Introduction

In multi-agent-based simulations designed to study complex social phenomena, agents are usually organised into some underlying structure that supports their contact, communication and interaction. To the purpose of facilitating simulation visualisation (which in turn is important for the experimenter in that it allows for a more intuitive view on the evolving patterns) most of the problems get represented into compressed systemic frameworks. This compression includes the reduction of the dimensionality involved, the use of abstraction to depict connections between agents, the uniformisation of the agent granularity, and other techniques. If this compression effort is adopted too soon in the modelling effort, it brings the risk of masking away the due complexity of the problems, and force the experimenter into solutions that are closer to his/her expectations (of what s/he can represent, of what can be sought after, of what can be allowed to come out of the simulations) than what the problem would demand for.

Whilst admitting that visualisation is an important desirable feature for social simulations of complex phenomena, we argue in favour of keeping the data as close to their original format for as long as possible. In this vein, we advocate the maintenance of richer representations of the complex set of relations in which the agents are involved in. These concomitant relations are closer to the representations traditional social scientists use in their research, and can be explored through simulation in a manner that can ultimately reveal so far hidden properties and phenomena. In particular, we are interested in exploring the permeability between contexts in these relations: how a particular phenomenon can disseminate in a qualitatively and quantitatively different manner through the communication that the permanence of multiple concomitant relations supports. Having in mind an abstract exploration of the potential of these ideas, we keep our definitions as clean and simple as possible: contexts are neighbourhoods in a given relation, and permeability (contact points between relations) is ensured by having the same agents present in several concomitant relations. For instance, consider two relations: family and work colleagues. Taking agent a, a context in relation family could be its spouse and children, whereas in relation work colleagues it could be composed by the agents sharing an office with a. Agent a ensures permeability between these two contexts.

In this paper, we further explore these ideas through the use of the new feature of NetLogo 4: links. Whereas relations would have an intricate representation in previous versions of this package, this new concept allows for a swifter and more transparent representation of relations between agents. We have tested out this framework through a very abstract game, the consensus game, so that not to overcommit to some application domain. Our results have showed that in certain circumstances, there is a qualitative difference in the structure of the convergence to a societal consensus. In this paper we show the results of experiments over the same games, but with the agents spanning over a variety of typical topologies.

The paper is organised as follows. In the next section we briefly address agent relations and their variety, focusing on the kind of social networks that are more frequent in the literature. Section 3 describes the idea of relations and contexts, and relates them to the concept of role. In Sect. 4 we argue in favour of simulation compared to mathematical or statistical analysis when addressing social networks data. The following section describes the consensus games we use to illustrate these ideas and examine their impact. Section 6 addresses the representation formalism used in the experiments, NetLogo 4 links. We finish the paper by mentioning the most important results of the experiments and drawing some conclusions.

Multiple and Multi-Modal Relations

In every relevant problem to be studied, agents will be involved in several concomitant relations. This complex setting gets only worse if we consider that even to define the relevant actors for a particular application may require a structure of intertwined levels of abstraction. When individuals are embedded in networks that are embedded in networks we have a “multi-modal” structure, such as students who together with a teacher form a classroom, of which in turn a school is composed of [8]. Surely the panorama can get more complicated, as these several networks of relations do not necessarily have such a “regular” meta-structure.

In an effort pointed at the study of decision change in agricultural networks, Amblard and Ferrand [2] propose a multi-agent system whose agents represent the model actors, the relations between them and the cliques formed by those actors. Actors are characterised by general, relational and decisional attributes. The relational attributes determine a “relational lacking” that can be considered a driving motivational force for the actor. If some general attributes seem to over-determine too soon the structure of the simulation (e.g. the division of socio-economical status in aristocracy, bourgeoisie and working class seems too restrictive even for an early XX century representation of a urban zone), this model possesses some self-reflective characteristics that render it quite interesting and general. The behavioural dynamics is based on relations, which allows for self-motivated agent re-structuring. This offers an alternative to data driven models [10], although possibly empirically less reliable. However, even data driven models must suffer from some model preconceptions since data necessarily have a source. The societal models organised into a complex ontology from isolated actor to domain-based systems, and including some time-aware models of relations is a good starting point for design explorations, since it is sufficiently detached from specific applications.

