1 In an undoubtedly harsh assessment, contemporary sociology, unlike classical sociology, has been criticized for excluding a certain number of basic theoretical and methodological problems from its agenda. Indeed, its current state of crisis (real or imaginary) is to some extent related to these oversights (Boudon 1998a, 13–14).
2 The subject of this article concerns one of the major issues that I feel contemporary sociology has effectively overlooked, namely, the place that the systematic construction of models should occupy in this field. In this respect, it is of particular note that a recent publication involving an analysis of the concept of the “model” as adopted in a number of disciplines (including the natural sciences as well as the social sciences) does not contain a single chapter on sociology (Nouvel 2002).
3 I believe—and this is the fundamental argument of this article—that sociologists should accord a significant place to “models.” This is due to the fact that, whether we like it or not, every social phenomenon relies on the multiple overlapping of actions, interactions, and structures, giving rise to chains (of variable length) of feedback loops. My impression is that sociologists will never be able to grasp the complexity of the flow of events caused by these loops. To attempt to reproduce it, often by focusing on analysis of the micro levels, would be senseless, and to ignore it by remaining at the societal level in terms of the relationships between aggregate quantities would be to give up trying to understand the sources behind these systematic regularities.
4 The “model” might, therefore, constitute a suitable cognitive and practical tool through which to master the complexity of reality, thus avoiding the two solutions noted above—solutions which, in a variety of more or less sophisticated forms, of course, have been used by sociologists in the past to try to resolve the difficulties encountered in the analysis of a given object. The use of “models” as an approach requires abstraction and simplification first of all and, secondly, a means of evaluating the discrepancy that exists between the results of the simplification operation and the actual explanandum. In addition, the logical sequence in “modeling” requires us to gradually advance with the construction of “local” models, followed by increasingly complex ones and, finally, by combining them (in a spirit that is highly reminiscent of Merton’s “middle-range theory” (1967)).
5 Might I be so bold as to quote the following passage in order to support my argument with an example from the past?
Generally speaking, we study things that vary by imperceptible degrees. The closer the representation we make of them to reality, the more it tends to become quantitative. We often express this by saying that, in perfecting themselves, the sciences tend to become quantitative. [...] For a long time, political economy was almost entirely qualitative; then, with pure economics, it became quantitative, at least in theory. We will, therefore, attempt to make similar progress in sociology and to substitute quantitative for qualitative considerations, for, although imperfect—even very imperfect—the former are always slightly more worthwhile than the latter. [back-translated from the French]
7 And yet, we now know that the revolution in economics referred to by Pareto has benefited precisely from the central place occupied by “modeling” in this discipline in the past (Armatte 2005, 99–102; Walliser 1999, 128–137; Walliser 2002). Ought we not therefore to expect that the progress advocated by Pareto will take even longer to achieve in sociology while a failure to grasp the importance of the concept of “model” persists?
8 We could immediately reply to this question in the negative, noting that this concept is in itself extremely ambiguous (Bunge 1973a), and, in addition, that it is used by sociologists in many different ways (Fararo 2005a). It might also be possible to raise the objection that we do not yet have the adequate tools to “‘implement” (with the dual meaning of formalizing and deductively analyzing) models that are sufficiently versatile for them to be of use to sociologists. In this respect, we would certainly be brought back to earth by being reminded of the difficulties connected with the use of mathematical formalism in sociology.
9 In short, this article constitutes an attempt to do away with these reservations. Firstly, my intention is to restrict the use of the concept of “model” to that of “generative model.” This concept has the advantage of clearly demonstrating the fundamental “ingredients” as well as the formal constraints that sociologists should use during the conceptual development of a “model.” Following this, I will run through some of the distinctive interpretations of the concept of “model” in sociology, with the intention of showing that this terminological pluralism refers, in reality, to different modalities of “formalization” and ways of studying theoretical models. The aim of this second part is to suggest that these different forms of “modeling” do not all have the same capacity for implementing a “generative model.” In this respect, I would particularly like to draw the reader’s attention to a recent type of “computational model,” agent-based modeling, which seems to be able to carry out this task in an exhaustive manner. On this specific point, my methodological reasoning will therefore refer to the concept of “multiagent systems” as “artificial microcosms,” an analogy that has also been formulated by eminent computer scientists, albeit in terms that are different from ours (see Ferber 2006).
10 Finally, I will put forward a schema that incorporates the numerous aspects of the concept of “model” referred to during the course of this article. I am, therefore, aiming to put forward a research grid that can assist in the design, development, and establishment of a type of sociology that is genuinely centered around the activity of “modeling.” 
1. Theoretical Development: “Generative Models”
11 I agree with Bunge in considering that the rigor of a discipline depends on the coherence with which it develops models with a specific purpose in mind namely, “mechanisms.” Indeed, this type of model aims to clarify how the observed phenomena arise. Models of mechanisms resemble a “translucent box” rather than a “black box.”
12 In sociology, models that have “mechanisms” as their object have been given a precise name, that of “generative models” (Boudon 1979a; Fararo 1969). As a first approximation, they might be defined as simplified conceptual representations, the consequences of which would include all the observations requiring explanation (c.f. also Schelling 1978, 89). 
13 As I shall attempt to demonstrate, a “generative model” can (and must) be characterized from two points of view: 1) according to its constituent parts (“mechanisms”); and 2) according to its structure (“complex methodological individualism”).
1.1 The Building Blocks of the “Generative Model”: “The Mechanisms”
14 The idea of a “mechanism” is not new. The epistemology and methodology behind many “hard sciences” (biology, physics, chemistry, or medicine, for example) have relied, at least since the seventeenth century, on the experimental demonstration and modeling of “mechanisms” (Machamer, Darden, and Craver 2000). Past traces of this approach are equally manifest in sociology, originating with the “classics” (Boudon 2005a, chap. 6; Cherkaoui 1998, chap. 3; Cherkaoui 2005a, chap. 1 and 2; Elster 2003, 44–48; Fararo 1989, 134–137, 345, 346). Although, undoubtedly, Merton (1949, 103, 106, chap. 11, 371–379; 1967, 43) has contributed to the introduction of some of the analytical characteristics of this idea into modern sociology, it certainly has its roots in the 1960s and 1970s, as much in terms of its epistemological background (Harré 1972; Harré and Secord 1972; Bunge 1997; 2004) as from the point of view of its application (Boudon 1973, 1976, 1979a; Davidovitch and Boudon 1964; Fararo 1969; Schelling 1971; 1978; Sorensen 1976). This long history should not be forgotten (c.f. also Philosophy of Social Science [2004, 34, 2 and 3]), despite the fact that the concept of “mechanism” has been attracting an increasing amount of attention since the 1990s and that a genuinely sociological approach—the so-called "analytical sociology" (Barbera 2004; Cherkaoui 2005; Hedström 2005; Hedström and Swedberg 1998a and b)—is currently developing in relation to it.
15 And yet, in order to counter the conceptual cloud currently surrounding the concept of “mechanism” (Mahoney 2001, 577–582), with the intention of defining it, we should, on the one hand, ask ourselves the purpose of a “mechanism,” while at the same time questioning its content.
16 Regarding the former, a mechanism has the epistemic function of clarifying how and why a relationship (or a structure of relationships) has been generated (Harré 1972, 6, 118). What is at issue is “mode of production of phenomena” (Cherkaoui 1998, chap. 3) so that by understanding mechanisms in this way, we are able to capture their distinguishing characteristic, that of “generativity” (Fararo 1989, 39–43). This idea inspires us to look for the emergence, production, and cause of what is observed. This is, moreover, the reason why mechanisms and “generators” are very often descriptively linked. Whether they are perceived as the world’s real entities (Harré 1972; Bunge 1997, 2004; Fararo 1989) or, on the contrary, as analytical structures (Stinchcombe 1991; Hedström and Swedberg 1998b), the shared premise is as follows: what is observed at level k must be explained as the effect of one or more instances—mechanisms —that are located more deeply at level k – n. If Y and X are any two phenomena, a mechanism does not act on the values or behavior of X and Y taken individually but on the process of the emergence of the relationship as such (form and nature). A “mechanism” should not therefore be confused with a "confounder" (Mahoney 2001, 508; Pawson 1989, 130–131). 
17 If we might venture to adapt a definition adopted from biology (Machamer, Darden, and Craver 2000), a “mechanism” could be defined, in view of its content, as a coherent body of entities (structures and agents), systematic relationships between these entities (a system of influence, interaction, and interdependence), and activities (choice, actions, exchanges, etc.). By consequence, the concept of mechanism may be defined by distinguishing three general sub-components (Hedström and Swedberg 1998b, 21–23; c.f. also Tilly 2001).
