  There has recently been an explosion of interest in how "higher-order"
structures emerge in complex systems. This "emergent" organization has been
found in a variety of natural and artificial systems, although at present the
field lacks a unified understanding of what the consequences of higher-order
synergies and redundancies are for systems. Typical research treat the presence
(or absence) of synergistic information as a dependent variable and report
changes in the level of synergy in response to some change in the system. Here,
we attempt to flip the script: rather than treating higher-order information as
a dependent variable, we use evolutionary optimization to evolve boolean
networks with significant higher-order redundancies, synergies, or statistical
complexity. We then analyse these evolved populations of networks using
established tools for characterizing discrete dynamics: the number of
attractors, average transient length, and Derrida coefficient. We also assess
the capacity of the systems to integrate information. We find that high-synergy
systems are unstable and chaotic, but with a high capacity to integrate
information. In contrast, evolved redundant systems are extremely stable, but
have negligible capacity to integrate information. Finally, the complex systems
that balance integration and segregation (known as Tononi-Sporns-Edelman
complexity) show features of both chaosticity and stability, with a greater
capacity to integrate information than the redundant systems while being more
stable than the random and synergistic systems. We conclude that there may be a
fundamental trade-off between the robustness of a systems dynamics and its
capacity to integrate information (which inherently requires flexibility and
sensitivity), and that certain kinds of complexity naturally balance this
trade-off.
