The efficiency of the information sharing mechanisms used by individuals during group decision processes determines to a large extent the fitness of the group decision. In nature, collective systems consist of a high number of individuals living in large and unknown environments, and needing to perform complex tasks to survive. Among the many examples of collective decision-making is choosing a new site to build their home (Richardson et al., 2018), or deciding among a number of foraging patches (Michelena et al., 2009). Despite the high diversity of tasks, uncertainty and complexity are common features. Hence, individuals apply information pooling to mitigate uncertainty and increase decision accuracy (Conradt, 2011). Achieving efficient opinion sampling depends to a large extent on the network topology that defines the interaction structure and opinion sharing of these individuals (Khaluf et al., 2018; Rausch et al., 2019b). The use of such network is fundamental for collective decision-making. It is generally exploited at two stages of the process (i) when spreading information on one or multiple stimuli that are initially perceived by a limited number of individuals that are able to trigger the collective decision process—e.g., a predator attack—; and (ii) when spreading the individuals' opinions or choices to achieve consensus (Khaluf et al., 2019a).

In artificial systems such as swarm robotics, collective decision-making is mostly designed in static environments (Bayındır, 2016), where options and their qualities are defined at the beginning and do not change over time. In these studies the focus is mainly on the design of efficient voting mechanisms that enable a high level of decision coherence within the shortest time possible (Khaluf et al., 2018). Alternatively, other studies were addressing the design of decision strategies that tackle the accuracy vs. speed trade-off (Valentini, 2017)—i.e., taking longer time to gather enough information and making more accurate decisions vs. exploiting the available information and taking the decision as soon as possible. In both cases, the speed of converging on a decision is a fundamental goal in the design of decision-making. The decision speed strongly depends on the interaction topology the individuals are part of, to spread stimuli or opinions during the decision-making process. Interactions in collective systems are frequently modeled using local (i.e., proximity) communication, where the neighborhood of an individual is defined spatially based on their interaction range, i.e., interacting with all peers within the individual's communication radius. Nevertheless, other interaction models such as scale-free networks were revealed in several real-world examples (Albert and Barabási, 2002; Holme, 2019). A comprehensive review on scale-free phenomena in a more general context can be found in Khaluf et al. (2017a). In various works, scale-free networks enable scalable, fast and efficient information transfer. For example, in Goh et al. (2001), authors showed how the betweenness centrality scales with the scale-free exponent. Other works showed how the ultrasmall diameter of the scale-free networks contributes to their efficiency in information transmission (Cohen and Havlin, 2003; Thivierge, 2014). Finally, scale-free topologies were studied in natural collective systems such as in Cavagna et al. (2010). In this work, the authors studied starlings flocks and suggest that collective response to predator's attacks may be achieved through scale-free behavioral correlations. Based on these studies, we extend the application of scale-free networks to artificial swarms in order to investigate the role these networks can play in improving a swarm's collective decision-making process.

A key aspect of scale-free networks is the presence of hubs—i.e., nodes with a comparably high connectivity degree—(Albert et al., 2000; Albert and Barabási, 2002). Hubs represent a small percentage of the network nodes, however, their high connectivity leads to a small network diameter. This facilitates efficient communication by enabling any two random nodes to share information over only few hops, resulting in fast information transfer (Cohen and Havlin, 2003). In this paper, we exploit this critical feature of scale-free networks to help collective systems to faster respond to changes in dynamic environments. In dynamic environments, conditions change over time and hence, the collective system needs to adapt its behavior within a short period of time in order to survive. We refer to this as the collective response time. In our study, this is the time required for the group to collectively change the intensity of its foraging activities as a response to a change in the availability of the food items.