18 a) “Situational mechanisms” or “macro-micro mechanisms” focus on the manner in which contextual elements constrain the autonomy of agents. In spite of the differences that exist between the numerous mechanisms that fall within the scope of this class, they all refer to the way in which elements that pre-exist a particular individual affect one or more components of his/her actions. They thus allow the clarification of how a link between a macro element and a micro element is created. 
19 In order to reject any “mechanistic” interpretation of a mechanism, we should point out here that “situational mechanisms” should be perceived as being based on numerous “filtration” processes put in place by actors in order to deal with elements that are imposed on them. Maintaining that a structural element, whatever it may be, affects the activity of the actor through the effects that it has on the components of his/her action (beliefs, preferences, and opportunities) is not the same as maintaining that it determines it. The structure is merely the “parameter” of the “action”; in the same way as, in an analytical equation of a straight line (Y = aX + b), for example, a relates only to the slope of the straight line and not to the final value of Y. If the structural element defines the range of possibilities, the actor is not, for all that, deprived of the strategic power of initiative, inventiveness, and creativity (Joas 1999). The capacity that actors have for regaining control of structural constraints (Archer 2004) makes the link between “structure” and “action” (a link which “situational mechanisms” claim to explain) merely probabilistic and conditional rather than deterministic and unconditional.
20 b) Next, if it is a matter of understanding how the beliefs, intentions, and opportunities of an actor come together in order to produce a particular action, “action-formation mechanisms” or “micro-micro mechanisms” would be brought into play. We might relate them to the stylized image of the actor that the sociologist decides to use and, more particularly, to the idea of rationality with which they wish to endow the individual. 
21 In this respect, above and beyond the variety of the interpretations available (Goldthorpe 2000, chap. 6), we might first allude to the instrumental and consequentialist notion of rationality. Here, the generative principle of individual choice relates to future consequences in terms of personal well-being (interest or utility) that agents are able to derive from the action being carried out, relative to the costs involved in completing it.  Then, we might call for an “adaptive,” “pragmatic,” and “progressive” concept of rationality (a concept that derives from the repeated and evolutionary games theory (Abell 1996), according to which actors will shape their actions, not according to the future consequences of that action, but in accordance with past strategies that have demonstrated a capacity to adapt to the environment (Macy 1997). We should also consider the notion of rationality, sometimes referred to as “sideward-looking” (Barbera 2004, 118), according to which actors will shape their actions in imitation of those carried out by others that have proved to be effective. Hence, the actions of others are understood to represent a source of information (Hedström 1998).
22 We should finally mention the “cognitive” notion of rationality, according to which actors have the ability to theorize the context in which they are acting and to develop systems in which their motives are subjectively perceived as justifiable from the point of view of the material, symbolic, and cognitive resources at their disposal (c.f. Boudon 2003, for a recent overview). This notion of rationality generally serves two purposes. On the one hand, since it accepts that the “motives” of actors may be essentially different, other notions of rationality might be included. On the other hand, cognitive rationality, as applied to the descriptive, as much as prescriptive, or normative domains, appears not only as a “means rationality,” but as an “ends rationality.” 
23 Thus, the construction of a “micro-micro mechanism” raises the problem of a choice of one concept of rationality among numerous possibilities. Since the fundamental purpose of a generative model is to represent regularity at the societal level, the degree of realism behind the notion of rationality to be retained is only part of the problem. Indeed, the intention behind the “micro-micro mechanism” is not to give a detailed reconstruction of the behavior of each individual. Therefore, the type of rationality selected should vary according to the complexity of the macrosocial regularity under investigation. I believe that this is the meaning of the principles of “decreasing abstraction” and “sufficient complexity” put forward by Lindenberg (1992; 1998; 2002).
24 Since it is now a matter of clarifying the modalities involving a combination of individual actions, we should finally turn to the mechanisms known as “transformational mechanisms” or “micro-macro mechanisms.” Their importance is paramount in sociological theory, owing to the fact that any outcome as far as society is concerned is reached through them (Boudon 1977; 1979b, chap. 4, 5, 6; 1984, 66–71).
25 We will now put forward a distinction between two general types of “transformational mechanism” namely, “simple aggregation mechanisms” on the one hand and “complex aggregation mechanisms,” on the other. These two types of mechanisms refer to two different modalities in which actors' actions are combined and thus to different ways of moving from the individual to the collective level.
26 In the case of the former, agents are not interconnected and act independently of one another. In this situation (solipsistic, one might say), there is a direct transition from the individual level to the level of the “society.” Actions are combined by a simple juxtaposition, or by their “addition.” The path from one level to the other presents no particular difficulty, with the “sum” being derived directly from the “parts.” Thus, if this type of path from “micro” to “macro,” realized through the simple aggregation of independent individual actions, allows for the passage from “action” to “structure,” as a general rule, and in the absence of more precise details, the structure in question will be of a very particular nature. It might be more correct to refer to an “aggregate level” rather than a “macro” level. 
27 It is quite another matter when “complex aggregation mechanisms” are present. These relate to any situation in which the action of one actor has a bearing on the beliefs, preferences, and/or opportunities of another agent.
28 This interdependence comes in many forms, which can be likened to a continuum covering a range from pure and simple imitation to strategic interaction (Abell 2000). Each of these modes of interdependence can take on a “direct” or an “indirect” form (Boudon 1979b, 130). In the former, the actors are networked. The modification of beliefs, preferences, and/or opportunities takes place as part of a personal interaction. These are adjacent interactions supportive of a potential “complex aggregation mechanism.” This is what is known as a “dyadic interaction effect” (Barbera 2004, 80–85). In the latter, on the other hand, the interdependence of agents is not as a consequence of their proximity and adjacency, but is rather the effect of aggregates stemming from past actions. Among these types of indirect interdependence, “parametric interdependence” (Abell 1996) and “strategic interdependence” (Abell 2000; Barbera 2004, 127–132) are particularly important. 
29 Whatever the type of interdependence in question, its presence implies that a knowledge of the characteristics of each person taken individually is insufficient to determine the result at a “societal” level. A “leap” comes between the initial action and its systematic result. The latter would be “original” in comparison with the former, owing to the play of interactions and mutual points of reference that exist between agents. It is only in this situation that we would speak of “emergence” (on this concept, c.f. Cherkaoui 1998; Sawyer 2001; Stephan 1999; 2002). The notion of “macro” should probably be reserved for this specific situation.
30 The distinction between “simple aggregation” and “complex aggregation” is of an analytical nature, however. We would be entitled to describe the “structural level” as “macro” or simply as “aggregate” depending on whether we are able to bring “mechanisms” relating to interdependence structures into action or not. I believe that emergence is therefore an analytical problem that derives from the type of theory being constructed by the sociologist rather than constituting a characteristic of social reality as such. In order to qualify this position, I am tempted to speak of an “analytical emergentism.”
31 Nonetheless, whether it is a question of a “situational mechanism,” an “action-formation mechanism,” or a “transformational mechanism,” an essential characteristic must now be made explicit. Every mechanism and, a fortiori, every combination of mechanisms (c.f. below) is generally indiscernible. Metaphorically speaking, mechanisms might be described as “invisible codes” (Cherkaoui 2005, 1). A mechanism can, therefore, only be conjectured (Bunge 1997, 420) or imagined (Harré and Secord 1972, 67, 71–73) by the scholar. In order to be able to study it, a theory must first be constructed, and then be given a stylized and abstract representation (Hedström and Swedberg 1998b; Hedström 2005, chap. 1), in other words, a model. As highlighted by Bunge, however, “there is no method, let alone a logic, for conjecturing mechanisms [...] [it] is an art, not a technique” (2004, 200).
32 This challenge is tentatively taken up by the “generative model.”
1.2 Structure of the “Generative Model”: “Complex Methodological Individualism”
33 I believe that it is unrealistic to suppose that mechanisms of a single type would be sufficient to reproduce the different facets of a social phenomenon. This is why it is more often a matter of surmising a “concatenation” of mechanisms or a “molecular mechanism” or a “generator,” depending on the terminology that one prefers (see Gambetta (1998), Elster (1998), or Fararo (1989, 75, 144), respectively).
34 This collection of mechanisms might have a regular basic form which we might represent by modifying the famous diagram usually attributed to Coleman (1986a, 1322; 1990, 5–21).
