (ISLs), or _cross-links_. For cross-
links, optical spectrum is also under active consideration, providing higher
rates than even THz channels, but simultaneously challenged by possible
pointing errors of extremely narrow (i.e., laser-formed) beams [7, 8].

One of the advantages of exploiting mmWave radio for both access and cross-
links is the partial reuse of the hardware, thus making the satellite simpler,
lighter, and cheaper. While THz radio is a longer-term goal [9], the band is
already actively exploited in satellite systems for sensing and imaging [10,
11]. THz communications are developing rapidly, with recent standardization
efforts [12] and multi-kilometer-long THz demos [13], while the estimated link
budget is sufficient for satellite-to-airplane [14]. Another advantage of
mmWave and THz cross-links is that terrestrial limitations (i.e., signal
absorption by atmosphere [9]) are not that profound in space.

An important challenge in developing mmWave and THz satellite communications
is related to the interference such solutions may produce to the existing
wireless systems, including networks, radars, Earth exploration satellites,
etc. [15]. Recently, numerous studies have been presented aiming to model
these effects (i.e., [16] and [17], among many others), as well as suggest
novel co-existence solutions [18, 19]. However, considering (tens of)
thousands of LEO communication satellites in the near future [4], interference
with terrestrial systems or satellite-based sensors is not the only challenge.
_Particularly, the interference between directional satellite cross-links
needs to be carefully accounted for._ The latter is important to guarantee
both reliable data exchange within a single satellite constellation, as well
as efficient co-existence of several LEO constellations in neighboring orbits.

Recent efforts here primarily comprise simulation-based and solution-centric
studies [20, 21, 22]. To the best of the authors’ knowledge, no comprehensive
mathematical model has been proposed to date for the interference among
directional mmWave/THz satellite cross-links as a function of radio and
orbital parameters. Meanwhile, the models for terrestrial mmWave and THz
networks [23, 24] are not directly applicable to satellites due to notably
different scenario geometry. _We aim to address the abovementioned gap in this
article._

The contributions of this work are summarized as follows:

  * •

_A mathematical framework_ capable to characterize the interference among
directional mmWave and THz satellite cross-links in various deployment
configurations. The framework is flexible and can be further tailored to other
deployment configurations of interest.

  * •

_A comprehensive numerical study_ illustrating the key trade-offs between the
constellation design parameters and the average cross-link interference in
different setups.

The paper is organized as follows. Section II introduces the system model and
the key assumptions. The mathematical framework to estimate the interference
among directional mmWave and THz satellite cross-links is detailed in Sec.
III. In Sec. IV, the developed models are numerically elaborated and also
cross-verified using extensive system-level simulations. The concluding
remarks are drawn in Sec. V.

(a) High-level deployment illustration

(b) Key LEO satellite orbital parameters

Figure 1: Interference modeling for LEO satellite THz directional cross-link
communications.

##  II System Model

####  II-1 Orbital parameters

LEO satellites may be deployed in multiple orbits of various configurations.
In this work, we target commonly used orbits, as illustrated in Fig. 1(a),
where the satellite placement is mainly determined by the four key orbital
parameters, $a$, $\gamma$, $\phi$, and $\Omega$, see Fig. 1(b). The first
parameter, $a$, is the _semi-major axis_ , which for circular LEO orbits is
equal to $a=h+R_{e}$, with $h$ being the orbit altitude above sea level and
$R_{e}$ being the Earth radius. The second one, $\gamma$, is the orbit’s
_inclination_ with respect to the Earth’s equator. The third parameter,
$\phi$, referred to as _true anomaly_ , corresponds to the angle between the
satellite position at a certain time and the orbit’s _ascending node_ , i.e.,
the crossing point between the orbit and the Earth’s equator when going from
lower to higher latitudes. The final key orbital parameter, $\Omega$,
corresponds to the angular position of the ascending node with respect to a
reference direction and is referred to as the _right ascension of the
ascending node_ (RAAN). Satellites on each of the modeled orbits are placed at
equal angular separation from each other.

