Ricci Flow, Economics, COVID outbreak prediction and Drug Design (Ricci Flow everywhere.. Hey Perelman & Poincare, what's up?)

Hey readers,

50,000 views in one night is a very cool thing! Thanks for appreciating my blog.. Very very cool... my fans, readers and followers!!! 

Again, the papers I like the most these days are -

1. Pratik Kothari on Job Postings & Investor Relations - https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3223141

Few times, Gita Gopinath has mentioned divergent economic and job recovery in her IMF blog - https://blogs.imf.org/2021/04/06/managing-divergent-recoveries/ 

This paper (on job postings & investor relations) is genius way of expressing 

- Gita's point of view on management of divergent economic & job recovery 

 

{from the last blog post - Here is an interesting & brilliant blog (April 2021) by Gita Gopinath on divergent recovery -
Managing Divergent Recoveries - https://blogs.imf.org/2021/04/06/managing-divergent-recoveries/ speaking about 

1. Financial risks, 

2. Financial stability risks using macro-prudential tools, 

3. Cross-country gaps, global poverty reduction, 

4. Emerging markets, 

5. Divergent recovery paths, 

6. High degree of uncertainty and developing economies 

7. International liquidity

8. Liquidity protection

9. Debt restructuring 

10. financial stability risks using macro-prudential tools

11. Withdrawn loan payments, firm insolvencies

12. Cross-border profit shifting

which reminded me of }

- And also Ricci Flow & market fragility paper I mentioned in previous blog - https://researchcircle.blogspot.com/2022/02/job-posting-paper-ricci-flow-economics.html 

{from last blogpost - network entropy, nodal entropy, geodesics curvature & divergence mentioned in the Sandhu's paper - Ricci curvature & Ricci Flow: An economic indicator for market fragility and systemic risk and my KDnuggets article on which intended to perform Ricci Flow on Social network analysis & information economics }

2. Spherical representations of C∗-flows II: Representation system and Quantum group setup - https://arxiv.org/abs/2201.10931 

This paper is perfect representation of 3 different works on Quantum groups across 3 different timelines 

- Ryosuke Sato’s approach to asymptotic representation theory for quantum groups
- Sir Atiyah work with TIFR in 1984 on Quantum groups & vector bundles (Yang mills & vector bundles has potential solution while working on graph neural networks because of Quantum Information Geometry connection. It's going to be one hell of an orchestration

- George Lusztig won Wolf Prize for representation theory & Quantum Groups

3. Using discrete Ricci curvatures to infer COVID-19 epidemic network fragility and systemic risk - https://www.medrxiv.org/content/10.1101/2020.04.01.20047225v2.full.pdf

The authors compute both Forman-Ricci and Ollivier-Ricci curvatures for real epidemic networks built from COVID-19 epidemic time-series available at the World Health Organization (WHO). Both curvatures allow them to detect early warning signs of the emergence of the pandemic. The advantage of this method lies in providing an early geometrical data marker for the pandemic state, regardless of parameter estimation and stochastic modeling. This work opens the possibility of using discrete geometry to study epidemic networks.. 

This analysis can be overlapped and Superimposed over Gita Gopinath's blog on Vaccine Inequality to make sure vaccines are provided first in the area where there can be immediate outbreak - 

4. Ricci Flow & Molecular Design - Ollivier persistent Ricci curvature (OPRC) based molecular representation for drug design - https://arxiv.org/pdf/2011.10281.pdf

The filtration process which authors have proposed in persistent homology is employed to generate a series of nested molecular graphs. Persistence and variation of Ollivier Ricci curvatures on these nested graphs are defined as Ollivier persistent Ricci curvature. Moreover, persistent attributes, which are statistical and combinatorial properties of OPRCs during the filtration process, are used as molecular descriptors, and further combined with machine learning models, in particular, gradient boosting tree (GBT). Our OPRC-GBT model is used in the prediction of protein ligand binding affinity, which is one of key steps in drug design. 

If you are still wondering what is Ricci Flow - 

Please don't forget to watch my video on Ricci Flow and Community detection - 

Firoozbakht's conjecture, BDS conjecture & deep learning, Lie Groups, Statistical Riemannian Framework, Ramanujan Machine..