Fig. 1. — Adaptation of the Coleman Boat – Static Form of a “Generative Model”
Fig. 1. — Adaptation of the Coleman Boat – Static Form of a “Generative Model”
35 Indeed, among the three general classes of mechanisms that I have just described, at least one micro-micro type of mechanism should be present. Since only actors are able to connect, transform, construct, or destroy aspects of social reality (Abell 2004, 293), the idea of “generativity” would be meaningless without a reference to individual actions (Bunge 1997, 447; Fararo 1989, 146). However, the shaping of action cannot be understood without referring to a pre-existing context (Esser 1998; Popper 1967) so that at least one macro-micro type of mechanism is also necessary. However, since, for sociologists, a reconstruction of the behavior of actors is not an end in itself (Coleman 1986a, 1321), we should provide ourselves with the means of passing from the level of action to that of “structure.” At least one “micro-macro mechanism” is therefore necessary.
36 Thus, the basic form of a “generative model” is based on a specific variety of methodological individualism, namely one that rejects the identification of the “micro-foundation” of the analysis with its “micro-reduction.” A “generative model” depends on the admission of the analytical primacy of the actors, as well as their motives and actions. In this sense, it is committed to methodological individualism. This is a form of non-reductionist individualism however, since the analytical primacy of the actor coexists with the admission of the importance of social “structures” first and foremost, followed by systems of the direct and indirect interdependencies that are continually being created among actors. 
37 As far as the first of these is concerned (the analytical primacy of the actor), many contemporary sociologists are increasingly in agreement with considering the intentionality and rationality of action as the most appropriate starting point (c.f. Marini 1992; Déchaux 2002; c.f. also European Sociological Review 1996; Sociologie et Société 2002). This hypothesis would, indeed, have an “explanatory privilege” in the sense that no additional question can be asked once it has been demonstrated that the phenomenon under investigation stems from a combination of intentional and rational actions (Coleman 1986b, 1). As I have already pointed out, since only actors are capable of “connecting” and “transforming” (Abell 2004, 293), there can be no real sources of causality other than individuals and their actions (Hedström and Swedberg 1998b, 11-13). In this sense, black boxes are then eliminated from the explanation (Boudon 1998b). However, it is also a question of a “privilege or logical priority” insofar as a hypothesis concerning the intentionality and rationality of action provides the analysis with a point of departure (Coleman and Fararo 1992, chap. 14–15; Goldthorpe 2000, chap. 6, 134).  This hypothesis would have an added advantage that could be described as “normative,” in the sense that the actors themselves appear to be rational, claiming this quality for their actions (Elster 2001, 12763). We should finally mention the necessity of referring to rational individual action in order to make progress with the conceptualization of the problem of the interconnection of the micro- and macro-levels (Abell 1992; Friedman and Hechter 1998). In order to clarify this point, we should effectively ask ourselves where and how a particular social structure came into being. The reference to individual action thus becomes indispensable. 
38 As far as the second is concerned (the admission of the importance of social “structures”), it would seem that the interest in action, revived since the late 1970s, comes in part from a more pronounced desire to combine action and structure and to articulate the micro and macro levels of the analysis (Giesen 1987). In my view, although there is a tendency to equate methodological individualism with its “atomistic” and “reductionist” forms, the numerous arguments found in literature suggest the existence of an alternative in which the systematic combination of “action” and “structure” is advocated, as is illustrated in figure 1 (c.f. also Boudon 1979a, 296).
39 It is found, for example, in some of the German and Dutch sociological writings of the 1970s, referred to as “structural individualism” (Lindenberg 1977; Wippler 1978; Raub 1982). It is also found in French sociology where it is called by the same name “individualisme structurel” (Boudon 1983, 3), or is referred to by slightly differently labels, such as “individualisme institutionnel [institutional individualism]” (Bourricaud 1977). Jean-Pierre Dupuy (1992) talks about “complex methodological individualism” to refer to the complex notion of methodological individualism, which “conflicts as much with simple methodological individualism as with holism. [...] Complex methodological individualism emphasizes the link that constantly unifies the individual and collective levels” (Dupuy 1992, 19).
40 Among British social theorists, the approach known as “morphogenetic” (Archer 1995) is overtly dependent on the mutual connection between these two levels of analysis. However, significantly, this “hybrid” form of methodological individualism is also present in the philosophy of science.  Bhaskar refers to a “transformational model” (Archer 1995, 137–141) and Bunge advocates a combination of holism and individualism that he describes as “systemism” (1997, 441; 2004). Philip Pettit (1993) suggests the notion of “holistic individualism.”
41 Thus, “Coleman’s Boat”—seen in figure 1—puts forward a form of methodological individualism that is less unusual than one might originally have supposed. As has already been noted (Udehn 2001, 307), the content of “Coleman’s Boat” actually equates to the formula “M=MmSM’,” through which Boudon (1984, 40) expresses his notion of methodological individualism.  I would like to suggest that Archer’s “morphogenetic” cycle (1995, 76), that is to say the sequence of “structural conditioning—social interaction—structural elaboration,” largely equates to the link “macro-micro-macro” proposed by Coleman and Boudon. Several authors have, moreover, recognized this basic similarity, extending it further to include Dupuy’s “complex methodological individualism” (Caillé 2004, 37).
42 The “generative model” therefore implies that “structure” and “action” should be considered as separable from an analytical point of view, which necessarily leads to hypotheses concerning modalities in which they reciprocally refer back to each other. “Macro-micro mechanisms,” “micro-micro mechanisms,” and “micro-macro mechanisms” form the supply source (for modeling) of these points of reference. In this respect, we might say that a “generative model” depends on a form of complex methodological individualism.
43 I believe that it is an original solution to the problem—recurrent in sociology (Cuin 2000)—of the relationships between “structure” and “action.” For those who have attempted to resolve it, the other interesting possibility concerns the consideration of “structure” and “action” as two indissoluble and mutually constituent sides of social reality. Giddens’ (1984) “theory of structuration” seems to me to represent the clearest example of this type of solution, along with Bourdieu’s “genetic structuralism” as Axel van den Berg (1998) has suggested. Some theorists of "complexity," such as Morin also frame the problem of the relationship between the individual and society in terms of inseparability and mutual co-production.  Both micro and “macro-reduction” are rejected from this particular point of view, in which both operations are deemed equally useless: “structure” and “action” continually co-produce each other, thus producing the social. In my view, the idea of the “duality of structure” poses the following problem: by perceiving “structure” and “action” as indissoluble and mutually constitutive, we cannot think of them as having reciprocal connections. Consequently, the problem of the relationship between the individual and the group remains unresolved: it is in reality blurred. This is why Archer (1994, chap. 4) describes this solution as “central conflation or elision.” Reduction is operated here through a combination of different levels of analysis (c.f. also Sawyer 2005, chap. 7).
44 However, according to the definition provided here, a “generative model” avoids this impasse, prompting the scholar to restore every social phenomenon with its dynamic character. This is the result of the fact that “structure” and “action” take place on two temporal scales that are out of step. In terms of an action that unfolds at given moment t, the structure is always further back in time: it is the result, most often unintentional, of interdependent actions that have taken place at given moment t-l (Abell 1996, 261; 2003; Archer 1995; Coleman 1993, 63). The construction of a “generative model” requires us to envisage all societal regularity as the consequence of a chain of “structure-action-structure” connections. Its aim is to unravel the labyrinth of mechanisms underlying these chains.
45 Thus, by adding complexity to figure 1, we can establish the basic form for a “generative model” with respect to its dynamic side in the following manner (see illustration on the following page).
46 It is important to note that the different levels of analysis that a “generative model” attempts to combine, all retain their relative autonomy.
Fig. 2 – Dynamic Form of a “Generative Model”
Fig. 2 – Dynamic Form of a “Generative Model”N.B: SM = “situational mechanisms”; AFM = “action formation mechanisms”; TM = “transformational mechanisms”
47 As far as “structures” are concerned, the possible presence of “complex aggregation mechanisms” will mean that the latter are the emergent products of individual actions. Although these actions constitute their ultimate founding principles, the presence of one (or several) interdependent structures filters actors' actions taken individually, thus introducing a “leap” between the “micro” and “macro” levels. We might then speak of an “ascending discontinuity” between “action” and “structure.” As far as “actions” are concerned, recognizing the unique character of structures that have come into being does not therefore necessarily mean, however, that we should deny the autonomy of the actors who act later on. The presence of “situational mechanisms” ensures that agents theorize “structure” by applying different sorts of “filter.” This makes the behavior of the actor an “emergent” product in relation to the “structure,” that is to say, one that is fundamentally original with regard to what we might expect to observe if the action was based solely upon it. Since we are positioned along the downward section of the “Coleman boat” we should therefore speak of a “descending discontinuity” between “structure” and “action.”
