11institutetext: Department of Engineering Science, University of Oxford, UK
22institutetext: Department of Medicine, University of Cambridge, UK
33institutetext: Nuffield Department of Clinical Neurosciences, University of
Oxford, UK 44institutetext: Department of Computing, Imperial College London,
UK 55institutetext: School of Computer Science, University of Birmingham, UK
55email<EMAIL_ADDRESS>
# Feasibility of Federated Learning from Client Databases with Different Brain
Diseases and MRI Modalities

Felix Wagner 11 Wentian Xu 11 Pramit Saha 11 Ziyun Liang 11 Daniel Whitehouse
22 David Menon 22 Natalie Voets 33 J. Alison Noble 11 Konstantinos Kamnitsas
114455

###### Abstract

Segmentation models for brain lesions in MRI are commonly developed for a
specific disease and trained on data with a predefined set of MRI modalities.
Each such model cannot segment the disease using data with a different set of
MRI modalities, nor can it segment any other type of disease. Moreover, this
training paradigm does not allow a model to benefit from learning from
heterogeneous databases that may contain scans and segmentation labels for
different types of brain pathologies and diverse sets of MRI modalities. Is it
feasible to use Federated Learning (FL) for training a single model on client
databases that contain scans and labels of different brain pathologies and
diverse sets of MRI modalities? We demonstrate promising results by combining
appropriate, simple, and practical modifications to the model and training
strategy: Designing a model with input channels that cover the whole set of
modalities available across clients, training with random modality drop, and
exploring the effects of feature normalization methods. Evaluation on 7 brain
MRI databases with 5 different diseases shows that such FL framework can train
a single model that is shown to be very promising in segmenting all disease
types seen during training. Importantly, it is able to segment these diseases
in new databases that contain sets of modalities different from those in
training clients. These results demonstrate, for the first time, feasibility
and effectiveness of using FL to train a single segmentation model on
decentralised data with diverse brain diseases and MRI modalities, a necessary
step towards leveraging heterogeneous real-world databases. Code will be made
available at: https://github.com/FelixWag/FL-MultiDisease-MRI

###### Keywords:

multi-modal MRI brain lesions federated learning

##  1 Introduction

Brain MRI segmentation is crucial for disease analysis, diagnosis and
treatment. Traditionally, brain MRI segmentation models are trained on data
for a specific disease, acquired with a predefined protocol and set of MRI
modalities [8, 9, 30, 28]. Models trained with this standard approach cannot
segment other types of diseases, and expect the same modalities in all
training and testing data. While a recent study [26] demonstrated the
feasibility of training a segmentation model across multiple brain MRI
databases with different diseases and sets of modalities, that study used a
centralised database. It did not address the practical challenge related to
the sensitive nature of medical data, which prevents data sharing between
institutions and therefore the creation of an appropriately diverse
centralised database due to legal and privacy concerns.

To address this, this study aims to develop a Federated Learning (FL)
framework [15] for training a single model on diverse brain lesion databases,
each with a different disease and set of MRI modalities. Central questions for
this study are: How to make such FL training feasible? Will it be beneficial
compared to training a disease-specific model? Will it result in a model that
can segment new data that have different combination of modalities than the
training data?

Related Work: While previous studies investigated training with heterogeneous
databases [27, 22, 16], they have not studied FL using decentralised
databases, nor data with diverse sets of MRI modalities. A recent study [26]
investigated heterogeneous databases with diverse sets of MRI modalities in a
centralized scenario. In contrast, our study focuses on the challenging
setting of decentralized FL, which has not been explored yet. Some works have
focused on FL of a segmentation model for a single, specific disease when a
specific set of multi-modal MRI is available in each client [11, 19, 17].
Other works studied FL for learning from data with different labels in
segmentation of CT [20] or classification in MRI [6]. In summary, there are
works that focus on FL with multi-modal MRI and FL for heterogeneous tasks;
however, the combination of both, especially with different sets of modalities
per client, has never been explored.

Learning from databases with varying sets of MRI modalities relates to
research on developing multi-modal models that perform well even when a
modality is missing at inference time [4, 18, 21, 5, 31, 24]. This line of
work focuses on learning a model for a specific disease, from data with the
same set of modalities during training. While addressing missing modalities is
an important capability of our method, it is not the primary topic of this
research. Instead, this study concentrates on the effectiveness and
practicality of learning from diverse databases with different sets of input
modalities and segmentation tasks.

Contribution: This study develops a simple but practical solution for training
via FL a single model that can process different sets of modalities and
segment brain lesions of different types. We show this can be achieved by a
standard U-net, enhanced such that input channels can receive the whole set of
unique modalities available across clients and using random drop of modalities
during training to prevent associations of modalities with specific
pathologies. Additionally, due to the high data distribution shift across
diverse databases, we present an analysis of the effects of feature
normalization techniques. Experimentation on 7 brain MRI databases with 5
different pathologies and 6 MRI modalities provides the following findings:

  * •

Feasibility: This study is the first to show feasibility of Federated training
of a single segmentation model on decentralised databases with different brain
diseases and sets of MRI modalities.

  * •

Knowledge Transfer Across Pathologies: This paper demonstrates effectiveness
of FL for jointly learning to segment multiple pathologies, achieving
performance that compares favorably to disease-specific models.

