Statistical Approaches to Structure and Activities of Dynamic Systems

We will use a spin-glass model to understand the relationship between structure and dynamics. The major goal is to establish a formal approach for inferring the internal structure of a system given its observed dynamics.

1. Given $\langle J\rangle$ and $\Delta J$, characterize resultant dynamics with eigenvalues $\lambda_n$ of covariance or correlation matrix $C$.
2. How much are the eigenvalues influenced by the realized coupling configuration? (With an ensemble created with different seeds of random number generator, do the eigenvalues change significantly? What are their variances?)
3. Fit the eigenvalues with $$\lambda_n\sim n^{-\alpha}.$$ Perform systematic calculations to find the relationship between $\alpha$ and $\langle J\rangle$, $\Delta J$.
4. Are there other ways to characterize the activity data beside $\alpha$?
5. Fixing $\langle J\rangle$ and $\Delta J$, does forms of the distribution $P(J)$ influence $\alpha$? For example, does having a long tail matters?

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