Álex Tsarev interesting question, which I would ask under a different angle: if we perform distributed computing on a set of nodes, each with (data, compute, network) functionality, what types of algebraic structures can be found, observed, crafted, for distributed computing efficiency, robustness, reliability or other criteria in this space?

Let me give you three very different illustrations below.

1. Home-made algebraic patent (application speech coding)

Here is a hands-on example which I have embedded in my first Patent.

Read this Patent Method of transmitting, at low throughput, a speech signal b...

Let me explain... You are probably familiar with Fast Fourier Transform (FFT) and Discrete Cosine Transform. I have used something similar, with butterfly structure, where a large number of multiplications (for a convolution) has been transformed into a small number of additions, because the vector set I was using was (x(1),...x(n)) with x(i) of the form+/- m with m ranging from 0 to 2k+1.

With your algebraic eyes you quickly see cyclic group Z/pZ (or GF(p)) emerge, and depending on p, we may get to finite fields, and finite fields are commutative as you know... Easy and nice.

2. Massively parallel computing

A second illustration on how algebraic structures lead to efficiencies on parallel computing architectures is given here

I. Ahmad and A. Ghafoor, "Semi-distributed load balancing for massively parallel multicomputer systems," in IEEE Transactions on Software Engineering, vol. 17, no. 10, pp. 987-1004, Oct. 1991, doi: 10.1109/32.99188.

https://surface.syr.edu/cgi/viewcontent.cgi?article=1123&context=eecs_techreports

3. Elementary non-commutative groups and Blockchains

Computing is currently moving fast away from centralised Cloud Computing to more distributed Edge Computing (which is actually a hybrid Edge/Cloud strategy in most cases of interest).

Therefore please look for Edge Computing use cases, and analyse the algebraic structures (property of graph of nodes, etc) which might help.

This is a promising domain of research.

Now, if you are looking for a "quick win", just an illustration of this very wide domain, you can read carefully this article which explores the construction of an elementary algebraic structure, a non-commutative group to be specific, as an overlay on top of a Distributed Ledger System (a.k.a. Blockchain).

Here is this example, of switches:

"Algebraic Structure of Blockchains: A Group-Theoretical Primer"

DONGFANG ZHAO, University of Nevada, Reno

https://arxiv.org/pdf/2002.05973

Let me know if it puts you on the track towards something interesting!

Preprint Big data, farmacoepidemiología y farmacovigilancia

Álex Tsarev

You can check out for data analysis:

Kelarev, A. V., Yearwood, J. L., & Vamplew, P. W. (2009). A polynomial ring construction for the classification of data. Bulletin of the Australian Mathematical Society, 79(2), 213-225.

Kelarev, A. V., Watters, P., & Yearwood, J. L. (2009). Rees matrix constructions for clustering of data. Journal of the Australian Mathematical Society, 87(3), 377-393.

For parrallel processing take a look at:

Raaphorst, S. (2004). Parallel Algebraic Algorithms.

Hope this helps

Álex Tsarev interesting question, which I would ask under a different angle: if we perform distributed computing on a set of nodes, each with (data, compute, network) functionality, what types of algebraic structures can be found, observed, crafted, for distributed computing efficiency, robustness, reliability or other criteria in this space?

Let me give you three very different illustrations below.

1. Home-made algebraic patent (application speech coding)

Here is a hands-on example which I have embedded in my first Patent.

Read this Patent Method of transmitting, at low throughput, a speech signal b...

Let me explain... You are probably familiar with Fast Fourier Transform (FFT) and Discrete Cosine Transform. I have used something similar, with butterfly structure, where a large number of multiplications (for a convolution) has been transformed into a small number of additions, because the vector set I was using was (x(1),...x(n)) with x(i) of the form+/- m with m ranging from 0 to 2k+1.

With your algebraic eyes you quickly see cyclic group Z/pZ (or GF(p)) emerge, and depending on p, we may get to finite fields, and finite fields are commutative as you know... Easy and nice.

2. Massively parallel computing

A second illustration on how algebraic structures lead to efficiencies on parallel computing architectures is given here

I. Ahmad and A. Ghafoor, "Semi-distributed load balancing for massively parallel multicomputer systems," in IEEE Transactions on Software Engineering, vol. 17, no. 10, pp. 987-1004, Oct. 1991, doi: 10.1109/32.99188.

https://surface.syr.edu/cgi/viewcontent.cgi?article=1123&context=eecs_techreports

3. Elementary non-commutative groups and Blockchains

Computing is currently moving fast away from centralised Cloud Computing to more distributed Edge Computing (which is actually a hybrid Edge/Cloud strategy in most cases of interest).

Therefore please look for Edge Computing use cases, and analyse the algebraic structures (property of graph of nodes, etc) which might help.

This is a promising domain of research.

Now, if you are looking for a "quick win", just an illustration of this very wide domain, you can read carefully this article which explores the construction of an elementary algebraic structure, a non-commutative group to be specific, as an overlay on top of a Distributed Ledger System (a.k.a. Blockchain).

Here is this example, of switches:

"Algebraic Structure of Blockchains: A Group-Theoretical Primer"

DONGFANG ZHAO, University of Nevada, Reno

https://arxiv.org/pdf/2002.05973

Let me know if it puts you on the track towards something interesting!

Article Big Data Algebra (BDA): A Denotational Mathematical Structur...

د. سليمان رجب الشيخ

Algebird : Abstract Algebra for big data analytics

https://www.slideshare.net/samkiller/algebird-abstract-algebra-for-big-data-analytics-devoxx-2014