  * •

Segmenting new combination of modalities: This study shows that models trained
with this FL paradigm can segment data from new databases with combination of
modalities not seen during training.

Figure 1: Method overview: (1) Initialization of model with distinct set of
modalities across clients. (2) Federated training of the model with random
modality drop. (3) Generalization to unseen clients with different input MRI
modality combination.

##  2 Method

###  2.1 Problem setting: Multi-Modal MRI FL for Multiple Tasks:

We consider $C$ multi-modal clients, where each client $c$ ($1\leq c\leq C$)
is associated with a set of MRI-modalities $\mathcal{M}_{c}$ and a dataset
$\mathcal{D}_{c}=\\{(\mathcal{X}_{ci},y_{ci})\\}_{i=1}^{|\mathcal{D}_{c}|}$,
where $\mathcal{X}_{ci}=\\{{x}_{ci}^{m}\\}_{m=1}^{|\mathcal{M}_{c}|}$ denotes
the multi-modal MRI image and $y_{ci}$ the corresponding segmentation mask of
the $i$-th sample. Here, each $x_{ci}^{m}\in\mathbb{R}^{w\times h\times d}$,
represents a 3D image. Furthermore, $\mathcal{M}=\bigcup_{i=1}^{C}M_{i}$
denotes the set that represents all unique modalities across all clients and
$p:=|\mathcal{M}|$ is the total number of unique modalities across all
datasets. Each client solves a specific segmentation task $\mathcal{T}_{c}$.
Our goal is train, without data sharing, a single global model parameterized
by $\theta$, that performs well on all tasks and clients. The global objective
is to minimize the sum of the client-local loss functions:

$\min_{\theta}[\mathcal{L}(\theta):=\frac{1}{C}\sum_{c=1}^{C}\mathcal{L}_{c}(\theta;\mathcal{D}_{c};\mathcal{T}_{c})],$
(1)

where $\mathcal{L}_{c}$ denotes the local loss function of client $c$. As loss
functions, we use a weighted combination of Dice and pixel-wise Binary Cross-
Entropy (BCE) losses
$\mathcal{L}_{c}=\alpha\mathcal{L}_{Dice}+(1-\alpha)\mathcal{L}_{BCE}$, where
$\alpha=0.8$ in our experiments. In the standard FL setting with $E$
communication rounds, a global server initializes a global model and
communicates it to every client. In each round, each client performs $\tau$
iterations of updates on its local data and loss function
$\mathcal{L}_{c}(\theta;\mathcal{D}_{c};\mathcal{T}_{c})$. Afterwards, every
client sends their updated model back to the server. The server combines these
updates, typically by averaging (FedAvg), into a new global model before
starting a new communication round [15]. This allows decentralised training
without data sharing. We want to highlight the differences from the standard
FL setting: this setting varies not only in the input space $|\mathcal{M}|>1$,
indicating multiple modalities, but also in the label space
$|\bigcup_{i=1}^{C}T_{i}|>1$, involving segmenting different brain pathologies
per client. Additionally, this setting considers different sets of modalities
per client, an aspect mostly overlooked in FL.

###  2.2 Training a Unified Model for Multiple Segmentation Tasks

Handling Different Sets of Modalities Across Clients: Since every client comes
with a different combination of MRI modalities, the challenge arises to find a
single model architecture capable of processing different sets of input
modalities. To address this, we augment each client’s input to cover the full
set of distinct MRI modalities across all clients, and fill missing modalities
$m\in\mathcal{M}-\mathcal{M}_{c}$ of each client $c$ with zeros. This
transforms the input space of each client to
$\mathcal{X}_{c}\in\mathbb{R}^{p\times w\times h\times d}$. Consequently, our
global model accepts the set of distinct MRI modalities as input and becomes a
function $f(\cdot;\theta):\mathbb{R}^{p\times w\times h\times
d}\to\mathbb{R}^{w\times h\times d}$, capable of processing each client’s
input modality combination, ensuring compatibility across all clients.

Modality Drop for Pathology Disassociation & Missing Modalities: To develop a
model that generalises to modality combinations not seen during training, we
introduce modality drop, where we randomly set input modalities to zero during
training. Therefore, we define a probabilistic masking function
$Z(\mathcal{X},\phi)$ where each input modality $x^{m}\in\mathcal{X}$ is set
to zero with probability $\phi$. This prevents the model from associating
specific modality combinations with specific pathologies, ensuring it can
accurately segment known pathologies in unseen clients with modality
combinations not seen during training. Furthermore, it also enhances the
model’s robustness against missing modalities during testing, as shown in our
evaluation.

