Figure 1
(a) Densely connected system composed of equilibrium (
system, blue nodes) and nonequilibrium (
system, red nodes) components. Blue and red edges represent positive and negative interactions, respectively. Interaction directions are not shown. (b) Top: Properties of sparse systems exhibiting equilibriumlike behavior. The degrees of freedom (blue nodes) are interacting on graphs with locally tree-like topology. A fraction of these nodes (red) are exposed to the changing environment (perturbations). Bottom: Comparing the equilibrium energy
(dashed line) and magnetization
(solid line) with the average energy
and magnetization densities
(symbols) measured in Monte Carlo simulations of a ferromagnetic Ising spin system defined on a random regular graph of size
with
, where a fraction
of sites are subject to the external time-dependent random binary fields
, with
; all Monte Carlo simulations results reported here have been carried out for a similar system size and connectivity degree. The measurements are taken only on sites not influenced by
. (c) Comparing thermal fluctuations of the equilibrium energy
(dashed line) and magnetization
(solid line) with those measured in Monte Carlo simulations (represented by symbols with error bars, much smaller than the symbol size, on the solid lines). The nonequilibrium simulation measurements (symbols with error bars) are taken only at sites not influenced by
. (d) Comparing the magnetization
(solid lines) calculated theoretically with values (symbols) measured in the Monte Carlo simulations of ferromagnetic Ising spin system defined on an asymmetric (an incoming edge with probability
) random regular graph show good agreement between the two.
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