2.1. Model

2.1.1. Random movement and the range rule.

In agent-based models, each of a population of agents follows an update rule every timestep. In this model, the model parameters are N, g and r. N is the population size, g is the grid size such that is the area of the coordinate space, and r is agent communication range. N agents are initialized and assigned a unique random coordinate value (x,y) within the grid.

Each timestep, every agent (1) moves, (2) connects, and (3) disconnects. To move, an agent chooses from a uniformly random distribution of its adjacent integer coordinates, including its current location. To keep agents within the bounded space and to prevent agents sharing the same position, adjacent coordinates with values below 0 or exceeding g, or currently occupied by other agents, are excluded from the random selection of a new position. If all adjacent coordinates are out-of-bounds or occupied, the agent remains in the current position.

To connect, after moving, each agent creates an undirected network link with all other agents that are currently in range. Range is calculated as Euclidean distance in the coordinate space, rounded to integer values for x and y. Euclidean distance uses the Pythagorean theorem that for right-angled triangles, (where a and b are the base and altitude, and c the hypothenuse of the triangle). Agent i checks whether agent j is in range by putting the difference in coordinates and as a and b, and calculating the distance to j as c; so . If the result is less than or equal to the value of r, agent j is within range. Finally, to disconnect, the agent deletes any links with agents that are no longer in range. The order in which agents are updated is randomized each timestep. Fig 1 shows the initialization and first timestep of a possible model run schematically.

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Fig 1. Demonstration of a possible initialization and timestep in the model for N = 4, r = 1, g = 4.

Agents are randomly assigned a unique (x,y) coordinate. The coordinate space is graphically represented here as a grid(left), and agents are shown with their coordinate values. At every timestep, every agent follows the range rule: if the Euclidean distance between the agents is less than or equal to r, a link is created between them. Thus, from the initial position shown on the grid, the graph shown in the middle of the figure is formed. At the start of the next timestep, agents move at random and follow the range rule again. When previously in range agents are now out of range, the link is cut. Hence, from every agent performing random movement and following the range rule, a new network is formed.

https://doi.org/10.1371/journal.pcsy.0000051.g001

A note on model ontology. For purposes of clear understanding, our model is intentionally very simple, varying only three parameters. One consequence is that the model is very generally applicable. Agents can be any group of moving components that make any type of proximity-constrained connection. One possible interpretation this leaves open is the 2D grid as an abstract problem space, in which agents explore adjacent solutions (such as in [36–38]). Similarly, the space could be seen as an attribute or trait space, in which agents are “proximate" when they have similar atttributes (akin to [39]). Though these are possible uses, the intended application is to represent the network of possible interaction partners available for individuals to choose from, as constrained by their embededness in a spatial environment, their ability to move in that environment, and the physical properties of communication. Nonhuman animals employ various communication media such as chemical, visual, tactile, and seismic signals, while humans additionally use technologies ranging from conversations to the Internet [30,40,41]. Each of these media has an effective range, that interacts with movement and position in an environment to constrain the possible networks of interactions between individuals. Though the intended application is human networks, this general interpretability is part of the point: we aim to model properties that human groups have in so far as they are also physical communication systems. We hope that as a consequence, our findings our also useful for nonhuman applications such as inferring interaction networks from animal movement and location data, and programming communication networks in swarm robotics based on relative positions alone [42,43].

2.2. Sensitivity analysis setup

Simulations consisted of several sensitivity analyses. In each such analysis, one of the model’s three input parameters N,g,r was chosen to vary while the others remained constant. For each integer value of the varying input parameter, both the range and the null model ran for 100 rounds of 100 timesteps, where at each timestep, all agents were updated in a random order and output measures were collected.