Feature Normalization for Heterogeneous Clients: Each client’s data contains
different pathologies and modalities, which increases the difficulty of
training a model due to data heterogeneity. To tackle this, we employ Batch
Normalization (BN) [7], but keep client-specific BN parameters and statistics,
and exclude them from the model averaging step, as proposed in [1, 12]. Our
experiments show that this method proves effective in improving the
performance of the training (source) databases. However, it is not clear how
to apply the source-specific BN layer parameters/statistics to unseen (target)
clients during inference. One way involves averaging the learned BN parameters
and estimating running statistics on the target data, which has the
disadvantage that target data is needed. Therefore, in our experiments, we
also explore alternative feature normalization methods for this training
framework with inherently high data heterogeneity. Specifically, the recently
proposed Normalization Free (NF) technique [32] uses re-parameterized
convolutional layers instead of explicitly normalizing statistics of feature
activations. This does not require target data to estimate the target
statistics, alleviating this requirement of keeping client-specific BN layers.
Empirical analysis (Sec. 3) also shows that this results in methods that
generalize better to unseen clients. However, it slightly decreases
performance on source clients. Therefore, the choice of feature normalization
method in this framework should be chosen based on the target application for
the trained model (generalization to new domains or application to source
domains) and whether data are expected to have been collected a-priori from
the target domain for estimating BN statistics.

Algorithm: To summarise, our simple yet effective algorithm is shown in Fig.
1. Initially, the central server receives a list of available MRI modalities
from each client’s dataset. Afterwards, the server initializes a model
architecture that accepts the set of unique MRI modalities $\mathcal{M}$ as
input and the FL algorithm starts (Sec. 2.1). During training, each client’s
input modalities are augmented to the whole set of unique MRI modalities and
modality drop is applied.

##  3 Experiments & Results

###  3.1 Datasets & Prepocessing

We use 7 annotated brain MRI databases, each with different MRI modalities and
segmentation tasks collected from different sites: ATLAS v2.0 [13] for stroke
lesion, BRATS 2016 [2] for tumor, MSSEG [3] for multiple sclerosis, TBI from
our site for traumatic brain injury, WMH [10] for White Matter Hyperintensity,
ISLES 2015’s SISS subtask [14] for stroke lesion, and Tumor2 for tumour
segmentation from our institution. The data is split into training and
validation data. Tab. 1 gives an overview of the data sizes and modalities. We
resample the images to 1x1x1 mm, skull-strip and apply z-score intensity
normalization, and combine all lesion types across databases into a single
’lesion’ label.

Table 1: MRI databases overview: ISLES and Tumor2 are used for evaluation
only, except for single center training, where a 70-30 train-validation split
is used.

Database | Modalities | Disease/Task | #Samples | #Val. Samples
---|---|---|---|---
T1 | T1c | FLAIR | T2 | PD | SWI
BRATS | ✓ | ✓ | ✓ | ✓ |  |  | Tumor | 484 | 40
MSSEG | ✓ | ✓ | ✓ | ✓ | ✓ |  | Multiple Sclerosis | 53 | 16
ATLAS | ✓ |  |  |  |  |  | Stroke Lesion | 654 | 195
TBI | ✓ |  | ✓ | ✓ |  | ✓ | Traumatic Brain Injury | 281 | 125
WMH | ✓ |  | ✓ |  |  |  | White Matter Hyperintensity | 60 | 18
ISLES | ✓ |  | ✓ | ✓ |  |  | Stroke Lesion | 28 | 28
Tumor2 | ✓ |  |  |  |  |  | Tumor | 57 | 57

###  3.2 Experimental Setup

Setup & Methods: We use the Residual U-Net [29] as network architecture and
Adam optimizer. We linearly scale the learning rate from 0 to $10^{-3}$ over
50 communication rounds, keep it constant until communication round 150, and
then linearly decay it. We set $\tau=58$ (equivalent to 1 epoch for the
largest database, ATLAS), $E=300$ (or 300 epochs for centralised training),
and use a batch size of 8. The Dice score on the validation set is reported
after the last epoch. For our federated experiments, both the standard FedAvg
algorithm [15] and, separately, FedBN [12] (to keep client-specific BN layers)
are used. As discussed in Section 2.2, in addition to BN, our experiments
investigate the impact of alternative feature normalization methods that do
not rely on client-specific normalization layers: Group Norm (GN) [25],
Instance Norm (IN) [23] and the NF [32] technique.

###  3.3 Results & Discussion

Unified Model - Improves Performance Across Databases: Results in Tab. 2 show
that, when keeping client-specific BN parameters, federated joint training
across diverse MRI databases with different brain diseases and different input
modalities improves the model’s performance on each database, either
surpassing or matching single center training. This highlights the benefits of
transferring knowledge between different brain diseases, especially for
databases with limited data (MSSEG, TBI, and WMH). Training with modality drop
and client-specific BN layers leads to the highest overall results and
improves over FedAvg using averaged BN parameters, demonstrating the benefits
of client specific BN layers for source clients. Furthermore, this joint
learning framework produces a single model instead of training 5 individual
models, demonstrating its practicality.

Table 2: Separate models are trained and evaluated on each database (single
center), with only BN results shown as they yield the best results. In the
federated setting the model is trained across ATLAS, BRATS, MSSEG, TBI, and
WMH with various feature normalization methods. Furthermore, results of
modality drop are reported.

