Causal Entropy, Control and Leadership Dynamics

TL;DR

The paper addresses how leadership and directional preferences affect fragmentation in decentralized collective motion. It develops a bottom-up FSM framework where agents maximize the entropy of their future visual states, augmented by a magnetisation-like energy term that encodes directional bias; leadership is implemented via two informed groups with opposite directions. The results show fragmentation depends on the bias strength $\\omega$, angular difference $\\Theta$, and the number of informed agents, with a DBSCAN-based metric $\\bar{C}$ quantifying cohesion. Importantly, introducing a four-neighbour alignment with coupling $J$ reveals a finite optimal value that maximizes cohesion, indicating that combining FSM with classical flocking rules yields robust, cohesive intelligent collectives. This hybrid approach provides a flexible model for understanding and designing decentralized, leader-influenced swarms in real-world contexts.

Abstract

Collective motion in animal groups provide examples of emergent, decentralised coordination. Here, we examine a bottom-up model of collective behavior based on Future State Maximisation (FSM). In this model agents seek to maximise the diversity of their future visual states over a finite time horizon. We further assume that a subset of agents have a directional bias, e.g. towards different destinations. We observe swarm fragmentation on increasing (i) the strength of these preferences, or (ii) the difference in preferred directions, or (iii) the number of biased agents. Depending on these factors, biased agents can leave the swarm alone, leaving behind all other agents, or they can entrain some fraction of the group to leave with them. We further study the role of a classical nearest-neighbor alignment term on cohesion. Notably, we identify the existence of an finite, optimal coupling strength that suppresses fragmentation and maximises the flock cohesion. Our results demonstrate that FSM can be successfully combined with classical flocking rules, offering a flexible framework for modeling intelligent collective systems.

Figures (7)

• Figure 1: A visual representation of the movement template. An agent, shown as a black disc, and its accessible positions in the future, as show in varying colours. The red discs are one action away, with other colours denoting more actions required to reach each position. Note that the orientation at each node is also important, since it affects how the the subsequent trees grow.
• Figure 2: A sketch showing how the visual state vector is defined. On the left is an example swarm configuration, as observed by the central white agent $i$. The thick lines represent the boundaries between open and obscured vision outwards. These boundaries are the union of the projections of the other agents, shown by the thin lines. On the right, we see the $2\pi$ input space surrounding agent agent $i$ is split into $n_s = 40$ equally-sised sensors, denoted by the dashed lines. The information that arrives at the sensors is completely defined by these boundaries of the projection. A sensor is considered active if more than half of the interval is obscured. The thick, black arcs show the visual footprint from the swarm, and the green arcs demonstrate which sensors are consequentially active.
• Figure 3: A visual representation to show the affect of truncation to due predicted collisions. At each safe node, a visual state is taken. A node is safe if there is no collisions with other agents prior to the decision. But how does an agent know if there will be a collision? To simplify computations, agent $i$ assumes all other agents are ballistic, travelling forwards linearly at nominal speed, maintaining their orientation. Here the grey disc is projected to collide with some of the decision nodes of the tree, indicated by a dotted red outline. A collision means that a visual state is not recorded and further actions are not explored. This means the tree grows asymmetrically, favouring "safer" root actions.
• Figure 4: Initial configuration with informed (biased) agents in colour.$N=50$ agents are initialised in infinite 2D space. Here 10 agents, in two groups of $n=5$, are informed (biased) and have a directional preference as shown in the compass in the upper right: red/blue agents have an alignment bias to the red/blue direction respectively.
• Figure 5: Heat maps showing the average cluster size of the informed agents, $\bar{C}$. The axes show the preferred angle, $\Theta$, against the directional strength, $\omega$. The splitting proportion, $\bar{C}$, is monotonic in both directions. For $n=1$, small $\omega$ and $\theta$, the whole collective stays together: the informed agents do not have enough influence over the group to cause splitting. The top right of each graph demonstrates the extreme effect of the directional preference; the FSM entropy is not enough to keep the informed agents within the collective and they leave regardless of whether other agents follow. Along the negative diagonals, we see a similar shading the same colour, suggesting that larger differences in directional preference require weaker strength to ensure the collective stays together. Squares with roman numerals are explored further below in Fig. \ref{['Fig: Ridgelines']}. The values at each location are averaged over time steps 490-500 for collectives of $N = 50$ agents.