####  II-2 Selected scenarios

We present the interference analysis in directional satellite communications
as a decomposed problem, with the aim of understanding the resulting
interference levels, the main sources of interference, and the interference
pattern across time. For this purpose, we have selected three practical sub-
scenarios of increasing complexity, which are the building blocks for the
near-future satellite mega-constellations under development:

  1. 1.

_Single orbit_ : The simplest option where the interference comes from the
satellites located in the same orbit.

  2. 2.

_Co-planar orbits_ : A more sophisticated case, where interference comes from
satellites with different altitude.111This case is of particular importance to
the co-existence in a multi-operator setup, where two constellations have
similar parameters, but different altitudes.

  3. 3.

_Shifted orbits_ : The most complex for the analysis setup, presenting a
typical configuration to obtain a “shell-like” pattern to cover a certain
range of latitudes [25].

The three scenarios are depicted in more detail in Sec. III.

####  II-3 Propagation, antenna, and routing assumptions

Our study focuses on LEO satellites at altitudes from $500$ km to $2000$ km,
where the effect of atmospheric absorption is already of secondary importance
compared to the spreading [9]. All the satellites are assumed to transmit with
the same Tx power, $P_{Tx}$. Hence, the received power, $P_{Rx}$ is primarily
determined by the transmit power, antenna gains, and the spreading loss, as
$P_{Rx}=\frac{P_{Tx}G^{2}}{(4\pi d/\lambda)^{2}}$, where $P_{Tx}$, $G$, and
$\lambda$ stand for the transmit power, antenna gain, and the signal
wavelength, respectively.

Analyzing the interference in a satellite constellation also requires certain
assumptions on the routing solutions in place. While there can be a wide
diversity of options here, it has been assumed that the satellites always
transmit to their immediate neighbors in the same orbit, if possible. This is
a feasible approach for the discussed deployment and all three modeled
scenarios.222Notably, the contributed model elements can be later recombined
in a different manner to account for a specific topology and routing protocol.
We also utilize an analytical _cone-shape_ antenna radiation pattern for all
the communicating nodes, where the main lobe gain, $G$, is given by
$G=2/(1-\cos(\alpha/2))$, where $\alpha$ is the pattern directivity
angle/beamwidth [24]. The delivered framework can also be further tailored to
other radiation patterns, both with and without strong side lobes.

####  II-4 Metrics of Interest

In this study, we particularly focus on the average interference power at the
target receiving node, $E[I]$, considering the contribution from each
potentially interfering satellite, and the signal-to-interference ratio (SIR),
$S$, at the receiver, hence putting the interference level in relation with
the power of the useful signal from the neighboring satellite.

##  III Interference Analysis

The discussed LEO satellite constellations feature a non-trivial geometry,
with complex and even temporarily-dependent mutual locations and orientations
of individual satellites. Therefore, for the sake of tractability and clarify
of the presentation, we first decompose the considered complex setup into
specific smaller elements, particularly modeling the interference coming: (i)
within a single orbit (analyzed in Sec. III-A), (ii) from a co-planar orbit
(analyzed in Sec. III-B), and (iii) from a shifted orbit (analyzed in Sec.
III-C). These studied cases allow us to explore directional cross-link
interference from different key perspectives.

###  III-A Single orbit

Fig. 2 illustrates the geometry for both the single orbit case and the co-
planar orbit case analyzed in the following subsection. When the interference,
$I_{1}$, comes from the same orbit, the model comes to $N$ satellites in the
same 2D plane forming a circle. We use index $i$ to number those satellites
($i\in[0;N-1]$), where $i=0$ is the target receiver (Rx), $i=1$ is the closest
satellite transmitting the useful signal (Tx, see Sec. II), while all other
satellites are potential interferers.

