Application of Graph Theory in Data Science

By Arbaz Sayed

Graph theory is a fascinating part of mathematical analysis that underpins not just intelligent automation, but also data science. A trillion-dollar company like Google would've been impossible even without capabilities generated by graph theory (e.g., PageRank builds some elementary insights about random walks on graphs). Several current data science problems are graph-related: Comprehending immense social connections, from Facebook and Twitter to LinkedIn and citation analysis in research journals, necessitates a deep grasp of graph theory. The planet would be filled with entropy if there was no suitable means of associating diverse interactions among any constituent in a system.

Beauty of Graph Theory

The examination of a single fundamental issue at the heart of all machine learning and artificial intelligence in order to illustrate the elegance and prominence of graph-theoretic concepts in cognitive computing, AI, and computer science: how to recover patterns from unpredictability. Without the capacity to filter information from noise, human cognition might not have existed. By its sheer nature, the current world is loud. The world is a "blooming, humming mess" to a child, and it is a marvel how they can sound right about this at all, as the renowned psychiatrist Henry James eloquently phrased it. If we think about it, a baby's mind is nothing more than graphs in motion. They collect and categorize many parts of the environment.

A Thought Experiment

Let's take a closer look at the challenge of pulling pattern from unpredictability using mathematical techniques as a lens. Picture oneself as a virtual insect moving over the internet, moving from one mutual acquaintance to their friends, either from one website to the next. What's the most effective technique to sound right about everything? Single graph edges could be captured, but how can you avoid getting lost in the intricacies and detect the massive building of large networks? For this, you'll have to know about spectral graph theory.

Spectral Graph Theory

Spectral mathematical models are another one of those charming mathematical subdisciplines that contain a miniature of all that is good regarding math, even without the frightening complexities of comprehensive mathematics. It is a mix of linear algebra and graph theory that may be used to reduce numerous complicated concepts from surfaces, Riemannian topology, group theory, and other advanced subjects. In its various versions, the Laplace operator is the most beautiful and central object in all of mathematics. This sun shines brightly in probability theory, mathematical physics, Fourier analysis, finite difference, Klein group theory, and finite difference, and its illumination penetrates even the most esoteric sections of mathematical logic and combinatorial geometry.

Graph Laplacian

The Laplacian takes on a very appealing and fundamental structure. To start, we'll need to understand how and where to convert a network to a table. That's the symmetric adjacency matrix. Remarkably, the adjacency matrix gives us very nothing on its own. We'll should do some tricks with adjacency matrix, then we'll have something which can contain a lot of data about such a graph. L = D - A is the simplest graph Laplacian, wherein D is the number of neighbors for every node. In this simple example, the graph Laplace transform is constructed using the adjacency matrix.

Data Science

In the context of data science, how about tackling challenges which aren't explicitly graph-based? Once more, the graphs Laplacian's exquisite structure is crucial. Manifold training is a method of unsupervised learning that seeks to determine the underlying architecture of the storage space for data. Regardless of the fact that these seem to be vectors in n dimensions, the much more interesting artificial intelligence and machine learning intelligence issues involve data which are not in Cartesian coordinates, but instead in a curved subdomain or continuum (e.g., faces, text documents). Algorithms like Interpolation eigenmaps and ISOMAP provide annotations which preserve the data's fundamental dimensionalities.