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.

翻译：近年来，复杂系统中“高阶”结构的涌现引发了广泛关注。这种“涌现性”组织已在多种自然与人工系统中被发现，但目前该领域尚缺乏对高阶协同与冗余对系统影响的统一理解。典型研究将协同信息的存在（或缺失）视为因变量，并报告系统变化时协同水平的变化。本文试图翻转这一范式：我们不再将高阶信息视为因变量，而是采用进化优化方法，演化出具有显著高阶冗余、协同或统计复杂性的布尔网络。随后，我们运用刻画离散动力学的成熟工具分析这些演化得到的网络群体，包括吸引子数量、平均瞬态长度及德里达系数。同时，我们评估系统整合信息的能力。结果发现：高协同系统不稳定且混沌，但具有较高的信息整合能力；相反，演化得到的冗余系统极为稳定，但信息整合能力可忽略不计。最后，平衡整合与分离的复杂系统（即Tononi-Sporns-Edelman复杂性）兼具混沌与稳定特征，其信息整合能力优于冗余系统，同时比随机系统与协同系统更稳定。我们得出结论：系统动力学的鲁棒性与其信息整合能力（本质上需要灵活性与敏感性）之间可能存在根本性权衡，而特定类型的复杂性自然能平衡这种权衡。