Network analysis plays a key role in social media platforms like Facebook and Instagram, where it assists in nominating potential friends or followers based on various factors like mutual interests and connections with other users. This recommendation system is quite familiar to all of us and has become ubiquitous in the digital era.

This essay aims to highlight the differences and similarities between network analysis and geometric deep learning.

## I have been struggling to understand the concept of a network.

A group of individuals or objects can be represented symbolically as a network, also known as a “graph” in mathematical terms, that highlights their most noteworthy characteristics.

If you are working as a project manager, your role will involve overseeing the smooth functioning of four machines using the assistance of two workers. This includes planning their work shifts, allocating their tasks, and monitoring their work progress to ensure accuracy. Although this job demands significant effort from your end, networks can be an effective tool to save time. Let’s take a closer look at an example to better understand this.

In this context, the relationships between the different nodes (depicted using circles) and objects (represented by vertices) can be demonstrated by the use of arrows, commonly known as “edges.” This visualised data representation allows us to efficiently analyse the concerned items and simplify our tasks.

An analysis of the social network of Instagram users, for instance, could include individuals as nodes and their connections like friendships as the edges.

## What makes network analysis significant?

Network analysis is a valuable tool for studying biological structures, social networks, and complex systems in several industries like banking, aviation, and the supply chain sector. To use network analysis, data is first collected and a model is constructed to evaluate and identify trends. After examining the model, the results can be employed to improve the network and optimise decision-making.

- An analysis of social networks is beneficial in identifying influential individuals within a group.
- The airline’s consumption of fuel or the distance travelled by a flight can be monitored using its network.
- Network analysis can assist in identifying the type, value, timing, and geographical position of financial transactions carried out in a network setting.
- An analysis of supply chain networks can determine their load capacity, the year of production, maintenance expenses, and more.

## What is the definition of geometric deep learning?

In 2017, Bronstein et al. presented the notion of Geometric Deep Learning in their research titled “Geometric Deep Learning: Going beyond Euclidean Data.” This term encompasses novel methodologies that aim to expand structured deep neural models to non-Euclidean areas, including graphs and manifolds. It serves as a significant progress in deep learning, creating opportunities for its application in non-Euclidean domains.

Deep geometric learning has the potential, in theory, to establish a unified mathematical foundation for a vast array of neural network designs, such as convolutional neural networks (CNNs), recurrent neural networks (RNNs), and transformers. Additionally, it could allow us to acquire distinct physics-based knowledge that can be incorporated into any architecture.

Geometric Deep Learning (GDL) is an up-and-coming subset of Machine Learning (ML) that enables researchers to examine and comprehend intricate data structures like networks and 3D point clouds. Thanks to recent improvements in computing power, ML techniques such as Convolutional Neural Networks (CNNs), Long Short-Term Memories (LSTMs), and Generative Adversarial Networks (GANs) have been delivering outstandingly precise outcomes on a wide range of tasks.

### Differentiating between Euclidean and non-Euclidean spaces

According to Wikipedia, a Euclidean space is a genuine vector space that comes equipped with an inner product. In essence, it is a space that is delineated by a group of n-dimensional functions that range from one-dimensional to n plus one-dimensional.

Non-Euclidean outlooks focus on the curvature of a surface, whereas the Euclidean perspective seeks to find a flat one. Non-Euclidean spaces include graphs, networks, manifolds, and any other relevant structures. Euclidean spaces, on the other hand, are characterized by content that can be depicted through text, sound, or visuals.

Machine Learning applications still predominantly employ conventional techniques that only accommodate Euclidean data. However, manifolds represent merely one possible three-dimensional representation of data. In this context, high-dimensional spaces provide a useful comparison to explore this concept. Similar forms are combined into a single point within a new space. This indicates that a single point in this space can represent any shape that is composed of multiple points.

When dealing with data that is not in Euclidean form, applying deep neural networks necessitates the use of alternative methods that take into account the challenges that this type of information presents. Graph-based neural networks, recurrent neural networks, and convolutional neural networks are some of the alternative methods that are used. By utilizing these methods, it is possible to successfully apply deep neural networks to non-Euclidean data.

- There is no universally accepted way to ascertain location.
- Lack of a precise vector-space structure
- There was no change in the displacement.

## What prompted the need for geometric deep learning?

As the number of dimensions increases, the intricacy of learning also rises, resulting in an overwhelming amount of data. To demonstrate this, consider a solid ball in three dimensions that is expanded to an infinite number of dimensions. The ball’s volume will stretch outwards from its center, creating a hollow sphere. Conversely, if the ball is shifted to a dimension below our own, the sphere reverts to its original form. This effectively shows that higher-dimensional data is less compact, making it more challenging to derive meaningful insights from it. This can make drawing conclusions from it more complex.

Due to the lack of information, it is not feasible to identify a solution that is equivalent to the other dataset. However, many tasks or functions that require estimation possess hidden patterns that can be explored through geometric concepts in lower dimensions, a process that can be assisted by Geometric Deep Learning.

Understanding or learning from graph-structured data is difficult for regular deep neural networks. This is because most modern neural networks rely on convolutions, which perform best on datasets that adhere to the principles of Euclidean geometry. Non-Euclidean datasets, which include graphs, manifolds, and other structures that lack a standard shift-invariant coordinate system, are examples of such datasets that fall outside of this category.

To address the challenges of non-Euclidean settings and broaden the capabilities of deep neural networks, researchers are investigating a new field known as geometric deep learning. This form of deep learning aims to overcome challenges that arise in non-Euclidean environments and explore methods to make deep neural networks more adaptable and relevant to various scenarios.

## Differentiating between Geometric Deep Learning and Network Analysis

## Examining Networks

- Refrains from utilizing deep learning techniques and instead focuses on memorizing network architecture.
- Offers a more efficient approach for managing abstract concepts such as connections and interactions.
- Graph theory can be used to analyze any type of network, such as social networks, medical networks, and supply chains. Social networks are likely the most well-known example of graph analysis in data science.
- Measures like centrality, modularity, and assortativity can be employed to examine the configuration of networks and acquire knowledge about their features. By using these metrics, it is feasible to achieve a more comprehensive understanding of network characteristics.

## The Potential of Geometric Deep Learning

- Applies deep learning techniques to non-Euclidean datasets.
- Primarily aimed at supplying graph or network-structured datasets as input for deep learning tasks such as regression.
- Graph representation and brain structure are two of the most extensively explored subjects.

The future of deep learning is closely tied to the shift from two-dimensional data to three-dimensional volumetric data. By harnessing the capabilities of machine learning and deep learning, AI can approach the efficiency level achieved by the human brain. Due to the increased accessibility of non-Euclidean data, geometric deep learning is making significant advances. For instance, technology can now match billions of atoms in order to identify potential novel treatments for current ailments. This has the potential to be a life-saving breakthrough for chronic illnesses that are currently difficult to treat.