# Is Contrastive Learning Necessary? A Study of Data Augmentation vs
Contrastive Learning in Sequential Recommendation

Peilin Zhou  Hong Kong University of Science and Technology (Guangzhou)
<EMAIL_ADDRESS>,  You-Liang Huang  Hong Kong University of Science and
Technology (Guangzhou<EMAIL_ADDRESS>,  Yueqi Xie  Hong
Kong University of Science and Technology<EMAIL_ADDRESS>,  Jingqi Gao
Upstage<EMAIL_ADDRESS>,  Shoujin Wang  University of Technology Sydney
<EMAIL_ADDRESS>,  Jae Boum Kim  Hong Kong University of Science and
Technology<EMAIL_ADDRESS>and  Sunghun Kim  Hong Kong University of Science
and Technology (Guangzhou<EMAIL_ADDRESS>
(2024)

###### Abstract.

Sequential recommender systems (SRS) are designed to predict users’ future
behaviors based on their historical interaction data. Recent research has
increasingly utilized contrastive learning (CL) to leverage unsupervised
signals to alleviate the data sparsity issue in SRS. In general, CL-based SRS
first augments the raw sequential interaction data by using data augmentation
strategies and employs a contrastive training scheme to enforce the
representations of those sequences from the same raw interaction data to be
similar. Despite the growing popularity of CL, data augmentation, as a basic
component of CL, has not received sufficient attention. This raises the
question: Is it possible to achieve superior recommendation results solely
through data augmentation? To answer this question, we benchmark eight widely
used data augmentation strategies, as well as state-of-the-art CL-based SRS
methods, on four real-world datasets under both warm- and cold-start settings.
Intriguingly, the conclusion drawn from our study is that, certain data
augmentation strategies can achieve similar or even superior performance
compared with some CL-based methods, demonstrating the potential to
significantly alleviate the data sparsity issue with fewer computational
overhead. We hope that our study can further inspire more fundamental studies
on the key functional components of complex CL techniques. Our processed
datasets and codes are available at https://github.com/AIM-SE/DA4Rec.

Data Augmentation, Sequential Recommendation, Contrastive Learning

††journalyear: 2024††copyright: acmlicensed††conference: Proceedings of the
ACM Web Conference 2024; May 13–17, 2024; Singapore, Singapore††booktitle:
Proceedings of the ACM Web Conference 2024 (WWW ’24), May 13–17, 2024,
Singapore, Singapore††doi: 10.1145/3589334.3645661††isbn:
979-8-4007-0171-9/24/05††ccs: Information systems Recommender systems

##  1\. Introduction

Figure 1. (a) Direct data augmentation for sequential recommendation; (b)
Contrastive learning for sequential recommendation. Figure 2. Eight widely
used sequence-level data augmentation strategies.

Sequential recommender systems (SRS) play a crucial role in various domains,
such as e-commerce (Chen et al., 2019; Yu et al., 2019; Cen et al., 2020; Lei
et al., 2023; Zhang and Kim, 2023), video (Lei et al., 2021; Gao et al.,
2017), music (Hansen et al., 2020; Pereira et al., 2019) and social media
(Jiang et al., 2016; Huang et al., 2018). The goal of these SRS is to predict
the next item that a user is likely to interact with based on his/her
historical behavior (Wang et al., 2022b, 2019a). One predominant obstacle in
developing SRS is the data sparsity issue, where user-item interaction data is
typically limited compared with a large number of users and items, leading to
insufficient training signals to learn informative item representations for
the downstream recommendations.

Recently, contrastive learning for recommendation has attracted increasing
attention due to its remarkable capability to enhance item representations
through the extraction of self-supervised signals from user-item interaction
data. Consequently, various contrastive learning-based SRS, such as CL4SRec
(Xie et al., 2020), CoSeRec (Liu et al., 2021a), ICLRec (Chen et al., 2022)
and DuoRec (Qiu et al., 2022), have been proposed. The core idea of these
methods can be summarized into two interrelated steps: (1) generating positive
views and negative views through data augmentation strategies; and (2)
minimizing (resp. maximizing) the distance between positive (resp. negative)
views using a contrastive loss function (such as InfoNCE (Oord et al., 2018)).
In these methods, data augmentation strategies are applied solely to the
auxiliary tasks (shown in Fig. 1 (b)) designed for contrastive learning rather
than directly applied to the recommendation task itself (shown in Fig. 1 (a)).
Such practice naturally raises two critical questions: Can the performance of
SRS be improved by solely relying on data augmentation (i.e., the first step)
without using contrastive learning paradigm (i.e., the second step)? Whether
existing CL-based SR methods consistently outperform direct sequence-level
data augmentation?

Answering these questions is crucial for revisiting the role of data
augmentation strategies in sequential recommendation tasks. However, so far,
the direct application of data augmentation to mitigate the data sparsity
issue in sequential recommendation has not received sufficient attention. Only
one study (Song and Suh, 2022) has explored the effects of four augmentation
strategies on sequential recommendation. Nevertheless, it still suffers from
following limitations: First, the augmentation strategies compared are not
comprehensive. For instance, strategies like reorder and delete, commonly used
in contrastive learning for sequential recommendation (SR), can also be
employed as standalone data augmentation methods. Second, comparison with
contrastive learning methods are not conducted, which is essential for
revisiting the effectiveness of contrastive learning in the sequential
recommendation research. Third, no insight or analysis is provided on the
factors that lead to varying performance results from different augmentation
strategies. Therefore, there is an urgent need to conduct a more systematic
empirical study to thoroughly investigate the effectiveness of data
augmentation in improving the performance of SRS.

To bridge this research gap, we conduct a comprehensive experimental study to
compare the performance of SRS based on data augmentation only and that of SRS
based on full contrastive learning. To be specific, we focus on investigating
the effectiveness of eight popular sequence-level augmentation strategies:
insert, replace, crop, delete, mask, reorder, subset-split, and slide-window.
Most of these augmentation strategies have been widely adopted in contrastive
learning-based SRS over the past five years. Specifically, we decouple these
sequence-level data augmentation strategies from contrastive learning methods
and directly apply them to augment the training sequences. Both the original
sequences and the augmented sequences are inputted together into backbone
models (such as SASRec (Kang and McAuley, 2018a)) for training. Afterward, we
benchmark these eight sequence-level data augmentation strategies and three
state-of-the-art contrastive learning-based SRS on four widely used datasets.
We also simulate different cold-start scenarios to further evaluate the
applicability of sequence-level data augmentation. Furthermore, we conduct in-
depth analysis on the impact of the size of data augmentations, as well as the
computational efficiency of various augmentation strategies and contrastive
learning methods.

The experimental results demonstrate that, when using SASRec as the backbone,
certain sequence-level augmentation strategies can achieve comparative or even
superior performance compared to some contrastive learning-based SR methods,
while requiring less training and inference time. This finding not only
validates the feasibility of directly using sequence-level augmentation to
alleviate data sparsity issue, but also suggests that the current research
community might underestimate the effectiveness of simple sequence-level
augmentation and overly emphasize the necessity of contrastive learning in
sequential recommendation tasks. Note that our claim does not negate the
effectiveness of contrastive learning in recommendation tasks. In fact, on
most datasets and metrics, specific contrastive learning based methods can
still achieve the best performance. However, we observe that the benefits of
these CL-based methods over direct data augmentation are not as pronounced as
expected, particularly when considering computational efficiency. From this
perspective, we believe that while contrastive learning is a popular means to
alleviate the data sparsity issue, it may not be the only choice, i.e., it is
not absolutely ”necessary” and direct data augmentation strategies are also
worth considering in practical scenarios.

The contribution of this paper can be summarized as follows:

  * •

We systematically benchmark eight popular sequence-level data augmentation
strategies and three representative contrastive learning methods for
sequential recommendation, providing insights into their performance and
applicability.

  * •

We explore the potential synergies in combining various data augmentation
strategies to improve sequential recommendation performance.

  * •

We find that employing specific sequence-level augmentation strategies can
effectively mitigate the problem of data sparsity in sequential recommender
systems. Moreover, these strategies typically demand less training time and
consume smaller GPU memory compared to CL-based methods.

##  2\. Problem Formulation

Consider two sets, $\mathcal{U}$ for users and $\mathcal{I}$ for items. For
each user $u$ belonging to $\mathcal{U}$, their past interactions are recorded
as a sequence
$\mathcal{S}^{u}=[v_{1}^{u},v_{2}^{u},...,v_{|\mathcal{S}^{u}|}^{u}]$. In this
sequence, each $v^{u}_{i}$, which is an item from $\mathcal{I}$, represents
the user’s $i$-th interaction, ordered by time, and $|\mathcal{S}^{u}|$ is the
total number of interactions for user $u$. All users’ interactions are
collectively represented by
$\mathcal{S}=\\{\mathcal{S}^{1},\mathcal{S}^{2},\dots,\mathcal{S}^{|\mathcal{U}|}\\}$,
with $|\mathcal{U}|$ indicating the total number of users. The purpose of
sequential recommendation systems is to use the sequence of interactions
$\mathcal{S}^{u}$ for a user $u$ to forecast the next item
$v_{|\mathcal{S}^{u}|+1}^{u}$ from $\mathcal{I}$, with which this user is
likely to engage at the next time step.

##  3\. Sequence-level Data Augmentation Strategies

Our experiments include multiple common rule-based sequence-level data
augmentation strategies. All of them can be viewed as operators that create
augmented sequences by performing certain transformations to the original
sequence. Fig. 2 illustrates how to augment each interaction sequence using
these operators.

