GRAPH THEORY AND STATS FOR THEM

Graph limits of Lovasz are currently being used for modeling the internet and other social nets that follow the Barbasi-Albert scale-free condition so have degrees that have a power law distributions. We can see this does not hold for our data of 188 stocks.

The major feature of our data that promises some characterization is the eigenvalues of the Laplacian (matrix) on a graph.

This spectrum is quite different than that of an Erdos-Renyi graph of the same number of nodes:

So we had the idea that perhaps the Lovasz graph limits and nonparametric estimation of graphon parameters of Wolfe and the limit ideas of Diaconis could all be brought to bear for models of market graph where we checked that the weight of the covariance off diagonals is significant compared to diagonals, i.e. volatility effects are definitely highly dependent on volatility of other assets and it’s pointless to model single asset volatility. We’ve argued that volatility is not a micro variable in the stat mechanics sense and in fact at a node gives us a dot on a grid where the high-dimensional (dimension=number of assets) chaotic dynamical system governs volatlity whose law must be sought by statistical models of (d/dt-Laplacian)Volatility and where we can apply Onsager’s reasoning for the creation of new vortices directly or by help from the theories and methods physicists have developed.

Volatility as a space object on a graph is one of our main ideas. We know something about this space which can be determined intrinsically. It’s a large network. We also know that hydrodynamic turbulence is not mere theory but tested and published in the late 1990s evan on Nature and PNAS. So our viewpoint is not a radical departure technically from some of the physicists attempt at these problems but it is a radical departure in asking for intrinsic laws of markets and finance in volatility as the primary object rather than returns. Global volatility is not necessarily caused by any events; its is the encapsulation of perhaps unconscious collective effects of the entire human psyche. It would not surprise me at all to find laws as simple as Newton’s law governing volatility. Such models are possible and should be the foundation of a new science of finance.

LOVASZ GRAPH LIMIT THEORY

Lovasz wrote an entire book since graph limits have natural applications to the internet. We’ll worry about staying away from scale-free assumption but the limit theory is general. Lovasz and his coworkers introduced graphons.

Patrick Wolfe shows how to do nonparametric estimation of graphons. We have to problem of understanding the situation of graph limits perhaps with prescribed Laplace spectrum.

## Leave a Reply