Hey Readers,

It's 4:30 AM Saturday night. And I am at it again. A blog I restarted which I used to write 12 years back to motivate new thinking in data, math and AI. The paper I like the most these days is Job Postings and Aggregate Stock Returns - https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3223141

1. If I ever write my next book, there will be special mention of Iranian mathematician Farideh Firoozbakht - almost an unsung hero of math - She studied pharmacology and later mathematics at the University of Isfahan, and later taught mathematics at that university (Talk about reinventing the career). 

Extremely famous for her Firoozbakht's conjecture (pattern in prime numbers in simple words) which was stated in 1982 - this conjecture achieved some fame 

- in 2011 when I wrote about it along with new orchestrations on dynamical systems (Naive Entropy, Nash Entropy etc), 

- also on Maryam Mirzakhani's work on topology, geometry and dynamical systems (specially her work on Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces - https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/Teichmuller%20Space/Mirz3.pdf And when I also wrote about possibility of GeoDesic Convolutional Neural Networks - Geodesic CNNs & 

- & on Laura De Marco's work on dynamical systems my datahulk blog & 

in 2015 when Alexei Kourbatov published a paper on it in International Mathematical Forum 10 (2015), 283–288. Verification of the Firoozbakht conjecture for primes up to four quintillion, (unproven till now and has a very strong affiliation with Riemann Hypothesis).

I have a very strong viewpoint that Andrica's conjecture & Firoozbakht's conjecture and how looking at prime numbers as a dynamical system can have a smarter way to solve this. I also have a way to look at it further in terms of How To use GANs - Generative Adversarial Networks (with optimum convergence while leveraging Nash entropy) to solve PDEs in Dynamical & Chaotic systems generated through prime numbers. Anyway, more to come later... some surprises in store along with my Ricci Flow Video..  

Firoozbakht passed away in 2019 without many people knowing it... Almost an unsung hero who could have been celebrated everywhere. I always found it bit disappointing that Terence Tao mentioned Cramér conjecture in his blog but not Firoozbakht's conjecture. But, hey, everyone makes their own choice (that's why Thaler's choice architecture right??? right????) 

2. Another person who is doing some cool work is Nina Miolane - Amazing work on Statistics on Lie groups (& abelian groups): A need to go beyond the pseudo-Riemannian framework - https://aip.scitation.org/doi/abs/10.1063/1.4905963

Again, I want to highlight 1984 paper by Sir Atiyah, TIFR and Sir Atiyah's Student George Lusztig and his work on Quantum Groups. I know Nina's GeomStats is focus on Biomedical Topology but there is one more way to grow GeomStats in topology

i.e. growing GeomStats like Ramanujan Machine where AI would solve complex group theory problems trained on Lusztig's group theory paper including lie groups, abelian groups, Weyl groups, Chevalley groups, semisimple p-adic groups, quantum groups etc..

Btw, The first suggestion I gave to Ramanujan Machine when I became their advisor was this paper on BDS conjecture & deep learning by Laura Alessandretti - Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer - https://arxiv.org/abs/1911.02008

Security, Topological Game Theory & Ricci Flow

Hey guys,

Thanks again for around 40,000 views of my last post. It's pretty cool to get these many views for a nerdy blog. And apologies, I have been super-busy at work, writing books, research papers, articles, giving guest lectures etc.

This post is for Perelman, Dr Yau, Melanie Weber, Cao and Zhu, Dr. Fefferman & all nerds who understand my Ricci Flow Video on youtube.. This is my entry into the Field prize Math Video Contest ..win win win.. - https://www.youtube.com/watch?v=z_pjsJisdHQ&t=6s  )...

Here you go... 