###  3.1. Item Insert

The ”insert” action begins by selecting an insertion position, followed by the
insertion of a chosen item from the item pool, resulting in an augmented
sequence. For the user’s interaction sequence $\mathcal{S}^{u}$, let $i$ be an
insertion position, and $t$ be a chosen item from the item set $\mathcal{I}$.
The augmented sequence $\mathcal{S}^{u^{\prime}}$ can be represented as:

(1)
$\mathcal{S}^{u^{\prime}}=[v_{1}^{u},\ldots,v_{i-1}^{u},t,v_{i}^{u},\ldots,v_{|\mathcal{S}^{u}|}^{u}].$

###  3.2. Item Delete

The ”delete” action randomly selects an item from the sequence for removal,
thereby generating an augmented sequence. For $\mathcal{S}^{u}$ and a randomly
chosen item at position $k$, the augmented sequence $\mathcal{S}^{u^{\prime}}$
is:

(2)
$\mathcal{S}^{u^{\prime}}=[]v_{1}^{u},\ldots,v_{k-1}^{u},v_{k+1}^{u},\ldots,v_{|\mathcal{S}^{u}|}^{u}].$

###  3.3. Item Replace

The ”replace” action begins by selecting the item in the sequence that will be
replaced, followed by selecting an item from the item pool to substitute the
chosen item, resulting in an augmented sequence. For $\mathcal{S}^{u}$, let
$j$ be a position of the item to be replaced, and $t^{\prime}$ be a chosen
item from $\mathcal{I}$. The augmented sequence $\mathcal{S}^{u^{\prime}}$ is:

(3)
$\mathcal{S}^{u^{\prime}}=[v_{1}^{u},\ldots,v_{j-1}^{u},t^{\prime},v_{j+1}^{u},\ldots,v_{|\mathcal{S}^{u}|}^{u}].$

###  3.4. Item Crop

The ”crop” action initially selects a cutoff position, from which a continuous
series of items with a specified length are extracted as an augmented
sequence. Let $c$ be the cutoff position and $l$ be the length of the cropped
sequence. The augmented sequence $\mathcal{S}^{u^{\prime}}$ is:

(4) $\mathcal{S}^{u^{\prime}}=[v_{c}^{u},v_{c+1}^{u},\ldots,v_{c+l-1}^{u}].$

###  3.5. Item Mask

The ”mask” action initially chooses an item from the sequence and subsequently
masks the ID of the selected item using a predefined mask symbol. Let $m$ be
the position of the chosen item from sequence $\mathcal{S}^{u}$. If $\mu$ is
the predefined mask symbol, the sequence after masking can be given as:

(5)
$\mathcal{S}^{u^{\prime}}=[v_{1}^{u},\ldots,v_{m-1}^{u},\mu,v_{m+1}^{u},\ldots,v_{|\mathcal{S}^{u}|}^{u}].$

###  3.6. Item Reorder

The ”reorder” action initially selects a sub-sequence of a specific length and
subsequently shuffles the order of the items within that sub-sequence. The
sub-sequence and the remaining parts of the original sequence are then
combined according to their original order, resulting in an augmented
sequence. Consider a sub-sequence of length $r$ starting at position $d$ from
$\mathcal{S}^{u}$. Let $\text{shuffle}(x)$ denote a function that shuffles the
order of the elements in $x$. The augmented sequence
$\mathcal{S}^{u^{\prime}}$ after shuffling this sub-sequence is:

(6)
$\mathcal{S}^{u^{\prime}}=[v_{1}^{u},\ldots,v_{d-1}^{u},\text{shuffle}(v_{d}^{u},\ldots,v_{d+r-1}^{u}),v_{d+r}^{u},\ldots,v_{|\mathcal{S}^{u}|}^{u}].$

###  3.7. Subset Split

Similar to the dropout mechanism (Srivastava et al., 2014), for subset split
method, each item $v^{u}_{i}$ in the original sequence $\mathcal{S}^{u}$ will
be included in $\mathcal{S}^{u^{\prime}}$ with a probability of $1-\theta$ and
the probability of discarding is $\theta$:

(7) $v^{u^{\prime}}_{i}=\begin{cases}v^{u}_{i},&P=1-\theta\\\
\text{discarded},&P=\theta\end{cases}$

Thus, the augmented sequence $\mathcal{S}^{u^{\prime}}$ is essentially a
”subset” of the original sequence $\mathcal{S}^{u}$, which can be
mathematically represented as:

(8)
$\mathcal{S}^{u^{\prime}}=[v^{u^{\prime}}_{1},v^{u^{\prime}}_{2},\dots,v^{u^{\prime}}_{|\mathcal{S}^{u^{\prime}}|}].$

Note that the length of the augmented sequence $|\mathcal{S}^{u^{\prime}}|$
can vary due to the random discarding process and is likely to be less than or
equal to $|\mathcal{S}^{u}|$.

###  3.8. Slide-window

The slide-window strategy generates a new segment of the sequence and its
subsequent item as the prediction target at each step by sliding a fixed-
length window along the user’s behavioral sequence $\mathcal{S}^{u}$. The
window initiates sliding with its right edge positioned to the left of the
first item, and the sliding process concludes when the right edge reaches the
last item. For a given window length $L$, at each step $t$, the cropped
sequence of augmented items from $\mathcal{S}^{u}$ is:

(9)
$\mathcal{S}_{t}^{u^{\prime}}=[v_{t}^{u},v_{t+1}^{u},\ldots,v_{t+L-1}^{u}],$

where the sliding process iterating until $t+L-1=|\mathcal{S}^{u}|$.

Note that for each sequence $\mathcal{S}_{t}^{u^{\prime}}$ obtained through
the slide-window, the training label also changes, typically to the next item
$v_{t+L}^{u}$ in the sequence. Consequently, the slide-window not only
increases the quantity of training data but also adds complexity and diversity
to the training process by creating new prediction targets for each sequence
segment. This distinctive characteristic has led to the widespread adoption of
the slide-window as a data preprocessing step in both traditional SR models
and CL-based SR models, setting it apart from other data augmentation
strategies that diversify model training by altering or perturbing specific
elements of the original sequence while preserving the same prediction target.

Table 1. Statistics of the datasets after preprocessing. Dataset | Beauty | Sports | ML-1m | Yelp
---|---|---|---|---
# Users | 22,363 | 35,598 | 6,041 | 30,499
# Items | 12,101 | 18,357 | 3,407 | 20,068
# Avg. Actions / User | 8.9 | 8.3 | 165.5 | 10.4
# Avg. Actions / Item | 16.4 | 16.1 | 292.6 | 15.8
# Actions | 198,502 | 296,337 | 999,611 | 317,182
Sparsity | 99.93% | 99.95% | 95.15% | 99.95%
Table 2. Comparison of data augmentation strategies and contrastive learning-
based SR methods. All the results are reported as percentages for clarity and
ease of reading. The best results are indicated in bold, while the runner-up
results are marked with an underline.

| Beauty | Sports | Yelp | ML-1m
---|---|---|---|---
| Recall | NDCG | Recall | NDCG | Recall | NDCG | Recall | NDCG
Model | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20
SASRec∗ | 4.52±0.04 | 6.47±0.13 | 2.24±0.03 | 2.73±0.05 | 2.32±0.08 | 3.41±0.08 | 1.18±0.05 | 1.46±0.05 | 3.51±0.07 | 4.68±0.09 | 2.32±0.03 | 2.61±0.04 | 5.63±0.40 | 8.45±0.30 | 2.79±0.15 | 3.50±0.12
\+ subset-split | 4.78±0.08 | 6.98±0.14 | 2.40±0.02 | 2.95±0.03 | 2.61±0.08 | 3.89±0.07 | 1.32±0.07 | 1.65±0.07 | 3.77±0.04 | 5.17±0.11 | 2.43±0.07 | 2.78±0.05 | 7.57±0.56 | 11.09±0.60 | 3.68±0.26 | 4.57±0.26
\+ crop | 5.28±0.10 | 7.72±0.05 | 2.61±0.05 | 3.22±0.04 | 2.86±0.03 | 4.28±0.04 | 1.41±0.03 | 1.77±0.03 | 4.17±0.14 | 5.79±0.13 | 2.69±0.05 | 3.10±0.03 | 9.08±0.42 | 13.51±0.48 | 4.23±0.18 | 5.35±0.19
\+ delete | 4.83±0.12 | 7.10±0.04 | 2.39±0.09 | 2.96±0.06 | 2.52±0.06 | 3.80±0.08 | 1.26±0.04 | 1.59±0.05 | 3.73±0.08 | 5.11±0.13 | 2.42±0.03 | 2.76±0.01 | 6.75±0.12 | 9.59±0.26 | 3.29±0.03 | 4.01±0.09
\+ mask | 4.26±0.06 | 6.29±0.21 | 2.07±0.05 | 2.58±0.08 | 2.13±0.04 | 3.17±0.04 | 1.06±0.03 | 1.32±0.03 | 3.28±0.11 | 4.38±0.13 | 2.30±0.05 | 2.58±0.04 | 6.05±0.10 | 8.86±0.08 | 2.87±0.03 | 3.58±0.04
\+ reorder | 4.67±0.08 | 6.87±0.08 | 2.28±0.06 | 2.83±0.06 | 2.45±0.05 | 3.67±0.08 | 1.22±0.01 | 1.53±0.02 | 3.60±0.07 | 4.95±0.10 | 2.42±0.07 | 2.76±0.05 | 5.84±0.19 | 8.54±0.33 | 2.8±0.08 | 3.47±0.10
\+ insert | 4.62±0.12 | 6.79±0.11 | 2.28±0.05 | 2.83±0.05 | 2.38±0.09 | 3.54±0.13 | 1.17±0.07 | 1.46±0.08 | 3.62±0.05 | 4.87±0.05 | 2.46±0.12 | 2.78±0.11 | 6.67±0.54 | 9.76±0.27 | 3.23±0.24 | 4.01±0.16
\+ replace | 4.26±0.09 | 6.20±0.09 | 2.08±0.05 | 2.57±0.04 | 2.05±0.07 | 3.07±0.09 | 1.01±0.03 | 1.26±0.04 | 3.23±0.06 | 4.22±0.05 | 2.33±0.04 | 2.57±0.04 | 5.90±0.15 | 8.64±0.16 | 2.79±0.05 | 3.48±0.04
\+ slide-window | 8.07±0.08 | 11.56±0.07 | 3.95±0.02 | 4.83±0.01 | 5.21±0.07 | 7.7±0.06 | 2.4±0.04 | 3.03±0.04 | 6.02±0.05 | 8.62±0.02 | 3.8±0.02 | 4.46±0.02 | 22.59±0.98 | 31.88±0.11 | 12.48±0.45 | 14.83±0.24
CL4SRec∗ | 5.20±0.13 | 7.87±0.15 | 2.65±0.02 | 3.32±0.03 | 3.32±0.07 | 5.10±0.09 | 1.66±0.05 | 2.11±0.05 | 4.70±0.08 | 6.42±0.13 | 2.89±0.08 | 3.33±0.06 | 5.65±0.23 | 8.51±0.27 | 2.73±0.11 | 3.45±0.16
CoSeRec∗ | 4.72±0.06 | 7.01±0.07 | 2.31±0.04 | 2.89±0.04 | 2.71±0.06 | 4.12±0.03 | 1.33±0.03 | 1.68±0.01 | 4.03±0.07 | 5.53±0.10 | 2.65±0.03 | 3.03±0.03 | 5.83±0.21 | 8.47±0.29 | 2.82±0.09 | 3.49±0.11
ICLRec∗ | 4.84±0.05 | 7.14±0.12 | 2.42±0.05 | 3.01±0.06 | 2.66±0.03 | 4.00±0.09 | 1.30±0.03 | 1.64±0.03 | 3.74±0.03 | 5.10±0.05 | 2.54±0.02 | 2.88±0.03 | 5.87±0.1 | 8.74±0.2 | 2.78±0.03 | 3.5±0.08

