Based on the ecological ideas presented so far and making use of LCC and the OpenKnowledge framework for the development and deployment of a multi-agent based simulation platform, we have designed and implemented ecological protocols of interaction for defining the relationships among agents in simulated ecosystems. These protocols are mainly focused on the ecological processes occurring at the interaction level between species in plant-animal mutualistic communities (see Methods).
We have found that interaction models for agents coordination thus engineered facilitate the emergence of network features such as those seen in ecological networks of interacting species in real communities, in our artificial societies of agents. In this section we present these results and we discuss how an ecological model of this kind can be used to study the ecological mechanisms behind the emergence of the network patterns we see in our agent societies and which are characteristic organisational features of the ecological communities they represent.
Our experiments consisted of the execution of a series of independent simulation runs following the model specifications outlined in the Methods section for parameter initialisation and runs configuration. During these runs, relationships among pairs of agents arose with different strengths (the number of times an agent interacts with any other relative to the number of times it has interacted during the entire simulation) and with different configurations.
We ran one hundred simulations where the configuration and properties of the networks of interactions between agents/species where similar among them. In order to be consistent throughout this paper we have selected fifteen samples from the simulated networks obtained, because, as we will see below, this was the number of natural communities that we were able to extract from a dataset of empirically obtained mutualistic networks that were more similar in number of species to our networks, and that we will employ for the analysis of similarities between the interactions networks obtained by the model and those observed in the real world. Changing the set of selected networks for analysis would not noticeably affect the results presented.
In this section we analyse the structure and features of this set of simulated networks. Apart from the identity of the agents, which is changed in every run based on the different configurations they can be initialised with (as explained in Methods), the architecture was, in general, similar among the obtained interactions networks; presenting constant patterns that are also found in ecological networks.
The mean and its standard deviation for the number of interactions occurring per node are plotted in Figure as the distribution of the frequency of each number of interactions among the nodes in our networks. Although the standard deviation from the mean value is high for small degree values, we can see in this plot that the frequency of the number of interactions among nodes is highly biased towards small values and that few nodes among all the networks possess high degrees (i.e. are highly connected). It is a common pattern in our simulated networks then, that the majority of nodes possess low degrees, or interact with only one or a few other nodes in the network, and only a small fraction of agents are well connected to others in the simulated ecosystem.
Distributions of frequencies of nodes degrees in simulated networks. Frequency distribution of the mean (with its standard deviation) of the number of interactions per node in fifteen of our simulated networks.
As we shall see, these are characteristic features that are believed to account for interesting properties of a particular kind of complex networks, which have come to be known as scale-free networks, and are important features observed in the mutualistic natural communities we are interested in analysing.
Figure shows an example, taken from one of the runs in our experiments, where the relationships among agents in our system are represented in a fashion similar to networks of interacting species described in the field of ecology (as we saw the Background section).
Figure 6 Example of the output of a simulation run performed using the model described. A network of interactions between artificial species in a simulated ecosystem. Host and visitor agents are represented by green and red nodes respectively. The thickness of (more ...)
As explained in Methods, we have defined two roles for agents to take in our simulations, in resemblance to the actors taking part in plant-animal mutualistic interactions in nature: the "host" and the "visitor". In our network representations we have represented "host" and "visitor" agents/species as green and red nodes in the graph respectively. Relationships between agents are represented by arcs (directed edges), where the direction of an arc represents the direction of the energy flow in the ecological system while its thickness represents its relative strength with respect to the other connections that depart from the same host (green node). Numbers on edges represent the number of times that particular interaction was observed in the simulation (i.e. the number of times the pair of agents linked by that arc completed a successful interaction). A link (arc) is thus generated between two agents whenever an interaction is successfully completed amongst them.
By representing the relationships between the agents in our artificial ecosystems in this way we are able to extract features, obtain descriptors, and perform analyses over the resulting network based on methods borrowed from network theory.
As introduced above (Figure ), we obtained networks that display a scale-free structure. The plots displayed in Figure show the data from the network in Figure , where it can be observed that the majority of nodes in this network have small degree (≤ 2), while a low fraction of them are highly connected, showing a distribution of the frequency of interactions (left plot) biased towards low values (1 and 2 interactions), and a distribution of degrees (right plot) with a decreasing slope in a fitted power law. Additionally, small-world properties are found in our networks: with short paths between any two nodes. Properties of the kind mentioned above, which are encountered in the networks of interactions among our agents in the simulated communities, are common patterns also found in different kinds of complex networks in nature and the artificial world [1
], and which differ significantly from the structure that we would expect from a randomly assembled network.
Plots corresponding to the network displayed in figure 6. The plots show the frequency distribution of the number of interactions and the distribution of degrees among the nodes on the network.
Another property seen in our networks, which is related to their scale-free character, is the preferential attachment displayed by visitor agents with low degrees (e.g. the five red nodes on the top right corner of Figure ) to host agents that are highly connected. This is a common feature encountered in mutualistic networks, where specialist species are more likely to interact with generalists [16
]. Patterns of this kind are important in practice because, as it has been argued, they can give us information about functional properties of the communities such as: disturbances propagation speed and robustness to species loss, which in turn provide us with a better understanding of the relationship between the complexity and stability of our ecological systems.
