Density Functional Theory (DFT) was used to elucidate the catalytic mechanisms in Pt-Co alloy catalysts for the CO$_2$ methanation reaction. Different ensemble sites of Co were constructed to compare the activity of these sites. The adsorption energies and reaction energy barriers show that Co is critical for catalytic activity and the Co-dimer sites are the most active sites. The activity of Co-dimer sites could be attributed to the binding configuration of CO$_2$, where each oxygen atom binds to one Co atom on the surface. Energy barrier results show that the formate pathway towards CH$_4$ production is most favored. To explain the increased CH$_4$ production with higher Co ratios, a Monte Carlo method was used to reveal that the number of active Co-dimer sites are proportional to the Co content.