In the single orbit scenario, the satellite of index $i$ causes interference
at the receiver if and only if the following three conditions are
simultaneously satisfied: (i) the interfering signal is not blocked by the
Earth; (ii) the interfering satellite is within the Rx beam; and (iii) the Rx
is within the interferer’s beam as well. We analyze those conditions below.

First, the interferer-Rx path is not blocked by the Earth if:

$\theta_{i}>\frac{\pi}{2}-\arccos\left(\frac{R_{e}}{R_{e}+h}\right),\
i\in[2,N].$ (1)

Second, based on the scenario symmetry in Fig. 2, Condition 2, $C2$, and
Condition 3, $C3$ are both determined by:

$\theta_{i}>\theta_{1}-\frac{\alpha}{2},\ i\in[2,N].$ (2)

Then, considering the satellites are evenly distributed in orbit and applying
some basic trigonometry we obtain:

$\theta_{i}=\frac{\pi}{2}-i\frac{\pi}{N},\ i\in[1,N].$ (3)

Figure 2: Modeling interference from same or co-planar orbit.

Substituting (3) into (1) and (2) thus leads to:

$i<\frac{N}{\pi}\arccos\left(\frac{R_{e}}{R_{e}+h}\right)\enspace{}\land\enspace{}i<1+\frac{N}{2\pi}\alpha,\
i\in[2,N].$ (4)

Hence, the number of interfering satellites, $N_{i}$ will be the minimum of
these two thresholds ($i=1$ is Tx, not interferer):

$N_{i}=\lfloor\min(\frac{N}{\pi}\arccos\left(\frac{R_{e}}{R_{e}+h}\right),1+\frac{N}{2\pi}\alpha)\rfloor-1.$
(5)

Applying the cosine theorem to the geometry in Fig. 2, we then derive the
distance to the interfering satellite $i$, $d_{i}$, as:

$d_{i}^{2}=2\left(R_{e}+h\right)^{2}\left(1-\cos\frac{2\pi}{N}i\right).$ (6)

Later, we can combine (6) with (5) and derive the average interference power
for the single orbit scenario, $E[I_{1}]$, as:

$E[I_{1}]=\sum_{i=2}^{N_{i}}\frac{\lambda^{2}P_{Tx}}{8\pi^{2}\left(1-\cos\frac{\alpha}{2}\right)^{2}\left(R_{e}+h\right)^{2}\left(1-\cos\frac{2\pi}{N}i\right)}.$
(7)

Finally, by calculating the Tx-Rx distance, $d_{0}$, similar to (6), we get
the average SIR in the single orbit setup, $S_{1}$, as333Notably, many input
parameters (i.e., the Tx power) cancel in numerator and denominator, so the
average SIR depends primarily on the number of satellites, not even their
altitude, as further elaborated on in Sec. IV.:

$S_{1}=\frac{1}{\left(1-\cos\frac{2\pi}{N}\right)}/\sum_{i=2}^{N_{i}}\frac{1}{\left(1-\cos\frac{2\pi}{N}i\right)}.$
(8)

###  III-B Co-planar orbits

We now proceed with the co-planar orbits setup (see Fig. 2), illustrating,
i.e., the co-existence of two satellite constellations by different service
providers. We particularly detail the analysis for a case, where the target Rx
is at a lower altitude, $h$, and the interference, $I_{2}$, comes from a
higher orbit, $h_{c}$, while the derivations for the opposite case are
obtained similarly. An illustration of the scenario geometry is given in Fig.
3 with $N$ satellites on the lower orbit and $N_{c}$ satellites on the higher
orbit. We introduce Cartesian axes $x$ and $y$ to simplify the further
derivations by using vector notation.

As the two orbits have different altitudes, they also have different orbital
speeds. Hence, there is a new time-dependent parameter $\Delta\beta$, referred
to as _relative angular offset_ , with the time period determined by Kepler’s
third law [26]. Similar to the single orbit setup, the three conditions from
the previous subsection have to be met for the satellite $j$ to interfere with
the Rx. However, as for co-planar orbits $h\neq h_{c}$, $C2$ and $C3$ are now
fulfilled separately.