Asymmetric specialisation (i.e. a specialist interacting with a generalist) has been found to be a pervasive feature of plant-pollinator interactions networks, and it is believed to be beneficial for the majority of species in these communities because it facilitates the avoidance of extinction risks when species are highly reciprocally specialised [19
]. This is another feature arising in the network of interactions among agents in the model presented.
The features encountered in the networks of relationships amongst our artificial agents are in many ways similar to those found in real mutualistic networks, as shown above; which are patterns that differentiate random networks from self-organised complex networks of relationships. This network architecture is an emergent property of our agent based system since the only mechanisms involved in agents' interactions are those specified by the protocol of interaction presented in the Methods section. The creation of such a complex and intricate pattern of relationships is not a hardwired property of the artificial communities arising from the simulations performed, but rather the product of many different agents interacting together for achieving their respective goals (gather resources and survive).
This conclusion is important because it means that the ecological and behavioural mechanisms that we are studying and that are translated into interaction models between agents in our computer model are directly and solely responsible for the system level attributes found in the artificial communities thus enabled, becoming in this way plausible causes for the emergence of these features in the studied systems.
Comparing with Empirical Data
In order to test the extent to which the networks of interactions found in our artificial systems are similar to mutualistic networks of interactions found in the real world (apart from the qualitative similarities introduced above), we have compared the architecture of the networks obtained from our simulations to some networks empirically collected from real communities and that have been compiled, analysed, and provided as supplementary material by Rezende et al in [31
]. Although in that paper they used the networks for different kinds of analyses, the datasets provided are useful for getting an idea of the common features encountered in mutualistic networks.
Because some of the properties of interest for analysing ecological networks are scale dependent, we have selected fifteen networks from this dataset, based on the number of species composing it and that were closer to the number of agents in our simulations, for comparison against fifteen of our simulated networks. Connectance (C
), the fraction of all possible links that are realised in a network, is an important property that is commonly employed in the analysis of ecological networks and which provides information about the degree of connectivity between their nodes. We use this network measure and the Nestedness metric based on Overlap and Decreasing Fill (NODF), introduced by Almeida-Neto et al in [32
], which provides a measure for community organisation in plant-animal mutualistic networks of interactions based on the overlapping diets of the species in the community studied; for comparing our simulated networks with fifteen empirically obtained plant-animal mutualistic networks. Additionally we perform qualitative comparisons between the structures obtained in one of our networks and one of the natural communities considered for this analysis.
In table we can see the connectance and NODF values derived from our simulated networks and the selected empirical networks obtained from natural communities, and in Figure we plot the NODF against the connectance values with the double purpose of analysing the behaviour of nestedness in relation to changes in the connectance of the network, and to compare these relationships in our simulated communities against their natural counterparts.
Connectance and NODF index of nestedness values from the artificial and real networks.
Figure 8 Connectance versus NODF nestedness values in natural and simulated communities. The values plotted are the connectance vs the NODF nestedness indexes obtained from the natural (blue dots) and simulated (yellow dots) communities presented in table 1 (see (more ...)
As we can see, in both types of networks there is a positive relationship between the connectance of the network and the nestedness value. This is not surprising, since the more connections a network has the more they can contribute to the nested diets observed in those networks. It is perfectly possible however, that more links could mean less nested communities; it is important to bear in mind then that in well organised communities, like the ones we consider in this work, the more connected networks are, the more nested they become.
We also can see from this plot (Figure ) that the data points corresponding to the naturally occurring communities are more distributed through the graph, while the points representing our simulated networks are more concentrated around connectances between 0.05 and 0.09, and nestedness indexes of 25 and 70. Apart from telling us about the variability encountered in real communities in this respect, this reinforces the fact that our simulated communities, although differing in some aspects from run to run, share a similar structure and enjoy features seen in natural communities.
These data (table and Figure ) also show us that the values of connectance and nestedness obtained from our simulated communities agree with the values of these measures commonly found in natural communities, where the connectance is normally between 0.03 and 0.1, and the nestedness values are usually found between 20 and 80. In our communities, the connectance values were greater than 0.05 and lower than 0.1; similarly, the NODF values for our communities were distributed along the 25-80 range of values. Apart from showing a connectance-nestedness relation comparable to that found in natural communities, these values are also in line with what is expected to find in these, which provides more evidence for the self-organised character of the communities modelled.
We want to further explore the similarities between our agent-based model communities and natural ones; for this we have selected, from the set presented above, one of our simulated and one of the empirically observed systems to perform a one to one comparison.
Two networks were thus selected based on their similitudes not only in connectance and nestedness values but also taking into account other features, as we shall see in the following paragraphs. The networks selected were: the one represented by the number 3 in table , with values 0.064 and 32.456 of connectance and NODF index for nestedness respectively, for representing our simulated communities; and the OFLO natural community, with 0.062 connectance and 35.961 NODF index, as a representative of the natural communities considered.
