  Many biological tissues feature a heterogeneous network of fibers whose
tensile and bending rigidity contribute substantially to these tissues' elastic
properties. Rigidity percolation has emerged as a important paradigm for
relating these filamentous tissues' mechanics to the concentrations of their
constituents. Past studies have generally considered tuning of networks by
spatially homogeneous variation in concentration, while ignoring structural
correlation. We here introduce a model in which dilute fiber networks are built
in a correlated manner that produces alternating sparse and dense regions. Our
simulations indicate that structural correlation consistently allows tissues to
attain rigidity with less material. We further find that the percolation
threshold varies non-monotonically with the degree of correlation, such that it
decreases with moderate correlation and once more increases for high
correlation. We explain the eventual reentrance in the dependence of the
rigidity percolation threshold on correlation as the consequence of large,
stiff clusters that are too poorly coupled to transmit forces across the
network. Our study offers deeper understanding of how spatial heterogeneity may
enable tissues to robustly adapt to different mechanical contexts.