48 Two elements seem to me to justify, therefore, the term “complex methodological individualism,” a term that I am giving to the basic form of every “generative model.” Firstly, these models attempt to disentangle the interplay of overlapping elements in a number of mechanisms related to different analytical levels and of which the effects are dynamically entangled. It could be said that the intention behind a “generative model” is to represent in a stylized form “the complexity of mechanisms” underpinned by any macrosocial uniformity that the sociologist wishes not only to describe, but also to explain.  Secondly, a “generative model” attaches particular importance to “complex aggregation mechanisms,” that is to say, mechanisms concerning the numerous systems of interdependence (direct and indirect) that link the actors together. In this respect, models of this type relate to one of the distinctive features of the so-called “complexity” approach (an approach that is definitely heterogeneous, c.f. Alexrod and Cohen 2000, 46–53), that is to say, the consistent attention that it devotes to the interdependence of constituent entities of a system, as well as to the emergent phenomena that result from it (Atlan 1991; Morgan 2005; Morin 1999, chap. 4; Simon 1996, chap. 7, 1999; Weisbuch 2003). On this point, we should note, however, that sociology does not lack pioneers. By the end of the 1970s, Boudon had already stated for example, that “societies should be considered as complex labyrinths of systems of interaction” (1979 b, 113).
2. The Formalization and Analysis of the “Generative Model”
49 If we agree to base theoretical activity in sociology on the systematic construction of “generative models,” there is, nevertheless, one important problem to resolve—that of its “implementation.” Since the model is conceived theoretically, it is effectively a matter of designing an appropriate form for it (the formalization of the model) in order to be able to work out its consequences (the analysis of the model).
50 Determining the degree of conformity between the results and the empirical regularities to be explained is a prerequisite for ascertaining a “presumption of validity and realism” in the posited mechanisms. Moreover, some authors have expressed doubts concerning the legitimacy of a process of reasoning that claims to explain phenomena observed by invisible entities, namely, mechanisms (Mahoney 2004; Sica 2004). In order to counter this criticism, I believe that it is important to tackle the problem of the “implementation” and study of the “generative model” in a manner other than the purely verbal.
2.1 “Statistical Models”
51 (Multivariate) statistical methods certainly constitute the most widespread mathematical application used in sociology. As Fararo (2005a) has noted moreover, one of the interpretations most frequently given to the concept of the model in our discipline is precisely that of “statistical model.” Given its omnipresence, it would be quite normal if we were to turn to statistical techniques in order to “implement” a “generative model.”
52 This solution would seem to have its not insignificant limitations, however. We should make clear from the outset that, despite its recurrent use, the very concept of a “statistical model” is not without ambiguity. In fact, on the one hand, through an undesirable linguistic shift, this concept blurs the distinction between the formalism used to express the model (mathematical language) and the tools used to evaluate it, which belong to statistics (particularly inferential statistics) –itself, in turn, merely a branch of mathematics (Barbut 1994, 2000). On the other hand, speaking of “statistical models” is also unfortunate, as the term “model” conceals the theoretical inadequacy of “statistical models” from the point of view of sociology (Esser 1996). Indeed, “statistical models” depend most often on hypotheses that take the form of relationships between variables (linear as opposed to non-linear; additive rather than multiplicative, etc.), the conditions that should be respected in order to evaluate relationships (the form taken by the distribution of variables and the distribution of errors, etc.), and the constraints needed to estimate the parameters (the sum of parameters equal to zero, the product of parameters equal to one, etc.). However, the first type of hypothesis is only loosely related to precise sociological hypotheses (Clogg and Haritou 1997, 88, 93; Goldthorpe 2000, chap. 8, 98; Sorensen 1998, 238, 239, 243–244) while the two others are difficultly justified in sociological terms (Freedman 1991a, 311). It is likely, therefore, that a “statistical model” will serve a purely descriptive purpose. 
53 Thus, the ambiguities contained in the very concept of “statistical model” suggest a first source of tension between it and the “generative model,” namely, that of their different objectives. Indeed, viewed in epistemological terms, a theoretical model centered around mechanisms is diametrically opposed to a model centered around variables, since the latter is quite simply unable to provide an explanation.
54 Sophisticated though they are, the parameters of “statistical models” are only expressive of the intensity, sign and, possibly, the form of the link between two or more aspects of reality, which are operationalized under the form of variables. On the other hand, in these models nothing is made of the sources behind this link (Boudon 1976; Bunge 1997; Hedström 2005, 31–36, chap. 5; Sorensen 1998). In the absence of a way of modeling the mechanisms underlying the observed relationships, the explanation is hollow. In this case, no characteristic of causality can be attributed to the action of X on Y. Neither can the attribution of the causality of an observed connection be justified by the temporal (or logical) precedence of X in relation to Y or by the resistance of their link to the introduction of a third set of variables, W. A “statistical model”—of which the basic logic has sometimes been described as “correlation analysis” (Mahoney 2001)—depends on a notion of causality which we might call “successionist causality” to adopt terms used by Harré (1972, 116, 121, 136–137). A “generative model” however, is based on what Harré has described as “generative causality” (c.f. also Goldthorpe 2000, chap. 7, 151, 154–155). 
55 My belief, however, is that a second source of tension between “statistical models” and “generative models” stems, not from epistemology, but in more concrete terms from the structure of multivariate statistical techniques themselves. It is likely that it is the intrinsic characteristics of these techniques that make it difficult, even impossible, to implement a “generative model” within it (Manzo 2006a).
56 a) A first point concerns the unobservable character of the “generative model.” Some of its constituent parts may certainly rely on empirical data, but the logic behind the functioning of the model in its entirety does not seem to be empirically definable. It would, therefore, be pointless to hope to understand its configuration through variables that rely on empirical factors (Mahoney 2001, 581). To assign “variables” the capacity of detecting mechanisms would necessarily be an artificial operation, since it is rhetorical and imaginary. At best, statistical techniques are able to capture the effects of a mechanism, but they cannot represent its detailed structure or its functioning.
57 b) A second essential point concerns the fact that the very definition of a “generative mechanism” implies that the observed relationships are produced by entities and activities below the level on which the relationship is observed. Whilst a “generative model” may be perceived as a “system” for the generation of data—in the sense that it allows for the temporary abstraction of empirical data and the artificial production of its own data—, this change of level is completely absent in a “statistical model.” All the constructions within it are based on the observed data. No operation relating to the “production of exogenous information” exists (Stinchcombe 1991, 371). No matter how sophisticated it is, a “statistical model” only reproduces the initial data in another form. It does not allow for the stylization of the generation of these structures, since it does not make use of any element that is external to the empirical data, as is the case with a “generative model.”
58 c) A third difficulty arises from the capacity of a “statistical model” to deal with the systems of multiple interdependencies that connect the actors. “Complex aggregation mechanisms” constitute one of the main components of a “generative model.” One of the objectives of the latter is the analytical decomposition of the different systems of direct and indirect influences that connect actors, and enable the passage from the individual to the societal level. Whatever the statistical technique, a “variable” derives only from the juxtaposition of information gathered at an individual level. Actors are regarded as separate from one another, variables only combining information concerning “solipsistic” actors. From the point of view of the transition from “micro” to “macro,” multivariate statistics can therefore only deal with simple forms of aggregation (those that operate through the immediate juxtaposition of individual actions), remaining silent in relation to complex forms of aggregation centered on the direct or indirect interdependence of actors (Esser 1996, 162). In other words, a “statistical model” is structurally unsuitable for the representation and direct modeling of the interaction between actors. 
59 d) A fourth point deserves to be raised. A “generative model” counts among its objectives the creation (and transformation) of the social “structures” of emergent macrosocial phenomena. Given our definition of the “macro” level, it would only be legitimate to describe societal regularity as “macro” on the condition that we can make of it the product of actors who are in a position of considerable interdependence. And yet, if, as I have just pointed out, a “statistical model” is unable to take into account the structures of interdependence in which the actors are immersed, it would also be impossible for it to take on board the transition from “micro” to “macro” (Cherkaoui 2005, chap. 6). Since multivariate statistics can only deal with forms of “simple aggregation,” meaning that they are based on the immediate juxtaposition of individual characteristics, they cannot lead to the “macro” since it is based, by definition, on the presence of structures of interdependence.
60 This is a sizeable consequence. If we are unable to “create” the elements related to “structure,” we are a fortiori unable to demonstrate the feedback effects that the latter can have on successive actions. In other words, a study of the dynamic concatenations of different levels of analysis is very hard to place within the framework of standard statistical techniques. The complex form of methodological individualism contained within every “generative model” therefore remains inaccessible to “statistical models.”