Figures and show the network of relationships between entities in the simulated community number 3 and the OFLO natural community, respectively.
Figure 9 Network representation of community number 3 in our simulations as presented in table 1. Host and visitor agents are represented by green and red nodes respectively. The thickness of the arcs represent the relative strength of that interaction relative (more ...)
Figure 10 Network representation of the OFLO natural community as introduced in table 1. Host and visitor agents are represented by green and red nodes respectively. The thickness of the arcs represent the relative strength of that interaction relative to other (more ...)
When closely inspecting these two networks we can easily realise a number of broad similarities such as the high proportion of visitor nodes (nodes in red) possessing only one interaction and also an important fraction of host nodes (nodes in green) with two interactions or less. It also attracts our attention the presence of one highly connected host node in both of the networks (green node in the bottom left corner in each of the networks), and also on the side of the visitor species/agents (red node on the top left). These nodes act as generalists/hubs, bringing cohesion and reachability to the whole network.
Another important feature captured by looking at the network of interactions and that is shared by our natural and artificial communities is the low fraction of strong dependences among nodes (solid dark edges in the graph) and the abundance of weak dependences, which in natural communities is believed to account for the stability and resilience of these systems, since the loss of a link can be easily adjusted for by recurring to other connections in the network.
Apart from the graphical representation of the interactions network, our simulation tool allows us to analyse certain properties derived from its structure and that can be employed to deepen our comparison among the selected networks.
Natural communities, as we have seen above, are somewhat less predictable than our simulated ones. This can be confirmed when comparing the interactions matrices of our simulated community number 3 and the OFLO natural community (bottom left plots of Figures and respectively): community number three presents a much more organised structure, with few interactions below the isocline of perfect nestedness, while the OFLO community presents not only more interactions below the isocline but also a few of them are actually far removed from it. Also the degree distributions in both communities, although agreeing in the overall pattern of power law fit, differ subtly by the fact that the values in our simulated community are closer to the fitted line (top right plots in the figures).
Properties derived from the network of interactions in community number 3 in our simulations. The community network is displayed in figure 9.
Properties derived from the network of interactions in the OFLO natural community. The community network is displayed in figure 10.
In spite of these differences, the communities present very similar distributions of the frequencies of the number of interactions (top left plots), with practically all the nodes possessing less than five interactions and the majority of them with two or one. Additionally, only three nodes in the case of community number 3 and two in the case of the OFLO community possess more than five interactions: reaffirming in this way the scale-free character displayed not only by the natural communities employed here as references, which is expected from this type of networks, but also by our simulated communities as an emergent feature of self-organisation in our agent-based simulations.
In the bottom right plots of Figures and we can observe the distribution of the dependence values (how dependent is a species on the others) for the simulated and the natural communities respectively. Again, there can be seen the similarities between these two instances of artificial and natural communities, with the majority of the dependence values located between 0.0 and 0.4 (agents/species not very dependent on other agents/species) and few values above this range. It is important to realise however that the probability of a dependence value of 1.0 (the highest value), which implies that there may exist a node which is entirely dependent on another, is high for the dependence of visitor on host agents, in comparison with the other values of dependence; and this happens in both the natural and the artificial community. This might be due to the fact that a considerable number of visitor species on the network are involved in only one interaction and this could potentially mean that, in the event of losing the resource they exploit and if they are not able to adapt to exploit a different one, they could fail to survive, becoming in this way part of the cascading effects of an extinction.
Figure shows the plots displaying the properties of community number 3 (Figure ) and the OFLO natural community (Figure ) overlaid, which facilitates the comparison between the datasets. Data shown in grey on the plots corresponds to the OFLO natural system while data in colour comes from the simulated community 3. In this picture the similarities and differences between these communities, as highlighted above, become more obvious and it can be seen how the interactions and dependence values distributions follow similar patterns, with slight differences; and how the interaction matrix for community number 3 is better organised, from the point of view of the nestedness isocline, than its counterpart in the OFLO community (grey squares in the bottom left plot of Figure ).
Figure 13 Properties of the communities number 3 (Figure 11) and OFLO (Figure 12) overlaid. In this plot we can see Figures 11 and 12 overlaid on each other, which facilitates the comparison between the properties of the community number 3 and the OFLO community. (more ...)
The similarities found between natural communities and our simulated ones, not only in terms of the network structure, but also in terms of their features and characteristics, let us see how the mechanisms implemented at the individual interaction level between artificial agents has allowed us to obtain system level properties that are commonly encountered in natural communities. This model can thus help us study and analyse the possible mechanisms by which these properties, and the features of stability and robustness they provide to natural ecosystems emerge; not only by employing the example interaction protocol we have used for demonstrating the viability of this approach, but also by incorporating to the model many other mechanisms and processes that could in theory provide these systems with the characteristics they display and that are awaiting to be experimentally tested.