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 interferencelike 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 treelike 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 and 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.

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• Fluctuations & noise
• Stochastic thermodynamics
• Thermodynamics of computation
• Graph theory