Figure 3: Interference from a co-planar orbit (zoomed).

Specifically, non-blocking by the Earth, $C1$, is satisfied if:

$\psi_{j}+\hat{A}>\frac{\pi}{2}-\arccos\left(\frac{R_{e}}{R_{e}+h}\right).$
(9)

In their turn, mutual antenna alignment conditions, $C2$ and $C3$, are
satisfied by $|\psi_{j}|<\frac{\alpha}{2}$ and
$|\psi_{j}^{\prime}|<\frac{\alpha}{2}$, respectively.

Using the vector notation, we then derive the angle $\psi_{j}$ as:

$\psi_{j}=\arccos\left(\langle\vec{AB},\vec{AC}\rangle/\|\vec{AB}\|\right),$
(10)

where

$\displaystyle\vec{A}=(0,\quad{}R_{e}+h),\
\vec{B}=(R_{e}+h_{c})(\sin\hat{O},\quad{}\cos\hat{O}),$ (11)
$\displaystyle\vec{AB}=\vec{B}-\vec{A}=((R_{e}+h_{c})\sin\hat{O},\quad{}(h_{c}-h)\cos\hat{O}),$
(12)
$\displaystyle\vec{AC}=(\sin\hat{A},\quad{}-\cos\hat{A}),\quad{}\hat{O}=\Delta\beta+2\pi
j/N_{c}.$ (13)

The angle $\hat{A}$ corresponds to the angle $\theta_{1}$ from the single
orbit scenario as $\hat{A}=\frac{\pi}{2}-\frac{\pi}{N}$.

Hence, (10) can be expanded into

$\psi_{j}=\arccos\left(\frac{(R_{e}+h_{c})\sin\hat{O}\sin\hat{A}-(h_{c}-h)\cos\hat{O}\cos\hat{A}}{\sqrt{(R_{e}+h_{c})^{2}\sin\hat{O}^{2}-(h_{c}-h)^{2}\cos\hat{O}^{2}}}\right).$
(14)

The final key angle here, $\psi_{j}^{\prime}$, is obtained similarly to
$\psi_{j}$ as:

$\psi_{j}^{\prime}=\arccos\left(\langle\vec{BD},\vec{BA}\rangle/\left[\|\vec{BD}\|\cdot\|\vec{BA}\|\right]\right),$
(15)

where

$\displaystyle\vec{BD}=\vec{D}-\vec{B},\quad{}\quad{}\vec{BA}=\vec{A}-\vec{B},$
(16)
$\displaystyle\vec{D}=(R_{e}+h_{c})(\sin\hat{O}^{\prime},\quad{}\cos\hat{O}^{\prime}),$
(17) $\displaystyle\hat{O}^{\prime}=\Delta\beta+\frac{2\pi}{N_{c}}((j-1)\mod
N_{c}-1).$ (18)

Once the main angles are specified, we can derive the set of interfering
satellites, $C$, as $C=C1\cap C2\cap C3$, where:

$\displaystyle C_{1}$ $\displaystyle=\\{j\in[0,N_{c}-1]\ |\
\psi_{j}<-\arccos\left(\frac{R_{e}}{R_{e}+h}\right)-\frac{\pi}{N}\\},$
$\displaystyle C_{2}$ $\displaystyle=\\{j\in[0,N_{c}-1]\ |\
|\psi_{j}|<\frac{\alpha}{2}\\},$ $\displaystyle C_{3}$
$\displaystyle=\\{j\in[0,N_{c}-1]\ |\
|\psi_{j}^{\prime}|<\frac{\alpha}{2}\\}.$ (19)

We now derive the distance from the potential interferer $i$ to the target Rx.
From the analysis above and the geometry in Fig. 3, this distance corresponds
to $\|\vec{AB}\|$, thus:

