A key aspect of social, epidemiological, and economic systems is collective decision-making, in which joint behavior influences individuals or autonomous components. Opinion evolution has a significant function in decentralized consensus formation systems for short-time scales. We conduct Monte Carlo simulations of the majority-vote opinion dynamics on random networks to understand how the average number of social interactions impacts emergent behavior. The characteristic exponents β/(νz), 1/(νz), and θ determine the critical slip behavior at the beginning of the opinion evolution. We show that relaxation time on random networks is significantly shorter than on regular lattices, incorporating that the exponent θ exhibits dependency on network size and connectivity. Our findings have substantial implications for decentralized consensus systems with critical applications in blockchain protocol technology.