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Piecewise-linear resistive networks can be characterized by the equation where is a finite positive number. The domain ( -dimensional Euclidean space) is divided into regions (closed convex polyhedrons). In each region is a constant matrix and is a constant -vector. In this paper, we derive necessary and sufficient conditions for the function to be a homeomorphism. Different formulations of network equations are investigated, and results in terms of the matrices 's are obtained. An algorithm with a new perturbation method is also developed which is capable of locating the unique solution in a finite number of steps. The work is different from the early work by Kuh and Fujisawa in many ways; comparisons are presented.