Intermediate interaction strategies for collective behavior

From bird flocks and fish schools to migrating cell sheets, collective motion is a ubiquitous biological phenomenon that inspires quantitative modeling through self-propelled particle (SPP) frameworks. Conventional SPP models prescribe either distance-based (metric) or rank-based (topological) interactions; however, empirical studies indicate that real groups may blend both types of interaction. Motivated by this graded perception, we introduce an extended three-dimensional SPP model in which metric and topological alignments act simultaneously and are weighted by a single tunable blending parameter called the interaction parameter. Large-scale simulations spanning a wide range of interaction parameters, densities, and neighborhood sizes revealed rich dynamics. Even when the global order parameter is low, cluster-level analysis with HDBSCAN shows that particles self-organize into several spatially distinct but internally well-aligned sub-flocks, exposing a hidden layer of order. Most importantly, an intermediate balance in which metric and topological cues contribute almost equally maximizes the global order parameter and markedly improves robustness to density variations. Numerical experiments supported by linear stability analysis demonstrate that activating metric and topological interactions in concert bridges the traditional modeling dichotomy and furnishes a more adaptive and resilient framework for collective motion. Therefore, the proposed model provides a versatile platform for exploring mixed-interaction effects in biological and engineered multi-agent systems.