61 In this respect, advanced techniques, such as time data analysis techniques (Blossfeld 1998), the numerous variations of multi-level analysis (Courgeau 2003, chap. 2), or indeed the methods of “optimal matching” (Abbott 1995; Abbott and Tsay 2000) certainly merit a close examination. These methods have as their objective the more efficient study of the connection between the micro and macro levels of analysis through a consideration of “time” and the introduction of different levels of aggregation, albeit by different routes. Moreover, they bring the dynamic nature of social phenomena back into the foreground, as well as the coherence of sequences of events that characterize the existence of both society and the individual. Even so, it might once again be a question of partial solutions. As these methods remain based on data that are structurally similar to those used by more classic techniques, they are unable, directly or dynamically (in the strict sense of the word) to represent the action of the numerous mechanisms underlying the chain of these events and that are responsible for their emergence. This is probably why Sorensen (1998, 265) considers that these advanced techniques have “become just another way of doing regression analysis with rich opportunities for controlling for everything.” 
62 Thus, the four problems that I have just outlined seem, at least for the time being, to present just as many obstacles to the formalization and analysis of a “generative model” by means of “statistical models.” Without necessarily rejecting their use (I will return to them in my conclusion), a real change of methodological perspective seems necessary.
2.2 “Computational" and “Simulation" Models
63 I would like to suggest that certain simulation methods should henceforth provide the sociologist with an adequate infrastructure for the formalization and, above all, for the analysis of a “generative model” (c.f. also Manzo 2003, 2005).
64 This involves referring to two more frequent interpretations of the notion of model—those of “computational" and “simulation" model—whose meaning, as with the concept of “statistical model,” requires prior clarification.
65 The expression of a “computational model” effectively operates as an amalgamation between the theoretical model itself and a specific way adopted to formalize it, namely, computational languages. The expression of a “simulation model,” on the other hand, operates as a direct leap from the theoretical model itself to the tool used to analyze it, namely, the simulation technique. However, if the concept of the “computational model” implies that of the “simulation model” (since a theoretical model expressed in computational language will be studied by simulation first and foremost), the reverse is not always true. A model analyzed by simulation might effectively be formulated in mathematical language.
66 Having clarified these linguistic shifts (which are potentially ambivalent), I should specify that my argument, according to which a “generative model” can be analyzed in the most complete manner using simulation methods, echoes previous suggestions. The idea that an advantageous link exists between “mechanisms” and “simulation” has been taking shape effectively since the 1960s, both in sociology (Boudon 1965, 1967, 1973, 1979a; Boudon and Gremy 1977; Coleman 1962, 1965; Davidovitch and Boudon 1964) and, more generally, in the social sciences (Guetzkow 1962; c.f. also Archives Européennes de Sociologie 1965; The American Behavioral Scientist 1965). It is, in fact, during this period, moreover, that Schelling (1971) provides us with a brilliant illustration of this viewpoint by creating and manually solving a sort of “cellular automaton” ahead of its time.
67 In consequence, the most recent lines of argument developed by Collins (1988) and Goldthorpe (2000, chap. 7, 158) in favor of simulation, the increasingly explicit links created between “artificial intelligence” and sociology (Carley, 1996), as well as the program of genuine “computational sociology” (Hummon and Fararo 1995), should be perceived as the most accomplished and palpable recent manifestations of ideas that have, however, never been entirely foreign to our discipline.
68 The recognition of this past link does not necessarily mean that the route taken during the period from the 1960s to the present should be underestimated. It is, in fact, undeniable that, since the start of the 1980s, we have been witnessing a diffusion and unprecedented development of thought relating to this perspective and its applications (Bruderer and Maiers 1997; Hummon 1990, 65; Troitzsch 1997, 45). The fact that a number of authors refer to the “simulation era” (Hartmann 1996, 77, 79, 84, 98) or a “new way of doing social science” (Gilbert 1999, 1486) is suggestive of the fact that a veritable “silent revolution” has been under way. Simulation methods currently seem to have come out of the marginal position they occupied when they were originally suggested in the 1960s (Halpin 1999). 
69 In this respect, technical developments undoubtedly represent the most far-reaching advance. Macy (2001) distinguishes three successive waves leading to the formation of a range of methods so widespread, varied, and disparate that a detailed explanation would fill an entire manual (such as Gilbert and Troitzsch 1999).
70 Over and above the characteristics of different techniques,  “simulation” can be generally defined as the execution of a program that translates the theoretical system in which the object under analysis (the model) is represented in the form of a set of algorithms written in specialized computerized language. Using these means, the behavior of this type of system can be studied and observed under different conditions as it evolves dynamically (Macy 2001, 14439; Moretti 2002; Troitzsch 1997, 46; Hanneman and Patrick 1997, 2–3; Hartmann 1996, 83). Simulation therefore presupposes a prior modeling without which it could not exist. We always simulate the behavior of an object—the model—, which needs to be designed in advance. This is why advocates of the “simulationist approach” in the social sciences often insist on the positive outcome of simulation relative to theory—sociological theory in particular (Collins 1988, 647–648; Hanneman 1995; Hanneman and Patrick 1997; Hanneman, Collins, and Mordt 1995, 3; Hegselmann 1996, 222–230; Jacobsen and Bronsen 1997, 98–99). The theoretical ambition is, moreover, one of the principal objectives of “computational sociology”, which is particularly well expressed in Carley’s writings (1994, 1999, 2000).
71 We can immediately see the gulf separating the model being treated by simulation from the “statistical model.” Instead of being bound by what is measurable and observable, in a “computational model,” the focus is primarily on theorizing the object of study. In this respect, these models apparently have a significant affinity with “generative models.” Before being examined in concrete terms, the “instances” of which the latter are composed must be theoretically modeled. Thus, if “statistical models” seem structurally unfit to accommodate a “generative model,” it is quite a different matter as far as simulation methods are concerned. The latter have been almost intrinsically created in order to implement them.
72 There are two main points in support of this idea. The first relates to the phase of “model translation” (Whicker and Sigelman 1991, 37), namely, the operation in which a computer program incorporating the theoretical model under investigation is written. Writing a set of algorithms that define how the variables are linked to each other is the same thing as positing a set of generative mechanisms. Unlike standard statistical techniques, algorithmic language is written explicitly in order to formalize the generative hypotheses of the data. The subsequent execution of the program—therein lies our second point—makes a veritable “animation” of the possible posited mechanisms. When the model is run, the computer is asked to scroll through all the instructions (algorithmic language) in turn, with the aim of observing the results of this dynamic. Using this procedure, we are therefore creating a virtual process resulting from the posited generative mechanisms as a whole.  Thus, as a number of authors have emphasized, “simulation” essentially equates to generating an information structure oneself from a set of theoretically significant rules which, in the eyes of the modeler, lie behind the phenomenon under analysis (Halpin 1999, 1500; Hanneman, Collins and Mordt 1995, 5). This precisely translates the idea at the essence of a “generative model,” namely, that of “generativity.”
73 The close link that exists between “mechanisms” and “simulation” is certainly quite general (Gilbert 1996a, 449; Gilbert 1999, 1485; Gilbert and Troitzsch 1999, 17).  In my view, however, a particular type of technique is even better suited to the implementation of a “generative model.”
74 This concerns the class of methods usually called “agent-based” simulation, the essence of which consists, on the one hand, of “networks of cellular automata” (derived from biology and physics, c.f. Hegselmann 1996; Weisbuch 1992) and, on the other hand, multiagent systems (derived from computer science, particularly the branch referred to as “distributed artificial intelligence,” c.f. Davidsson 2002). A considerable body of literature concerning this type of technique has rapidly built up over the last few years (c.f. among others, Doran 1998; Moss 1998; Sichman, Conte and Gilbert; Terna 1998). This undoubtedly represents the most recent “wave” in the history of simulation methods (Macy 2001).
75 Increasing attention is being paid to “multiagent systems” in particular as a tool for the modeling, formalization, and analysis of the dynamics underlying social phenomena (Axtell 2006; Gilbert 2006; Sistemi Intelligenti 2005). There is an increasing amount of interest being shown to them in geography (Sanders 2006), political science (Axelrod 1997; Cederman 2001; Johnson 1999), and economics (Bourgine and Nadal 2004; Phan 2004; Phan and Pajot 2005). A fact that is of more interest to us here is that a similar tendency is in evidence in sociology. The potential of “multiagent systems” with respect to sociological analysis is beginning to be recognized (Gilbert 1996b; Hedström 2005, chap. 6; Macy and Willer 2002; Moretti 2004, chap. 4; Sawyer 2003, 2004a, 2004b, 2005; Squazzoni and Boero 2005).