For the time being, in an attempt to control complexity and focus on the study of dynamic consequences of the topological structures underlying social simulations, we will opt for what we can call a “first order” approach. We will take actors and relations between them as givens of the problem, in a similar way as what is done in [12]. Here the authors select relations among scholars in a series of scientific conferences, namely meets, knows, and collaborates.

Our agents will be the atomic individuals of the simulations, but our relations will be kept abstract, so that we can concentrate on the dynamics they induce. We can think of the relations in our simulation as reasonably stable, such as family ties, or work colleagues, while we study the consequences of their mutual connectedness. In the final section we enumerate some of the possible ways to proceed this study into more complex scenarios, following an incremental deepening of the concepts recommended by the e*plore methodology [4, 9].

Relations, Roles and Contexts

The concept of role has been traditionally adopted in artificial intelligence to account for multiple engagement into several activities. An extensive survey of this notion can be found in [11]. Masolo et al. describe the account of roles adopted in several disciplines, including knowledge representation (role as a binary relation), knowledge engineering (roles as task ontologies representing individuals), object-oriented and conceptual modelling (roles as places in a relationship). The approaches that is more interesting to us is when roles are considered from a multi-agent system or a sociological/philosophical standpoint. In multi-agents systems, roles are seen as restrictions to behaviour, an abstract description of an entity’s expected functioning. This assumes many times a deontic characterisation that includes time dynamics and a structure of dependencies and relations between roles and individuals that fulfil them. In the sociological approach, roles are seen as behaviours specific to a set of persons in a context, including sets of rights and duties, acting parts of expected patterns. Masolo et al. proceed to provide a first-order approach to the notion of social roles, which is quite interesting: roles are properties, can be predicated, roles are anti-rigid and have dynamic properties (temporally evolving, or more generally considering other modalities), roles have a relational nature, roles are related to contexts.

The notion of context has also remained a hot issue in the literature over the years. McCarthy proposed to clarify the notion some years ago [13], but did not go much further than an ambiguous idea involving some structure in which first-order formulas could be evaluated and related to each other. Contexts can be transcended, and evaluation can be relatively decontextualised. Contexts provide a referential basis for linguistic processing, and can be related to each other to provide concept synchronisation (for instance in databases).

Many of these concepts seem to be basilar for the others, and their definition and properties are far from consensual in the literature. The notion of social relation as naïvely described in the social networks literature seems too dry and simplistic. Roles do seem to suffer from an overly heavy deontic character that might render them attractive for a logician but impractical for the simulator. The logic underlying a usable theory of roles would have to include a complex structure of contexts to provide grounding of the concepts, behaviours, and intricate relations intra- and inter-agents (let alone them and institutions, since we are postponing multi-modality). That structure of contexts would constitute an ontological challenge in itself, with complete references to the symbols in the agent’s mind, making it unbearably difficult to obtain even the simplest coherent behaviour. This is the type of difficulties that usually drives modellers into the use of simplistic theories such as utility theory, where numbers and friendly mathematics can make uniform and simple what is inherently and utterly complex.

In our case, simplicity (arguably excessive, or, as modellers defensively prescribe, necessary) comes from the eyes and experience of the modeller. It has been proven useful to use different researchers to accomplish the several phases of the development of simulations [4]. Different views from different disciplines can ­contribute to avoid the formal-conceptual prejudice that often computer ­programming approaches carry-[10]. Relations drawn by social scientists can help to gradually focus on a simpler world of representations, by proceeding towards the direction of answering the relevant research questions. Granted, perhaps the precocious shaping of those questions in the representational framework prevents other perspectives to be adopted, but hopefully the multi-disciplinary dialogue can open the problem before narrowing towards shaped solutions.

Generic arbitrary roles seem to carry additional disadvantages, such as some impermeability between each other, which implies lack of grounding with essential (corporeal/bodily/motivational) features of the agent that is currently fulfilling. This impermeability carries through several modalities, including time, which is fundamental to simulation. On the other hand, relations such as the ones found in social networks, can be subject to dynamical analysis (for instance, stochastic), can be built over themselves by drawing arbitrarily complex structures of relations deemed relevant to the simulation (as shown in [2]). Graph theory and equilibrium laws can be searched for through conventional techniques, but we are more interested in analysis aggregate behaviours together with individual trajectories, and the reasons for both [5].

