We consider a finite structured population of mobile individuals that strategically explore a network using a Markov movement model and interact with each other via a public goods game. We extend the model of Erovenko et al. (2019) from complete, circle, and star graphs to various random networks to further investigate the effect of network topology on the evolution of cooperation. We discover that the network topology affects the outcomes of the evolutionary process only for networks of small average degree. Once the degree becomes sufficiently high, the outcomes match those for the complete graph. The actual value of the degree when this happens is much smaller than that of the complete graph, and the threshold value depends on other network characteristics.