Table 3. Comparison of slide-window (SW) combined with different data
augmentation strategies or contrastive learning methods. All the results are
reported as percentages for clarity and ease of reading. The best results are
indicated in bold, while the runner-up results are marked with an underline.

| Beauty | Sports | Yelp | ML-1m
---|---|---|---|---
| Recall | NDCG | Recall | NDCG | Recall | NDCG | Recall | NDCG
Model | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20 | @10 | @20
SASRec∗ \+ SW | 8.07±0.08 | 11.56±0.07 | 3.95±0.02 | 4.83±0.01 | 5.21±0.07 | 7.7±0.06 | 2.4±0.04 | 3.03±0.04 | 6.02±0.05 | 8.62±0.02 | 3.8±0.02 | 4.46±0.02 | 22.59±0.98 | 31.88±0.11 | 12.48±0.45 | 14.83±0.24
\+ subset-split | 8.48±0.14 | 12.04±0.2 | 4.2±0.08 | 5.1±0.1 |  5.54±0.02 |  8.32±0.08 | 2.57±0.02 | 3.27±0.02 | 6.28±0.06 | 9.05±0.12 | 3.91±0.02 | 4.6±0.04 | 22.89±0.38 | 31.78±0.31 | 12.66±0.3 | 14.9±0.28
\+ crop | 7.13±0.02 | 10.24±0.06 | 3.59±0.02 | 4.38±0.02 | 4.4±0.01 | 6.68±0.04 | 2.07±0.01 | 2.64±0.01 | 5.23±0.07 | 7.38±0.08 | 3.44±0.04 | 3.98±0.03 | 20.59±0.21 | 29.55±0.22 | 11.35±0.12 | 13.61±0.1
\+ delete |  8.33±0.07 | 12.02±0.09 |  4.11±0.06 | 5.04±0.06 | 5.19±0.14 | 7.84±0.21 | 2.38±0.06 | 3.05±0.08 | 6.1±0.07 | 8.83±0.03 | 3.84±0.02 | 4.52±0.01 |  23.29±0.58 |  32.12±0.55 |  12.96±0.29 |  15.19±0.31
\+ mask | 7.5±0.16 | 10.9±0.08 | 3.66±0.07 | 4.52±0.05 | 4.5±0.02 | 6.82±0.03 | 2.05±0.02 | 2.64±0.03 | 5.74±0.13 | 8.29±0.15 | 3.64±0.06 | 4.28±0.07 | 22.93±0.25 | 32.05±0.17 | 12.75±0.09 | 15.06±0.1
\+ reorder | 7.95±0.07 | 11.53±0.07 | 3.88±0.05 | 4.78±0.05 | 4.96±0.06 | 7.53±0.2 | 2.26±0.05 | 2.91±0.08 | 6.03±0.04 | 8.64±0.06 | 3.79±0.02 | 4.45±0.03 | 23.15±0.02 | 32.11±0.2 | 12.9±0.17 | 15.16±0.2
\+ insert | 8.45±0.18 |  12.05±0.17 | 4.17±0.1 |  5.07±0.07 | 5.34±0.16 | 8.1±0.22 | 2.48±0.08 | 3.17±0.1 | 6.21±0.09 | 9.16±0.1 | 3.87±0.03 | 4.61±0.03 | 22.87±0.12 | 31.84±0.1 | 12.7±0.1 | 14.96±0.07
\+ replace | 7.25±0.09 | 10.46±0.11 | 3.52±0.02 | 4.33±0.02 | 4.3±0.13 | 6.53±0.05 | 1.96±0.05 | 2.52±0.02 | 5.33±0.07 | 7.76±0.2 | 3.44±0.05 | 4.05±0.09 | 22.58±0.21 | 31.6±0.54 | 12.62±0.3 | 14.9±0.39
CL4SRec∗ \+ SW | 7.68±0.13 | 11.4±0.1 | 3.78±0.08 | 4.72±0.07 | 5.16±0 | 7.98±0.05 | 2.51±0.04 |  3.23±0.04 | 6.37±0.03 | 9.56±0.07 | 3.57±0.06 | 4.37±0.06 | 21.66±0.55 | 31.98±0.77 | 11.37±0.33 | 13.97±0.37
CoSeRec∗ \+ SW | 8.24±0.04 | 12.07±0.18 | 3.99±0.08 | 4.96±0.03 | 5.5±0.05 | 8.26±0.03 | 2.52±0.02 | 3.21±0.01 | 6.93±0.11 | 10.08±0.1 | 4.13±0.06 | 4.91±0.06 | 21.85±0.41 | 31.48±0.88 | 10.82±0.7 | 13.25±0.57
ICLRec∗ \+ SW | 8.28±0.05 | 12±0.05 | 3.93±0.02 | 4.87±0.02 | 5.65±0.01 | 8.34±0.02 |  2.55±0.01 |  3.23±0.01 |  6.78±0.07 |  9.92±0.08 |  4.02±0.02 |  4.8±0.02 | 24.04±0.35 | 32.78±0.23 | 13.71±0.25 | 15.91±0.31

##  4\. Experiment

In this section, we carry out extensive experiments to answer the following
main research questions:

  * •

RQ1: How do different sequence-level augmentation strategies compare against
state-of-the-art contrastive learning based SR methods?

  * •

RQ2: How do sequence-level augmentation and contrastive learning methods
perform in cold-start scenarios?

  * •

RQ3: How do sequence-level augmentation and contrastive learning methods
perform under varying levels of item popularity?

  * •

RQ4: Does the size of augmentations affect the performance of sequence-level
augmentation?

  * •

RQ5: How is the efficiency of sequence-level data augmentation compared to
contrastive learning methods?

###  4.1. Experimental Settings

####  4.1.1. Dataset.

Experiments are carried out on four well-known benchmark datasets with diverse
distributions: Beauty and Sports are derived from Amazon review
datasets111http://jmcauley.ucsd.edu/data/amazon/ (McAuley et al., 2015);
Yelp222https://www.yelp.com/dataset is a renowned business recommendation
dataset; ML-1m333https://grouplens.org/datasets/movielens/1m/ is a popular
movie rating dataset containing 1 million ratings from 6,000 users on 4,000
movies. We preprocess these datasets uniformly following the methodology in
(Kang and McAuley, 2018a; Zhou et al., 2020; Yuan et al., 2021; Tang and Wang,
2018a; Yuan et al., 2019; Liu et al., 2021a) by eliminating items and users
with less than 5 occurrences. Table 1 presents the dataset characteristics
after preprocessing.

####  4.1.2. Evaluation Metrics.

In our experiment, we select Recall@$K$ and NDCG@$K$ as the evaluation metrics
for different methods, where $K$ can be either 10 or 20. These two metrics are
widely adopted in existing sequential recommendation research (Sun et al.,
2019; Xie et al., 2022; Liu et al., 2021a; Chen et al., 2022). Regarding
dataset partitioning, we utilize a leave-one-out approach: the final two items
within each user interaction sequence are allocated to the validation and test
sets, respectively, while the remainder of the items is utilized for model
training. To ensure fair comparison, we follow the suggestion of (Krichene and
Rendle, 2020; Dallmann et al., 2021) to calculate the ranking results over the
complete item set rather than a sampled subset.

####  4.1.3. Baseline Models.

The performance of eight data augmentation strategies is evaluated based on
SASRec (Kang and McAuley, 2018a). The details of these data augmentation
strategies have been described in Sec. 3. In addition, we select three
representative SR methods as our baselines, all of which are based on
contrastive learning: (1) CL4SRec (Xie et al., 2020): The first work to employ
contrastive learning for sequential recommendation, it utilizes three
sequence-level augmentation methods for generating positive pairs. (2) CoSeRec
(Liu et al., 2021a): Different from CL4SRec, this contrastive learning
approach further considers the associations between items to enhance the
quality of the positive pairs. (3) ICLRec (Chen et al., 2022): A contrastive
learning method that adopts clustering technique to capture users’ latent
intent.

