The RC model (Figure 2a) is based on the simulation of a swarm of robots that self-distribute across multiple rooms in proportion to the amount of collectible objects in each room, as in Zahadat (2023). Each robot acts as a micro-agent equipped with sensors that provide it with the lowest scale of information related to the number of robots and objects in the rooms based on the sensing of the environment; this forms the micro-scale ( S 0 ) . Over time, robots abstract their micro-scale information into the meso-scale ( S 1 ) , consisting of estimations correlated with the current number of robots and objects in the rooms. This is further abstracted into robots’ estimations of each room’s demand for robots, constituting the system’s macro-scale S 2 . Based on that, the robots decide their motions, forming a feedback loop. The system dynamics are simulated in discrete time steps, and the robots are synchronized: at each time step, all robots update their information at every scale, starting with S 0 and ending with S 2 , followed by a motion action. The process of abstracting micro- into meso-information occurs over time. In contrast, the abstraction of meso- into macro-information happens in a single step. Robots’ motions also unfold over time. As a result, feedback cycles are inseparable from each other, extended in time, and occur at variable intervals depending on the system’s state. We present our measurements either at consecutive timesteps or over a series of intervals.

The CD model simulates a system in which a group of agents makes a collective decision about the size of two tasks by observing their environment. The environment is composed of a fixed set of information sources, each showing one of the two tasks: A and B . This is the micro-scale ( S 0 ) . Agents are placed on the grid of information sources and each agent forms an individual opinion O i about the relative size of A by observing the environment around them and averaging what they see. The meso-scale S 1 is formed by the set of individual agent opinions. In the second abstraction step, agents go through a consensus process to form a collective opinion O coll of the size of A , constituting the macro-scale S 2 . Then, O coll generates feedback on the micro-scale—that is, the environment of information sources. If agents underestimate the size of A , the task will grow until the next feedback cycle, and vice versa. If task A becomes too big or too small, the environment collapses. The goal of the system is to avoid collapse for as long as possible. Timing is discrete: the environment of information sources ( S 0 ) changes at each timestep, and the length of the feedback cycle depends on the system variables and algorithms. We focus on measuring the value of the amount of information used by agents to observe their immediate environment, the number of agents scanning the environment, and the algorithm used to reach a consensus.

This system aims to distribute a set of tasks of two types across a set of workers. We consider a small system with three scales for traceability reasons (Figure 2c). Larger instances were presented elsewhere (Diaconescu et al., 2021a). At the micro-scale ( S 0 ) , workers select an active task of one of two types. The initial choice is random, and then it is driven by feedback from the upper scale ( S 1 ) . Managers at the meso-scale ( S 1 ) collect, aggregate, and forward workers’ task choices to the macro-scale ( S 2 ) and then retrieve, split, and distribute the feedback from the macro-scale ( S 2 ) to the workers ( S 0 ) . The top manager (at S 2 ) receives the task distribution goal as initial input and provides feedback to the meso-scale ( S 1 ) about the difference between the goal and the workers’ current task choices. Information processing and transmission at each scale depend on the task distribution strategy employed in the system. This forms a feedback cycle c , taking two simulation steps. The workers’ state information is sent to the top in one step and then feedback returns one scale at each step. As soon as meso-managers detain new control information, workers adapt to it and send their new states up. Hence, there is a two-step delay between workers sending their state ( t i ) and adapting to feedback obtained from that state ( t i + 2 ) . We calculate information measures at each step t i considering the information flows over the preceding feedback cycle from t i − 2 to t i .

The HO case study simulates a system of communicating oscillators whose goal is to achieve synchronized oscillation. This system is based on the differential equation model for coupled biochemical oscillators presented in Kim et al. (2010), extended to a hierarchical structure (Mellodge et al., 2021). A single oscillator consists of two interacting components X and Y (e.g., mRNA and protein) in which X inhibits its own synthesis while promoting that of Y , and Y inhibits its own and X ’s synthesis, resulting in a feedback relationship within a single oscillator that induces oscillations in the concentrations of both X and Y . In the HO system, multiple scales of oscillators communicate by means of their X concentrations. The oscillators at the micro-scale S 0 send their X concentrations to the oscillators at the meso-scale S 1 which average that information. These oscillators in turn send their X concentrations to the next highest scale until the macro-scale S M − 1 is reached, which contains the average of all micro-scale X concentrations. Information is also passed down the system one scale at a time through feedback. The control information contains time-delayed X concentration averages. At each scale S m , oscillators update their own X and Y concentrations based on a combination of current averages from the lower scale S m − 1 and time-delayed averages from the higher scale S m + 1 . Information measures are used to compare the behavior and performance of two system structures: a two-scale and a three-scale hierarchy.