$d_{i}^{2}=(R_{e}+h_{c})^{2}\sin\hat{O}^{2}+(h_{c}-h)^{2}\cos\hat{O}^{2}.$
(20)

Finally, the average interference power, $E[I_{2}]$, and the average SIR,
$S_{2}$, for the co-planar scenario are derived as:

$\displaystyle E[I_{2}]$ $\displaystyle=\sum_{j\in C_{1}\cap C_{2}\cap
C_{3}}\lambda^{2}P_{Tx}/[(h_{c}-h)^{2}\cos\hat{O}^{2})+$
$\displaystyle+4\pi^{2}\left(1-\cos\frac{\alpha}{2}\right)^{2}((R_{e}+h_{c})^{2}\sin\hat{O}^{2}],$
(21) $\displaystyle S_{2}=\frac{1}{2Z\left(1-\cos\frac{2\pi}{N}\right)},$ (22)

where

$\displaystyle Z=\sum_{j\in C_{1}\cap C_{2}\cap
C_{3}}\frac{1}{(R_{e}+h_{c})^{2}\sin\hat{O}^{2}+(h_{c}-h)^{2}\cos\hat{O}^{2}}.$
(23)

###  III-C Shifted orbits

The third component of the model covers the case where the interference,
$I_{3}$, comes from an orbit with the same altitude and the same inclination,
but a different $\Omega$ [25], as in Fig. 4.444Many practical constellations
employ multiple _shifted orbits_ to grant continuous coverage between the
latitude range $[-\gamma,\gamma]$, as in Fig. 1(a) [25]. The angular shift
between the orbits is termed as $\Delta\Omega$.

Figure 4: Modeling interference from a shifted orbit.

As the satellites in shifted orbits are not always within the same 2D plane,
we have to use a 3D Geocentric Equatorial Coordinate (GEC) system fixed with
respect to the background stars [26]. The $z$-axis corresponds to the Earth’s
rotation axis, the $x$-axis corresponds to the direction of the vernal
equinox, and the $y$-axis follows a right-handed convention with the other
two. With this coordinate system, we can identify every satellite’s position
and pointing directions as 3-element vectors, thus also determining the angles
between them.

For a given satellite $j$ out of $N_{s}$ in orbit 2 (Orbit 1 contains the
target Rx), we begin by identifying its position inside its own orbital plane.
Concretely, we first compute its true anomaly (angular position inside the
orbital plane) as:

$\phi_{j}=\Delta\beta+\frac{2\pi}{N_{s}}j,\ \forall j\in[0,N_{s}-1],$ (24)

where $N_{s}$ is the number of satellites in orbit 2 and the relative angular
offset $\Delta\beta$ is constant through time since both orbits have the same
altitude and, therefore, the same orbital period.

We then get the satellite’s position in the orbital plane as:

$\vec{r}_{0j}=(x_{0j},y_{0j},z_{0j})=(R_{e}+h)(\cos\phi_{j},\sin\phi_{j},0).$
(25)

Next, we define the conversion matrix from the orbital coordinate system to
the GEC system as:

$M_{k}=\begin{bmatrix}\cos\Omega_{k}&-\sin\Omega_{k}\cos\gamma&\sin\Omega_{k}\sin\gamma\\\
\sin\Omega_{k}&\cos\Omega_{k}\cos\gamma&-\cos\Omega_{k}\sin\gamma\\\
0&\sin\gamma&\cos\gamma\\\ \end{bmatrix}$ (26)

where $k=0$ corresponds to the orbit with Rx and $k=1$ to the shifted orbit,
where the interference comes from. $\Omega_{k}$ is the orbit’s RAAN, and
$\gamma$ is the orbit’s inclination. We then obtain the $j$th satellite
position in orbit $k$ in the GEC system as $\vec{r}_{j}=M_{k}\vec{r}_{0j}$. We
use the same procedure to find the transmitter’s and receiver’s position
vectors, $\vec{r}_{Tx}$ and $\vec{r}_{Tx}$, respectively, and validate the
interference conditions.

