  For any graph~\(G,\) a set of vertices~\({\cal V}\) is said to be dominating
if every vertex of~\(G\) contains at least one node of~\(G\) and separating if
each vertex~\(v\) contains a unique neighbour~\(u_v \in {\cal V}\) that is
adjacent to no other vertex of~\(G.\) If~\({\cal V}\) is both dominating and
separating, then~\({\cal V}\) is defined to be an identification code. In this
paper, we study strong identification codes with an index~\(r,\) by imposing
the constraint that each vertex of~\(G\) contains at least~\(r\) unique
neighbours in~\({\cal V}.\) We use the probabilistic method to study both the
minimum size of strong identification codes and the existence of graphs that
allow an identification code with a given index.