The information processing described above forms our main strategy in this study. We use two configurations for this strategy: main strategy with sensing horizon of M = 100 , and main (short memory) strategy with M = 10 . The two configurations reflect the effects of memory resource usage on the various information measures. In addition to these, two other strategies are implemented: ground truth and random strategies. In both, all processing that leads to the estimated counts is omitted. In the ground truth strategy, the actual counts of robots and objects, normalized across the relevant rooms, are directly copied into the estimated counts.

Comparing the measures between the main and ground truth strategies reflects the effects of the estimation algorithm, particularly regarding sub-optimal accuracy and time delays. In the random strategy, estimated counts are generated by sampling from a uniform distribution, normalized across rooms. Since this strategy lacks feedback loops, it allows us to evaluate the benefits of feedback in achieving the system’s goal.

The CD model is partially based on Di Felice and Zahadat (2022). It is a simplified representation of decision-making in small, consensus-based groups ( N < 30 , where N is the number of agents), where agents react to their environment to avoid collapse. Agents estimate the size of two tasks, A and B . The environment is modeled as a polarized grid of 58 × 58 information sources, each showing task A or B , with the sum of A and B remaining fixed. The size of each task is measured by the weight of A W A ∈ [ 0.0 , 1.0 ] , determining the share of total information sources showing task A . The feedback cycle includes two abstraction steps. First, agents scan R information sources around them, each showing either 0 (task B ) or 1 (task A ) and average them out to form an individual opinion O i ∈ [ 0.00 , 1.00 ] of W A , with a memory component between subsequent feedback cycles. Second, agents follow a consensus process to reach a collective opinion O coll ∈ [ 0.00 , 1.00 ] , representing the average of O i . The maximum opinion divergence between agents and the number of agents N determines the consensus time T C N . For full details on the decision-making algorithm, see the Supplementary Material.

After T C N , O coll generates feedback on the environment W A , which grows at each time step until the next feedback cycle if O coll < W A , and shrinks if O coll > W A . If W A reaches either 0.10 or 0.90, the environment collapses.

We refer to the decision-making strategy described above as the consensus strategy. To evaluate it, we implement three reference strategies, randomizing different abstraction steps: opinion formation, consensus formation, or both.

• The random _OP strategy randomizes opinion formation, where agents form a random O i [ 0.00 , 1.00 ] without perceiving the environment within one time-step. They then form O coll normally, reaching consensus among those random O i . This reflects the extreme situation where opinions are completely disconnected from the environment.

Comparing the consensus strategy with random _OP highlights the value of evidence-based opinion formation. Comparison with random _CN shows the advantage of the consensus process once opinions have been formed, and comparison with random _TOT evaluates the usefulness of the entire feedback cycle.

Parameters N and R are both varied between [1,30]; the starting W A is set at 0.6, each simulation is capped at 5,000 timesteps, and 40 simulations are run for each N × R combination, leading to 36,000 simulations per strategy and a total of 144,000 simulations. Results show the average value across all simulation runs, including a visual representation of uncertainty ranges, when possible. For each simulation, agents are placed randomly on the inner 50 × 50 information grid at t = 0 , with their position not changing, and the environment W A changing at each time step. Multiple feedback cycles are executed sequentially until the environment collapses or the maximum time is reached.