####  4.1.4. Implementation Details.

We use RecBole (Zhao et al., 2022), a unified platform widely adopted in
academic research, to implement and evaluate all of these baselines. In
RecBole, slide-window is enabled by default and the augmented data is used for
training both SASRec and CL baselines. To facilitate a unified and fair
evaluation of different data augmentation strategies, we turn off the slide-
window for these models and adds an ”$*$” after the model name to distinguish
them from their original setting444 For example, SASRec∗ only uses the last
item of the training instance as a supervision signal, while SASRec∗+SW can be
seen as an approximation of the original SASRec, wherein each item in the
training sequence serves as a label to guide model training.. The models are
trained with the Adam optimizer for 300 epochs, employing a batch size of 1024
and a learning rate of 0.001. For Beauty, Sports, and Yelp datasets, the
maximum sequence length is set to 50, while for the ML-1m dataset, it is set
to 200 due to its longer average sequence length. For attention-based methods,
we conduct a grid search on hyper-parameters to identify the optimal
combination. The searching space is: number of self-attention layers
$\in\left\\{2,3\right\\}$, number of self-attention heads
$\in\left\\{2,4\right\\}$,dropout rate on the embedding matrix and attention
matrix $\in\left\\{0.1,0.2,0.3,0.5\right\\}$, hidden size
$\in\left\\{64,128,256\right\\}$ and embedding
$\in\left\\{64,128,256\right\\}$. Regarding data augmentation approaches, the
slide-window has a length of 50 for Beauty, Sports, and Yelp, and 200 for
ML-1m to accommodate the varying average sequence lengths. Other
hyperparameters for data augmentation are as follows: for insert, replace,
delete, and mask, a single item is inserted, replaced, deleted, or masked; the
dropout factor $\theta$ of subset split is set to 0.25; the length of sub-
sequence in crop and reorder is set to 2. The selection of augmentation
positions and augmented items in the aforementioned data augmentation
strategies is obtained through random sampling according to a uniform
distribution. For the slide-window strategy, the size of augmentations depends
on the original sequence length and window length. For other augmentation
strategies, their size of augmentations is regarded as a hyperparameter
$n$555Note that $n$ counts both the original sequence and its $n-1$
augmentations for each individual sequence. . We discuss the impact of the
size of augmentations $n$ in Sec. 4.4.2. All baselines and our backbone are
carefully tuned on the used datasets for best performance. To ensure the
reliability of the results, each baseline in the benchmark is trained 5 times
with different random seeds, and the mean value and standard deviation (Mean ±
std) are reported in main tables (Tab. 2 and Tab. 3).

###  4.2. Overall Performance (RQ1)

####  4.2.1. Performance of single data augmentation strategy

In this section, we explore the impact of eight sequence-level augmentation
strategies on recommendation performance and compare them with three classic
contrastive learning-based SR models. Specifically, for each sequence-level
augmentation strategy, we select SASRec∗ as the backbone. Each instance in the
training set undergoes augmentation twice using the corresponding strategy.
The results are presented in Tab. 2, and we draw the following observations:

Most sequence-level data augmentations can improve the performance of SASRec.
Among the eight augmentation strategies, slide-window yields the best results,
followed by crop. Specifically, slide-window achieves average relative
performance improvements of 96.2% and 85.1% in terms of Recall@20 and NDCG@20,
respectively, on datasets with shorter average sequence lengths (Beauty,
Sports, and Yelp). Furthermore, on the ML-1m dataset with longer
sequences,slide-window demonstrates more significant improvements, with
Recall@20 and NDCG@20 increasing by 2.8x and 3.2x. Conversely, mask and
replace perform poorly as data augmentation methods, reducing the performance
of SASRec on Beauty, Sports, and Yelp datasets. This can be attributed to the
detrimental impact of noise introduced by these methods on model training,
particularly in shorter sequences.

Some sequence-level data augmentation strategies outperform contrastive
learning-based SR models. Among them, slide-window performs better than all
contrastive learning methods, while cropping achieves performance close to or
even surpassing that of some contrastive learning methods in most cases. It is
worth noting that, on the ML-1m dataset with longer sequence lengths, most of
the sequence-level data augmentation strategies can achieve performance
comparable to or even superior to contrastive learning-based methods.

####  4.2.2. Performance of combined data augmentation strategy

Furthermore, we evaluate the performance of combining slide-window with other
sequence-level augmentation strategies or contrastive learning methods, and
summarize the results in Tab. 3. We observe that, in most cases, the
performance of slide-window + crop/mask /replace is inferior to that of using
slide-window alone. However, the combination of slide-window with the other
four augmentation strategies, namely subset-split, delete, reorder, and
insert, leads to an improvement in recommendation performance, highlighting
the synergistic effect between these strategies. Particularly, on the Beauty
and Sports datasets, the slide-window + subset-split achieves the highest
performance among all augmentation combinations, with an average relative
improvement of 6.0% in Recall@20 and 6.8% in NDCG@20 compared to using slide-
window alone.

Contrastive learning methods also exhibit notable performance improvements
when integrated with the slide-window strategy. For instance, on the Yelp
dataset, the combination of CoSeRec and the slide-window strategy outperforms
the use of CoSeRec alone, achieving increases of 82.3% in Recall@20 and 62.1%
in NDCG@20. Similarly, on the ML-1m dataset, the combination of ICLRec and the
slide-window strategy achieves the best performance in most cases, leading to
improvements of 2.8x in Recall@20 and 3.6x in NDCG@20 compared to using ICLRec
alone. These results indicate that integrating the slide-window strategy with
various sequence-level augmentation techniques or contrastive learning-based
SR methods can further boost recommendation performance. It is noteworthy that
contrastive learning methods exhibit significant performance improvements over
pure data augmentation methods only when employed in conjunction with the
slide-window strategy. Otherwise, contrastive learning methods demonstrate
comparable or even inferior performance compared to certain data augmentation
strategies.

Figure 3. Performance improvements (Recall@20) of each data augmentation
strategy over backbone model (i.e. SASRec) on Amazon Beauty dataset. Figure 4.
Performance ranking variations of two data augmentation strategies and three
contrastive learning methods in various cold-start scenarios. Figure 5.
Performance comparison of different data augmentation and contrastive learning
methods under different item popularity. For each dataset, we select the top-
performing three data augmentation methods for comparison. Baseline denotes no
augmentation is utilized. Figure 6. Performance comparison of SASRec with and
without slide-window augmentation on target items with different popularity
levels. The x-axis denotes the average number of target item from test set as
the target item in the training set. The text next to the scatters denote
which popularity interval it belongs to. For example, for the Amazon Beauty
dataset, ”0” denotes popularity interval 0-50, ”1” represents interval 50-100,
and so on.

###  4.3. Cold-start Performance (RQ2)

To compare different data augmentation strategies and contrastive learning
methods in the cold-start scenario, we first simulate different levels of
cold-start by randomly sampling the training data at proportions of [0.1, 0.2,
0.3, 0.4, 0.5]. Then, we apply each data augmentation strategy or contrastive
learning method to the sampled training set and train the models using the
augmented data. Finally, we assess the performance of all models on the
original test set. Fig. 3 illustrates the relative performance improvement
ratios of the different methods compared to SASRec (no augmentation) on Amazon
Beauty dataset. We observe that in most cases, the less training data
available, the more significant the relative performance improvement brought
by different augmentation strategies.

Furthermore, we rank different contrastive learning methods and sequence-level
augmentation methods (slide-window and crop) based on their performance
(Recall@20) in various cold-start scenarios and depict the changes in their
rankings in Fig 4. It is evident that slide-window consistently outperforms
other sequence-level augmentation methods and contrastive learning methods
across all five cold-start settings. Interestingly, while crop performs less
favorably than contrastive learning methods on the original training set
(i.e., sampling ratio equals 100%), it surpasses all contrastive learning
methods when the sampling ratios are reduced to 10% and 20%.

###  4.4. In-depth Analysis (RQ3$\&$4$\&$5)

####  4.4.1. Performance under different levels of item popularity (RQ3)

We compare the top-performing data augmentation methods and contrastive
learning-based methods under varying item popularity on four benchmark
datasets, using Recall@20 as the performance metric. In this work, the item
popularity is defined as the frequency of each item as the target item in the
training set. With such standard, the partition of sequences with different
popularity is fixed and all data augmentation methods can be evaluated and
fairly compared using the same test sequences. As shown in Fig 5, we can
observe that all these methods, no matter direct data augmentation or
contrastive learning-based ones, tend to perform better when the target item
has higher popularity. Actually, most items in recommendation datasets are
less popular, and the higher the popularity, the fewer test data in that
group. In low popularity groups (0-50 and 50-100), which contain most items,
we observe significant performance improvement with the slide-window in all
four datasets. For other direct data augmentation strategies, such as subset
split and insert, they can also achieve comparable or even better performance
on these low popularity groups.

To further explore how simple data augmentation methods can have significant
performance gains, we use the slide-window method as an example to test the
effect of item popularity change on the model performance. In Fig 6, the ticks
on the x-axis denote the average number of target item from test set as the
target item in the training set. Addtionally, the text next to the scatters
denotes which popularity interval it belongs to, and scatter with same number
indicates they belong to the same target item popularity interval (e.g.,
scatters with text ”0” in Amazon Sports denotes they come from the first
popularity interval, namely 0-50 of Amazon Sports shown in Fig 5). As shown in
Fig 6, the variation in model performance with the number of target items is
approximately linear in the beginning, and then we can observe marginal
utility of the performance improvement (in ML-1m dataset). This result, to
some extent, indicates the model performance improvement brought by slide-
window is obtaine by increasing the number of target item from test set as the
target item in augmented training set.

####  4.4.2. Impact of the size of data augmentations $n$ (RQ4)

We compare the Recall@20 for different augmentation strategies when $n$ is set
to 2, 3, 5, and 10. Fig 7 presents the results on the Amazon Beauty dataset,
where the left plot shows the results using a single sequence augmentation,
and the right plot shows the results of slide-window augmentation combined
with other augmentation strategies. We observe that when using a single
strategy, the Recall@20 for most augmentation strategies increases with the
number of augmentations. However, when combined with slide-window
augmentation, more than half of the augmentation strategies show a decrease in
Recall@20 with larger size of augmentations. This could be attributed to the
fact that excessive random augmentations introduce too much random noise
during model training, resulting in the model’s inability to accurately
capture user interests.

Figure 7. Impact of the size of data augmentations.

