Our interest is in real symmetric conservative matrices. \[ \mathbf{W} = \left(\matrix{ -\gamma_{1} & w_{12} & w_{13} & \cdots & \cdots \cr w_{12} & -\gamma_2 & w_{23} & \cr w_{13} & w_{23} & -\gamma_3 & \cr \vdots & & & \ddots\cr \vdots & & & & \ddots}\right) \] The "conservative" means that each row's sum is zero, or: \[\gamma_n \ \ \equiv \ \ \sum_{m\neq n} w_{mn} \]

Such matrices appear in rate equations: \[ \frac{d}{dt}{\mathbf p} \ \ = \ \ \mathbf{W}\cdot\mathbf{p} \] in Newtons II law for systems of masses and springs, which are used as heat transport models: \[ \frac{d^2}{dt^2}{\mathbf q} \ \ = \ \ -\mathbf{K}\cdot\mathbf{q} \] and in SchrÃ¶dinger's equation \[ \frac{d}{dt}{\mathbf \psi} \ \ = \ \ -i {\mathcal{H}}\cdot\mathbf{\psi} \]

We focus on sparse/glassy networks, where by sparse we mean that a small number of elements is larger by orders of magnitude than the rest.

As two examples we use:

#### Response of driven weakly chaotic systems

#### Mott random site model

In this model, the sites are randomly distributed in space, and the rates depend on the distance between sites: \[ w_{nm} \ \ =\ \ w_0 \cdot \exp\left(-\frac{\left|r_n-r_m\right|}{\xi} - \epsilon_{nm}\right)\] Where \(\xi\) and \(w_0\) are system parameters (defining the time and space units), and \(\epsilon\) a bond specific parameter. In the*degenerate*version, \(\epsilon=0\) for all the bonds, while in the*non-degenerate*version \(\epsilon_{nm} = \textrm{uniform}[0,\infty]\).The dimensionless parameter defining sparsity in this model is \[ s \ \ =\ \ \frac{\xi}{r_0} \] where \(r_0\) is the typical distance between sites.

- The parameter \(n_c\) is the average number of bonds required to get percolation. For \(d=1\) \(n_c=2\) and for \(d=2\) \(n_c\approx 4.5\), based on studies of disk-percolation.
- If we disregard the percolative nature (by setting \(n_c=0\)), we get the linear estimate back.
- In the limit \(s_\textrm{eff}\ll 1\) we obtain VRH-like behavior: \[ D\sim e^{-1/s_\textrm{eff}} \]

The long term behavior of the system is characterized by a diffusion coefficient \(D\). It is defined by the long term spreading: \[ S(t)\quad = \quad \left\langle r^2(t)\right\rangle \quad \sim \quad D t\] And it is also related to the long time survival probability: \[ \mathcal{P}(t)\quad \sim \quad \frac{1}{(Dt)^{1/2}} \] And to the low eigenvalue distribution (by Laplace transform): \[ \mathcal{N}(\lambda) \quad \sim \quad \left[ \frac{\lambda}{D}\right]^{d/2} \]

For a dense system, (large \(\xi\) compared to typical site distance), the diffusion coefficient can be estimated by a linear equation (linear in the rates): \[ D_{\textrm{linear}} = \frac{1}{2d}\sum_r w(r) r^2 \] We present the ERH procedure to estimate \(D\), based on resistor network analysis, with a smooth cross-over between the dense and sparse regimes.

\[ D \quad =\quad \textrm{EXP}_{d+2}\left(\frac{1}{s_\textrm{eff}}\right) \mbox{e}^{-1/s_\textrm{eff}} D_{\textrm{linear}} \\ s_\textrm{eff}\quad=\quad \left(\frac{d}{\Omega_d} n_c\right)^{-1/d}\frac{\xi}{r_0} \\ \textrm{EXP}_{l}(x) \quad = \quad \sum_{k=0}^l \frac{1}{k!}x^k\]

Y. de Leeuw, D. Cohen, Phys. Rev. E 86, 051120 (2012).