Social Network Representations and Analysis

Most graph representations of real world social networks follow patterns that have only recently being revealed. Such is the case of scale-free networks [6]. Scale-free networks can be defined as a connected graph in which the number of links (k) connected to a given node follows a power law distribution, P(k) ∼ k-γ. Scale-free networks are to be found in a great variety of real situations, and display the property that most of the nodes have little connections, whereas some network nodes (usually called “hubs”) are highly connected. This is depicted by a right-skewed, of “fat tail” distribution. Barabási and colleagues proposed a simple generative mechanism called “preferential attachment” (cf. [1]). Although these mechanisms only generate a subset of the universe of scale-free networks, this is what we used for our experiments, and with γ fixed to 3 (most real data exhibit γ on the range [2, 3], although sometimes, smaller exponents can arise [14].

Scale-free networks have the additional property of being “small-world ­networks” (although there are other small-world networks other than scale-free). This means that while most nodes are not neighbours to one another, they can still be connected by a small number of connections. Scale-free networks have other interesting properties, such as a close to constant diameter as the number of nodes grow (d ∼ ln ln n), or a certain “fault-tolerant” behaviour (problems affecting random nodes will hardly fall on the critical hub nodes). In small-world networks, even though there is a high incidence of cliques (or subgraphs close to cliques), there prevails many times a popular notion (the famous “six degrees of separation” ­property) that it is easy to link any two people together by a path with only a few connections. While true in theoretical terms, this notion is based on the idea of ­connecting two nodes that have some kind of relation between them, when in reality the social world is much more complicated than that: people have all kind of relations linking them (family, work, acquaintances, etc.) and moreover know a lot about (what they know about) those relations. This is what often causes the “It’s a small world!” utterance in real life, and must not be masked away by hasty ­simplifications on the modeller’s side [3].

Most of the analysis of social networks is done in quite statical terms. Mathematics and statistics tools only start to provide the possibility of dynamical analysis [7]. Given the purpose of multi-agent-based social simulation, it is fundamental that useful dynamical properties, even some ones linking individual behaviours to global behaviours, can be derived from the network analysis. With our approach, we aim to contribute to such research endeavour, by bringing the fields closer together and feeding on each other. Our approach is to use simulation to explore the design space, not only of agents, but also of societies and even experiments. The key point in these simulations is to understand to what extent the structure of connections the agents engage in simultaneously can have a role in the shape of convergence towards a simple collective common goal, an arbitrary consensus.

Consensus Games

To try out some of the theories without commit ourselves to applications that could demand for shaping of the relevant relations we picked up a really simple example, a consensus game [3, 15]. In this case we select a variant of what is called the majority game. Each agent has a current choice towards one of two possible options (say, green and red), which are for all purposes arbitrary (no rationality or strategic behaviour here). Every time an agent meets another one, each of the agents has the chance to either keep its current choice, or change towards the other agent’s choice. In the variant we use in these experiments, agents keep track of previous encounters and when engaged in a new encounter it calculates the total number of agents of each colour it has seen before and adopt the colour of the majority of those.

Experimental Setting

Results are taken over 80 simulations. In each simulation we have 100 agents. In each simulation, agents are deployed in a number p of “planes,” representing different relations (or different views over a more complex relation, which is equivalent). Each agent is present in every plane, but its connections are possibly different in every plane.

Before the simulation starts, the agents are set up for the planes involved. Experiments are run with agents organised either over a “regular graph” or over a scale free graph. This initial set up determines whom the agents can have contact with. For the multiple-plane experiments, all agents are present in every plane (every relation), but are initially set up independently. We run simulations with one, two, and three planes, with all possible combinations of the following base agent distributions: k-regular (with k = 1, 2, 3, 4, 5, 10, 20, 30, 40, 50), and scale free (with g = 3). For regular networks we run experiments with four planes. The parameter k-in regular networks means that each agent is connected with exactly 2k other agents. This means that for k = 50 the graph is fully connected.

