We make use of multiplicative degree-based topological indices $X_Π(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_Π(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_Π(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $F_V(x)$ and $F_E(x,y)$ are functions of the vertex degrees. Specifically, we find analytical inequalities involving these multiplicative indices. Also, we apply $X_Π(G)$ on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. We show that $\left< \ln X_Π(G) \right>$, normalized to the order of the network, scale with the corresponding average degree; here $\left< \cdot \right>$ denotes the average over an ensemble of random networks.