The vulnerability value of a communication network shows the resistance of the network after the disruption of some centres or connection lines until the communication breakdown. In a network, as the number of centres belonging to sub networks changes, the vulnerability of the network also changes and requires greater degrees of stability or less vulnerability. If the communication network is modelled by a graph G, deterministic measures tend to provide a worst-case analysis of some aspects of the overall disconnection process. Differently from other measures, in the neighbour-integrity is considered that any failure vertex affects its neighbour vertices. Neighbour-integrity is very important measure in stability of security networks and spy networks. It replies three questions: How many vertices can still communicate? How difficult is it to reconnect the graph? How can we design an optimal network? In this paper we discuss the concept of neighbour-integrity. Firstly, we give some definitions and notation and then we calculate some stability numbers of two-dimensional mesh and torus graphs, which are used in computer sciences.