Setting | Mod.Drop | Norm | ATLAS | BRATS | MSSEG | TBI | WMH | Average
---|---|---|---|---|---|---|---|---
Single Database | ✗ | BN | 52.8 | 91.9 | 68.3 | 56.1 | 71.5 | 68.1
Federated (client spec. BN params) | ✗ | BN | 50.9 | 91.8 | 70.2 | 55.8 | 74.8 | 68.7
Single Database | ✓ | BN | 52.8 | 91.7 | 66.7 | 53.9 | 69.1 | 66.8
Federated (avg. BN params) | ✓ | BN | 43.1 | 89.4 | 63.2 | 51.4 | 67.3 | 62.9
Federated | ✓ | IN | 49.0 | 90.7 | 66.6 | 52.9 | 71.0 | 66.0
Federated | ✓ | GN | 48.1 | 91.0 | 67.3 | 53.7 | 70.4 | 66.1
Federated | ✓ | NF | 51.2 | 91.1 | 64.8 | 54.4 | 72.3 | 66.8
Federated (client spec. BN params) | ✓ | BN | 54.5 | 91.8 | 69.1 | 56.2 | 73.7 | 69.1

Modality Drop Improves Robustness Against Missing Modalities: Our initial step
towards developing a model capable of segmenting pathologies with modality
combinations different from those seen during training involves robustness
against missing modalities. Results in Tab. 3 show the need for modality drop
to improve the model’s robustness against missing modalities. When using
modality drop during training, the Dice score decreases by only 3.9 when
evaluating with missing modalities compared to all modalities, versus a 21.3
decrease without modality drop. Without modality drop, the model relies on
having all modalities for a specific disease present during inference.
Therefore, the following experiments focus on models trained with modality
drop.

Table 3: Results from jointly training a model using client-specific BN layers on ATLAS, BRATS, MSSEG, TBI, and WMH, both with and without modality drop and evaluated on the same databases by randomly excluding modalities during testing to mimic missing modalities, as well as with all modalities present during testing. Training with modality drop is important so that at test-time the model can work with different combinations of modalities. The average Dice score across all databases is reported. Training Method | Evaluation (Dice) | $\Delta$ Dice
---|---|---
| Modality Drop | No Modality Drop |
No Modality Drop | 47.4 | 68.7 | 21.3
Modality Drop | 65.2 | 69.1 | 03.9

Zero-Shot Generalization to Unseen Modality Combinations: Tab. 4 presents
generalization results on unseen databases with modality combinations not seen
during training. It is remarkable that despite not seeing any of the new
client’s data, the best federated model’s average Dice score reaches only 4.1%
below the upper bound of single center training on the target database and
comes close to centralized joint training (for which only results with IN are
shown as it achieved best performance). Traditionally, models are trained on
databases specific to a disease and evaluated on unseen clients with the same
disease, as we trained with ATLAS (for stroke lesion) and evaluated on ISLES,
and trained with BRATS (for brain tumors) and evaluated on Tumor2, setting
this as baseline. Both are only trained and evaluated on T1 since it is the
only overlapping modality. Federated joint training achieves a performance
gain of 49.3 on ISLES and 8.1 on Tumor2, reinforcing the benefits of joint
training across multiple MRI databases compared to pathology specific models.

Additionally, modality drop proves essential to dissociate specific modality
combinations with certain pathologies, as all jointly trained models with
modality drop show improved average performance on segmenting pathologies with
modality combinations not seen during training. For Tumor2, there is extensive
improvement (both in centralised and federated settings), whereas for ISLES,
the federated models performed unexpectedly slightly worse, likely due to
gradient stochasticity in federated optimization.

Comparing client-specific BN layers to non-client-specific normalization
approaches, the results indicate that the NF technique outperforms all other
techniques. Using the average of client-specific BN layers and estimating the
running statistics on the validation data gives the second-best results.
However, this estimation is less practical as it requires unlabeled data from
the unseen client.

Table 4: Generalization results on unseen clients ISLES and Tumor2, each with
modality combinations not seen during training (reported in Dice score):
Separate models were trained on ISLES and Tumor2 and evaluated on the same
database (Upper Bound). Baselines involve training on stroke lesion database
ATLAS, evaluating on ISLES, and training on tumor database BRATS, evaluating
on Tumor2. These results are compared to FL, using different normalization
techniques, across 5 training databases (ATLAS, BRATS, MSSEG, TBI, and WMH)
and evaluated on ISLES and Tumor2. Centralised training on all 5 databases
together is also presented.

Setting | Mod. Drop | Norm. | Needs Targ. Data | ISLES | Tumor2 | Average
---|---|---|---|---|---|---
| ✗ | BN | N/A | 57.9 | 75.3 | 66.6
| ✗ | IN | N/A | 55.7 | 76.7 | 66.2
Upper Bound: Trained on Target only | ✗ | NF | N/A | 60.6 | 72.6 | 66.6
Trained on ATLAS (T1) | ✗ | IN | N/A | 5.3 | - |
Trained on BRATS (T1) | ✗ | IN | N/A | - | 63.7 | 34.5
Centralised | ✗ | IN | No | 50.7 | 58.1 | 54.4
Centralised | ✓ | IN | No | 55.5 | 72.2 | 63.9
Federated (avg. BN params) | ✗ | BN | No | 53.8 | 0.0 | 26.9
Federated (avg. BN params) | ✓ | BN | No | 43.1 | 68.0 | 55.6
Federated | ✗ | IN | No | 48.3 | 59.1 | 53.7
Federated | ✓ | IN | No | 44.2 | 72.7 | 58.5
Federated | ✗ | NF | No | 54.6 | 31.1 | 42.9
Federated | ✓ | NF | No | 52.8 | 72.1 | 62.5
Federated (client spec. BN params) | ✗ | BN | Yes | 50.2 | 36.8 | 43.5
Federated (client spec. BN params) | ✓ | BN | Yes | 49.9 | 70.1 | 60.0

Investigating the Effects of Normalization Layers on Source Clients: Tab. 2
shows the effects of feature normalization on the source databases. The
findings indicate that keeping client-specific BN layers yields better
performance compared to other normalization techniques on the source clients,
but pose difficulties when dealing with unseen clients (as shown in Table 4).
The NF method outperforms other non client-specific feature normalization
layers in both source client and generalization results. Consequently, a
trade-off must be made when designing a model for training with this type of
heterogeneous data: either improve the source clients’ performance or
prioritizing the development of a model that generalizes to unseen clients.