76 This fast diffusion seems to me to be largely due to the outstanding characteristics of the very infrastructure of this technique. A “multiagent system” effectively allows for the individual modeling of entities (the agents), their introduction into network structures, the study of their dynamic evolution, and thus the establishment of connections (possibly recursive) between the behavior of these entities, their interactions, and the system-level regularities that arise from these interactions.
77 On the “ascending” side (that which goes from “micro” to “macro”), this method allows for the treatment of the essential theoretical problem of the systems of interdependence that connect individual actions. Above all, one of the chief benefits of the technique lies in the ability to construct a veritable topology of connections between agents. Sociology, therefore, has at its disposal a concrete medium for the formalization and study of the effects of interactions between actors. “Complex aggregation mechanisms” can thus be represented. The “descending” side (that which goes from “macro” to “micro”), namely, the relationships between “structure” and “action” can also be modeled efficiently. Since a multiagent system enables us to place the agents within multiple network structures and to “store” the partial results of these interactions, progress can also be made in the understanding of a second vital theoretical problem, that of the means by which the behavior of an agent is constrained, both by the presence of another agents and by aggregates that are constantly being created by this “generalized interdependence.” “Situational mechanisms” are thus also represented within a multiagent system. In addition, since there is a possibility of representing even the cognitive components of agents in a stylized manner (as in “cognitive agent” modeling, c.f. Castelfranchi 1998), “action-formation mechanisms” might also have a place within a multiagent system.
78 My impression is, therefore, that “computational models”, agent-based models in particular, probably represent the best framework through which to formalize and analyze “generative models” from the point of view of both their content (mechanisms) and their form (complex methodological individualism). The simulation of this type of computational model allows for the numerous “macro-micro-macro” loops on which all generative models are based (c.f. §1, fig. 2). Sociologists therefore have access to a method that allows for the conception and study of theoretical models which, while falling far short of the complexity of reality, nonetheless respect its basic configuration. 
2.3 “Mathematical Models”
79 In order to complete the discussion of modalities for the implementation of a “generative model,” we should ask whether mathematical language might not also play an important role in the realization of this objective.
80 It is important to point out first of all that, as with the expression of “computational model,” that of “mathematical model” involves an implicit shift from the theoretical model itself to the language through which it is expressed (mathematical language). Moreover, in many fields of knowledge, the very concept of model is almost automatically identified with that of “mathematical model.” We often read that for mathematicians “formalization” is “mathematization” (Barbut 1994).
81 “Mathematical models” have a definite advantage over “statistical models.” The former cannot exist without the prior theoretical modeling of the phenomenon under investigation. The stylized representation of a given reality is always what is mathematized rather than reality as such (Barbut 1994, 8–10; 200, 206–207; Bunge 1973b, 131). This is why a number of authors have maintained that statistical techniques should be used primarily in order to work out the parameters of a “mathematical model” intended to represent precise ideas concerning the generative mechanisms of social phenomena. This is an old tradition inherited from Coleman (1964) and Sorensen (1976, 1998), the reverberations of which continue to be felt today (Backman and Edling 1999).
82 The relationship between mathematics and sociology are complex and historically varied (Barbut 2000; Edling 2002; Martin 2002).
83 “Mathematical sociology” as a specialized form of sociology is an undeniable historical reality (Scott 1997; The Journal of Mathematical Sociology 1984; Sociological Forum 1997; Sociological Theory 2000). It is built precisely on the necessity of “mathematical models” to theorize before being able to formalize. In spite of the variant that I have just mentioned (directed towards the empirical evaluation of a mathematical model), mathematical sociology is essentially a type of “theoretical sociology” (Collins 1988, appendix; Edling 2002, 202; Fararo 1997, 91; Hayes 1984, 325). Mathematical language is chosen here because of its precision, power, and parsimony in comparison with natural language. Mathematical language thus ensures the clarification of theory, and the systematic and logical study of all its consequences, thus enriching the theory's content and making its structure more articulated. Thus, a sociology that is centered on mathematical modeling has ambitions and objectives other than a sociology based on “statistical models” that involve description and empirical quantification (Fararo 2005a, b). 
84 However, in spite of the fact that “mathematical models” are better equipped to represent ideas concerning the mechanisms that generate social phenomena, we might ask whether they necessarily constitute the most effective solution when used to formalize and analyze really complex “generative models.” In my view, the answer is to be found in recent developments within mathematical sociology itself.
85 For several years now, there has been a growing recognition that it is in a state of crisis (Fararo 1997, 89–95; 2005b, 441). Its consolidation, and even survival, seems less certain today than it did in the past. The section of the American Sociological Association devoted to “mathematical sociology” only has 185 members (Edling 2002, 2) and, in addition, the members come from an assortment of backgrounds, not a single one being a qualified “mathematical sociologist” in the strict sense of the word (Edling 2002, 211 n.). It would seem that mathematical sociology has relinquished its ambition as a specific subfield of sociological analysis (Scott 1997). However, what seems to me to be particularly interesting is that, in order to remedy this impasse, a number of authors are starting to combine “mathematical sociology” with simulation methods (Fararo and Butts 1999). It is, moreover, precisely as part of mathematical sociology that the program of “computational sociology” came into being (Hummon and Fararo 1995). These authors justify the idea of turning to “computational models” examined using simulation methods on the basis of the following argument: when the process to be modeled is too complex, mathematical language no longer allows a model to be drawn up and, when this is still possible, it may well be that it cannot provide the tools to solve the model analytically (Fararo and Butts 1999, 35; c.f. also Collins 1998, appendix, fig. A.1).
86 Similar conclusions have been reached by the branch of heterodox economics that is dependent on “multiagent systems.” Since they come from a very mathematically-orientated discipline, these economists have asked themselves whether “multiagent systems” should be considered as a substitute (or as a complement) to the analytical treatment of classic mathematical models (c.f. among others, Axtell 2000; Epstein 2006, chap. 1, 2; Phan 2006).
87 The answer that has been put forward is that agent-based computational models only constitute a viable alternative to the mathematical treatment of a model in particularly complex situations in which mathematical formulation has proved to be incapable of solving the problem. This is particularly the case in situations in which it is a matter of representing numerous inextricably connected interactions of heterogeneous entities in terms of behaviors and preferences. However, as Epstein (2006, chap. 2) notes, even in these complex cases, “the issue is not whether equivalent equations exist, but which representation (equations or programs) is most illuminating.”
88 In this sense, it would be an advantage in terms of the “readability” of a model (particularly of some of its constituent parts—the interaction between agents, for example), which might lead to a preference for computational over mathematical formulation.
89 Thus, in view of the current state of research, we should not rule out a priori the possibility of the mathematical expression and, above all, the analytical treatment of a “generative model.” When possible, it appears to me that it would be useful to go through a phase of mathematical formalization owing to the precision, parsimony and elegance of its language. Next we would move on to computational language in order to treat particularly complex “generative models” containing numerous inextricably-woven systems of interdependence between agents. In this case, “multiagent” simulation proves to be of paramount importance for the treatment of these models and to restore their underlying dynamic dimension. The computational formalization and treatment of a “generative model” by simulation should, thus, probably not be considered as a substitute for its mathematical formalization. Far from representing two mutually exclusive stages, these two formalization operations are rather more representative of two complementary and successive phases of the same research procedure, enabling the alignment of a model with the empirical data to be explained.
90 This article is in unequivocal favor of the concept of “model” and suggests that it should become one of the cornerstones of knowledge production in sociology. Generally speaking, I have understood a “model” as both a partial representation of a given social phenomenon and a modality for the partial rediscovery of that phenomenon, thanks to the deductive treatment of its fundamental hypotheses (c.f. also Armatte 2005, 112–118). It is a question of using these “models” as “sophisticated analogies” that are indispensable for considering and mastering the complexity of reality (Edmonds 2005).
91 The polysemy and the numerous uses to which the concept of “model” appears to be condemned do not really constitute obstacles to such a proposal. Provided that we are aware of the assimilation effected by certain expressions between different operations (conception, formalization, analysis, etc.), it is less a matter of establishing the “correct” meaning than of integrating its different uses within a cohesive research framework. By way of conclusion, I am putting one forward by reorganizing the essential elements of the preceding analysis.
92 Firstly, I have suggested that theoretical modeling should be conceived in terms of the construction of “generative models.” These models formulate hypotheses on the mechanisms that are likely to produce the observed regularities and they aim to combine a set of mechanisms connecting several levels of analysis. In spite of its specific content, in a “generative model” social phenomena are assumed to derive from a dynamic chain of “macro-micro-macro” loops. Secondly, I have noted that this type of theoretical model is not easily “implemented.” Its formalization and the analysis of its results are not instantaneous tasks. In this respect, I have called attention to the difficulties that we might expect to encounter if we persist in carrying them out within the framework of traditional “statistical modeling.” “Statistical models” are, in fact, particularly ill equipped to take into account the “complex aggregation mechanisms” inherent within generative models, as well as to “animate” the process powerfully contained within them. Owing to its flexibility, “computational language” appears to be, on the other hand, a particularly appropriate modality of expression for a generative model and, consequently, some simulation methods, notably “multiagent systems,” appear to be a powerful tool in determining the results of these models.
