We study the relationship between microscopic structure and viscosity in non-Brownian suspensions. We argue that the formation and opening of contacts between particles in flow effectively leads to a negative selection of the contacts carrying weak forces. We show that an analytically tractable model capturing this negative selection correctly reproduces scaling properties of flows near the jamming transition. In particular, we predict that i) the viscosity η diverges with the coordination number z as η(z cz) (3+θ)/(1+θ), ii) the operator which governs flow displays a low-frequency mode that controls the divergence of viscosity, at a frequency ω min(z cz) (3+θ)/(2+2θ), and iii) the distribution of forces displays a scale f* that vanishes near jamming as f*/f(z cz) 1/(1+θ) where θ characterizes the distribution of contact forces P(f)f θ at jamming, and where z c is the Maxwell threshold for rigidity. © Copyright EPLA, 2012.