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Despite its widespread use in nanocomposites, the effect of embedding graphene in highly viscoelastic polymer matrices is not well-understood. We add graphene to a lightly crosslinked polysilicone, often encountered as Silly Putty, changing its electro-mechanical properties significantly. The resulting nanocomposites display unusual electromechanical behavior such as post-deformation temporal relaxation of electrical resistance and nonmonotonic changes in resistivity with strain. These phenomena are associated with the mobility of the nanosheets in the low-viscosity polymer matrix. By considering both the connectivity and mobility of the nanosheets, we develop a quantitative model that completely describes the electromechanical properties. These nanocomposites are sensitive electromechanical sensors with gauge factors >500 which can measure pulse, blood pressure and even the impact associated with the footsteps of a small spider.