A recurring data mining task in complex networks is to determine how individual nodes contribute to system behavior. Existing approaches rely on either static-graph centralities or control-theoretic quantities such as controllability Gramians, which assume linear, time-invariant dynamics. Estimated systems, however, are typically nonlinear and time-varying. We define "emergent contribution (EC)," a finite-horizon measure of a node's dynamical leverage: the metric-weighted energy of its impulse response accumulated along the system trajectory. Computed from the Jacobians of any differentiable model, EC is estimator-agnostic and reduces exactly to average controllability in the linear, time-invariant limit. Our contribution is a characterization of when the two measures agree and diverge. Using a controlled synthetic family with known ground-truth contribution, we construct a phase diagram spanning nonlinearity, regime structure, persistence, and perturbation amplitude. EC and average controllability agree under static or smoothly drifting dynamics and both track ground truth. Divergence emerges under persistent regime switching, is strongest under persistent sign reversal, and disappears when the sign reversal is removed. At extreme perturbation amplitudes, both measures degrade, identifying the limits of local linearization. We place five estimated real systems from several domains within this phase space. Their placement serves as a diagnostic of when EC provides information beyond static controllability and therefore justifies its additional computational cost. On one panel examined in depth, a twenty-seed retraining ensemble reveals a robust variance--leverage dissociation: nodes whose perturbations propagate widely despite low within-system variance, which is not recovered by static centralities nor variance-based summaries.

翻译：在复杂网络中，一项常见的数据挖掘任务是确定单个节点如何影响系统行为。现有方法要么依赖静态图中心性，要么依赖诸如可控性格拉姆矩阵等控制论量，这些方法假设系统是线性时不变的。然而，实际估计的系统通常是非线性和时变的。我们定义了“涌现贡献”（Emergent Contribution，EC），这是一种衡量节点动力学杠杆作用的有限时间度量：即其脉冲响应沿系统轨迹累积的度量加权能量。通过任何可微模型的雅可比矩阵计算得出，EC与具体估计方法无关，并在线性时不变极限下精确退化为平均可控性。我们的贡献在于刻画了这两种度量何时一致、何时分歧。利用一个已知真实贡献的受控合成系统族，我们构建了一个涵盖非线性、体制结构、持久性和扰动幅度的相图。在静态或缓慢漂移动力学条件下，EC与平均可控性一致，且两者均能追踪真实值。分歧在持续体制切换时出现，在持续符号反转时最为强烈，而当符号反转被移除时分歧消失。在极端扰动幅度下，两种度量均会退化，这揭示了局部线性化的极限。我们将来自多个领域的五个真实估计系统置于该相空间中，其位置可作为诊断依据，判断EC何时能提供超越静态可控性的信息，从而证明其额外计算成本的合理性。在对其中一个面板进行的深入分析中，一个基于二十个种子点的重训练集成揭示了稳健的方差-杠杆分离现象：某些节点的扰动会广泛传播，尽管其系统内方差较低，而静态中心性及基于方差的总结无法发现这一现象。