  Several networks occurring in real life have modular structures that are
arranged in an hierarchical fashion. In this paper, we have proposed a model
for such networks, using a stochastic generation method. Using this model we
show that, the scaling relation between the clustering and degree of the nodes
is not a necessary property of hierarchical modular networks, as had previously
been suggested on the basis of a deterministically constructed model. We also
look at dynamics on such networks, in particular, the stability of equilibria
of network dynamics and of synchronized activity in the network. For both of
these, we find that, increasing modularity or the number of hierarchical levels
tends to increase the probability of instability. As both hierarchy and
modularity are seen in natural systems, which necessarily have to be robust
against environmental fluctuations, we conclude that additional constraints are
necessary for the emergence of hierarchical structure, similar to the
occurrence of modularity through multi-constraint optimization as shown by us
previously.