##  4 Conclusion

This study shows the feasibility, practicality and effectiveness of using FL
to train a single model across decentralised brain MRI databases, each with
different input modalities and segmentation tasks. Our results show that
federated joint training on multiple databases results in a _single_ model
that can segment _all_ diseases and sets of modalities seen in training
clients as well as client-specific models trained for a specific set of
modalities and disease. Furthermore, it shows promising generalization to
unseen clients with modality combinations not seen during training, and
highlight the importance of the feature normalization method’s choice to
address data heterogeneity. We believe this study highlights the benefits of
leveraging heterogeneous brain MRI databases for federated training and opens
up future research directions in this area.

##  5 Acknowledgement

FW is supported by the EPSRC Centre for Doctoral Training in Health Data
Science (EP/S02428X/1), by the Anglo-Austrian Society, and by an Oxford-Reuben
scholarship. PS is supported in part by the UK EPSRC Programme Grant
EP/T028572/1 (VisualAI), in part by the UKRI grant reference EP/X040186/1
(Turing AI Fellowship: Ultra Sound Multi-Modal Video-based Human-Machine
Collaboration), and a UK EPSRC Doctoral Training Partnership award. ZL is
supported by scholarship provided by the EPSRC Doctoral Training Partnerships
programme [EP/W524311/1]. VN, NIHR Rosetrees Trust Advanced Fellowship,
NIHR302544, is funded in partnership by the NIHR and Rosetrees Trust. NV is
supported by the NIHR Oxford Health Biomedical Research Centre (NIHR203316).
The views expressed are those of the authors and not necessarily those of the
NIHR, Rosetrees Trust or the Department of Health and Social Care. The authors
also acknowledge the use of the University of Oxford Advanced Research
Computing (ARC) facility in carrying out this work
(http://dx.doi.org/10.5281/zenodo.22558).

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# Growth principles of natural hypergraphs

Alexei Vazquez<EMAIL_ADDRESS>Nodes & Links Ltd, Salisbury House,
Station Road, Cambridge, CB1 2LA, UK

###### Abstract

Several systems can be represented by hypergraphs, an extension of graphs with
associations between any number of vertices. These natural hypergraphs doe not
appear at once. They are generated by some dynamical process of hypergraph
evolution. Here I investigate what are the minimal growth principles of
natural hypergraphs. I postulate edge duplication and vertex addition at edge
duplications as the key principles of hypergraph growth. The implementation of
these two principles induce the emergence of preferential attachment, power
law degree distribution, the small-world property, high clustering coefficient
and the founder effect. This work clarifies the distinction between
principles, emergent properties and context specific details in the context of
hypergraph growth dynamics.

##  I Introduction

In recent years there has been an increased interest in higher order
associations beyond pairwise interactions Battiston  _et al._ (2020, 2021).
Higher order associations have been represented by hypergraphs and simplicial
complexes. Hypergraphs are combinatorial objects that generalizes graphs to
include associations between more than two elements. Simplicial complexes are
families of higher order associations that are closed under taking subsets
(e.g., nodes, edges, triangles, …) and they are the building blocks of
topological spaces. These mathematical constructions have been around for
quite some time. The novelty is in how they appear in natural and human
related systems Wang  _et al._ (2010); Petri and Barrat (2018); Iacopini  _et
al._ (2019); Coutinho  _et al._ (2020); Civilini  _et al._ (2021); St-Onge
_et al._ (2021); Neuhäuser  _et al._ (2021), how they can be generated by
random ensembles Chodrow  _et al._ (2021); Young  _et al._ (2021) and how they
impact the behavior of process on such structures Vazquez (2009); Ghoshal  _et
al._ (2009); Courtney and Bianconi (2018).

When studying models of natural or human related systems we should make a
distinction between principles, emergent properties and details. Let us take
as an example preferential attachment in the context of networks Barabási and
Albert (1999). There is no doubt preferential attachment is a distinctive
feature in the dynamics of real networks. Yet, preferential attachment is an
emergent property of local evolution rules on networks Vazquez (2003). When
navigating a network the transversed links bias the walk to nodes with a
higher number of neighbors (degree), with a bias proportional to the degree.
New links generated by local walks on an existing network lead to preferential
attachment Vazquez (2001). Making copies of existing nodes (e.g., web pages or
proteins) generates preferential attachment as well Kleinberg  _et al._
(1999); Pastor-Satorras  _et al._ (2003); Vázquez  _et al._ (2003); Chung  _et
al._ (2003). The local rules do more, they generate the small world property,
high clustering coefficient and degree correlations Vazquez (2003).