For the first set of analyses, range was varied between r = 0 and r = 10 while N was kept at constant values, varying between tests from N = 1 to N = 100 (see Fig 2 for results for N = 20; see S2 Appendix, S2 Fig and S3 Fig for, 40, 60, and 80). Varying range up to r = g ensures that in terms of network structure, the complete meaningful range of r-values is tested. At r = 0, no agents are connected; at , where the diameter of each agent’s range covers the whole length of the coordinate grid, networks are always fully connected. In the null model, the connection probability parameter P(connect) was varied between 0 and 1 with intervals of 0.1, similarly ensuring that the simulations start at disconnected agents and end at fully connected networks.

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Fig 2. Network properties of range model and null model networks for varying range, N = 20, g = 10.

Range model results are shown in blue; null model results in red. Range was tested for each integer value from 0 to 10, shown on the bottom labels of the x-axis. The null model P(connect) parameter was tested at r/g for each of these values, shown on the top labels of the x-axis. At each of these values, both the range and the null model were simulated for 100 rounds, each of 100 timesteps. Each network measure was collected at each timestep, and the dots shown in the figures represent time-averaged network measures for each round. Lines are drawn through the means of these time averages purely as visual aid, while transparent areas show 1.5 standard deviations. At low range settings (when spatial constraints are most articulated), networks are much more clustered, have slightly higher degree, higher average shortest path length, a lower number of connected components and a higher small-world index than the null model (for N = 40, 60, 80 and 100, see S2 Appendix, S2 Fig, and S3 Fig).

https://doi.org/10.1371/journal.pcsy.0000051.g002

In the second set of analyses, in each test N in both the range and the null model was varied between 1 and 49, while g = 7 and r varied at constant values between tests, from 0 to 7. N varies up to 49 because when g = 7 there are unique positions in the environment. Since agents avoid sharing the same position, as N increases towards , the space becomes increasingly “crowded" until it is completely saturated and agents remain in place, each in a unique position. For this parameter, then, the meaningful range is determined by population density , varying between 1/49 at N = 1 and 49/49 = 1 at N = 49. Since it lacks spatial positions, the null model (or any other nonspatial model) has no analog to g. We therefore show results here only for varying N; for varying g, see the S3 Appendix and S4 Fig. In the null model, N was varied over the same values while P(connect) was set to r/g. This way the null model mimics the connection probability of the range model, in which an agent’s probability to connect is proportional to the range relative to the environment size. Table 1 summarizes the model parameters and the ranges in which they were tested.

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Table 1. Overview of model parameters and ranges tested.

https://doi.org/10.1371/journal.pcsy.0000051.t001

2.3. Output measures

Each timestep, 6 network properties were calculated in both models. The network properties calculated are average degree, clustering coefficient, average shortest path length, number of connected components, size of largest component, and small-world index. Here, we briefly explain how these metrics are calculated, motivate why we include them, and what they can show about the properties of a network. We chose measures that give a clear general overview of network structure. Since the model attempts to capture pathways for information transmission, we focus on macro-level properties that are known to have relevant impacts on the diffusion of information. To an extent, the choice for these measures is arbitrary, and future research stands to benefit from studying the effects of our model parameters on more individual and meso-level measures such as degree distributions and community formation.

Networks are mathematical representations of relationships (edges or links) between objects (nodes or vertices). Degree is a property of single nodes: the amount of connections a node has in the network. The average degree K is simply the population average of node degrees k_i: . In this model, agents are network nodes. Agents do not connect to themselves, and redundant connections are not possible. Therefore, the maximum amount of connections one agent can have is N–1. For a population of 10 agents, average degree will therefore be a value between 0 and 9. Note that if the population average degree is equal to N–1, this indicates a complete graph: the network where each agent is connected to all other agents. This measure, then, gives a simple and intuitive indication of where the network is on the spectrum between completely disconnected and fully connected, while retaining information about the number of connections of each agent.

