  We consider a class of random, weighted networks, obtained through a
redefinition of patterns in an Hopfield-like model and, by performing
percolation processes, we get information about topology and resilience
properties of the networks themselves. Given the weighted nature of the graphs,
different kinds of bond percolation can be studied: stochastic (deleting links
randomly) and deterministic (deleting links based on rank weights), each
mimicking a different physical process. The evolution of the network is
accordingly different, as evidenced by the behavior of the largest component
size and of the distribution of cluster sizes. In particular, we can derive
that weak ties are crucial in order to maintain the graph connected and that,
when they are the most prone to failure, the giant component typically shrinks
without abruptly breaking apart; these results have been recently evidenced in
several kinds of social networks.
