We study the spectral properties of sparse random graphs with different
topologies and type of interactions, and their implications on the stability of
complex systems, with particular attention to ecosystems. Specifically, we
focus on the behaviour of the leading eigenvalue in different type of random
matrices (including interaction matrices and Jacobian-like matrices), relevant
for the assessment of different types of dynamical stability. By comparing the
results on Erdos-Renyi and Husimi graphs with sign-antisymmetric interactions
or mixed sign patterns, we introduce a sufficient criterion, called strong
local sign stability, for stability not to be affected by system size, as
traditionally implied by the complexity-stability trade-off in conventional
models of random matrices. The criterion requires sign-antisymmetric or
unidirectional interactions and a local structure of the graph such that the
number of cycles of finite length do not increase with the system size. Note
that the last requirement is stronger than the classical local tree-like
condition, which we associate to the less stringent definition of local sign
stability, also defined in the paper. In addition, for strong local sign stable
graphs which show stability to linear perturbations irrespectively of system
size, we observe that the leading eigenvalue can undergo a transition from
being real to acquiring a nonnull imaginary part, which implies a dynamical
transition from nonoscillatory to oscillatory linear response to perturbations.
Lastly, we ascertain the discontinuous nature of this transition.Comment: 55 pages, 17 figure