1. Amazing Amazing book by Milind Tambe on Security and Game Theory: Algorithms, Deployed Systems - https://www.amazon.com/Security-Game-Theory-Algorithms-Deployed/dp/1107096421 

The book provides detailed overview on

1. Game theoretical framework for security measures,
2. Intelligent Randomization over routes,
3. Bayesian Stackelberg Games,
4. Robust Game Theory,
5. Bayesian Nash Equilibrium,
6. Strategic Security Allocation in Transport networks,
7. Randomization with partial adversary model,
8. Stackelberg vs Nash in security games

2. I want to push this book and concept further in terms of Ricci Flow, Evolutionary Game theory & Security measures - 

Evolutionary game theory is a continuous model with interactions (frequency dependent selection). e.g. the classical Lotka-Volterra predator-prey system fits into this framework. Another example is an infinite population of people playing the game rock-paper-scissors in continuous time. Obviously, if almost everyone is playing rock, the trend will be for more people to play paper. Asymptotic behavior, Nash equilibria and stability of fixed points are studied.

This framework may also be used to analyze the evolution of geometric structures. The Ricci flow is exactly a “replicator equation of quadratic type” for evolutionary game theory. New evolutionary models for various types of geometric flows are put forward.

Hence application of Evolutionary game theory, the classical Lotka-Volterra predator-prey system with The Ricci flow as exactly a “replicator equation of quadratic type” for evolutionary game theory for defense strategy. 

For Example - 

Defense Strategy Selection Model Based on Multistage Evolutionary Game Theory - https://www.hindawi.com/journals/scn/2021/4773894/

The existing network attack and defense analysis methods based on evolutionary games adopt the bounded rationality hypothesis. However, the existing research ignores that both sides of the game get more information about each other with the deepening of the network attack and defense game, which may cause the attacker to crack a certain type of defense strategy, resulting in an invalid defense strategy. The failure of the defense strategy reduces the accuracy and guidance value of existing methods.

3. Ricci Flow and Community Detection for defense and fraud detection. My video is self explanatory -

https://www.youtube.com/watch?v=z_pjsJisdHQ&t=6s 

The community detection application is not only limited to fraud detection but also for applications on biological networks, protein-protein networks, metabolic networks, and gene networks, etc. 

Ricci Flow, Neural Networks, Nash Entropy, Ricci Yang-Mills flow

Hey readers,

It's Presidents day and Monday night. It's dark outside.. Interestingly, Around 40,000 reads on yesterdays blog. Thanks for continued readership. My memory jogs back to my 2013-14 Poincare Conjecture and Ricci Flow article followed by the lecture I gave at SDSU in 2017 followed by . One of the topics in the lecture apart from Convolutional Neural Networks, Generative Adversarial Networks etc. was Ricci Flow & Neural Networks. 

Ricci Flow is partial differential equation (PDE) for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.

Lot of research was inspired by this lecture including this paper -

1. RicciNets: Curvature-guided Pruning of High-performance Neural Networks Using Ricci Flow - https://arxiv.org/abs/2007.04216 (University of Cambridge)

This model combines the principle of Ricci curvature with ML to carry out neural architecture search. It successfully identifies salient computational paths, and demonstrates a reduction in computational cost for no degradation in baseline performance. It outperforms pruning via lowest-magnitude weights on randomly wired neural networks

2. Thoughts on the Consistency between Ricci Flow and Neural Network Behavior - https://arxiv.org/pdf/2111.08410.pdf (Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, China) 

In this paper, construct the linearly nearly Euclidean manifold as a background to observe the evolution of Ricci flow and the training of neural networks. It suggests that In view of the convergence and stability of linearly nearly Euclidean metrics against perturbations, we observe that the training of the neural network under the weakly approximated gradient flow is consistent with the evolution of the Ricci flow.

3. A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy (https://arxiv.org/pdf/2107.01419.pdf  UCSD  ) - 

A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy - Ricci flow with uniformly bounded Nash entropy must also have uniformly bounded ν-functional. Consequently, on such an ancient solution there are uniform logarithmic Sobolev and Sobolev inequalities.

4. This is the best paper I have seen so far -
Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network - https://www.mdpi.com/1099-4300/23/3/292

Since graph-structured data are inherently non-Euclidean, we seek to use a non-Euclidean mathematical tool, namely, Riemannian geometry, to analyze graphs (networks). In this paper, we present a novel model for semi-supervised learning called the Ricci curvature-based graph convolutional neural network, i.e., RCGCN. The aggregation pattern of RCGCN is inspired by that of GCN.. 

