Steady-state distributions of probability fluxes on complex networks

Abstract

The methodology based on the random walk processes is adapted and applied to a comprehensive analysis of the statistical properties of the probability fluxes. To this aim we define a simple model of the Markovian stochastic dynamics on a complex network extended by the additional transition, called hereafter the gate. The random skips through the gate, driven by the external constant force, violate the detailed balance in the network. We argue, using a theoretical approach and numerical simulations, that the stationary distributions of the probability fluxes emergent under such conditions converge, regardless of the network topology, to the normal distribution. This result, combined with the stationary fluctuation theorem, permits to show that its standard deviation depends directly on the square root of the average flux. In turn, the central result of our paper relates this quantity to the external constant force and the two parameters that entirely characterize the normal distribution of the probability fluxes both close to as well as far from the equilibrium state. Also, the other effects that modify these parameters, such as the addition of shortcuts to the tree-like network, the extension and configuration of the gate and a change in the network size studied by means of the computer simulations are widely discussed in terms of the rigorous theoretical predictions.

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“…From ( 9) its follows that standard deviations ∆ 1 and ∆ 0 are inversely proportional to the square root of t. For the lack of correlations, ρ = 0, the solutions to (9) reconstruct our earlier result [50].…”

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“…From 9, it follows that standard deviations ∆ 1 and ∆ 0 are inversely proportional to the square root of t. For the lack of correlations, ρ = 0, the solutions to 9 reconstruct our earlier result [48].…”

“…For given node l, the transition probability to any of the neighboring nodes per one random walking step is one over the number of links and equals (p eq l τ int ) −1 , where p eq l is the equilibrium occupation probability of the given node and τ int is the mean time to repeat a chosen internal transition, counted in the random walking steps. This time is determined by the doubled number of links minus one 54 , τ int = 2 (100 + 3 − 1) = 204 random walking steps for the 100 node tree network with 3 shortcuts assumed.…”

“…To provide the network with stochastic dynamics, we assume the probability of changing a node to any of its neighbors to be the same in each random walking step [19,48]. Then, following the detailed balance condition, the equilibrium free energy of a given node is inversely proportional to the number of its links (the node degree).…”

“…We assume τ ext = 20 random walking steps, one order of the magnitude shorter than the mean time of internal transition τ int , which means that the whole process is controlled by the internal dynamics of the system. Identifying a time unit with a computer step still requires establishing the probabilities of staying, different on network and gate nodes [48].…”