To verify no blockage by the Earth, we get the rise and set between the
candidate satellite $\vec{r}_{j}$ and the receiver $\vec{r}_{Rx}$ as [27]:

$\displaystyle\begin{split}R_{j}=&\langle\vec{r}_{j},\vec{r}_{Rx}\rangle^{2}-\|\vec{r}_{j}\|^{2}\|\vec{r}_{Rx}\|^{2}+\\\
&+(\|\vec{r}_{j}\|^{2}+\|\vec{r}_{Rx}\|^{2})R_{e}^{2}-2R_{e}^{2}\langle\vec{r}_{j},\vec{r}_{Rx}\rangle.\end{split}$
(27)

and thus establish the set of visible satellites as:

$C_{1}=\\{j\in[0,N_{s}-1]\ |\ R_{j}\leq 0\\}.$ (28)

To verify $C2$ and $C3$, we compute the distance vector between the Rx and the
interfering satellite, and their pointing directions, respectively, as:

$\displaystyle\vec{r}_{Rx\rightarrow j}$
$\displaystyle=\vec{r}_{j}-\vec{r}_{Rx},$ (29)
$\displaystyle\vec{r}_{Rx\rightarrow Tx}$
$\displaystyle=\vec{r}_{Tx}-\vec{r}_{Rx},$ (30)
$\displaystyle\vec{r}_{j\rightarrow j-1}$
$\displaystyle=\vec{r}_{j-1}-\vec{r}_{j}.$ (31)

and compute angles $\psi_{j}$ and $\psi_{j}^{\prime}$ as:

$\displaystyle\psi_{j}$
$\displaystyle=\arccos\frac{\langle\vec{r}_{Rx\rightarrow
j},\vec{r}_{Rx\rightarrow Tx}\rangle}{\|\vec{r}_{Rx\rightarrow
j}\|\cdot\|\vec{r}_{Rx\rightarrow Tx}\|},$ (32)
$\displaystyle\psi_{j}^{\prime}$
$\displaystyle=\arccos\frac{\langle\vec{r}_{j\rightarrow
j-1},\vec{r}_{j\rightarrow Rx}\rangle}{\|\vec{r}_{j\rightarrow
j-1}\|\cdot\|\vec{r}_{j\rightarrow Rx}\|},$ (33)

where $\vec{r}_{j\rightarrow Rx}=-\vec{r}_{Rx\rightarrow j}$.

Similar to the previous subsection, the set of satellites satisfying $C2$ and
$C3$ will then be:

$\displaystyle C_{2}$ $\displaystyle=\\{j\in[0,N_{s}-1]\mod(N_{s})\ |\
|\psi_{j}|<\frac{\alpha}{2}\\},$ (34) $\displaystyle C_{3}$
$\displaystyle=\\{j\in[0,N_{s}-1]\mod(N_{s})\ |\
|\psi_{j}^{\prime}|<\frac{\alpha}{2}\\}.$ (35)

These derivations finally lead to the equations for the average interference
power, $E[I_{3}]$, and average SIR, $S_{3}$, for the case where the
interference comes from a shifted orbit:

$E[I_{3}]=\sum_{j\in C_{1}\cap C_{2}\cap C_{3}}\\\
\frac{\lambda^{2}P_{Tx}}{4\pi^{2}\left(1-\cos\frac{\alpha}{2}\right)^{2}\|\vec{r}_{Rx\rightarrow
j}\|^{2}},$ (36)
$S_{3}=1/\left(2\left(1-\cos\frac{2\pi}{N}\right)\sum_{j=0}^{N_{j}}\frac{1}{\|\vec{r}_{Rx\rightarrow
j}\|^{2}}\right).$ (37)

##  IV Numerical Results

In this section, the mathematical models developed above get numerically
elaborated. We also complement the analytical results with those produced
within an extensive system-lever simulation campaign. For this purpose, the
in-house developed satellite communications simulator from [28] has been
utilized. The framework is implemented in Python, it accurately models massive
satellite constellations with communications and sensing capabilities deployed
in most of the practical orbits [6].