5. This is super awesome thesis guided by Abel Prize winner - Karen Uhlenbeck - 

Modified Ricci flow on a Principal Bundle - https://www.proquest.com/openview/9d0358256cc058132f31dc4b712b3ea9/1?pq-origsite=gscholar&cbl=18750 - 

Let M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow 

Ricci Flow, Geomstats, Lie Groups on Julia & EconoPhysics (and belated Happy Valentines day)

Hey Readers,

Again... thanks for reading my blog. I have been super-busy with lot of work. But,  30,000 views in single night is pretty cool. It is 9 PM in the evening, I just came from my walk in Maryland and I am on a mission to motivate everyone to learn more about data, AI, Math, Topology and so on. I should give more guest lectures when I get time.. but, for now please have a look at wonderful lecture by Geometric Statistics in Machine Learning GeomStats with Nina Miolane - https://www.youtube.com/watch?v=3yNInHuIWlY&t=1251s 

Hence, the next blogpost in the night on Ricci Flow (the topological concept that Grigori Perelman used to prove Poincare Conjecture) and Network Geometry.

1. Network geometry and market instability -https://royalsocietypublishing.org/doi/10.1098/rsos.201734

This paper talks about Time series of log-returns over a 32-year period (1985–2016) with network of stocks with 

1. Ollivier–Ricci (ORE), 
2. Forman–Ricci (FRE), 
3. Menger–Ricci (MRE), 
4. Haantjes–Ricci (HRE) 
5. Algebraic topological aspects, such as the homology groups and Betti numbers

minimum risk Markowitz portfolio of all the stocks, to better understand tipping points, systemic risk and resilience in financial networks, and enable us to develop monitoring tools required for the highly interconnected financial systems and perhaps forecast future financial crises and market slowdowns with use of Python package NetworkX (Yeah.. I used to use a lot of R, SAS, SQL, Cytoscape, Python in 2006 but Julia became my favorite language in 2012).

You can also check out few packages such as 

1. TheanoGeometry ( https://arxiv.org/abs/1712.08364 )- Riemann, Ricci and scalar curvature & geodesics  
2. Geomstats ( https://arxiv.org/abs/2004.04667 )   
3. McTorch  - McTorch, a manifold optimization library for deep learning ( https://arxiv.org/abs/1810.01811 ) 
4. Pymanopt: A Python Toolbox for Manifold Optimization using Automatic Differentiation ( https://arxiv.org/abs/1603.03236 )
5. Topological Entropy for Geodesic Flows under a Ricci Curvature condition: Jacobi field, Ricci curvature, topological entropy, tangent bundle, Negative Ricci curvature, isometry group and Injectivity radius - ( https://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03780-5/S0002-9939-97-03780-5.pdf )
6. And how can I forget my new favorite Julia for manifold Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds - a fast and easy to use library of Riemannian manifolds and Lie groups - ( https://arxiv.org/pdf/2106.08777.pdf ) 

Ricci Flow, Economics & a brilliant blog by Gita Gopinath

Hey readers,

It's 2:40 AM on Saturday night. Thanks for reading my DataHulk blog. I was glad to see 20,000 views in a single night on my last post - Yang Mills, Vector bundles, Quantum Information Geometry & Fisher-Bures Adversary Graph Convolutional Networks etc. And Yes, I noticed Terence Tao wrote his blog on the same day talking about Sphere Pinching paper (Smart people think alike I guess) - https://terrytao.wordpress.com/2022/02/08/perfectly-packing-a-square-by-squares-of-nearly-harmonic-sidelength/ 

I used to write this blog 12 years back to motivate everyone to learn math, data & AI folks. By the way, I have an exciting blogpost today which might change the way some of my followers look at economics (I know.. I know.. I promised to write about Spin Neural Networks, diffusion and capsule networks in my last blog.. which I will write later). 