Clustering indicates the fraction of a node’s network neighbors that are themselves connected to each other. To illustrate, say agent A is connected to three others B, C and D. B and C are connected, but C and D are not and neither are B and D. In this case, ABC forms a fully connected triangle, while ABD and ACD do not. Clustering can be indicated as the ratio of actual to possible fully connected triangles, in this case 1/3. This is the clustering coefficient. This coefficient is calculated for the whole network each timestep, resulting each time in a value between 0 and 1. Note that a value of 1 for this coefficient corresponds to the complete graph at which average degree is N-1; if all possible triangles are fully connected, then the whole network is fully connected. Clustering is an indicator of how “local" and redundant a network’s connections are. The opposite of a very clustered network we might call a fully ‘hierarchical’ or ‘efficient’ network, where all nodes connect back to a single node or small group of nodes [44]. A signal across such a network spreads very efficiently, taking 1 step to reach the ‘leader’ node(s) and 1 step back to reach all others. In contrast, a signal across a very clustered network can take many steps to travel between any 2 given nodes if it lacks long ties between local ‘neighborhoods,’ and the traveling signal can become “stuck" in local clusters. More clustered networks have therefore been shown to slow down processes of diffusion across populations, but also to increase performance in the exploration of more complex task spaces, by slowing down conformity to a single central solution [4–6].

The shortest path length between two nodes is the minimum amount of links that need to be traversed in order to get from one to the other. Average shortest path length is the average taken over all node pairs. Here, too, an average value of 1 means that the networks are fully connected, since it indicates that all agent pairs have a direct path of 1 link between them. In terms of diffusion across the network, average shortest path length has a comparable but inverse effect to clustering: spreading processes will be slower when shortest path lengths are high. In fact, this measure is a direct indicator of the time any type of diffusion will take to spread across a network. Note that since networks in this model often form several disconnected components, the average shortest path length is often calculated seperately for each component, and then averaged.

In this model, the agents will not always form a single network. More often, especially at low range (and at low connection probability in the null model), the population will consist of several mutually unconnected local networks, or single isolated agents. To keep track of this, we calculated the number of such connected components and the size of the largest connected component. When the number of components is 1, the population forms a single connected graph; when it is equal to N, no agents are connected. Conversely, when the largest component is equal to N, the population forms a single connected graph, and when it is 1, no agents are connected. This measure is useful to include if we’re interested in diffusion processes on the network: whatever the structure indicated by the previous three measures, as long as networks do not form a single component, nothing that starts in single agents will spread across the whole network. Note that a population that forms a single connected network, where there is at least one path between all agents, is not necessarily a fully connected network, where there is a direct link between all agents.

Finally, we calculated the small-world-index of networks. Strictly speaking, a small world network is one defined by the parameters n,k and p, where n is the population size, k is the number of neighbors on a ring lattice that each node connects to (therefore each node has the same degree k). Parameter p is the probability that a link is “rewired": replaced by a link to a node chosen at random from the whole population [44]. A direct comparison between a small-world networks and range model networks, however, does not make much sense, for two reasons. First, the range model is a spatial model: in a small-world network, nodes have a fixed position on a ring lattice, while in the range model, agents move around through a 2D coordinate space. Second, in a small-world network k specifies a global degree value. When rewiring probability p is 0, this means that all nodes have degree k. This is very rare in a dynamic, spatially explicit model with random movement; it can only occur when agents happen to form an equidistant ring in space.

In order to still have an indication of the occurrence frequency of small-world-like structures, there is the small-world index [45]. The small-world index S of a graph G is the ratio of C to L. C is equal to CG/CR, where CG is the clustering coefficient of G, and CR is the clustering coefficient of a random graph with the same number of n nodes and m edges as G. Similarly, L is equal to LG/LR, where L is the average shortest path length. S, then, is calculated as . This measure captures the feature of small-world graphs that most nodes are locally connected, while some are rewired to form ‘longer’ ties outside of local clusters. Adding only a few of such long ties connects relatively clustered local communities, creating much shorter average shortest path lengths while keeping clustering high [44]. If, therefore, clustering is low and average shortest path length is high compared to a random model, S will be low; if the opposite is the case, S is high. Small-world networks were designed specifically to generate properties specific to human networks, which exhbit high clustering and low average shortest path length [44]. They have also been studied in terms of impact on information diffusion [2,9]. Having an indication of small-world-ness in our output measures helps pinpoint the circumstances under which there could be spatial contributions to the generation of these well-studied network structures. Note that at low range and low population density, both the null and the range model can often generate a population of completely isolated agents: an empty graph. In such a situation, both the clustering coefficient and the average shortest path length can be 0 in some runs. Since small-world index uses these measures in the denominators of fractions, such cases would require division by zero, leading the index to be undefined. We chose to simply exclude these runs, leading to the apparent lack of data at r = 1 in the lower right panel of Fig 2, and at lower values of N (see below).