####  4.4.3. Comparison of efficiency (RQ5)

As shown in Tab. 4, although the utilization of single or combined data
augmentation strategies inevitably increases the volume of training data,
their training time is still lower than contrastive learning methods. This is
because contrastive learning methods usually introduce auxiliary tasks and
complex positive view construction strategies, which increase computational
overhead. Additionally, sequence-level augmentation strategies do not increase
the model’s inference time, as they only affect the training phase of the
backbone model.

In terms of memory usage, our experiments, using Amazon Beauty as a case study
with a batch size of 256, reveal substantial disparities among the methods.
Direct data augmentation strategies, employing SASRec as the base model,
maintains a memory usage of 0.36GB, consistent with SASRec’s standalone
implementation. In contrast, the three contrastive learning baselines require
significantly higher memory allocations (CL4SRec: 0.78GB, CoSeRec: 1.37GB, and
ICLRec: 2.35GB), highlighting the trade-off between improved performance and
increased memory requirements in these approaches.

Table 4. Comparison of training time and inference time.

| Alone | +SW |
---|---|---|---
Methods | Recall@20 | Training time (s) | Recall@20 | Training time (s) | Test time (s)
No aug. | 0.065 | 335.76 | 0.114 | 691.72 | 0.20
slide-window | 0.114 | 691.72 | - | - | 0.20
subset-split | 0.070 | 463.32 | 0.120 | 1232.58 | 0.20
replace | 0.062 | 495.05 | 0.103 | 1443.56 | 0.20
reorder | 0.069 | 453.19 | 0.114 | 1766.25 | 0.20
mask | 0.063 | 510.61 | 0.109 | 1064.83 | 0.20
insert | 0.068 | 431.12 | 0.118 | 2023.55 | 0.20
delete | 0.071 | 467.31 | 0.119 | 2642.98 | 0.20
crop | 0.077 | 539.55 | 0.105 | 1894.35 | 0.20
CoSeRec∗ | 0.070 | 4650.78 | 0.117 | 13800.46 | 0.25
CL4SRec∗ | 0.079 | 3321.12 | 0.114 | 8941.15 | 0.20
ICLRec∗ | 0.071 | 1436.71 | 0.117 | 4055.04 | 0.26

##  5\. Related Work

###  5.1. Sequential Recommendation

Intensive studies about recommender systems of various real-world scenarios
found that sequential behaviors are important signals to model user
preferences. And many efforts have been devoted to leveraging sequential
behaviors to better capture behavior patterns. The very early and the most
intuitive method is adopting the Markov Chain assumption for sequential
recommendation (Rendle et al., 2010; He and McAuley, 2016), where the next
interaction is conditional on the past few interactions. Later, with the
population of deep learning, many DL-based models were proposed to model
sequential behaviors. GRU4Rec (Hidasi et al., 2016) is one of the most well-
known SR models, of which Gated Recurrent Unit (GRU) is first introduced to
model sequential behaviors. In addition, many other deep learning models were
also introduced to seek better performance, such as Recurrent Neural Network
(RNN) (Li et al., 2017; Wang et al., 2020a), Convolutional Neural Network
(CNN) (Tang and Wang, 2018b; Jiang et al., 2023), Graph Neural Network (GNN)
(Chang et al., 2021; Guo et al., 2022; Zhang et al., 2023b; Wang et al.,
2020b, 2022a), and Multilayer Perceptron (MLP) (Zhou et al., 2022). Except for
the aforementioned models, attention-based models have also been intensively
studied and widely adopted in sequential recommendation tasks (Kang and
McAuley, 2018b; Sun et al., 2019; Zhang et al., 2023a). Besides, there are
many interesting ongoing works focusing on other techniques like contrastive
learning (Xie et al., 2020; Liu et al., 2021a; Chen et al., 2022; Zhou et al.,
2023b, c), reinforcement learning (Xin et al., 2020), multi-interest learning
(Xie et al., 2023), large language model (Liu et al., 2023a, b; Zhou et al.,
2023a) and relation awareness (Ji et al., 2020).

###  5.2. Contrastive Learning for Recommendation

Contrastive Learning (CL) aims to improve the quality of representations by
reducing the distance between positive views generated from the same data
instance while separating them from negative views in a latent space. In the
field of sequential recommendation, sequence-level data augmentation or
feature-level data augmentation is often used to create positive views, with
augmented views of other data instances in the same training batch serving as
negative views. For instance, CL4SRec (Xie et al., 2020) employed three
sequence-level data augmentation techniques, namely cropping, masking, and
reordering, to construct positive views. Subsequently, CoSeRec (Liu et al.,
2021a) proposed to generate robust augmented sequences based on item
correlations. To mitigate the representation degradation, DuoRec (Qiu et al.,
2022) utilized feature-level augmentation based on dropout to better maintain
semantic consistency between positive views. Despite these methods claiming
that contrastive learning can significantly enhance the performance of
recommender systems, they do not consider direct data augmentation as a
baseline and thus cannot ascertain whether contrastive learning has a distinct
advantage in mitigating data sparsity compared direct data augmentation.

###  5.3. Data Augmentation for Recommendation

Data augmentation is an effective method to improve the performance of DL-
based models, particularly when the training data is scarce. In CV and NLP,
data augmentation has drawn much attention and is widely adopted in model
training. However, as for recommender systems, compared with CV or NLP,
studies regarding data augmentation are still at a rather rudimentary stage.

For sequential recommendation tasks, basic data augmentation approaches create
augmented sequences out of the original sequences themselves through simple
transformations (e.g., crop, reorder), small perbulation (e.g.,
noise/redundancy injection, synonym replacement (Song and Suh, 2022)), or
subset selection (e.g., slide-window (Tang and Wang, 2018b), subset split (Tan
et al., 2016), and item masking (Song and Suh, 2022)). Recent works regarding
the aforementioned approaches focus on time-aware approaches, which better
retain time coherence between the augmented sequences and the original ones,
and further enhance the model’s performance (Dang et al., 2022; Petrov and
Macdonald, 2022). Except for the aforementioned basic approaches, some data
augmentation approaches choose to create highly plausible sequences by
synthesizing and injecting/prepending fake samples into the original sequence
(Huang et al., 2021; Liu et al., 2021b; Jiang et al., 2021), or modeling
counterfactual data distribution (Zhang et al., 2021; Wang et al., 2021).
Apart from being applied in the sequential recommendation, data augmentation
techniques are also applied in collaborative filtering to alleviate the data
sparsity problem (Wang et al., 2019b) or bypass negative sampling (Lee et al.,
2021) during training.

The work most related to ours is (Song and Suh, 2022), which explored the
impact of four augmentation strategies on sequential recommendation, namely
noise injection, redundancy injection, item masking, and synonym replacement.
Different from it, our work benchmarked various direct data augmentation
methods and contrastive learning methods, providing comprehensive analysis of
the effectiveness of sequence-level data augmentation in sequential
recommendation research, and offers insights into the improvements achieved.

##  6\. Conclusion and future work

In this paper, we benchmark eight widely used sequence-level data augmentation
strategies, as well as three state-of-the-art contrastive learning SR methods,
on four real datasets under both full data and cold-start settings. The
results reveal that the performance of SRS can be improved by solely relying
on some data augmentation strategies without using contrastive learning
paradigm. Moreover, certain sequence-level augmentation strategies can achieve
comparative or even superior performance compared to some contrastive
learning-based SR methods, while requiring less computational resources. In
the future, we will extend the scope of benchmarking to include a broader
range of data augmentation strategies and contrastive learning methods,
providing theoretical justification for the effectiveness of direct data
augmentation.

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# Technosignatures longevity and Lindy’s law

Amedeo Balbi  Dipartimento di Fisica, Università di Roma Tor Vergata, Via
della Ricerca Scientifica, 00133 Roma, Italy  Claudio Grimaldi  Laboratory of
Statistical Biophysics, Ecole Polytechnique Fédérale de Lausanne - EPFL, 1015
Lausanne, Switzerland  Centro Studi e Ricerche Enrico Fermi, 00184 Roma, Italy

###### Abstract

The probability of detecting technosignatures (i.e. evidence of technological
activity beyond Earth) increases with their longevity, or the time interval
over which they manifest. Therefore, the assumed distribution of longevities
has some bearing on the chances of success of technosignature searches, as
well as on the inferred age of technosignatures following a first contact.
Here, we investigate the possibility that the longevity of technosignatures
conforms to the so-called Lindy’s law, whereby, at any time, their remaining
life expectancy is roughly proportional to their age. We show that, if Lindy’s
law applies, the general tenet that the first detected technosignature ought
to be very long lived may be overruled. We conclude by discussing the number
of emitters that had to appear, over the history of the Galaxy, in order for
one of them to be detectable today from Earth.

##  1 Introduction

In the modern search for extraterrestrial intelligence (SETI), the term
“technosignature” refers to any remotely detectable evidence of technological
activity (Tarter, 2006; Socas-Navarro et al., 2021). This is not limited to
the fingerprints of (intentional or unintentional) interstellar communication
that have constituted the goal of traditional radio (Margot et al., 2019) and
optical (Schwartz & Townes, 1961; Wright et al., 2018) searches, but can
include many other possible evidences. For example, the infrared glow emitted
by Dyson spheres or other astroengineering megastructures harvesting the
energy of stars (Dyson, 1960; Wright, 2020), the signature of stellar transit
from such structures (Wright & Kipping, 2019) or from artificial satellites
(Socas-Navarro, 2018), and the electromagnetic radiation associated to such
technological activities as asteroid mining (Forgan & Elvis, 2011), artificial
illumination on planetary night-sides (Lingam & Loeb, 2017a), atmospheric
pollutants (Lin et al., 2014; Kopparapu et al., 2021; Haqq-Misra et al.,
2022a, b), interstellar propulsion (Lingam & Loeb, 2017b), leftover artifacts
(Haqq-Misra & Kopparapu, 2012; Davies & Wagner, 2013), etc.