In line with the timeline of growth models of networks, preferential
attachment has been extended to the context hypergraph growth models Wang  _et
al._ (2010). Yet again, as in the context of networks, I would expect that
preferential attachment is an emergent property of dynamical systems with a
hypergraph structure. The basic mechanism should be rooted on some local
dynamics.

Here I investigate the principles of natural hypergraphs growth dynamics. I
have gotten inspiration from previous work on the topic Courtney and Bianconi
(2018); Kovalenko  _et al._ (2021). My contribution is to establish a clear
separation between principles and consequence, cause and effect. I start in
Section II postulating the principles of hypergraph evolution. I follow in
Section III with an analysis of hypergraph properties derived from those
principles. The emphasis is on the emergent properties: preferential
attachment Barabási and Albert (1999), scale-free degree distribution Albert
_et al._ (1999), small-world and high clustering coefficient Watts and
Strogatz (1998). In Section III.3 I elaborate on the founder effect as another
emergent property. Finishing with concluding remarks in Section IV.

##  II Principles

Preferential attachment and scale-free degree distributions are consequencse
of network dynamics with local evolution rules Vazquez (2003). In a nutshell,
whenever we transverse an edge we are performing a bias sampling towards
vertices with higher degrees. Selecting an edge, rather than a vertex, is what
drives the preferential attachment to vertices with higher degree. Since many
of the networks studied in the past are actually projections of latent
hypergraphs (e.g., co-autorships and metabolic networks), I expect that
similar principles apply to the dynamics of natural and human related
hypergraphs.

A hypergraph $H(V,E)$ is composed by a set of vertices $V$ and a set of edges
$E$, where edges are sets with one or more vertices. When there are not two
edges with the same vertex composition we call it a simple hypergraph, or a
multi-hypergraph otherwise. Following the principles of network evolution, I
focus on local and edge centric evolution principles of hypegraphs. Local
means that the evolution rules are based on one or a few edges in a local
neighborhood of the hypergraph. The simplest local rule of hypergraph growth
is edge duplication

$e\rightarrow 2e\ .$ (1)

There are several natural and human related scenarios that are represented in
this rule. Starting with the obvious resemblance with asexual reproduction by
cell division. If vertices are genes and edges are cells then Eq. (1) is a
model of cell division. In the context of human behavior an edge can be an
instance of a recurrent activity involving a group of people (the vertices).
Examples include scientific publications, scientific meetings and working days
at institutions departments. Notice that Eq. (1) generates multi-hypergraphs
or temporal hypergraphs if we add a time stamp label to each edge.

We need a second rule to model the addition of new vertices. The simplest of
local edge centric rules is duplication with vertex addition at the duplicate

$e\rightarrow e+e\cup\\{v\\}\ .$ (2)

In asexual reproduction this rule represents gene duplication, an event that
can happen during the genome replication preceding cell division. In the
context of human behavior Eq. (2) represents a new individual joining a group
at their next group meeting. We could spell further evolution rules involving
two or more edges that are adjacent (thus local) leading to edge composition
mixing. Here I will stop at the two simplest rules (1)-(2) and investigate the
structure of hypergraphs they generate.

Figure 1: Hypergraph instance generated with the implementation of rules
(1)-(2) (App. A), using as input $N=10$ vertices, vertex growth rate $\mu=0.5$
and a starting hypergraph consisting of one edge with one vertex. Visualized
with Python package HyperNetX.

##  III Emergent properties

To investigate the emergent properties from the hypergraph growth principles I
introduce a specific model. At time $t=0$ we start with hypergraph $H_{0}$. At
each subsequent time step, we select an edge $e$ sampled with equal
probability among all current edges. To this edge we apply (1) with
probability $1-\mu$ or (2) otherwise. When (1) is applied, we create a new
edge $e^{\prime}=e$ with the exact same vertex composition. When (2) is
applied, we add a new vertex $n\rightarrow n+1$ and a new edge
$e^{\prime}=e\cup\\{v(n+1)\\}$ with the composition of $e$ plus the new
vertex. We continue until the hypergraph has $n=N$ vertices. Since new
vertices are added with probability $\mu$, $\mu$ represents the vertex growth
rate per edge duplication. Each hypergraph ensemble is parametrized by
$\\{H_{0},\ \mu,\ N\\}$.

A Python implementation is listed in Appendix A. A hypergraph example
generated with this code is shown in Fig. 1. The star-like shape already
anticipates the heterogeneity in the vertices connectivity and the small world
property.

###  III.1 Preferential attachment

The degree of a vertex is the number of edges containing that vertex and it
will be denoted by $k$. At each step one edge is selected for duplication,
with or without vertex addition. The probability that a vertex with degree $k$
is in the selected edge, and therefore increases its degree, is

$\Pi_{k}(t)=\frac{k}{E(t)}$ (3)

where $E(t)$ is the number of edges at step $t$. That is preferential
attachment: new edges are added at vertices with a bias by their degree.

