The intrinsic specificity obeys a Gaussian distribution near the mean and an exponential distribution near the tail.

It is also Gaussian distributed:

Because the underlying interaction energy is Gaussian distributed , the functional form of the distribution of the resulting free energy can be obtained by carefully studying the moments of the partition function of the resulting random energy model [36, 37].

Where f(F) is a Gaussian distribution of the free energy F around its mean 1—3 (See details in 81 Text) With the variance of the distribution A132 2 AB2 (The Width of the energy distribution AE here has the meaning of the roughness of the underlying energy landscape above the trapping transition temperature TC.

Notice that the distribution of free energy develops an exponential tail Which decays slower than the Gaussian distribution .

In contrast to the gaussian distribution , the log-normal distribution has a longer tail representing the higher frequencies of equilibrium constant when plotted in the original scale without taking the logarithm.

This leads to a random energy model with the interaction energy follows a Gaussian distribution [19, 33, 34].

In particular we obtained the analytical functional form of the probability function of the binding free energy to be Gaussian distributed near the mean and exponential-like distributed in the tail.

The statistical distribution of the intrinsic specificity ratio is also Gaussian distributed near the mean and exponentially distributed near the tail.

For both models, the natural frequencies of the oscillators in our simulation are given as a Gaussian distribution to simulate alpha with mean at 10 Hz and standard deviation 1, making wj around 10211 rad/ s. Time delay is (a) given an identical value between 2ms and 50 ms for all edges (for model networks as well as Gong et al.’s and Hagmann et al.’s human brain networks), or (b) given proportional to the physical distances for each edges with propagation speed of between 5 to 10m/ s (for Gong et al.’s human brain network) [60, 61].

The natural frequency of each node was randomly drawn from a Gaussian distribution with the mean at 10 Hz and standard deviation of 1 Hz, simulating the alpha bandwidth (8-13HZ) of human EEG, and we systematically varied the coupling strength 8 from 0 to 50.

For simulations, we expanded our conditions further: we used a Gaussian distribution for natural frequencies of the oscillators and distance-varying time delays between the oscillators for Gong’s anatomical network.

Product mutations were drawn from a Gaussian distribution with o = 0.1, element-wise mutation rate

Mutation values were drawn from a Gaussian distribution (unless otherwise stated).

(G + 86) H, where 8,- are independent noise realizations drawn from a Gaussian distribution N(1, o) with different values of o varying between 0.001 to 0.2.

For multivariate Gaussian distributions , zero partial correlations indicate conditional independence of the pair, implying a lack of direct interaction [40, 52].

We used these covariance matrices as the ground truth in multivariate Gaussian distributions with zero means and drew samples of various sizes.

Conveniently, the Ising model has equivalent mathematical form to the Gaussian distribution , but the Ising model is defined on the multivariate binary domain rather than the continuous domain.