93 It is important to note in this respect that the epistemology of “generativity” was almost spontaneously developed by authors, who, outside sociology, had long been interested in “multiagent systems” (Epstein 2006, chap. 1 and 2; c.f. also Cederman 2005).
94 And yet, the importance that we assign “computational models” and their “animation” by “multiagent” simulation should not conceal the usefulness believed to belong to more traditional “statistical models.” The fact that they are not well equipped to implement “generative models” does not mean ipso facto that they have no use at all. I will indicate these two uses here.
95 Firstly, in the absence of any concrete descriptions of empirical realities, a “generative model” remains without explananda. Without firm statistical analyses, sociologists are doomed to the construction of phenomena requiring explanations that are only relevant within their own view of the world. In addition, empirical research of mediocre quality (from the point of view of statistical techniques) risks providing the sociologist with pure fictions in the place of facts. “Statistical models” thus serve a vitally important descriptive function. Seen from this perspective, the role of “computational models” may be understood in the following manner: “Agent-based models use simulation to search for causal mechanisms that may underlie statistical associations” (Macy and Willer 2002, 162).
96 Secondly, we must first equip ourselves with empirical data that have been correctly drawn up from the statistical point of view, so that we are able to compare the simulated outcomes with data derived from observation. The comparison between simulated and empirical data can, in effect, constitute an essential part of the procedure to validate the results of a multiagent system and the relevance of the generative models that it attempts to “animate.” Compared to the understanding of “computational models” as pure “virtual laboratories” (Carley 1999), a subsequent alliance between descriptive statistical techniques and “generative models” with an explanatory purpose animated through simulation might, thus, satisfy, at least in part, the increasingly felt need for the better “validation” of “simulated models” (Boero and Squazzoni 2005; Moss and Edmonds 2005; Hedström 2005, chap. 6).
97 Lastly, if we consider the interest that we have identified in using mathematical formalization to clarify the analytical structure of a “generative model,” the reader will have all the necessary elements to understand the configuration of my vision of the “combined” use of the concept of the “model.” An illustration of this is found in figure 3.
Fig. 3. — Integration of Operations and Languages Implicit in a Sociology Centered on Modelling
Fig. 3. — Integration of Operations and Languages Implicit in a Sociology Centered on Modelling
98 A model-centered sociology would thus hinge on four principal theoretical stages.
99 First of all, the scholar should focus on an empirical “description.” This involves identifying systematic regularities for which the origins are unclear. Appropriate “statistical models” can be called upon here. Secondly, we should “model” the generative mechanisms that are likely to bring these regularities about. This is an intrinsically theoretical activity, involving methods such as abstraction, simplification, and stylization. The purpose of this phase is the construction of a “generative model.”
100 Thirdly, the sociologist should turn his attention to “formalization,” an activity that involves “building” the model in a specific language, different from natural language (mathematical language and/or computational language). Finally, scholars should concern themselves with the concrete tools necessary for studying the results of the generative model (in my view multiagent simulation) with the ultimate aim of coming closer to the empirical referent that the model claims to elucidate. This comparison could take place by means of an inductive statistical study of the data produced by the formal generative model in order to determine their structural proximity with the empirical data.
101 This combination of operations and languages must of course be taken in the sense of an ideal-type schema that I am opening to debate. I myself am only in the initial stages of its application (Manzo 2006b; 2007). I am perfectly well aware, moreover, that each of these stages could be realized more or less in its entirety depending on the particular object under study and the state of development of knowledge available in the relevant domain. In addition, depending on research conditions, their coexistence and connection will be more or less linear.
102 However, I would like to emphasize here that this type of integration of operations and languages involves combining several sociological perspectives which, at least with regard to contemporary research practices, seem to center exclusively on one of the stages outlined above. Notably, the type of sociology that treats “variables” has a tendency to stop at the descriptive phase, relying on increasingly refined “statistical models” (Esser 1996). “Analytical” sociology, on the other hand, currently specializes in the theoretical construction of models centered on mechanisms (Barbera 2004; Cherkaoui 2005; Hedström 2005; Hedström and Swedberg 1998a and b). Very rarely does it carry on to the successive phases of formalization and analysis (Hedstöm 2005, chap. 6). Next, in mathematical sociology, priority is given to the stage that concerns mathematical formalization, and the establishment of a relationship between the formalized model and the empirical data is very rarely sought (Edling 2002; Fararo 2005b). Finally, there is the same tendency in computational sociology, although computational language is substituted for mathematical language (Carley 1994; 1999; 2001; Fararo and Butts 1999; Hummon and Fararo 1995; Macy and Willer 2002).
103 By contrast, my suggestion for integration subscribes to the idea that has been concisely formulated by Halpin (1999, 1501, 1503), according to which an “interface between statistics, simulation, and sociological theory is critically important for the development of a sociology that is both theoretically sound and empirically founded, particularly when it comes to dealing with issues that are inherently complex.”
105 I would like to thank Marie Duru-Bellat, Jean-Jacques Paul, and Alexandre Steyer for their valuable comments on the first version of this article that I presented at the workshop on “the notion of model” organized in April, 2006, by the doctoral school “Gestion-Économie-Formation” of the University of Bourgogne. I would also like to express my gratitude to Jacques Lautman and Pierre Demeulenaere for the help that they have given in the final stages of preparation of this article.
I would like to point out here that I have already provided an example of the application of this type of research program (see Manzo 2006b; 2007).
It is already clear that a model can only be regarded as “theoretical” in the sense that it is always the product of our thinking on an empirical reference. Only the inadequacy and/or limited abstraction of this thinking could justify relatively ambiguous expressions such as “empirical models.”
As I have noted above, we are now in a better position to understand the reason why Bunge describes “theoretical models” with mechanisms as their object as “translucent boxes.” Since these models aim to explain the production of a relationship, they are neither restricted to ascertaining its existence (which would constitute a “black box”) nor to clarifying its original existence (which would constitute in the words of Bunge, a “grey box”).
Adaptive and counter-adaptive preferences are an example of a macro-micro mechanism that affects the desires of actors (Elster 1989; 2003; Hedström 2005, 60–61, 88–89). “Vacancy chains” are an example of a “macro-micro mechanism” affecting the opportunities that are available to actors (White 1970; Sorensen 1976; 1998). Things that are described either as an “endogenous social effect,” on the one hand, or a “structural effect” (Barbera 2004, 80–85, 85–91) on the other, constitute interesting examples of situational mechanisms that affect the beliefs (or both beliefs and opportunities) of the actors.
We should note, however, that it would be appropriate to include cognitive intrapsychic mechanisms, such as “cognitive dissonance” (Kuran 1998) among the “micro-micro mechanisms,” as well as mechanisms that are more strictly linked to the emotional component of actions (Elster 2003; Scheff 1992).
The hegemonic ambitions of certain large economic programs, such as the one put forward by Gary Becker (1993), together with the agreement concerning this model that has come out in sociology (Coleman 1990), undoubtedly explain why the rational choice theory is currently the most widespread notion of rationality, accepted almost unequivocally in large parts of the discipline (Abell 2001; Demeulenaere 1996; Revue Fran?aise de Sociologie 2003). Finally, we should note that, in spite of the new perspectives that it is in the process of opening up, Simon’s suggestion (1979; 1996, chap. 2), which puts forward the idea that the criterion “satisfying” should be substituted for the maximization principle remains within the bounds of this type of rationality.
Lindenberg (2000) engages in a stimulating discussion concerning Boudon’s model, in which he puts forward a different version of the cognitive notion of rationality. For all this, without its being incompatible with Boudon’s model, Lindberg’s suggestion introduces a notion (that of “framing”) which, in my view, gives rise to an element of difficulty. Cuin (2005) has also recently put forward an interesting argument concerning the cognitive notion of rationality, an argument to which Boudon (2005c) swiftly responded.
