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# Poincare Conjecture, Perelman way, and Topology of social networks

We examine the connections between the \$1 million proof of Poincare conjecture by a reclusive math genius and the topological behavior and information diffusion over social networks.

Guest post by Prasad Kothari, May 3, 2014.

A few years ago a brilliant mathematician Grigori Perelman solved  a very famous 100 years old problem in topology called Poincare Conjecture.  Then, Perelman cut his all contacts with the world and started living lonesome life in his mother’s apartment in Russia . He  rejected Fields Medal, a "Nobel"-equivalent prize for Mathematics, and also a \$1 million award by Clay institute for solving this problem.

There was lot of media coverage on this issue and many said that even though it is a brilliant solution, it does not mean anything in the practical world.  It kept me thinking for years. Here, I present a small article on how Perelman’s work can be used to better information diffusion analysis in social networks:

Perelman’s tool (Ricci Flow entropy calculation – from Wikipedia):

In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.

The Ricci flow, named after Gregorio Ricci-Curbastro, was first introduced by Richard Hamilton in 1981 and is also referred to as the Ricci–Hamilton flow. It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture.

Perelman’s equation gives rise to importance of entropy in diffusion equations through geometric entities. What Perelman did not know at that point was that the similar concept of Shannon entropy will be used in social networks 10 years later to calculate information diffusion.

Connection:

In simple words, the geometric flow defined by Poincare and proved by Perelman, can be measured by entropy using Ricci flow equations (these are non-linear diffusion equations). The social networks which have evolved over time tend to have similar geometric properties and adhere to similar rules and assumptions defined by Perelman. When you take into account degrees of freedom of node of the social network and its association with geometric flow measured by entropy, it pretty much gives you reach of that node and defines the virality of the given post in social network.

This measures the gossip flow through social networks (See: Efficient Reconciliation and Flow Control for Anti-Entropy Protocols, Robbert van Renesse, Dan Dumitriuy, Valient Gough, Chris Thomas, Proceedings of LADIS'08 Workshop)

To expand this discussion, we can say that social networks might be the best tool to validate Perelman’s and other topology proofs. Not only that, but we can apply the same transformational formulas used in topology to transform social networks and control the flow of information through given nodes. Some nodes work as catalysts of the information flow through social networks and some nodes behave as dampening agents of the same information which is similar to the properties exhibited by geometrical objects considered for Navier Stokes equation and Perelman’s work.

Topological behaviour of social networks is the topic less researched because there can be few application of it as of now. But, none the less social networks work as a proxy for all the real life communication and are governed by some of the topological phenomenon we ought to know to improve the existing structures and come up with more catalysts and less dampening agents for the increased flow of information given by the marketing agency using social networks.

The following figure represents one of the visualization of Facebook (Please see: www.emilio.ferrara.name/2012/10/09/large-scale-community-structure-analysis-facebook/). This pretty much converges into topological properties of spheres and geometric flows Perelman worked on.