In graph signal processing, the shift-enabled property of an underlying graph is essential in designing distributed filters. This article discusses when a random network graph is shift-enabled. In particular, popular network graph models Erdős-Rényi (ER), Watts-Strogatz (WS), Barabási-Albert (BA) for both weighted and unweighted are considered. Moreover, both balanced and unbalanced signed graphs constructing using ER are considered. Our results show that the considered unweighted connected random network graphs are shift-enabled with high probability when the number of edges is moderately high. However, very dense graphs, as well as fully connected graphs, are not shift-enabled. Interestingly, this behaviour is not observed for weighted connected graphs, which are always shift-enabled unless the number of edges in the graph is very low. Finally, we evaluate the shift-enabled property of nine real-world graphs. The experimental results are consistent with our findings on randomly generated data. The results provide the basis for the filter design in a graph network.