In communication networks, ``vulnerability'' indicates the resistance of a network to disruptions in communication after a breakdown of some processors or communication links. We may use graphs to model networks, as graph theoretical parameters can be used to describe the stability and reliability of communication networks. If we think of a graph as modeling a network, the average lower independence number of a graph is one measure of graph vulnerability. For a vertex $v$ of a graph $G=(V,E)$, the lower independence number $i_v(G)$ of $G$ relative to $v$ is the minimum cardinality of a maximal independent set of $G$ that contains $v$. The average lower independence number of $G$, denoted by $i_{av}(G)$, is the value $i_{av}(G)=\frac1{|V(G)|}\sum_{v\in V(G)}i_v(G)$. In this paper, we defined and examined this parameter and considered the average lower independence number of special graphs and theirs total graphs.