Activation of the fibroblast growth factor (FGFR) and melanocyte stimulating hormone (MC1R) receptors stimulates B-Raf and C-Raf isoforms that regulate the dynamics of MAPK1,2 signaling. Network topology motifs in mammalian cells include feed-forward and feedback loops and bifans where signals from two upstream molecules integrate to modulate the activity of two downstream molecules. We computationally modeled and experimentally tested signal processing in the FGFR/MC1R/B-Raf/C-Raf/MAPK1,2 network in human melanoma cells; identifying 7 regulatory loops and a bifan motif. Signaling from FGFR leads to sustained activation of MAPK1,2, whereas signaling from MC1R results in transient activation of MAPK1,2. The dynamics of MAPK activation depends critically on the expression level and connectivity to C-Raf, which is critical for a sustained MAPK1,2 response. A partially incoherent bifan motif with a feedback loop acts as a logic gate to integrate signals and regulate duration of activation of the MAPK signaling cascade. Further reducing a 106-node ordinary differential equations network encompassing the complete network to a 6-node network encompassing rate-limiting processes sustains the feedback loops and the bifan, providing sufficient information to predict biological responses.