Among many examples of collective tasks in natural systems, we select foraging (Liu et al., 2007) and perform our study using a simulated population of swarming robots. Foraging is a complex task used by many species to retrieve food to their homes, but beyond that it is a metaphor for many real-world robotics tasks such as search and rescue, retrieve materials for collective construction and others. In foraging, individuals (robots) need to continuously make a decision between staying at their base or leaving to forage for food items. A large body of literature has been dedicated to investigate foraging in artificial systems such as swarm robotics. These studies have addressed various research questions such as the foraging performance under the influence of physical robot interference (Lerman and Galstyan, 2002; Khaluf et al., 2016), the multi-foraging task (Campo and Dorigo, 2007)—i.e., the foraging for different types of items—or consensus achievement (Hoff et al., 2013; Khaluf et al., 2017b). Additionally, some studies have focused on how to optimize the task allocation in foraging using cost functions (Pini et al., 2013; Khaluf et al., 2019b). Also how to investigate simple probabilistic models that rely on the foraging success probability in achieving an efficient foraging behavior (Pinciroli et al., 2012). Other studies have gone further to investigate whether the performance of swarms in the foraging tasks bears a particular characteristic distribution (e.g., a power law) for any of its time or space features (Khaluf and Dorigo, 2016; Rausch et al., 2019a). Despite this intensive research effort, foraging of robot swarms in dynamic environments and the influence of different interaction models are still not well understood. However, these questions are paramount, given the prevalence of scale-free phenomena in real-world systems and admitting that most real environments are dynamic. Therefore, in this paper, we focus on the fundamental question of how the integration of a scale-free interaction structure may influence the collective response of simulated swarms to changes in food density within the foraging environment. We approach this question by analyzing the speed and coherence of the collective response to those changes. We begin with defining the robot (microscopic) and the swarm (macroscopic) behaviors in sections 2.1, 2.3, respectively. The details on generating scale-free networks from local neighborhoods are given in section 2.2. In section 2.4, we describe the experimental setup. Thereafter, in section 3 we compare the collective response of the swarm in presence and absence of scale-free interactions. We discuss our findings that suggest that the use of scale-free interactions can be advantageous due to (i) reduced correlations between a robot's decisions and those of its spatial neighbors and (ii) enhanced information spread through long-range interactions and frequent rewiring of communication links. These insights are obtained by comparing the influence of scale-free networks to scenarios with alternative random networks as well as scenarios that include packet loss. Conclusions are drawn in section 4.

The goal of this study is to investigate the influence of the scale-free topology on the collective performance and response of a swarm foraging in a dynamic environment. The dynamics of the environment is modeled in terms of single and periodic changes in the food density. In robot swarms, the interaction among individuals is mostly modeled using local communications, where each robot has a limited communication range. The communication range is usually much smaller than the dimension of the world. The robot's neighborhood is defined as the set (or a subset) of robots that is located within its communication range. In this study, besides local interactions, we make use of the well-known preferential attachment mechanism (applied in Algorithm 1, see section 2.2) to construct a scale-free topology that accelerates information sharing. Hence, we investigate whether it may improve the efficiency of the swarm collective response to environmental dynamics.

As mentioned above, we define the collective response in terms of the number of resting robots and measure it as the change in this number over time. In our experiments, initially, the entire swarm is in the resting state. In the following, a transient period begins, during which the swarm displays oscillations at the group level. First, almost all robots begin foraging during the first 500 time steps (ts)—Note that a simulated time step is one second, with one tick per second. Within the subsequent ≈ 500 ts most of the swarm individuals come back to the nest and switch to resting. Even though such collective behavior oscillates over several following time periods—due to the probabilistic nature of the robot controller—the coherence increases rapidly and the swarm converges on a relatively stable number of resting robots. The duration of this transient period is mostly shorter than 5·10³ ts, after which we begin our measurements. Finally, based on preliminary results, we set the swarm size to N = 950, which balances physical interference with swarm performance and delivers a sufficiently large number of samples for statistically sound analysis.