It has long been recognized that the probability of detecting a
technosignature depends on its longevity, $L$, that is, the timespan over
which it manifests; in essence, the probability of detection grows with $L$,
because long-lasting technosignatures are visible over a larger volume of
space than short-lived ones, and thus have a higher chance of being
intercepted from our location at the present epoch (Grimaldi, 2017; Balbi,
2018). Since the event of detection weighs more favorably technosignatures
with large $L$, it has been recently proposed that first contact will be with
older civilizations than our own (Kipping et al., 2020). More generally, the
longevity of detected technosignatures could exceed millions of years, as it
is decoupled from both the epoch of their first appearance and the duration of
the civilization that produced them (Balbi & Ćirković, 2021).

Of course, inferring the longevity of a detected technosignature relies on the
underlying probability distribution function (PDF) of $L$, denoted
$\rho_{L}(L)$: this is independent of whether a technosignature is detected or
not. On the other hand, $L$ is an unknown parameter that has been the subject
of much speculation since the early days of SETI. The difficulty lies mainly
in the lack of an a priori understanding of what would be the typical duration
of technosignatures, even at the order-of-magnitude level. Actually, it is not
even clear that the notion of “typical” can be applied, since the possible
values of $L$ could be so widely distributed that their average may not be
descriptive at all. Because of such lack of knowledge about the underlying
distribution of $L$, $\rho_{L}(L)$ is usually taken to be as uninformative as
possible. For example, Balbi & Ćirković (2021) used a log-uniform distribution
of $L$ ($\rho_{L}(L)\propto 1/L$) which reflects the lack of information on
$L$ even at the order-of-magnitude level, whereas Kipping et al. (2020)
derived a uninformative PDF of the longevity by adopting an objective
Jeffrey’s prior for the shape parameter of an exponential PDF of $L$.

However, a complete agnosticism on the prior distribution of $L$ may not
always be justified. For example, it can be argued that, for a broad class of
technosignatures, longevity is necessarily limited by the energy required for
their operation. We term this subclass of technosignatures technoemissions, as
they imply some sort of activity necessary for their operation and
maintainance, as opposed to “passive” technosignatures (such as various forms
of relic artifacts, pollutants, etc.) that could in principle last for a long
time without direct intervention. An obvious example of this sort are radio
transmitters, which, for a given power required for interstellar transmission,
would consume an amount of energy at least proportional to $L$ (here we take
in consideration also the energy required for the maintenance of the
transmitter and of all the infrastructures required for its functioning).
High-gain antennas or intermittent isotropic transmitters require less power
(Benford et al., 2010; Gray, 2020), but the energy used still grows with
longevity. Dysonian megastructures may in principle radiate for millions of
years or more, but their functioning requires nevertheless that energy is
spent in maintenance over such long periods of time (even though they may
still be remotely detectable long after the extinction of their constructors;
see Ćirković et al., 2019).

These considerations suggest that, for technosignatures meeting the
aforementioned condition (specifically, energy consumption increasing with
longevity in some unspecified way), we should not expect long-lived ones to
outnumber or be as common as short-lived ones. The rationale here is that
long-lasting technoemissions will be disfavored because of high energy
demands, and their production will require to overcome substantial
technological challenges (for example by improving efficiency, by exploiting
new sources of energy or new physics discoveries).

While we do not expect all extraterrestrial species to have overcome such
technological challenges, those that have successfully done so and managed to
generate technoemissions for a long time will likely be able to produce them
for a long time in the future as well.

The latter argument suggests that the probability distribution of longevities
can be shaped by the so-called “Lindy effect”, or “Lindy’s law” (Mandelbrot,
1984; Taleb, 2012; Eliazar, 2017; Ord, 2023): this is a common (albeit
counterintuitive) phenomenon observed in technology, whereby the future life
expectancy of an innovation is proportional to its current age, leading to a
longevity that increases with time. In other words, technologies that have
been proved to function for a long time are expected to also have a long
projected duration (roughly proportional to their current age), and outlive
newly adopted ones. This generates a power-law, fat-tailed distribution of $L$
of the form:

$\rho_{L}(L)\propto L^{-\alpha},$ (1)

where the positive definite exponent $\alpha$ weights the population of long-
lasting technoemissions compared to short-lived ones. For example, $\alpha=1$
reduces Eq. (1) to the uninformative log-uniform PDF mentioned above, which
claims complete ignorance about even the order of magnitude of $L$, whereas
$\alpha>1$ implies that longer longevities are disfavoured over to shorter
ones, in line with our argument that $L$ is constrained by energy demands.

Of course, in shaping the tecnoemission longevities in terms of a Lindy
process we are implicitly assuming that alien technologies follow similar
evolutionary paths that terrestrial ones. However, as pointed out by Ord
(2023), the Lindy effect arises from a wide variety of mechanisms, and can
provide useful epistemic guidance in cases when little information on
lifespans exists.

Our main purpose here is to study how the adoption of such power-law models
for $\rho_{L}(L)$ reflects on the inferred longevity of technosignatures after
a detection. Our focus on power-law probability distributions also stems from
the observation that they are found to govern a variety of situations
involving ranking, such as city population, number of citations, word
frequency, earthquake magnitude, wealth distribution, and so on (see, e.g.,
Newman, 2005). In addition, as we will see, power-law distributions are
flexible enough to model a wide range of behaviors for $L$.

Next, we describe in more detail our statistical model and explore its
consequences.

##  2 The model

In formulating our model, we assume that technoemissions are generated by
statistically independent sources located in the Galaxy at random positions
$\mathbf{r}$ with respect to the galactic center. In addition, we adopt the
rather reasonable hypothesis that the generation rate of technoemissions is
independent of the spatial distribution of the emitters, so that the expected
number of technoemissions per unit time and volume can be factorized as
$\Gamma(t)\rho_{E}(\mathbf{r})$, where $\Gamma(t)$ gives the technoemission
rate at a time $t$ before present and $\rho_{E}(\mathbf{r})$ is the PDF of the
emitter position, defined such that $\rho_{E}(\mathbf{r})d\mathbf{r}$ is the
probability of an emitter being within the element volume $d\mathbf{r}$
centered around $\mathbf{r}$. Here, we assume that $\rho_{E}(\mathbf{r})$ is
proportional to the distribution of stars in the galactic thin disk and adopt
the following axisymmetric PDF (Grimaldi, 2017; Grimaldi & Marcy, 2018):

$\rho_{E}(\mathbf{r})=\frac{\exp(-r/r_{s})\exp(-|z|/z_{s})}{4\pi
r_{s}^{2}z_{s}}$ (2)

where $r$ is the radial distance from the galactic center, $z$ is the height
from the galactic plane, $r_{s}=8.15$ kly, and $z_{s}=0.52$ kly.

Since our analysis will focus on the temporal dependence of $\Gamma(t)$ rather
than on its absolute value, we introduce the (unconditional) PDF of $t$
defined as $\rho_{t}(t)=\Gamma(t)/\bar{N}$, where
$\bar{N}=\int_{0}^{T_{G}}\\!dt\,\Gamma(t)$ gives the average number of
emitters ever existed since the birth of the Galaxy $T_{G}\sim 10^{10}$ yr
ago. Here, we mainly adopt two models for $\rho_{t}(t)$: a PDF uniform in $t$,
which corresponds to assuming a stationary rate in the interval $0\leq t\leq
T_{G}$, and a PDF that is uniform in the logarithm of $t$, $\rho_{t}(t)\propto
1/t$, in the interval $t\in[T_{\textrm{min}},T_{G}]$, where
$T_{\textrm{min}}\neq 0$ makes the log-uniform prior integrable. Since
$T_{\textrm{min}}$ is not expected to be smaller than the travel time of a
photon from the nearest extrasolar system, $\approx 1$ yr, in the following we
take $T_{\textrm{min}}=1$ yr.

As already pointed out in previous studies (Grimaldi, 2017; Balbi, 2018;
Grimaldi & Marcy, 2018; Balbi & Ćirković, 2021), a necessary condition for any
technoemission to be potentially detectable is that it must be located within
our past light cone, that is, for a signal generated in $\mathbf{r}$ at time
$t$ to be observable, the following condition must be fulfilled:

$t-L\leq|\mathbf{r}-\mathbf{r}_{o}|/c\leq t,$ (3)

where $|\mathbf{r}-\mathbf{r}_{o}|$ is the distance of the emitter from the
Earth, located at the vector position $\mathbf{r}_{o}$ (with
$|\mathbf{r}_{o}|\simeq 27$ kly), and $c$ is the speed of light. We note that
in the case of isotropic technoemissions, Eq. (3) is equivalent to requiring
that at present time the Earth is within a spherical shell region of outer
radius $ct$ and thickness $cL$ filled by the emitted electromagnetic radiation
(Grimaldi, 2017; Grimaldi & Marcy, 2018).

The causal constraint of Eq. (3) introduces a selection effect for the
observable technoemissions, which can profoundly alter the expected longevity
of a detected signals relative to the expected $L$ coming from the underlying
distribution $\rho_{L}(L)$. Indeed, as pointed out in Balbi & Ćirković (2021),
from Eq. (3) it follows that only technosignals of longevity greater than
$t-|\mathbf{r}_{o}-\mathbf{r}|/c$ are detectable in principle, which leads to
two important consequences. First, it introduces a correlation between the
otherwise statistically independent variables $L$ and $t$ and, second, it
filters out those signals that are too short-lived to cross our planet,
depending on the emitter distance and the time of emission. The very event of
detection, therefore, promotes long-lived techoemissions over short-lasting
ones, as quantitatively confirmed by Monte Carlo simulations in Balbi &
Ćirković (2021) and by a Bayesian analysis in Kipping et al. (2020), thereby
implying that the first contact will likely be with an older technology than
our own.

As remarked in the Introduction, however, there are arguments suggesting that
energy and cost constraints and expected lifetimes of technologies might
actually make long-lived technoemissions a priori less likely than short-lived
ones, leading one to wonder what would then be the longevity of a detected
signal and the age of its emitter. To address this issue, we assume that a
search for technosignatures has been successful in detecting a signal coming
from an emitter within a distance $R_{o}$ from Earth. Although $R_{o}$ depends
on both the technoemission characteristics and detector specifications
(Grimaldi & Marcy, 2018), in the following we treat it as a free parameter
representing the maximum distance the hypothetical search is able to sample.