3.1. Spatial constraints create networks with higher clustering and lower average shortest path length

The most conspicious difference between the range model and the null model is the difference in clustering. As can be seen in the upper middle panel of Fig 2, clustering is much higher in the range model at all settings except fully connected networks. This difference is particularly visible at lower values of r. Part of this difference in clustering is accounted for by the difference in average degree. As range and connection probability increase, agents form more connections, leading to higher population average degree. As more of the possible connections between agents are made, more connected triangles form, increasing the clustering coefficient. In the non-spatial null model, this effect seems to account for the increase in clustering, which closely follows the increase in degree (Fig 2 top-left and upper middle panels). In the range model, however, this straightforward relationship between degree and clustering is not observed, especially at lower values of r. This is because there is an additional effect increasing clustering, introduced by range-constrained network formation. Since movement is random, the probability of two agents A and B being within range and therefore connected is well-modelled as a random process. If we simplify B’s random walk up until this timestep by assuming its current location is drawn uniformly at random from the coordinate space, its probability to connect to A is proportional to the percentage of the area covered by A’s range: (see for a formal treatment in unbounded space [35]). Given a connection between A and B, a third agent C follows the same probability of connecting to A. However, once both A and C are connected to B, we can limit our assumption of their possible current positions, from the entire coordinate space , to the area covered by A’s range , making it much more likely for B and C to connect as well and form an ABC triangle. Fig 3 illustrates this difference with network visualizations, showing many more complete triangles in the range model compared to the null model at comparable settings.

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Fig 3. Four snapshots comparing null model to range model networks, N = 18, g = 10.

Each snapshot is a timestep taken from a simulation round. In order to make visual comparison easier, the spatial layout created by the positions of the agents in the range model graphs are copied in the null model comparisons (which do not have positions themselves). At each timestep, the null model (left) is compared to the range model (right) at r/g. Upper left: r = 1, P(connect) = 0.1. Lower left: r = 2, P(connect) = 0.2. Upper right: r = 3, P(connect) = 0.3. Lower right: r = 4, P(connect) = 0.4. At similar connection probabilities, agents in the range model only connect to closer agents, creating more connected triangles.

https://doi.org/10.1371/journal.pcsy.0000051.g003

This effect leads to networks with highly localized structure when r:g is low, which is reflected in much higher clustering as well as higher average shortest path length. Low range “assorts" agents into local clusters of physical proximity. As r increases and the proximity constraint softens, these differences dissolve until resembling the null model. As a direct result of the large difference in clustering and average shortest path length, the range model networks have much higher small-world index when range is low. Introducing spatial constraints therefore produces more ‘small-world-like’ networks, most notably so when r:g = 0.2 (This values scales with N; see S1 Fig). Although the non-spatial model often shows lower average shortest path length, the difference in clustering at low r is high enough for the range model to show a clear spike in small-world index (Fig 2, lower right panel). For discussion of results at other constant values of N, see S1 Appendix and S1 Fig.