Figure 5: Average SIR with interference from the same orbit.

(a) SIR evolution in time for an orbit separation of $10$ Km

(b) Expected interference for different orbit separation

Figure 6: Average interference and SIR for co-planar orbits.

###  IV-A Single orbit

We start with Fig. 5 presenting the average SIR for the single orbit setup,
$S_{1}$, as a function of the number of satellites in the orbit, $N$. First,
while the altitude for this figure is set to $500$ km, it has been observed in
(8) and later confirmed with simulations that the orbit altitude, $h$, does
not impact the average SIR, as the altitude-dependent parameters cancel each
other. Then, Fig. 5 illustrates that for a fixed beamwidth, $\alpha$, a higher
number of satellites, in general, leads to greater interference and,
consequently, lower average SIR. However, this effect is non-monotonic.
Particularly, there are specific _drop_ points in Fig. 5, where the SIR
notably decreases from $k$ to $k+1$ satellites. These are the points when the
number of interfering satellites, $N_{i}$, gets incremented.

Notably, the impact of every new interfering satellite on the average SIR is
smaller and smaller, i.e., for $\alpha=30\degree$ the drop in SIR from 24 to
25 satellites is $>$$1.5$ dB, while the corresponding SIR decrease from 95 to
96 satellites is $<$$0.2$ dB. We also observe that the intuitive dependency of
SIR increasing with antenna directivity (demonstrated in [23] and [24], among
other studies) holds for satellite deployments as well. Nevertheless, even for
narrow-beam links of $5\degree$ only, the SIR can be as low as $6$ dB, thus
challenging the reliability and performance of the cross-link if not properly
accounted for. We finally notice a close-to-perfect match between the
analytical curves and the performed Monte-Carlo simulations.

###  IV-B Co-planar orbits

We then study the interference from a co-planar orbit, $E[I_{2}]$.

####  IV-B1 Time-dependent interference

Due to non-stationary mutual positions and orientations of the satellites in
different orbits, the average SIR, $S_{2}$, evolves in time. This effect is
illustrated in Fig. 6(a), modeling two co-planar orbits with $N=N_{c}$ and the
altitudes of $500$ km and $510$ km (hence, with only $10$ km separation
distance). Despite a relatively small separation distance between the orbits,
the average SIR fluctuates between $5$ dB and $-30$ dB by following a periodic
pattern. We observe that the period depends on the number of satellites in the
orbits, particularly becoming shorter with more satellites. This is because
the higher the number of satellites per orbit, the lower the angular
separation between them, and the rotation required to achieve a redundant
satellite relative position is longer. Additionally, we again observe that the
beamwidth plays a critical role, with the curves for wider beamwidth always
staying below 0 dB (interference is higher than the received signal). These
results, however, are influenced by the close separation between orbits.

####  IV-B2 Time-averaged interference

We proceed with exploring the impact of the separation distance between the
orbits (the difference in their attitudes) on the time-averaged SIR, as
illustrated in Fig. 6(b). Here, we first notice a certain separation distance,
starting from which the satellites do not interfere with each other anymore.
This is an important practical outcome from our study, allowing, i.e., full
frequency reuse among the satellite constellations deployed with an altitude
difference greater than $X$.

(a) SIR evolution in time

(b) The effect of beamwidth

(c) The effect of inclination

Figure 7: SIR for the interference coming from a shifted orbit.

The second key observation from Fig.6(b) is the fact that the interference
from a co-planar orbit is higher for a greater number of satellites, but drops
faster after a certain orbit separation leading to the crossing of the
corresponding curves. This effect is caused by the fact that the satellites
pointing directions depend on the number of satellites in orbit. As the
satellites communicate with their direct neighbors, a larger number of
satellites in a co-planar orbit leads to more interferers but also steers
interfering signals away from the Rx. At a certain orbit separation, the
latter effect starts dominating the former one.

