One of the most intriguing features of spin-ice materials is that their elementary excitations behave as emergent magnetic monopoles. While the properties of spin ices suggest applications in magnetism and spintronics, $a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}f\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l$ spin ices---metamaterials built of interacting ferromagnetic nanowires---also exhibit unusual properties that could be utilized in reconfigurable magnetoresistive devices. The circuit model presented in this study provides a simple picture for understanding these systems, underscoring the many-body origin of their complex magnetotransport phenomena. This model also provides a guiding principle for designing devices based on these metamaterials.