Before revision:  More specifically, we consider a bipartite stochastic graph G = (U,V,E) where U is the set of offline vertices, V is the set of online vertices and G has edge probabilities (p_{e})_{e \in E} and vertex weights (w_{ u })_{ u \in U }. Additionally, G has patience values (\ell _{v})_{v \in V}, where \ell_v indicates the maximum number of edges adjacent to an online vertex v which can be probed.
After revision:  More specifically, we consider a bipartite stochastic graph G = (U,V,E) where U is the set of offline vertices, V is the set of online vertices and G has edge probabilities (p_{e})_{e \in E} and edge weights (w_{ u })_{ u \in U }. Additionally, G has patience values (\ell _{v})_{v \in V}, where \ell_v indicates the maximum number of edges adjacent to an online vertex v which can be probed.