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## Wolfe-Perry models

I continue my study of Wolfe and other statisticians on attempting to have a clear understanding of statistical models of graphs which for us needs understanding for the world financial graph, the x variable for any dynamical laws that can describe volatility.  I have now understood clearly the types of stats models that exist from the attached paper of Perry and Wolfe.  The main models they clearly describe and generalize as follows: first their generalization.  Let G=(V,E) be a graph with n vertices and consider the independent Bernoulli biased coin probability p_ij for each edge.  The Perry-Wolfe class of models are parametrized by a function epsilon(x,y) and they have the form
log(p_ij) = alpha_i + alpha_j + epsilon_ij(alpha_i, alpha_j)

This can be seen to generalize the logistic, log-linear and other models.  They prove that the likelihood equation for the entire class of models can be solved with some quantitative control of convergence by Newton’s method.  The interesting feature of their analysis is that they employ optimal error bounds on Newton’s method via Kantorovich theorem.  Attached is an example of the type of bounds that have been proved for Newton’s method in quite general setting of Banach spaces.  I am paying more attention to hard analysis as I proceed study since I would like to careful with the mathematics as I make progress toward a scientific model of volatility based on the space-time approach to volatility.  So the snapshot looks at the bounds on Newton’s method and a paper that proves this.  I have looked at the theoretical analysis of interior point methods before and am generally quite familiar with numerical methods of this type;  I’ve also worked with iterative solvers for convex optimization.  In this case, the interest in the technique of employing optimal bounds for Kantorovich-Newton theorem for theory. I will turn to estimating the basic model with epsilon_ij==0. next.

The serious question here is not mathematical but scientific:  are these models fair models for the world financial network or graph?  The independent Bernoulli assumption allows ease of maximum likelihood estimation, and this is very important but the real advantage to these n-parameter models is that they can capture the degree heterogeneity in some sensible way.  It is this degree heterogeneity-matching that makes these models valuable for us as we cannot fall into parametric degree models.  At the same time, my next problem is to examine the Laplacian spectrum as well and try to understand whether these models help us with the setting we have for graph limits — we’d like the limit object to have spectral constraints.
NOTE ON ESTIMATING THE LOGISTIC MODEL IN R:

glm( ,family=binomial(logit),)

solves the estimation problem in practical terms.  The purpose of trying to understand these better is to assess their value for the volatility dynamics problem where the graph model needs to be stochastic because we will never have the resources to gather data on a complete market network.

NOTE ON MARKET NETWORKS

There is an approach of modeling trading in markets in terms of buyers and sellers with bid and ask prices being helped by the market-maker.   It is pure delusion that this bid-ask level DATA DETAIL provides any deep insight into volatility.  The volatility is a SCIENTIFIC OBJECT of study that is global.  It represents a collective state of the entire human collective and it is silly to make arbitrary assumptions about what drives humans.  I believe that just I discovered by considering global geometry of the universe which mathematically implies quantization of energy and have argued that when we look with too close and speculate on the nature of reality in the small we lose perspective and create giant fairy tales parading as serious thought, the same could happen when we seek some sane science by highest frequency bid-ask assuming that the DECISIONS to trade happened anywhere close to the execution of the trade.  Machiavelli was a great genius and had repeated throughout his work that an invariant of man is that we are driven by passions but not without reason.  We will discover, I believe, deeper explanations of human nature in the global volatility dynamics with no simpleminded rush to explain than the microscope thinking that now predominates hedge fund thinking.

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