We use two experimental settings. In the first setting, after the system converges on a number of resting robots N_rest (number of foraging robots is then N_forg = N − N_rest), a single external stimulus is applied. This stimulus represents an increase in the number of food items N_items by the factor of 10 (from 30 to 300 items) at a particular time point t_crit. In the second experimental settings, we challenge the swarm further by applying a periodic change in the density of the food items, hence the benefit of a quicker response becomes clearer. The periodic change is applied over periods of 2500 ts and can be of two types, either increasing or decreasing the number of food items N_items, always by a factor of 10.

In each of the two experimental settings, two interaction networks are implemented, proximity network (emerging from local interactions), and scale-free network (generated using preferential attachment). As mentioned above, for the construction of scale-free networks, the connected components of the robots resting at the nest site are used to impose the network topology. Over these networks the robots exchange specific information about their success or failure of the latest foraging attempt seeking an accurate estimation of the current situation in the foraging environment.

According to our experiments, there are two main cases, in which the influence of the communication topology is negligible. These are (i) small social cues (i.e., with s_f and s_i values smaller than 0.01), and (ii) small number of resting robots N_rest. The first case is straightforward, as the social cues decrease, the impact of the information obtained from other robots decreases, and hence the impact of the interaction network on the emergent dynamics vanishes. The second case is associated with the particular implementation of the scale-free communication network in the nest. Since the construction of this network relies on the connected components present in the nest at every time step, small numbers of resting robots result in scaling down the size of such connected components and hence topological contribution becomes negligible. Therefore, as we aim to investigate the influence of the interaction network on the emerging dynamics, we consider those cue configurations in which the social feedback of the robot's neighborhood has a distinguishable role in shaping its decision. This is achieved by setting the social cues to have a clear advantage over the individual cues—i.e., s_f ≫ i_f, s_r ≫ i_r. For an extensive discussion on the impact of cue values on swarm behavior in a similar settings of the foraging task the interested reader is referred to Liu et al. (2007) and Rausch et al. (2019a). For the reasons mentioned above, we set the cue values to s_f = 0.25, s_r = 0.25, i_f = 0.01, i_r = 0.01. Nevertheless, further below we will additionally compare our results to those obtained with more extreme values of the social cues, i.e., s_f = 0.01, s_r = 0.01 and s_f = 0.99, s_r = 0.99.

The plots in Figure 3 depict results obtained over 30 runs. They compare the emergent collective response of the swarm to a single stimulus (i.e., change in food density) as well as to multiple stimuli when individuals interact locally in comparison to interacting via scale-free topologies. Firstly, our results reveal a clear impact of the network structure on the robot activation level across all types of stimuli (i.e., increasing or decreasing food item density). This is illustrated through the number of resting robots being considerably smaller when using the scale-free network as opposed to the proximity network throughout the entire simulation time (see Figures 3A,B). Proximity networks in Figure 3B show a non-adaptive swarm behavior that is largely due to the very low number of foraging robots. When there are too few foraging robots, the system tends to approach a global absorbing state in which robots cease to switch to foraging. In case of proximity networks in Figure 3B, this tendency toward the global resting state is due to the initial low density in food items (i.e., N_items = 30). Low N_items leads to a large number of failed attempts to find and retrieve them. Consequently, this increases P_f→r up to its maximum P_f→r = 1, pushing the robots to keep resting. Thus, the subsequent increase in items to N_items = 300 is not sensed by the swarm. As an example, this behavior is evident at t = 7, 500 ts when N_rest did not decrease in response to the increasing N_items.