3.2. Range constrains the influence of population density on network structure.

For constant values of r and g, varying population size N also impacts network structures in the range model. In both the null model and the range model, as population size increases, networks become more densely connected, and average degree increases. As has been shown for static Erdos-Renyi graphs, both models also show a critical value of N after which networks form a single connected ‘giant’ component [46]. The mechanisms, however, differ. In the null model, the increase in degree is a matter of increased possible links, which, with the same P(connect) but an increased number of trials, will lead to more links overall as well as per agent. In the range model, population size increases population density N/g², which, much like increasing r:g, makes randomly moving agents increasingly likely to encounter other agents. Moreover, the range model exhibits an additional ‘crowding’ effect that leads to higher average degree as well as a higher rate of increase of average degree with N (as can be seen in the upper left panel of Fig 4). This crowding effect works as follows. In an unbounded space, it would be more accurate to assume an agent’s probability of randomly encountering and therefore connecting to another agent remains approximately constant. In such a model, N would only increase degree by increasing the number of times the model draws form this probability distribution, therefore increasing the number of successes, as is the case in the null model. In a bounded space, however, where agents cannot share positions, increasing population density means positions become less randomized by movement, and the area covered by any given agent’s range is more likely to contain more agents, until the coordinate space is fully ‘saturated’ and agents can no longer move. The result of this effect of population density is consistently higher degree than the null model, as well as a consistently lower ‘critical’ value of N at which the network forms a single component (see upper left and lower middle panel of Fig 4).

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Fig 4. Network properties of range model and null model networks for varying population size, r = 2, g = 7.

Range model results are shown in blue; null model results in red. In both models, population size was tested for all integer values between 1 and 49, shown on the x-axis. At each of these values, both the range and the null model were simulated for 100 rounds, each of 100 timesteps. Each network measure was collected at each timestep, and the dots shown in the figures represent time-averaged network measures each round. As visual aid, lines are drawn through the means of these time averages, while transparent areas show 1.5 standard deviations. Ranged networks show higher clustering, average degree, and average shortest path length, as well as a lower number of connected components. For r = 1,3,4,6 and 8, as well as varying g and higher N, see S2 Fig, S2 Appendix, S3 Appendix, S3 Fig and S4 Fig.

https://doi.org/10.1371/journal.pcsy.0000051.g004

The fact that, as discussed above, the range model creates more locally structured graphs, is reflected in these results as well: the range model networks have consistently higher clustering and average shortest path length, and as a result, a higher small-world index (upper middle, upper right, and lower right panels of Fig 4). Range model networks do, however, show a much lower number of connected components (lower left panel of Fig 4): connections may be local, but high clustering as well as a bounded space mean that connections are redundant enough to be more likely than a random model to connect to larger portions of the population. Range model networks at low r form long, clustered chains of agents that form a large share of the population, such as shown in Fig 3 for r = 3 (upper right panel). One conspicuous difference, however, from the results of varying r, is that the increases in these measures with N each show a clear ‘plateau’ value of N after which they remain constant. This is due to the model still being constrained by a constant value of r. Though the space becomes increasingly crowded, agents are still limited to connect to the agents in their covered area , limiting both the possible closed triangles and the shortest possible path between distant agents that population density can achieve. A similar bounding effect of r is visible in average degree: though average degree increases with population size, the values stay far from their maximum N-1. In general, as population density tends towards the point where each coordinate is occupied by an agent, N/g² = 1, agents tend toward a uniform state characterized by a number of connections equal to the number of coordinates in their ranged area, . The possible degree, complete triangles, and shortest paths are set in this situation by the size of this area, which is defined by r. For discussion of results at other constant values of r and of varying g, see S2 Appendix, S2 Fig, S3 Appendix, and S4 Fig.

3.3. Information diffusion on range model networks replicates known results.

In addition to tracking network properties outputs, our simulations included four commonly modelled processes of information diffusion on the networks. We simulated a simple contagion process, where ‘susceptible’ agents can be infected by connected ‘infected’ agents with varying probability P(infect); a complex contagion, where P(infect) increases with the number of connected infected agents; a basic cultural evolution model, where two traits each have their own probability of transmission; and a model of the more specific ‘potion task’ process. In a potion task, based on an experiment by Derex and Boyd [4], individuals start with an item inventory, can combine some items into progressively more complex items, and can share known combinations. In the first three processes, we track the timesteps until the state has spread to all agents. In simple and complex contagion, this is reached when the entire population reaches the “infected" state. In the cultural evolution case, this means that one of the traits stochastically comes to dominate and reaches ‘fixation’. In the potion task, we instead track whether a ‘crossover event’ happened after 100 timesteps. A crossover event is a term specific to the potion task, in which there are two ‘trajectories’ of increasingly complex combinations of items, and the most complex item of each trajectory combines into the final ‘crossover’ item. More details on implementation as well as figures are found in S4 Appendix, S5 Fig and S6 Fig. Here, we briefly summarize the expected and observed results.