Figure 7 shows examples of the main types of behavior that can be observed in each simulation. Figure 8 shows average behaviors over the 36,000 simulations for each strategy. Long-lasting simulations (with t avg = 5,000 timesteps) show two patterns. In the first, O coll follows W A with some delay, leading to an oscillation of both values (Figure 7a). The general strategy behaves this way when N < 15 and 3 < R < 15 (see darker area in Figure 8e), surviving for 5,000 timesteps. In the second pattern, O coll rapidly oscillates between polarized values, forcing W A to remain stable with tight oscillations around its starting point of 0.6 (Figure 7d). This is what is observed in the high performance area of random _CN (Figure 8f). Here, T C N is reduced to one time step; the O i randomly chosen as O coll is either 0 or 1 for R = 1 and is likely to remain so for low R , leading to a very fast feedback cycle with drastic fluctuations in O coll . In other instances, W A and O coll both enter an oscillatory behavior, but W A crashes before 5000 t (Figure 7b). This can happen in the consensus strategy when a slower feedback cycle cannot keep up with changes in W A . Finally, if W A does not oscillate, it crashes before t reaches 400 timesteps (Figures 7e–h), as observed in both random _TOT and random _OP .

Figures 8c and d show the average delta of truth Δ t h , a v g —that is, the average difference between O coll and W A —for each parameter combination. A lower Δ t h shows that agents are better at collectively perceiving the environment. The relation between Δ t h and t avg highlights the weight of the timing mechanisms in the model. The consensus strategy has the lowest Δ t h , a v g , and it decreases and stabilizes as N increases (Figure 8c): poor performance at high N is not due to incorrect O coll but to a long feedback cycle that prevents W A oscillations. For random _CN , Δ t h , a v g is highest for low R and even higher than random _TOT and random _OP (Figure 8d). In this range, high t avg is not the result of a good O coll but of an oscillating O coll that stabilizes W A .

This model is a simplified version of the ABM in Diaconescu et al. (2021a), aiming to achieve a given task distribution among a set of agents. We exemplify a small system to facilitate the formal analysis of information flows. The system contains N = 3 scales S s , where s = 02 , populated by agents a s , j at scale S s and index j . S 0 includes four workers ( | S 0 | = 4 ) that must select two task types, k 0 and k 1 (sometimes noted as 0 and 1), according to given proportions g 0 , g 1 , where g 0 + g 1 = | S 0 | . Workers coordinate their task selection via managers, each handling C = 2 children (Figure 2c). At t 0 , the top-manager a 2,0 receives the goal v g = g 1 , and workers a 0 , j , where j = 0.4 , are assigned an active task k a ∈ { k 0 , k 1 } . Time is modeled in discrete steps t i . For simplicity, we only consider information about k 1 , k 0 being complementary to it.

Each manager a s , j , s > 0 , holds two variables concerning its children: the number of workers that execute k 1 (abstraction v s , j A ) that should switch to k 1 (control-error v s , j Δ ). For workers, v 0 , j A = v 0 , j k a (their active task) and v 0 , j Δ = v 1 , l Δ (control from parent a 1 , l ).

The system uses the Blackboard (BB) strategy (simplified from Diaconescu et al. (2021a) —see Supplementary Material for further details), forming a multi-scale feedback cycle via the following information flows.

• Abstraction: managers sum up their children states (active tasks), v s , j A = ∑ c = 01 v s − 1 , c A

The BB simulation repeats this cycle recursively until the goal is reached, and it is maintained for N = 3 steps.

We define several alternative strategies for reference.

• Random switch (RS): at each step, workers switch states with probability p c h = 0.15 (same as BB).

We selected these alternative strategies based on the following considerations.

• St provides an extreme case where the system does not adapt, whereas RB provides the other extreme where the system adapts completely randomly.

In the initial state, all workers perform k 0 ( v S 0 k = 0000 ) , whereas the goal v g = 4 indicates that they should perform k 1 ( v g , S 0 k = 1111 ) . Experiments rely on an analytic simulation to facilitate information tracking. We employ two experimental alternatives.

• Probabilistic analysis E x all for all strategies considers all possible system behaviors over 50 steps. We use this to estimate probabilities for the syntactic content and pragmatic values.

We define information measures through generic equations and then illustrate them in specific cases. We consider all measures for BB and Md and only syntactic content and pragmatic values for RS, RB, and St, as they lack multi-scale coordination. Table 1 summarizes results for syntactic ( E x all ) , semantic ( E x scenario ) , and pragmatic delta ( E x scenario ) measures. Figure 11 compares pragmatic values of all strategies ( E x all ) (see the Supplementary Material for further results and calculations).