Figure 2: Degree distribution of hypergraph instances generated with the
implementation of rules (1)-(2) (App. A), using as input $N=10000$ vertices,
the vertex growth rate $\mu$ indicated in the legend and a starting hypergraph
consisting of one edge with one vertex. The dashed line is the expected degree
distribution $p_{k}=1/(k(k+1))$. Figure 3: Number of nearest neighbors
distribution of hypergraph instances generated with the implementation of
rules (1)-(2) (App. A), using as input $N=10000$ vertices, the vertex growth
rate $\mu$ indicated in the legend and a starting hypergraph consisting of one
edge with one vertex. The dashed line is the expected degree distribution
$p_{k}=1/(k(k+1))$.

###  III.2 Power law degree distribution

For large hypergraphs ($t\gg 1$), the rate equation describing the evolution
of the number $n_{k}(t)$ of vertices with degree $k$ is

$\displaystyle n_{k}(t+1)$ $\displaystyle=$ $\displaystyle
n_{k}(t)+\Pi_{k-1}(t)n_{k-1}(t)-\Pi_{k}(t)n_{k}(t)$ (4)
$\displaystyle+\mu\delta_{k1}\ .$

Since an edge is added at each step and vertices are added at rate $\mu$,
$E(t)\approx t$ and $N(t)\approx\mu t$ for $t\gg 1$. In such a case Eq. (4)
has the steady state solution $n_{k}(t)=\mu tp_{k}$, where $p_{k}$ satisfies
the recursive equation

$p_{k}=(k-1)p_{k-1}-kp_{k}+\delta_{k1}$ (5)

Iterating this equation with the boundary condition $p_{-1}=0$ I obtain

$p_{k}=\frac{1}{k(k+1)}$ (6)

The degree distribution of hypergraphs generated by (1)-(2) have a power low
tail with exponent -2. This is corroborated by numerical simulations (Fig. 2).

In many applications the hypergraph is projected into a graph, where two
vertices are connected if there is at least one edge containing them (on-
vertex projection). The vertex degree in this graph is the number of nearest
neighbors, the number of other vertices that can be reached via the edges
containing the vertex. Although I do not have an analytical derivation for the
number of neighbors distribution, I expect it is characterized by the same
power law tail with exponent -2. This seems to be the cased based on numerical
simulations (Fig. 3)

Figure 4: Hypergraph instance generated with the implementation of rules
(1)+(7) (App. B) with maximum edge size $s_{c}=2$, using as input $N=10$
vertices, vertex growth rate $\mu=0.5$ and starting from a hypergraph
consisting of one edge with one vertex. Figure 5: Scatter plot of the vertices
degree as a function of the time stamp when they were added, for a hypergraph
instance generated with the implementation of rules (1)+(7) (App. B). The
input parameters were $N=100$ vertices, vertex growth rate $\mu=0.5$, maximum
edge size $s_{c}=2$ and starting hypergraph consisting of one edge with one
vertex. The red symbol is the founder.

###  III.3 Founder effect

Visual inspection of Fig. 1 indicates that the seeding vertex is part of all
edges. This is indeed the case. If we start with one edge containing one or
more vertices then all other edges will contain those starting vertices. The
founders persist in all edges.

This behavior can be relaxed if we specify further rules. For example, during
the growth of organizations large departments are split and new managers are
brought in to maintain communication between the new departments. This can be
modeled by duplication with edge split and vertex addition

$e\rightarrow\left\\{\begin{array}[]{ll}e+e\cup\\{v\\}&{\rm if}\ |e|<e_{c}\
,\\\ e+e_{1}\cup\\{v\\}+e_{2}\cup\\{v\\}&{\rm otherwise}\ ,\end{array}\right.$
(7)

where $e_{1}+e_{2}=e$. A specific implementation with random and balance edge
split is found in Appendix B. Figure 4 shows an example with edges constrained
to size 2. The vertex in the far right does not share an edge with the founder
(in red). Yet, the founder is still connected to most vertices.

A founder is basically a member that joined the system early in its evolution
and it has a high connectivity at later times. This is illustrated in Fig. 5,
with the founder highlighted in red. In fact, that plot is an operational
method to identify putative founders. They are found in the upper left corner.

Figure 6: Semi-logarithmic scaling of the average shortest path distance
between vertices with the number of vertices. The hypergraphs instances were
generated generated with the implementation of rules (1)+(7) (App. A), using
as input $N=10000$ vertices, vertex growth rate $\mu=0.5$, the maximum edge
sizes $s_{c}$ indicated in the legend and a starting hypergraph consisting of
one edge with one vertex.

###  III.4 Small-world property

The founder effect induces the small-world property. The hypergraphs generated
by (1)-(2) have diameter 2. Any two vertices are at most two edges away, each
edge containing the founder. This singular behavior of constant diameter is
relaxed after adding the edge split rule (7). When a cap is imposed in the
edge size there is a logarithmic increase in the average shortest path
distance with increasing the number of vertices (Fig. 6), the typical behavior
of small-world networks Amaral  _et al._ (2000). Furthermore, the larger the
maximum allowed edge size the smaller the hypergraph, getting closer to the
unconstrained case (Fig. 6, blue symbols).

###  III.5 High clustering coefficient

The hypergraph extension to include edges with size larger than 2 induces a
high clustering coefficient in the associated graph on-vertex projection. Not
surprisingly hypergraphs evolving through the growth principles introduced
here have a high clustering coefficient. The numerical estimates of the
average clustering coefficient are above 0.5 for the range of parameters
investigated (see Table 1).

