  We study the stability of an inverse problem for the fractional conductivity
equation on bounded smooth domains. We obtain a logarithmic stability estimate
for the inverse problem under suitable a priori bounds on the globally defined
conductivities. The argument has three main ingredients: 1. the logarithmic
stability of the related inverse problem for the fractional Schr\"odinger
equation by R\"uland and Salo; 2. the Lipschitz stability of the exterior
determination problem; 3. utilizing and identifying nonlocal analogies of
Alessandrini's work on the stability of the classical Calder\'on problem. The
main contribution of the article is the resolution of the technical
difficulties related to the last mentioned step. Furthermore, we show the
optimality of the logarithmic stability estimates, following the earlier works
by Mandache on the instability of the inverse conductivity problem, and by
R\"uland and Salo on the analogous problem for the fractional Schr\"odinger
equation.