In each cycle, we select 100 pairs agent-relation (i, j), with i ∈ {1,..., n} and j ∈ {1,..., p}. So the same agent can get selected more than once in on cycle, if a different relation gets picked. Each agent in one selected pair (i, j) has one meeting (or encounter) with the agents that are directly connected to it. In this meeting, the agent plays the majority game: it updates its record of the total meetings it had with agents of each colour, and then its own colour is updated to the colour that has the majority. The update policy of the agents is sequential in the chosen 100 pairs. The simulation stops after 3,000 cycles (and so 300,000 meetings) or whenever total consensus is achieved.

Analysis of Simulation Outcomes

The following tables show the results of our simulations. Table 1 shows the percentage number of times that convergence was achieved in the 3,000 cycles. We can see clearly that for some networks (regular with small k and scale-free) consensus is never achieved in one single plane. However, as soon as we provide context permeability by adding more planes, consensus is achieved in a significantly greater number of occasions. These results are especially exciting for the scale-free networks, so common in real world domains: as soon as we add more planes, we can achieve consensus in a lot of occasions. For instance, with three planes we already have consensus in 82% of the cases. Figure 1 illustrates the results.

Table 1 Percentage of times consensus was achieved with all planes equal in kind
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Fig. 1
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Percentage of times consensus was achieved

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Moreover, Table 2 shows that to add more planes has a dramatic reduction effect in the total number of meetings necessary to achieve the consensus. This is true for all the networks except for those that are almost fully connected (k ≥30) and for the fourth plane in some other cases. Scale-free networks display numbers similar to regular networks with small k (2 or 3), although convergence is quite slower for more than two planes. We should note however that convergence is achieved in a rather reasonable amount of meetings, even in the worst cases, while in the vast majority of cases the number of meetings in very low. Figure 2 gives us a visual grasp of these results.

Table 2 Average number of meetings to achieve consensus with all planes equal in kind
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Fig. 2
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Average number of meetings to achieve consensus

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On top of these experiments with several planes, we also run experiments with combinations of scale-free with regular networks. So, for two planes we run simulations with a regular network and a scale-free network, while making k span over 1, 2, 3, 4, 5, 10. The results obtained are approximately what we would expect by interpolating the corresponding results. For instance, when with two planes, one is a regular network with k = 2, and the other is a scale-free one, we obtain a 45% of convergence, which is between 34% (s-f/s-f) and 65% (reg/reg).

For three planes we show the results in Table 3. It is apparent that the connections between agents allowed by the permeability with regular networks yields a significant improvement on the consensus numbers for scale-free networks. Since scale-free networks are frequent in real-world situations, this fact can help to enhance the effectiveness and speed of dissemination of phenomena, for instance for deployment of policies.

Table 3 Percentage of consensus achievement, average and standard deviation of the number of meetings for heterogeneous combinations of networks
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Conclusions and Future Work

In this paper we have defended the use of explicitly represented multiple relations in multi-agent-based social simulations. Our approach is still quite simplistic in keeping a “first-order” view of the relations: agents only know their own ­connections, and not of the relations themselves, and the granularity of the society is centred on the agents only (no groups or institutions). On top of that, we assume that every agent is present in every plane (relation), that the several relations are homogeneous: similar in structure, as in the kind of connections between agents. In [3] we explored other possibilities, such as having agents placed in one plane only, and allowing them to change planes. Nevertheless, we show through extensive experimentation with a simple game that the permeability between contexts allows for more nodes to be reached and causes more effective and quicker diffusion. In scale free networks, the achievement of consensus proves impossible with one relation only, whereas it significantly improves with the increase of the number of relations. In regular networks, the introduction of more relations always ensures more and quicker convergence. Finally, in experiments with different types of networks we notice that regular networks of any degree greater than two always induce more convergence to consensus in situations where scale-free networks are involved. If only one scale-free is involved, this convergence is quite faster, while with two it is not significantly slower. Overall, the behaviour of scale-free networks is comparable to a regular network with a small k (2 or 3). Future work will focus on further exploration of this experimental setting, namely by running simulations with other types of networks, by studying alternative policies for the individual agents update, and by increasing the heterogeneity of the networks considered in each run. We will also consider the use of dynamic networks.