We follow Balbi & Ćirković (2021) and equate the probability of detecting an
emission of longevity $L$ generated at time $t$ with the joint probability
that the technoemission fullfils the causal condition (3) and that
$|\mathbf{r}-\mathbf{r}_{o}|\leq R_{o}$:

$P(t,L;\mathcal{D})=\int\\!d\mathbf{r}\rho_{E}(\mathbf{r})\theta(R_{o}-|\mathbf{r}-\mathbf{r}_{o}|)f_{t,L}(\mathbf{r}-\mathbf{r}_{o}),$
(4)

where $f_{t,L}(\mathbf{r}-\mathbf{r}_{o})=1$ if Eq. (3) is satisfied and
$f_{t,L}(\mathbf{r}-\mathbf{r}_{o})=0$ otherwise, $\theta(x)$ is the Heaviside
step function, and $\mathcal{D}$ indicates the event of detection, whose
probability of occurrence is obtained by marginalizing Eq. (4) over the
underlying distributions $\rho_{L}(L)$ and $\rho_{t}(t)=\Gamma(t)/\bar{N}$:

$P(\mathcal{D})=\int\\!dt\rho_{t}(t)\int\\!dL\rho_{L}(L)P(t,L;\mathcal{D}).$
(5)

Finally, partial marginalization of Eq. (4) with respect to $t$ and $L$ and
the use of Bayes’ theorem allow us to write the PDFs of, respectively, $L$ and
$t$, conditional on the detection:

$\rho_{L}(L|\mathcal{D})=\frac{\rho_{L}(L)\int\\!dt\rho_{t}(t)P(t,L;\mathcal{D})}{P(\mathcal{D})},$
(6)
$\rho_{t}(t|\mathcal{D})=\frac{\rho_{t}(t)\int\\!dL\rho_{L}(L)P(t,L;\mathcal{D})}{P(\mathcal{D})}.$
(7)

Note that the above equations can be viewed as a Bayesian inference of $L$ and
$t$, where $\rho_{L}(L)$ and $\rho_{t}(t)$ are identifiable as the prior PDFs
from which the posteriors $\rho_{L}(L|\mathcal{D})$ and
$\rho_{t}(t|\mathcal{D})$ are inferred, given the event of detection.

Figure 1: PDF of technoemission longevity $L$ (top row) and starting time $t$
of emissions (bottom row) calculated under the assumption that the
unconditional PDF of $t$ is uniform over the interval $[1,10^{10}]$ yr. Panels
(a)-(c) show the conditional, Eq. (9) and unconditional, Eq. (8), PDFs of $L$
for three values of the exponent $\alpha$, whereas panel (d) shows the
corresponding values of $L^{*}$ such that $L\leq L^{*}$ with 95 % probability.
Panels (e) and (f) show the PDF of $t$ calculated numerically for different
values of $\alpha$ and (e) for $R_{o}$ large enough to encompass the entire
galaxy and (f) for $R_{o}=100$ ly. Panel (g) shows the values of $t^{*}$ such
that $t\leq t^{*}$ with 95 % probability calculated for different values of
$R_{o}$.

As mentioned in the Introduction, we take the underlying PDF of the longevity
to be shaped by the Lindy effect and adopt for $\rho_{L}(L)$ the following
power-law distribution:

$\rho_{L}(L)=\left\\{\begin{array}[]{ll}\lambda
L^{-\alpha}&L_{\textrm{min}}\leq L\leq L_{\textrm{max}}\\\
0&\textrm{otherwise}\end{array}\right.$ (8)

where
$\lambda=(1-\alpha)/(L_{\textrm{max}}^{1-\alpha}-L_{\textrm{min}}^{1-\alpha})$
is a normalization constant, and $L_{\textrm{min}}=1$ yr and
$L_{\textrm{max}}=10^{9}$ yr are lower and upper limits on $L$ which we
introduce to make $\rho_{L}(L)$ normalizable to unity for any choice of
$\alpha$. Recall that $\alpha>1$ implies a scenario in which small values of
$L$ are favored over large ones, as should be expected if the longevity of
technoemissions is constrained by energy demands. In the following, however,
we will treat the exponent $\alpha$ as a free, positive parameter to
illustrate its effect on the longevity distribution inferred from a detection
event.

Figure 2: PDFs of $L$ (top row) and $t$ of (bottom row) calculated numerically
using an unconditional distribution of emission starting times that is log-
uniform over the interval $[1,10^{10}]$ yr. In panels (a) and (b) the
conditional PDF of $L$ (solid lines) is plotted different $\alpha$ and $R_{o}$
values. Panel (c) shows the resulting $L^{*}$ for several observation radii
$R_{o}$. Panels (d) and (e) show the PDF of $t$ calculated numerically for
different values of $\alpha$ and (d) for $R_{o}$ large and (e) for $R_{o}=100$
ly. Panel (f) shows the values of $t^{*}$ calculated for different values of
$R_{o}$.

##  3 Results

We start our analysis by first exploring the scenario in which $\rho_{t}(t)$
is uniform in $t$ ($\rho_{t}(t)=\rho_{t}$), since in this case
$\rho_{L}(L|R_{0})$ can be calculated analytically. The integration over $t$
appearing in Eqs. (5) and (6) can indeed be performed exactly, yielding:

$\rho_{L}(L|R_{o})=\frac{\rho_{L}(L)L}{\langle
L\rangle}=\frac{(2-\alpha)L^{1-\alpha}}{L_{\textrm{max}}^{2-\alpha}-L_{\textrm{min}}^{2-\alpha}},$
(9)

where $\langle L\rangle=\int\\!dLL\rho_{L}(L)$ is the unconditional average
longevity and the second equality stems from using for $\rho_{L}(L)$ the
power-law form of Eq. (8). It is apparent from Eq. (9) that the probability of
detecting a technoemission is proportional to its longevity, regardless of the
observational radius $R_{o}$ and the spatial distribution of the emitters.
This is a consequence of using a stationary emission rate, whereas, as shown
in the following, a log-uniform $\rho_{t}(t)$ does not yield such simple
scaling.

As a consequence of $\rho_{L}(L|R_{o})\propto L\rho_{L}(L)$, the event of
detection can entail a significant weight shift of $\rho_{L}(L|R_{o})$ from
small to large $L$ compared to the unconditional PDF, as shown in Figs.
(1)(a)-(b) for $\alpha=1$ and $\alpha=2$. However, for $\alpha=3$ (Fig. 1(c))
the unconditional PDF becomes so skewed toward small values of $L$ that the
causal constraint has little effect on $\rho_{L}(L|R_{o})$. Borrowing a
Bayesian terminology, we can say that for sufficiently large $\alpha$ the
prior $\rho_{L}(L)$ is so informative that completely determines the posterior
$\rho_{L}(L|R_{o})$.

To see more quantitatively how $\alpha$ affects the longevity of a detected
signal, we introduce $L^{*}$ defined as the longevity such that $L\leq L^{*}$
with $95$ % probability. For a detected signal, this is obtained by requiring
that the integral of Eq. (9) in the interval $[L_{\textrm{min}},L^{*}]$ be
equal to $0.95$, which by isolating $L^{*}$ yields:

$L^{*}=\left[L_{\textrm{min}}^{2-\alpha}+0.95(L_{\textrm{max}}^{2-\alpha}-L_{\textrm{min}}^{2-\alpha})\right]^{\frac{1}{2-\alpha}}.$
(10)

The formula for the unconditional $L^{*}$ is obtained by replacing $2-\alpha$
with $1-\alpha$ in Eq. (10). Figure 1(d) shows that for $\alpha\lesssim 1$
both the unconditional and conditional $L^{*}$ are equally large ($\sim
10^{9}$ yr), whereas in the interval $1\lesssim\alpha\lesssim 2$ the
conditional $L^{*}$ exceeds by several orders of magnitude the rapidly
decreasing unconditional $L^{*}$. It is in this range of $\alpha$ that the
event of detection has its strongest selection effect. Finally, upon an
increase of $\alpha$ beyond $\alpha=2$, the conditional $L^{*}$ rapidly drops
toward $L_{\textrm{min}}=1$ yr, following asymptotically
$L^{*}=20^{1/\alpha}L_{\textrm{min}}$.

At this point, it is worth noting that Lindy’s law with $\alpha>2$ arises from
assuming that the future lifetime expectancy of technoemissions is
proportional to their current age (see the Appendix). Should the duration of
technoemissions meet this requirement, therefore, we should not expect first
contact to occur with very long-lived signals. For example, already for
$\alpha=3$, we find from Eq. (10) that the longevity of a detected signal is
smaller than $20$ yr, with $95$ % probability (to be compared to a
unconditional longevity smaller than about $4.5$ yr). However, as shown by
Eliazar (2017), an alternative mathematical definition of the Lindy effect can
be formulated through the median (instead of the average) future lifespan
being proportional to the current age. In this case, the power-law
distribution of $L$ would have an exponent $\alpha>1$.

We turn now our attention to the conditional PDF of $t$ and its evolution with
$\alpha$, shown in Fig. 1(e) for $R_{o}$ large enough to encompass the entire
galaxy and in Fig. 1(f) for $R_{o}$ of only $100$ ly. Two features are
apparent. First, for $\alpha=1$ and regardless of the observation distance
$R_{o}$, the starting time of a detected signal is broadly distributed around
$3\times 10^{8}$ yr, as found by the Monte Carlo simulations of Balbi &
Ćirković (2021). This means that among the technoemissions that do not have
preferred scale on $L$, those that can be detected likely have very large
longevities and were generated several millions of years ago. The second
result is that by increasing $\alpha$, the age $t$ of a detectable signal can
be considerably reduced, as the weight of $\rho_{t}(t|\mathcal{D})$ is
gradually transferred to much smaller times, eventually forming a single peak
centered around $3\times 10^{4}$ yr for $R_{o}$ large and $\sim 10^{2}$ yr for
$R_{o}=100$ ly. This is understood by recognizing that for $\alpha>2$ the
technoemissions are so short lived (as shown by the dashed line in Fig. 1(c))
that the causal constraint of Eq. (3) selects only values of $t$ near
$|\mathbf{r}-\mathbf{r}_{o}|/c$, whose most probable value for the emitter
distribution of Eq. (2) is at $\min(3\times 10^{4}\textrm{yr},R_{o}/c)$.