Therefore, it is important to consider the robustness of the swarm behavior to initial conditions, prior to the external stimulus. To this end, we inverted the changes of N_items, starting with N_items = 300, reducing it to N_items = 30 at t = 7, 500 ts, then increasing it back to N_items = 300, etc… Under this specific setting, foraging robots have a higher likelihood to find items than when the initial item density is as low as N_items = 30. Consequently, the returning robots broadcast a larger number of “success” messages, increasing the robots' probability to switch to foraging (P_r→f). Figure 3C shows that this configuration of the initial conditions led to an adaptive swarm behavior for the case of proximity networks. This adaptive behavior comes with a reduced time to convergence (see Figure 3F vs. Figure 3E) and a significantly higher number of retrieved items (see Figure 3I vs. Figure 3H). Nevertheless, with scale-free networks the collective response not only remained more rapid but also appeared to be more robust to the initial conditions of the system, as the trajectory of N_rest in Figure 3C is qualitatively similar to Figure 3B. Nevertheless, the scale-free networks display higher fluctuations of N_rest compared to the relatively coherent decision achieved when using proximity networks (Figures 3A–C). This is due to the high impact that a single hub can have on a large population of the swarm.

The key contribution of the network topology is reflected in the time the swarm requires to build up its collective response. When using scale-free networks, hubs—i.e., robots with an exceptionally high connectivity degree—help accelerate the information propagation in two manners: (i) due to their high connectivity degree, their individual experience is shared with a large number of robots within one time step. (ii) Their presence creates a shorter average path of the network compared to proifbximity networks, which allows any two robots to exchange information over a smaller number of hops (i.e., within fewer time steps). As mentioned above, we use the settling time defined in Equation (4) to compute the swarm's convergence time after each stimuli—i.e., change in the items' density. Figure 3D shows the time it took the swarm to converge to a steady number of resting/foraging robots after increasing the items at the foraging area from N_items = 30 to N_items = 300 at time step Δt_crit = 7, 500. Figures 3E,F show the same measure for the repeated stimuli of items increase and decrease, starting from N_items = 30 (Figure 3E) and N_items = 300 (Figure 3F). In all three findings, Figures 3D–F, we can notice the significantly shorter convergence time when robots in the nest are communicating using the scale-free network in comparison to the proximity network. These results suggest a higher level of swarm adaptivity to dynamic environments under scale-free communications. Furthermore, as shown in Figures 3G–I, using scale-free networks the cumulative sum of the retrieved items is either considerably higher from the beginning or at the later stages of the experiment, compared to the scenarios with proximity networks.

An important aspect to notice is the physical division between the site at which the information is to harvest (i.e., the foraging environment), and the site at which the information is to exchange (i.e., the nest). Usually, the communication speed is considerably higher than motion speed. However, specifically in the foraging scenario, the communication speed is limited by the motion speed, since it is necessary for the robot to travel across the foraging environment to reach the nest, where it can start communicating. One of the clear consequences of this important remark is that even for the case of scale-free networks where the collective response is accelerated, there is a considerably faster swarm reaction to an increase in the food density compared to the reaction to a decrease (see the blue line in Figure 3B). Before the increase of food items, there were few foraging robots. Those robots consumed time to return to the nest, switch to resting, inform their neighbors about their foraging experience, and, ultimately, convince more robots to leave the nest in case of a successful foraging attempt. For scale-free networks this resulted in a rapid activation of resting robots. Differently, collective reaction slowed down when the environmental change was a decrease in food items. This behavior can be explained as follows: the large number of robots foraging while the food density was high experienced the drop in the food density through their failed foraging attempts. Upon returning to the nest, these robots led to considerably higher crowding at the nest entrance. This prolonged the time that the robots needed to enter the nest and start communicating. Moreover, the higher N_rest the higher the likelihood that there is one, giant, connected component inside the nest, spanning over a large number of robots. If such a network is scale-free, the hubs have a high chance of influencing many robots to switch to foraging. By contrast, a low N_rest often led to fragmented networks, reducing the influence of hubs, lowering the number of switching robots and, thus, slowing down the collective response compared to a high N_rest. Hence, the collective response time—even when using scale-free networks—is longer when there are many robots foraging.