“Suicide rates” are a notable example of this type of variable. As Durkheim’s commentators (Cherkaoui 1998; c.f. also Udhen 2001, 179–185 for an overview) have shown, the “suicide rate” in itself is not a social fact. It is merely a statistic deriving from the simple aggregation of actions (the action of committing suicide) carried out individually. Numerous economic (such as savings, investments, or unemployment levels), or demographic (such as fertility levels, birthrate, etc.) variables usually considered to belong to the “macro” are also part of this category. The aggregate demand (or offer) is nothing more than another example of the simple path from the “micro” to the “macro,” a passage that is justified by means of a metaphor, that of the “representative agent” (Barbera 2004, 124–125).
In the case of the former, when yet another actor acts, he suffers the consequences of actions that have been carried out by previous agents through an accumulation of effects caused by these same previous actions. In the case of the latter, the actor is aware—albeit vaguely—of the effects of the actions of another and seeks to pre-empt them with the intention of choosing an action more suitable to this changing environment. If truth be told, these two forms of interdependence contain a third form, namely, “processual interdependence” (Esser 1996, 161). Here, the action of the actor who acts at given moment t is affected by the configuration of successive actions (and their effects) carried out previously by other actors. In certain respects, therefore, “processual interdependence” is established in all the processes that unfold over time.
The notion of “structure” is notorious for its diversity of meaning and ambiguity. I am keeping here to a definition of “social structure” as a “collection of contextual parameters,” thus limiting the margin of autonomy of an individual or, in accordance with the object under study, of a group or a network of groups (Boudon 2003, 160; Hedström 2005, 10).
This idea is, moreover, a familiar one in sociology (Weber 1903–1906, 69; 1913, 306, 309; 1917, 426–427; 1922, 32) as well as in the philosophy of science (Popper 1967, 145–146).
This idea has already also been clearly formulated by Weber (1922, 46–47).
This is also the case in economics, particularly in the branch currently known as the “economics of convention” (c.f. among others, Lazega and Favereau 2002) as well as in the school referred to as “neo-institutionalism” (for a critical review see: Udehn 2001, chap. 9).
This is, in fact, why Bunge refers to the “Coleman-Boudon diagram” (Bunge 1997, 454). We should note, in passing, that historically speaking, it was probably not Coleman who first created this diagram. Lindenberg (1997) describes it in words in an article written in German referred to above and never translated into English (Barbera 2004, 38, n. 14). In the light of this detail, it is understandable that Abell (1996) often speaks of the “Coleman-Lindenberg diagram,” without ever citing any of Lindenberg’s writings.
The “inseparability” hypothesis appears to be integral to the very conception of “complexity” proposed by Morin. According to Morin, this approach effectively refers back to an intellectual attitude which aims to keep together what is ordinarily considered as unconnected and separate and to unify opposites. As Berthelot (1996, 11, n. 1) has rightly pointed out, the criticism of the “analytical principle” (which is our source of inspiration) constitutes one of the essential themes running throughout Edgar Morin’s work. Morin's conception of society as a complex system is exposed in Morin (1984, in particular part 2); an overview can be found in Morin (1990). In this respect, Morin and Le Moigne (1999, chap. 4) are also a useful source.
The idea of the “complexity of generative mechanisms” differentiates me from some of the few authors who have questioned the use of the notion of “complexity” in sociology. For Havelange (1991), this notion relates to the understanding of the co-production of the “social” and the “individual.” However, this type of notion of complexity merely ties up with Giddens’s “theory of structuration.”
This is probably why Grusky and DiCarlo (2001) have attempted to defend their use by putting forward the idea of “descriptive theorizing,” which, I believe, is actually a contradiction in terms.
We should note that an explanation based on “mechanisms” also runs contrary to the deductive-nomological [DN] model (Hempel 1965a, b) of explanation, variable-centered explanations being a variant of it, according to which explanation implies making observed empirical regularities an instance of general relationships (c.f. on this point, Bunge 1997; Cherkaoui 2005, chap. 4; Hedström 2005, chap. 2).
The difficulty related to multivariate statistics is undoubtedly fueled by the dominance of protocols concerning the collection of data that consist in reconstructing individual characteristics outside of any information related to the numerous contexts of interdependence in which the actors are rooted (Hedström, 2005, 159). In this respect, the accumulation of strictly relational databases can only serve to open stimulating avenues of research. However, I believe that this might constitute a partial solution to the inability of multivariate statistics to implement a process involving mechanisms. This effectively necessitates a direct representation of the interdependence of agents as well as the analysis of what is created by its dynamic constitution over time. These relational data would therefore only serve to provide the scholar with another type of explanandum but they cannot provide the “complex aggregation mechanisms” that probably created them. Moreover, this idea seems to be reinforced by the conclusions reached in the work of Tom Snijders (1996; Snijders and Van Duijn 1997) concerning the statistical treatment of longitudinal data of social networks. According to Snijders, available statistical techniques, albeit useful, should effectively be coupled with “agent-oriented” simulation methods.
Beyond the radical nature of Sorensen’s assertion, I can see three principal reasons at its basis. In the first place, concerning its use in the techniques in question, time is related to the duration of an “episode” or its equivalent, in other words, the speed at which a given state comes to an end. Thus, individual trajectories have a dynamic character in the sense that they are understood as a sequence of “permanent/exit points” of a series of connected events. In this sense, the technique is unsuitable for the modeling of mechanisms and, above all, for the concrete animation of their action. Once again, it is only their effects that can conceivably be captured. We remain, therefore, on the surface of things. In second place, the only form of interdependence that can be described there is that which exists between the states of things experienced by individuals, for instance does the fact of getting married affect the probability of having a child? If this is the case, does it affect men and women in the same way, and so forth. Once again, we remain on the surface. Concrete interdependence structures between the individuals responsible for the existence of a given structuring of events are not modeled. Finally, even in the most sophisticated of multilevel versions, the “micro” and the “macro” are reduced to the co-presence of individual and aggregate variables. The contextual is not produced by specific mechanisms constructed by the scholar and there are no dynamic representations for the multiple feedback loops established between the individual and their context through the mediation of the diverse interdependence structures that connect actors one to another.
Special issues that important journals have devoted to this question have proliferated over the past fifteen years (c.f. for the most recent at the time this article was written: American Journal of Sociology 2005). A number of specialized journals have also come out (c.f. the electronic journal JASSS). Edited books, usually resulting from conferences, symposiums, working groups, and forums, have multiplied over the space of a few years (c.f. among others: Gilbert and Conte 1995; Hegselmann, Mueller, and Troitzsch 1996; Sichman, Conte, and Gilbert 1998; Troitzsch et al. 1996). The European Social Simulation Association has come into being (Moss et al. 2002) and complete manuals concerning simulation methods have finally appeared (Gilbert and Troitzsch 1999).
It should be noted in passing that this profusion of methods nonetheless involves a certain lack of unity and the absence—for the time being—of homogenous and standardized research procedures. This is why a number of authors have wasted no time in referring to the “art” of simulation (Axelrod 2005; Marney and Tarbert 2000; Whicker and Sigelman 1991, chap. 8).
In order to visualize this extremely important point, the reader should refer to figure 3 (c.f. conclusion). The text boxes numbered “4” and “5” clarify that the real “animation” of a “generative model” is set in motion only at the moment in which the computer program that translates the analytical structure is executed, that is, from the moment that the first instruction is read by the computer, once the algorithmic language originally formulated in a specialized computer language has been transformed into “machine language” (the only language that is truly understood by a computer) by means of a “compilation” operation. It is only from this moment that the simulation as such is activated and that the dynamic action of the mechanisms can unfold. Thus, a technical distinction between the “writing,” “compilation,” and “execution” of the program allows us to clarify a conceptual distinction that is vital for sociology. “Mechanism” and “process” cannot be considered as synonymous: the “process” constitutes the dynamic aspect of the “mechanism,” which the latter sets in motion. A clear description of this essential point is found in Simon (1996, 170).
It seems to me that this is proved by the recurrent use of terms such as “mechanism,” “process,” “underlying process,” “causal processes,” “underlying generative mechanisms,” and “underlying causal mechanisms” in numerous studies examining simulation (c.f. for example, Grémy 1977, 60, 77, 82; Hanneman 1995, 458; Hanneman, Collins, and Mordt 1995, 3, 4, 28, 29, 40; Hartmann 1996, 77, 83, 91, 98; Novak and Lewenstein 1996, 255, 277, 278, 279).
In this respect, the following passage appears to me to be of interest: “Artificial society simulations are dynamic models, because they are simplified representations of postulated real-world processes. They are mechanistic because they represent the causal structure of social mechanisms” (Sawyer 2004 a, 225).
This “dualism” is moreover identified with a certain specialization both in scientific journals and research networks. If the reader would like confirmation of this, it suffices to leaf through, firstly, The Journal of Mathematical Sociology and, secondly, Sociological Methodology, Sociological Methods and Research, or Quantity and Quality.