We investigate a special case of hereditary property that we refer to as {\em robustness}. A property is {\em robust} in a given graph if it is inherited by all connected spanning subgraphs of this graph. We motivate this definition in different contexts, showing that it plays a central role in highly dynamic networks, although the problem is defined in terms of classical (static) graph theory. In this paper, we focus on the robustness of {\em maximal independent sets} (MIS). Following the above definition, a MIS is said to be {\em robust} (RMIS) if it remains a valid MIS in all connected spanning subgraphs of the original graph. We characterize the class of graphs in which {\em all} possible MISs are robust. We show that, in these particular graphs, the problem of finding a robust MIS is {\em local}; that is, we present an RMIS algorithm using only a sublogarithmic number of rounds (in the number of nodes $n$) in the ${\cal LOCAL}$ model. On the negative side, we show that, in general graphs, the problem is not local. Precisely, we prove a $Ω(n)$ lower bound on the number of rounds required for the nodes to decide consistently in some graphs. This result implies a separation between the RMIS problem and the MIS problem in general graphs. It also implies that any strategy in this case is asymptotically (in order) as bad as collecting all the network information at one node and solving the problem in a centralized manner. Motivated by this observation, we present a centralized algorithm that computes a robust MIS in a given graph, if one exists, and rejects otherwise. Significantly, this algorithm requires only a polynomial amount of local computation time, despite the fact that exponentially many MISs and exponentially many connected spanning subgraphs may exist.