###  IV-C Shifted orbits

The last part of the numerical study focuses on the interference coming from a
shifted orbit, as explored in Fig. 7. For comparison, we present the results
in all the subfigures for the same altitude of $500$ Km and the same relative
offset of $0^{\circ}$.

####  IV-C1 Time-dependent interference

Similar to a co-planar setup, the impact of the interference is time-variant,
as in Fig. 7(a). We illustrate the setup, where an inclination is $3^{\circ}$,
$N$$=$$N_{s}$, and $\alpha=30^{\circ}$. The time duration of the results
corresponds to two orbital periods. Again we observe periodicity, this time
corresponding to the orbital period, with two clear SIR drops per period (four
in total in the figure). These drops correspond to the moments when the two
orbits cross.

####  IV-C2 Time-averaged interference

We then proceed with Fig. 7(b), presenting the time-averaged SIR, $S_{3}$, as
a function of beamwidth, $a$. We first observe that there is a “threshold”
beamwidth from which interference starts to appear (for smaller $\alpha$, SIR
goes to infinity). For $3^{\circ}$ inclination, this threshold is
$\alpha$$\approx$$6^{\circ}$. We also observe a step-like decreasing behavior
of the curves, meaning that larger beamwidths result in a reduction of SIR.
Hence, from this perspective, the interference from a shifted orbit behaves
similarly to the one from the same orbit, as in Fig. 5.

The results also notably depend on the orbit inclination angle, as explored in
Fig. 7(c). Here, higher inclination angles, in general, yield better SIR
values. Certain thresholds are again present, after which there is no
interference (around $\gamma=9^{\circ}$ to for $\alpha=20^{\circ}$ and
$\gamma=18^{\circ}$ for $\alpha=30^{\circ}$, respectively). We also observe
that there are specific small regions with an opposite trend, i.e.,
$\gamma=14^{\circ}$ to $\gamma=17^{\circ}$ for $\alpha=30^{\circ}$ and $N=60$.
This effect is primarily due to the fact that in these inclination regions,
the particular satellites interfering at the moment get closer to the Rx,
while still within the Rx beam, thus decreasing the average SIR. When the
inclination grows further (i.e., over $17^{\circ}$), some potential
interferers are still close to the Rx, but not anymore within the Rx beam. So,
the SIR keeps growing further. The latter is a useful finding for practical
constellations design, allowing to reduce the impact of interference even with
a high number of satellites per orbit.

##  V Conclusions

Directional cross-links over mmWave and, in the future, THz bands are
important technology enablers for high-rate LEO satellite connectivity
services as an integral part of 5G-Advanced and 6G networks. In this paper, we
have presented a novel mathematical framework to model the interference among
these cross-links in selected prospective scenarios. We have also complemented
our mathematical analysis with an extensive simulation campaign, identifying
both temporal and time-averaged interference-related characteristics. The
delivered models can be extended further by accounting for additional orbits,
such as polar orbits, imperfect beam alignment, and more sophisticated
specific antenna radiation patterns.

Our numerical results indicate that, despite the use of directional antennas,
cross-link interference can be non-negligible in some of the possible
deployment configurations. The observed effect may challenge the reliability
and performance of cross-links within the single LEO satellite constellation,
as well as the efficient co-existence of neighboring satellite constellations.
Hence, the interference among mmWave and THz cross-links cannot be always
neglected, but rather should be explored further and accounted for in the
design and deployment of next-generation LEO satellite networks.

## Acknowledgment

This work has been supported in part by the projects CNS-1955004 and
CNS-2011411 by the National Science Foundation (NSF), as well as by the U.S.
Air Force FA8750-20-1-0200. This project was also supported in part by a
fellowship from “la Caixa” foundation (ID 100010434, LCF/BQ/AA20/11820041).

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