The starting emission times of a detectable signal are summarized in Fig 1(g),
where we show the computed time $t^{*}$ such that $t\leq t^{*}$ with $95$ %
probability. Two regimes are evident: for $\alpha\lesssim 2$, where the
longevity is expected to be very large, $t^{*}$ conditioned to the event of
detection is about $10^{9}$ yr, whereas for $\alpha\gtrsim 2$ the conditional
$t^{*}$ rapidly drops by several orders of magnitude toward times that depend
on the sampled distance $R_{o}$. From the evolution of $L^{*}$ and $t^{*}$
with $\alpha$ shown in Figs. 1(d) and 1(g), we conclude therefore that the
event of detection per se does not necessarily imply that first contact will
be with very long lasting and very old technoemissions, and that relatively
young and short-lived signals have the highest probability of being detected
once $\alpha\gtrsim 2$.

The assumption of a stationary emission rate (that is, a constant
$\rho_{t}(t)$) is questionable in several respects, not least the fact that
the habitability of the galaxy is itself a function of time. Adopting a PDF of
$t$ that reproduces the main feature of the temporal dependence of the
galactic habitable zone of Lineweaver et al. (2004) (i.e., the age of planets
suitable to host complex life is normally distributed, with mean $5.5$ Gyr and
standard deviation of $2$ Gyr, with a fiducial interval of $4$ Gyr for the
appearance of a technological species; see also Balbi & Ćirković (2021))
essentially replicates the results of Fig. 1, whereas assuming a log-uniform
PDF of $t$ leads to some quantitative changes but does not substantially alter
the conclusions reported above. First, in contrast with the case of a constant
$\rho_{t}(t)$, the conditional PDF of the longevity depends on the sampled
distance $R_{o}$, as summarized in Figs. 2(a)-(c). In addition, we find that
the selection effect of Eq. (3) is less effective in promoting large
conditional longevities, as seen by the $L^{*}$ versus $\alpha$ plot of Fig.
2(c). Second, compared to the case $\rho_{t}(t)=\textrm{constant}$, an
increase of $\alpha$ entails a more rapid weight shift of the conditional PDF
of $t$ toward $\min(3\times 10^{4}\textrm{yr},R_{o}/c)$, as shown in Fig.
2(d)-(f).

Figure 3: Number of technoemissions ever generated over the history of the
Galaxy as a function of the longevity exponent $\alpha$. $\bar{N}$ is
calculated from Eq. (11) using a uniform in $t$ (a) and a log-uniform in $t$
(b) probability disribution of the time of occurrence of technoemissions.
Results are shown for different values of the sampled radius $R_{o}$.

##  4 Discussion and conclusions

The main conclusion of our analysis is that assuming different power-law
shapes for the probability distribution of technosignature longevities has a
sizable effect on the most likely duration of detected technosignatures. We
confirmed previous findings that showed that the expected longevity of
detected technosignatures will generally be very large (Kipping et al., 2020;
Balbi & Ćirković, 2021), but we showed that this is strictly true only for
rather uniformative priors, with $\alpha$ not too different from 1. In
particular, we found that in the regime $1\lesssim\alpha\lesssim 2$ the most
likely technoemissions to be detected have $L\sim 10^{9}$ yr (with $95\%$
confidence).

However, when we make the reasonable assumption that short-lived
technoemissions vastly outnumber the long-lived ones (as it is the case if
their operation has an energy or maintenance cost that increases with time),
then the first to be detected will likely have a relatively short $L$. This
starts to become apparent for power-law distributions with $\alpha\gtrsim 2$
(characteristic of a Lindy effect behavior); for $\alpha>3$ the preference for
large $L$ essentially disappears, with the most likely to be detected having
$L$ much smaller than $\sim 10^{2}$ yr (at $95\%$ confidence; Fig. 1(d)). This
is a significant departure from previous studies.

The selection effect is further reduced if we assume that the epoch of
appearance of technosignatures over the history of the Galaxy has a log-
uniform (rather than uniform) distribution in $t$. In this case, already in
the regime $\alpha\gtrsim 1$ the most likely detected longevity is well below
$\sim 10^{9}$ yr, regardless of the radius of the surveyed volume (Fig. 2c).

Figure 4: Technoemission rates calculated from $\Gamma(t)=\bar{N}\rho_{t}(t)$
for uniform, log-uniform, and normally distributed PDFs of $t$. For all cases,
$\bar{N}$ is calculated for the entire galaxy and for $\alpha=3$.

A final point is worth discussing; namely, what is the number of emitters
$\bar{N}$ that had to appear, over the history of the Galaxy, in order that,
typically, one of them satisfies the condition of Eq. (3), and is therefore
detectable today from Earth? The answer, again, depends on the assumed
distribution of $L$, and is obtained by requiring that
$\bar{N}P(\mathcal{D})=1$, where $P(\mathcal{D})$ is the detection probability
given in Eq. (5). Figure 3 shows

$\bar{N}=\frac{1}{P(\mathcal{D})}$ (11)

calculated as a function of the exponent $\alpha$ for uniform in $t$ (Fig.
3(a)) and log-uniform in $t$ (Fig. 3(b)) distributions of the unconditional
starting times of emission. In both cases, $\bar{N}$ increases by several
orders of magnitude as $\alpha$ goes from $\alpha\lesssim 1$, where it takes
only a few long-lived technoemissions to have typically one detection, to
$\alpha\gtrsim 2$, where there must be many short-lived ones for one to be
detectable. In particular, if the epoch of appearance is uniformly distributed
over the history of the Galaxy (Fig. 3(a)), then $\bar{N}\approx 10^{10}$ for
$\alpha\gtrsim 2$ and $R_{o}$ such that the entire Milky Way is sampled. This
would imply a constant rate of about one technoemission per year generated
over the last $\approx 10$ Gyr, as seen in Fig. 4 for $\alpha=3$, where we
show that a similar rate would also occur during the last $\approx 3$ Gyr if
the PDF of $t$ followed a normal distribution. Although high, rates of this
magnitude do not exceed those of some natural astrophysical phenomena, such as
star formation and novae in the Milky Way. For a log-uniform emission rate,
Fig. 3(b), we find that $\bar{N}$ can be reduced by several orders of
magnitude compared to the previous scenarios. However, since in this case
$\rho_{t}(t)\propto 1/t$, the rate would be such that $20$ % of the total
number of emissions ever produced, $\bar{N}\approx 5\times 10^{5}$, must be
generated only in the last $100$ yr.

Reducing the sampled volume inevitably increases $\bar{N}$, as seen in Fig. 3
for different values of $R_{o}$, leading to exceedingly high emission rates.
This is because $P(\mathcal{D})$ becomes small for reduced $R_{o}$, thus
leading $\bar{N}$ to increase in order to have one detectable technoemission,
especially if $\alpha$ is large. It is worth pointing out, however, that if
the emitters were less uniformly distributed than the density profile of Eq.
(2), different estimates of $\bar{N}$ would then be expected. Assuming for
example that our planet belongs to a more or less localized cluster of planets
harboring technological species, then $\bar{N}$ could be greatly reduced, in
analogy to what would happen to the number of biospheres in a panspermia
scenario (Balbi & Grimaldi, 2020).

##  Appendix A Lindy’s law

Lindy’s law derives from the empirical observation that the future lifetime of
non-perishable things such as technology increases with its current age. Here
we show that if we assume that the future life expectancy of technoemissions
is proportional to their current age, then the PDF of their longevities
follows the power-law of Eq.(6) with $\alpha>2$. In the following derivation,
we adapt the formulation of Eliazar (2017) to our notation.

We denote $\rho_{L}(L)$ the unconditional PDF of the longevity, which we treat
as a random variable $X_{L}$ defined in the interval
$[L_{\textrm{min}},L_{\textrm{max}}]$. The corresponding complementary
cumulative distribution function (also known as survival function) is
$P_{L}(L)=\int_{L}^{\infty}\\!ds\rho_{L}(s)$, from which we have also that
$\rho_{L}(L)=-P_{L}^{\prime}(L)$. In terms of the random variable $X_{L}$,
Lindy’s law can be formulated as:

$\mathbf{E}[X_{L}-L|X_{L}\geq L]=aL$ (A1)

where $a$ is a positive constant of proportionality. The left-hand side of
(A1) can be written in terms of $\rho_{L}$ and $P_{L}$ as follows:

$\displaystyle\mathbf{E}[X_{L}-L|X_{L}\geq L]$ $\displaystyle=$
$\displaystyle\dfrac{\int_{L}^{\infty}\\!ds(s-L)\rho_{L}(s)}{\int_{L}^{\infty}\\!ds\rho_{L}(s)}$
(A2) $\displaystyle=$
$\displaystyle-\dfrac{\int_{L}^{\infty}\\!\\!ds\,sP_{L}^{\prime}(s)}{P_{L}(L)}-L.$

Equating the right-hand sides of Eqs. (A1) and (A2), and derivating the
resulting equation with respect to $L$ we obtain:

$LP_{L}^{\prime}(L)=(1+a)P_{L}(L)+(1+a)LP_{L}^{(}L),$ (A3)

which can be rearranged to obtain the following differential equation for
$P_{L}$:

$\frac{P_{L}^{\prime}(L)}{P_{L}(L)}=-\frac{1+a}{a}\frac{1}{L},$ (A4)

whose solution is $P_{L}(L)\propto L^{-(1+1/a)}$, and hence using
$\rho_{L}(L)=-P_{L}^{\prime}(L)$:

$\rho_{L}(L)\propto L^{-\alpha},$ (A5)

where $\alpha=2+1/a$. Since $a>0$, we obtain therefore $\alpha>2$.

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