The expectations from prior research [6,9,44,47] are as follows. As discussed above, high clustering and average shortest path length describe networks that have a delaying effect on transmission. This is true specifically for transmission processes that are based on individual-to-individual transmission probabilities: in this case, the simple contagion, cultural evolution, and potion task processes. Previous research also shows that in the potion task, network properties that delay transmission lead to higher rates of succesful crossover events because the population is less likely to get stuck in a single trajectory of possible combinations [3–5]. Finally, in the case of complex contagion, higher clustering is known to speed up rather than delay diffusion, since there is a higher chance of individuals being exposed to multiple same-state neighbors [1]. Accordingly, in the range model and especially at lower r, where we observe higher average shortest path length and higher clustering than in the non-spatial model, we expect slower time to full diffusion in the simple contagion process, and slower trait fixation in cultural evolution. Furthermore, we expect higher crossover event rates and faster complex contagion.

Our simulations replicate these known effects. In simple contagion, the proportion of infected agents increases slightly faster in the null model at each parameter setting tested, from r = 1 and to r = 9 and (), showing higher differences when range is low. Similarly, traits reach fixation faster at most settings in the cultural evolution model, especially so at higher values of r and P(connect), where we observe more connected (higher average degree) networks. The network structure properties created by range constraints have an especially large effect on complex contagion, where at lower values of r (, , N = 10, g = 10) the proportion of infected individuals increases with more than double the rate observed in the null model. Finally, in the potion task, the expected result is achieved at (). On average, across 100 rounds of 100 timesteps, the range model population achieves crossover events in 72 rounds at r = 1, versus 32 in the null model (). The range model maintains a slight advantage (74 versus 70) at r = 2. Interestingly, however, at higher settings, the null model performs better (by 9 events on average). A possible explanation could be found in the higher number of connected components observed in the null model. For a more detailed discussion of network forces that can influence the potion task and how they relate to our simulations, see the S4 Appendix. Overall, our diffusion tests illustrate and quantify the possible consequences of this paper’s main results: that spatial constraints have a major influence on our predictions of network structure between individuals. It follows that spatial constraints impact our predictions of the collective processes that take place on these networks.

3.4. Results summary

To summarize the simulation results, we found three major network formation mechanisms introduced by a coordinate space, random movement, and ranged interaction. First, for any constant population size, the ratio of r to g determines how densely connected graphs are, from disconnected agents at r = 0 to fully connected networks when r>g (not precisely at r = g because the circular Euclidean ranges do not overlap exactly with the square grid). Second, independently from range, increasing the population density N/g² also increases graph density, as agents ‘crowd’ the coordinate space, making connections more likely. The impact of increasing population density on graph structures is still constrained by r, each node relying on movement for connection when population density is low, but reaching the maximum degree and clustering possible under the current value of r when the population density reaches 1. Range (relative to coordinate space size g) therefore sets the slope of the increase of average degree with population density, as well as ‘plateau’ maximum values to the increase in clustering, decrease in average shortest path length, and decrease in number of connected components. Finally, range model graphs are more clustered and have lower average shortest path length overall due to the higher probability of connecting to proximate agents, especially at lower r:g. In other words, in this model of spatially constrained randomly moving agents, when range constraints are very strict, much of the work of connections is done by movement, which is in turn restricted by the population density. This situation generates highly clustered graphs. Conversely, as range constraints are loosened, movement and position starts to matter less, until they become irrelevant when r>g. Finally, we confirm that these differences in network structures impact the behavior of diffusion processes in ways predicted by prior literature.