1. Ricci Flow & Economics (And why Gita Gopinath's sentence in her blog is genius) - 

Here is an interesting paper followed by 2013-14 Ricci Flow & Poincare Conjecture in topology of Social Networks & Information Geometry article ( https://www.kdnuggets.com/2014/05/poincare-conjecture-perelman-topology-social-networks.html ).... Ricci curvature: An economic indicator for market fragility and systemic risk - https://www.researchgate.net/publication/303600815_Ricci_curvature_An_economic_indicator_for_market_fragility_and_systemic_risk which is an analysis based on geometric feature extraction on network data of daily returns from a set of stocks & market topology comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature, is negatively correlated to increases in network fragility. 

network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily

returns from a set of stocks comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight

the fact that corresponding changes in Ricci curvature constitute a financial “crash hallmark.”

  

Average Ricci curvature over a 15-year span of the S&P 500 - Choosing a window of T = 22 days, we see that curvature captures several financial crashes and show that, on average, market behavior is fragile.

2. Gita Gopinath - 

Here is an interesting & brilliant blog (April 2021) by Gita Gopinath on divergent recovery -
Managing Divergent Recoveries - https://blogs.imf.org/2021/04/06/managing-divergent-recoveries/ speaking about 

1. Financial risks, 

2. Financial stability risks using macro-prudential tools, 

3. Cross-country gaps, global poverty reduction, 

4. Emerging markets, 

5. Divergent recovery paths, 

6. High degree of uncertainty and developing economies 

7. International liquidity

8. Liquidity protection

9. Debt restructuring 

10. financial stability risks using macro-prudential tools

11. Withdrawn loan payments, firm insolvencies

12. Cross-border profit shifting

which reminded me of network entropy, nodal entropy, geodesics curvature & divergence mentioned in the Sandhu's paper - Ricci curvature & Ricci Flow: An economic indicator for market fragility and systemic risk and my KDnuggets article on which intended to perform Ricci Flow on Social network analysis & information economics. 

Yang Mills, Vector bundles, Quantum Information Geometry & Fisher-Bures Adversary Graph Convolutional Networks etc.

Hello guys,


It's been a long time since my last post & its Tuesday night.. Yes.. I am still DataHulk.. When I get angry I work on Big Data... :) 

- And off course.. Maryna Viazovska (one of the researcher who might win field prize as per my first blogpost based on Graph Mining, NLP, Reinforcement Learning & Profiling along with Bhargav Bhatt etc. - Graph Mining, Network Science + Topology & Predicting Field Prize - https://researchcircle.blogspot.com/2021/12/graph-mining-network-science-topology.html) has published new paper on Hyperbolic Fourier series - https://arxiv.org/abs/2110.00148

....... while Terence Tao has published new paper on Perfectly packing a square by squares of nearly harmonic sidelength - https://arxiv.org/abs/2202.03594 (Another signal that Sphere Packing problem is flavor of the season and Maryna Viazovska might win Field Prize for her effort on Sphere Packing in N = 8 & 24) 

There are 4 papers which I suggest reading everyone who is interested in intersection of Yang Mills, Vector Bundles & Graph neural networks -

1. Fisher-Bures Adversary Graph Convolutional Networks (quantum information geometry & Fisher information of the neural network) -

http://proceedings.mlr.press/v115/sun20a.html

You can enhance this Graph Neural Network further with reconstructing Quantum geometry from Quantum Information from the following paper. 

2. Reconstructing Quantum Geometry from Quantum Information 

https://arxiv.org/pdf/gr-qc/0501075.pdf

2. The Yang-Mills α-flow in vector bundles over four manifolds and its applications (Gang Tian) - 

https://arxiv.org/abs/1303.0628

4. Vector Bundles On Algebraic Varieties (This is the most important paper by Dr. Atiyah & team from 1984 in collaboration with TIFR Mumbai & Oxford University) -

http://www.math.tifr.res.in/~publ/studies/Vector-Bundles-On-Algebraic-Varieties.pdf

One of these days, I will write in detail why Yang mills & vector bundles has potential solution while working on graph neural networks because of Quantum Information Geometry connection. It's going to be one hell of an orchestration. 

For now, look at the beauty of Sir Atiyah's 1984 paper in collaboration with TIFR. (BTW, Update 8 days after this blogpost was published.. on 17th Feb Sir Atiyah's Student George Lusztig won Wolf Prize for representation theory & Quantum Groups) 

My next blog is going to be about Spin Networks.