To obtain a closer look at the interaction network topology, we can analyze the degree distributions of the resting robots interacting inside nest. We draw the degree distributions for different time steps that are selected when the item density was both high (i.e., 300 items) and low (i.e., 30 items). As we can see in Figure 4A, scale-free networks strongly resemble a power-law distributed degree for all time steps at which the networks are recorded. Similar consistent is the degree distribution of the proximity networks in Figure 4B for all tested time steps. However, the degree distribution here appears closer to a Gaussian distribution which is more symmetrical around the mean than the scale-free network and has fewer outliers. To get a clearer look at the outliers, in Figures 4C,D, we show the communication degree using boxplots. For the scale-free networks the density of outliers is notably large, the most extreme among those are the hubs in the network. We can also notice a clear trend of a higher number of hubs when the number of resting robots N_rest is higher due to low N_items. This density of outliers changes periodically between the external stimuli S_i together with N_rest. In the case of proximity networks, the boxplots show a relatively low density of outliers and negligible changes with S_i.

Additionally, it is worthwhile considering the effect of rewiring on the collective response. As elaborated in section 2.2, Algorithm 1 is applied at every time step as the robots are in motion. However, because Algorithm 1 has a stochastic component, the resulting network at time step t is very likely to be different from t − 1. Such dynamic rewiring increases the probability that two remote robots share a link. Consequently, a random robot is more likely to obtain information from spatially uncorrelated sources, i.e., it obtains a sample that is more representative of the swarm opinion. This resembles the common “random mixing” paradigm often found in swarm robotics, stating that an encounter probability between two robots is the same for any pair of robots. Thus, the adaptive behavior that follows from using Algorithm 1 could be largely attributed to this rewiring-induced opinion mixing.

To examine whether this may indeed be the case, we ran simulations with a modified version of Algorithm 1 where we replaced the preferential attachment component ds/∑idi by a real number ρ ∈ {0.01, 0.1}. Note that while this modification aims at altering the network topology, the resulting alternative networks are still regenerated at each time step, similar to scale-free networks, i.e., the notion of rewiring is preserved. The results are shown in Figure 5. The similarity to the scale-free networks scenario is particularly striking for ρ = 0.01. When N_rest is low, it becomes difficult to separate a scale-free network (where the degrees are power law distributed) from a small-world network (where the degree distribution is much less skewed, i.e., more symmetric around the mean value). Therefore, for low N_rest the impact of the preferential attachment component in Algorithm 1 can be well-approximated by a constant such as ρ = 0.01. More importantly, it shows that the strong effect that dynamic rewiring has on swarm adaptivity and collective response.

A feature that frequently occurs in realistic communication is the packet loss. It occurs when a robot fails to receive a message broadcast by a neighbor, due to radio-frequency interference or due to overflow of a robot's receiver queue. We implemented packet loss events by allowing the robots to ignore incoming messages with probability p_pl. Figure 6 shows the results for the proximity and scale-free networks with p_pl ∈ {0.1, 0.5}. Surprisingly, the swarm adaptivity considerably improves in case of proximity networks, while with scale-free networks the swarm remains more robust to the influence of the packet loss. Higher probabilities of packet loss appears to shorten the time to convergence and slightly increase the number of collected items. One possible explanation for this behavior could be that by probabilistically ignoring incoming messages the robots become to some extent able to reduce the correlation between their behavior and that of their spatial neighbors. Synthetically generated networks, such as the scale-free networks considered in this study, represent an extreme case of such spatial decorrelation. In contrast, in proximity networks and absence of packet loss, spatial correlations are very high, leading to feedback mechanisms that reduce sensitivity to new information. The presence of packet loss appears to create a middle ground that bolsters the adaptive behavior at the swarm level. However, we only tested two values of p_pl and it is possible that for p_pl > 0.5 inverse effects could be observed. Finally, when resting state can be associated with low energy consumption, the behavior of the system in the presence of here considered p_pl may demonstrate a high level of efficiency, in terms of increasing N_rest while preserving the high number of retrieved items. Nevertheless, as mentioned above, the detailed investigation of the influence of packet loss is beyond the scope of the current study and future research is needed to confirm the generality of our findings⁵. Moreover, here we consider constant values of p_pl that are the same for every robot in the swarm and that do not change based on the location of the robot or the number of communication links. In contrast, in more realistic settings not only the packet loss but also p_pl itself may have fluctuating values depending on the situation and both could be profoundly difficult to control.

