  In multiplex networks with a large number of layers, the nodes can have
different activities, indicating the total number of layers in which the nodes
are present. Here we model multiplex networks with heterogeneous activity of
the nodes and we study their robustness properties. We introduce a percolation
model where nodes need to belong to the giant component only on the layers
where they are active (i.e. their degree on that layer is larger than zero). We
show that when there are enough nodes active only in one layer, the multiplex
becomes more resilient and the transition becomes continuous. We find that
multiplex networks with a power-law distribution of node activities are more
fragile if the distribution of activity is broader. We also show that while
positive correlations between node activity and degree can enhance the
robustness of the system, the phase transition may become discontinuous, making
the system highly unpredictable.
