The viscosity of colloidal dispersions is a highly sensitive function of the microstructure of the dispersion and the nature of the interactions among the particles. While the viscosity of solutions of colloidal hard spheres is well understood, the viscosity in a broader array of colloidal solutions remains poorly studied. Here, we use immersed boundary simulations to investigate the viscosity of a prototypical colloidal solution, adhesive hard spheres, which under the right conditions can percolate to form sample spanning networks. We examine how rigidity of the particle-particle contacts in these dispersions can enhance the dispersion viscosity. By rigidly constraining a fraction of the adhered particles, we establish a reaction coordinate, the extent of rigidity, along which we can track the development of a gelled state characterized by a diverging viscosity. The fraction of rigid bonds is analogous to the extent of reaction for cross-linking in chemical gels, which suggests a close connection between chemical gelation with polymers and physical gelation with adhesive hard spheres. A critical gel point is identified when particles connected by rigid bonds begin to percolate the sample. In the vicinity of critical gel point, we observe a diverging viscosity and critical scaling behavior in agreement with previous studies on adhesive hard spheres. We use these results to formulate empirical equations for the viscosity across a broad range of volume fractions as a function of the extent of rigidity. Most importantly, through these simulations, we show that rigid constraints among the bonded particles are essential to producing the large viscosities measured experimentally in dispersions of adhesive hard spheres. Hydrodynamic interactions alone are insufficient for reproducing experimental observations. (c) 2019 The Society of Rheology.