Finally, we compare the intensity of the collective response resulting from different social cues. As mentioned above, social cues are the main driver of the dynamics to build up a faster response over the interaction network. Our results show that higher social cues lead to a higher activation of the resting robots, see Figure 7 that shows the activation of the resting robots when setting s_f = 0.99, s_r = 0.99 in comparison to the setting s_f = 0.01, s_r = 0.01 (results are averaged over 30 runs). High social cues activate considerably more resting robots (i.e., reduces number of resting robots) than low cue values (Figure 7A). However, the convergence time with high cue values is comparable to the previously considered default case of s_f = s_r = 0.25 (see Figure 7B). The number of collected items overlaps for all three cue values (see Figure 7C).

The goal of this study is to investigate the role of network topology in influencing the propagation of information in a foraging scenario with changing the availability of food items. Therefore, we have addressed scenarios with dynamic environments, a realistic aspect of most real-world applications. We considered two types of changes: a single abrupt change (referred to a single stimulus) and periodic changes (multiple stimuli). We aimed to examine how scale-free networks, in particular, may accelerate the spreading of information and hence enable a quicker collective response than proximity networks to the global changes.

We have implemented scale-free networks across the robots resting in the nest, as the nest is usually the part of the environment in which communication takes place. We applied the well-known preferential attachment technique to construct the scale-free topology. Following preferential attachment, the probability of connecting to a robot is proportional to its current connectivity degree. Therefore, a number of robots emerge to have a relatively high degree of connectivity, those are referred to as the hub robots. When the density of food items changes at the foraging environment, and this change is reflected in the robots' experience, scale-free networks enable a faster spreading of this information in the nest. This led to a faster collective response compared to the scenarios in which interactions between the resting robots were implemented using proximity networks.

Our results suggest that the use of scale-free networks can improve the collective response of the swarm to changes in their dynamic environment, by improving the spread of shared information and reducing the spatial correlation in the robots' decisions. These two desired features in collective systems are achieved due to the introduced possibility to communicate over long distances, as well as due to the dynamic rewiring of the interaction network at every time step as a consequence of robot motion. These insights were obtained by comparing the swarm behavior in scenarios with and without systematic packet loss, in addition to comparing the swarm performance between scenarios with scale-free networks and with alternative random networks. Furthermore, our findings showcase the effect of social cues on the intensity of the collective response in presence of scale-free networks. Our results show that higher social cues lead to a higher activation of the resting robots, due to the increased influence of their neighbors' experience.

Although scale-free networks have shown to equip the swarm with a quicker reaction to changes in dynamic environments—studied for the collective foraging task—this came at the cost of the coherence of the collective response. Scale-free topologies led to more fluctuations of the swarm decision (whether to rest or to forage). These fluctuations can be explained in terms of the high influence of particular individuals (i.e., the hubs) on the opinions of a large population of the resting robots. Two particularly promising research directions for future work include the design of self-organized algorithms to implement scale-free topologies in robots swarms. Additionally, the design of efficient individual decision mechanisms that helps the collective response to demonstrate a higher stability. Finally, generalizing this study to other collective tasks such as site selection, flocking, and others may also lead to new interesting insights.

The original contributions presented in the study are publicly available. This data can be found here: https://osf.io/48b9h/.

All authors contributed conception and design of the study. PS provided funding and administered the project. YK and PS supervised the study. IR implemented the simulation setup, performed the simulations, and the formal data analysis. YK wrote the first draft of the manuscript. YK and IR continuously discussed and elaborated on the design parameters and obtained results, which led in some cases to further implementations and analysis. YK and IR wrote most sections of the manuscript and analyzed the experimental data. IR and YK critically reviewed and edited the manuscript. All authors approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.