  Hierarchically structured materials, which possess distinct features on
different length scales, are ubiquitous in nature and engineering. In many
cases, one structural level may be ordered while another structural level may
be disordered. Here, we investigate the impact of structural disorder on the
mechanical properties of hierarchical filamentous structures. Through
simulations of networks with two hierarchical levels, we show that disorder
does not change how stiffness scales with the mean coordination number - the
average number of bonds per node - on large and small length scales. However,
we find that network rigidity and stiffness depend strongly on the presence or
absence of disorder on the small length scale, but not on the large length
scale. In fact, the amount of material necessary for a fully connected, network
ordered on the small scale is insufficient to create even a marginally rigid
network with small-scale disorder. We trace these phenomena back to a
difference in the maximum mean coordination number on the small scale. While
single length scale ordered and disordered networks have similar mean
coordination numbers in the interior and on surfaces, we find that disorder
strongly impacts the structure of surfaces, resulting in a larger fraction of
surface nodes on the small scale. While this effect increases in strength as
large scale bonds become narrower, it persists even for bonds that are wider
than they are long (i.e., aspect ratios < 1).
