Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation

A graph-theoretic analysis of the steady-state current noise in master equations under a finite thermodynamic force (affinity) is presented. The incidence matrix twisted by a finite affinity is not orthogonal to the standard cycle space, motivating the introduction of twisted circuit matrices to restore the orthogonality. The resulting twisted-cycle matrix yields an interference-like effect, enabling us to express the signal-to-noise ratio as a quadratic optimization problem in terms of twisted-cycle currents. We apply this framework to a Brownian computation model on a tree-like state-transition diagram with exponential backward branching, finite affinity at each step, and a single reset cycle. In the limit of an infinitely long intended computation path $\ell$, the Fano factor of the reset current undergoes a transition from noiseless to Poissonian behavior at an affinity equal to the logarithm of the number of immediate predecessors $\alpha$. This corresponds to an easy-hard transition in the computational time complexity [K. Okajima, K. Hukushima, arXiv:2512.24728 ], which is not captured by the thermodynamic uncertainty relation. This transition point precisely characterizes the thermodynamic costs of logically irreversible computation: in the absence of affinity, the reset cost scales as $\ln \ell$, whereas reaching the transition point requires a thermodynamic force of order $\ln \alpha$ per step to counteract backward branching.