$H_{0}$ | $s_{c}$ | $\mu$ | $\langle d\rangle$ | $\langle c\rangle$ | $r$
---|---|---|---|---|---
H1 | 2 | 0.1 | 2.25 | 0.55 | -0.53
H1 | 2 | 0.5 | 4.54 | 0.76 | -0.07
H1 | 2 | 0.9 | 4.69 | 0.74 | -0.07
H1 | 0 | 0.1 | 2.00 | 0.55 | -0.59
H1 | 0 | 0.5 | 1.99 | 0.91 | -0.22
H1 | 0 | 0.9 | 1.99 | 0.92 | -0.15
H10 | 2 | 0.1 | 2.14 | 0.93 | 0.07
H10 | 2 | 0.5 | 3.31 | 0.80 | 0.05
H10 | 2 | 0.9 | 4.25 | 0.75 | 0.18
H10 | 0 | 0.1 | 1.98 | 0.98 | -0.01
H10 | 0 | 0.5 | 1.97 | 0.97 | -0.03
H10 | 0 | 0.9 | 1.97 | 0.95 | -0.03
Table 1: Properties of hypergraph instances generated with the implementation
of rules (1)+(7) (App. B), using as input $N=10000$ vertices, different vertex
growth rates $\mu$, different maximum edge sizes $s_{c}$ and starting
hypergraph $H1$ or $H10$. H1 is a hypergraph with an edge containing one
vertex. H10 is is a hypergraph with an edge containing 10 vertices.

###  III.6 Degree correlations

Degree correlations inform about degree mixing between connected vertices
Pastor-Satorras  _et al._ (2001); Newman (2002). Degree correlations are
quantified by the Pearson correlation of the degree of connected vertices,
denoted by $r$ Newman (2002). When $r>0$ we say there is assortative mixing,
whereby connected vertices tend to have similar degrees. When $r<0$ we say
there is disassortative mixing, whereby connected vertices tend to have
dissimilar degrees. For hypergraphs we can use the same quantity, interpreting
connected vertices as vertices sharing at least one hyper-edge.

The hypergraphs generated by (1)-(2) or (1)+(7) have variable values of $r$
ranging from negative to positive values (Tab. 1). $r$ is modulated by the
founder effect. For example, when the starting hypergraph has one edge
containing one vertex then $r<0$ for all values of $\mu$ and the maximum edge
size $s_{c}$ that I have simulated. In contrast, when the starting hypergraph
has one edge containing 10 vertices then $r>0$ for $s_{c}=2$ but $r<0$ for
$s_{c}=\infty$, for all simulated values of $\mu$. Therefore degree
correlations are sensitive to parameters and initial conditions. They are
context depending.

##  IV Conclusions

Edge duplication and vertex addition at edge duplication are natural
principles of hypergraph evolution. The universal properties of real world
networks that are projections of hypergraphs are emergent properties from
these two growth principles.

The evolution of natural or human related hypergraphs is richer. I have
illustrated the impact of adding a constraint on the maximum edge size. Higher
order evolution rules can involve two or more edges. These variations will
modulate the emergent properties described above. They could modify the
exponent of the power law tail in the degree distribution or increase the
diameter. Yet, as long as the hypergraph growth principles of edge duplication
and vertex addition at edge duplication are part of the system dynamics the
emergent properties will persist.

The founder effect modulates the hypergraph properties. The founder effect is
a typical property of natural systems that grow from a small initial seed.
Biological populations of organisms is the canonical example. The founder
effect is observed during the creation of new academic fields or institutions.
Based on this work, the founder effect is an emergent property of the
hypergraph growth principles.

## Acknowledgements

Nodes & Links did not have any role in the study design, data collection and
analysis, decision to publish, or preparation of the manuscript.

## Appendix A

Implementation of (1)-(2) in Python

    import numpy as np
    import hypernetx as hnx
    def HGP(n, mu, E0, seed):
        # hypergraph by growth principles
        np.random.seed(seed)
        E = E0
        i = max([j for e in E for j in e]) + 1
        while i<n:
            e = E[np.random.randint(len(E))]
            if np.random.random() < mu:
                E.append(e + [i])
                i += 1
            else:
                E.append(e)
        return E
    E = HP1(10, 0.5, [[0]], 100)
    H = hnx.Hypergraph(E)
    hnx.drawing.draw(H)

## Appendix B

Implementation of (1)+(7) inPpython

    import numpy as np
    import hypernetx as hnx
    def HGPSC(n, mu, E0, seed, sc):
        # hypergraph growth principles
        #   with edge size constraint
        np.random.seed(seed)
        E = E0
        i = max([j for e in E for j in e]) + 1
        while i<n:
            e = E[np.random.randint(len(E))]
            if np.random.random() < mu:
                if len(e)<sc:
                    E.append(e + [i])
                else:
                    split = np.array_split(
                        np.random.permutation(e),2)
                    E.append(list(split[0])+[i])
                    E.append(list(split[1])+[i])
                i += 1
            else:
                E.append(e)
        return E
    E = HGPSC(10, 0.5, [[0]], 100, 2)
    H = hnx.Hypergraph(E)
    hnx.drawing.draw(H)

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