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Geometries of Desire: Simulating René Girard’s Mimetic Theory

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In this paper, I develop the first computational model of René Girard’s mimetic theory, an influential account of the social psychology of imitation. Girard argues that many forms of desire are socially learned and ought to be understood in terms of a triangular relationship between a desiring subject (S), a mediator (M), and a desired object (O). Mimetic theory is widely applicable to advertising, social influence, identity formation, character psychology, financial markets, and geopolitical soft power. I begin by translating Girard’s framework into the formalism of directed graphs—what I call “mimetic networks”—by representing mediation and desire relationships between agents as directed out-links. I use simulation to explore network dynamics, demonstrating the conditions under which the model arrives at stability—what I term a “Nash Equilibrium of Desire.” I also explore the effect of cascades, narcissists, and self-mediating agents and show that mimetic networks have globally emergent properties—notably, the tendency for all agents to converge on a single, universal object of desire (“Convergence of Desire”). In the appendix, I develop a set of mathematical definitions and theorems regarding mimetic networks. This paper is aimed at an intentionally multi-disciplinary audience, including social scientists, complex systems and network theorists, philosophers, narratologists, and literary and cultural critics.

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  1. 1.

    This conceptual correspondence invites empirical testing. Future studies could measure the degree to which “follows” and “likes” on Instagram and similar platforms obey the rules of triangular desire.

  2. 2.

    The recent rise of meme stocks, such as last year’s GameStop mania, is an especially salient example of the role of imitation and the replication of desire in asset bubble formation.

  3. 3.

    Gender theorist Eve Kosofsky Segdewick uses Girard’s mimetic theory to motivate her concept of homosociality, defined as “as a form of male bonding with a characteristic triangular structure. In this triangle, men have intense but nonsexual bonds with other men, and women serve as the conduits through which those bonds are expressed” [14].

  4. 4.

    The simulations used in this paper were programmed in NetLogo, an interactive development environment for agent-based modeling.

  5. 5.

    The initial state at time t = 1 is unstable because A desires B but A’s mediator, C, desires A. Since A cannot desire itself, A switches its object to C. The time-step at t = 2 is also unstable: C desires A but C’s mediator, B, desires C. Since C cannot desire itself, so C switches his object to B, etc.

  6. 6.

    Just as there are only eight possible configurations of mediation, there only are eight possible configurations of desire. Within a short time, the simulation has tried all of them and failed to achieve equilibrium. By t = 15, it has returned to its initial state at t = 1, and then repeats.

  7. 7.

    Throughout this article, I explore the special case of mimetic social networks, in which any node-agent can be a desiring subject, mediator, and/or desired object. In this context, the concept of narcissism (an agent desiring itself) is intelligible. An alternative version of the model would differentiate between nodes representing persons, who can be subjects and/or mediators, and nodes representing things, which can only be objects of desire. Such a distinction would obviate narcissism. I choose not to make this differentiation. Social influence networks such as Instagram offer an interesting real-world example of this elision: products such as Nike sneakers or Gillette aftershave can have profiles in the same way that persons do and may like, follow, or be followed by other profiles. In this sense, both persons and things can be subjects, mediators, and/or objects.

  8. 8.

    For example, initially at t = 1 agent C desires agent A, who is his protégé. However, C’s mediator is B, who in turn desires C (his respective protégé). Imitating his mediator’s desire, C therefore changes his object of desire to himself at t = 2, thereby becoming a narcissist.

  9. 9.

    To ground this scenario in a concrete example, consider the dynamics of desire and mediation in Oscar Wilde’s The Picture of Dorian Gray—perhaps the preeminent European novel on the topic of narcissism. At the outset of the story, Dorian is a fresh-faced and impressionable ingenue mentored by an artist, Basil Hallward, and a hedonistic aristocrat, Lord Henry, both of whom desire Dorian for his extraordinary beauty. Dorian internalizes the affection and praise of his mentors and becomes a narcissist obsessed with the preservation of his youth and utterly indifferent to the needs and desires of others. The novel climaxes with Dorian murdering his mentor, Basil, shortly after he confesses to loving Dorian. With the extinguishing of the mediator that supported and enabled his self-desire, Dorian’s narcissism quite literally implodes: he destroys the portrait that symbolizes his self-obsession and abruptly, mysteriously dies.

  10. 10.

    These results are strictly true for the “triadic” case of three agents. In larger networks—quadrangular, pentagonal, etc.—non-narcissistic Nash equilibria are possible provided the mediator relationships are separable into disjoint sets. In this case, we may also have multiple objects of desire. This result is discussed further below.

  11. 11.

    Although Girard’s discussion unfolds in the context of literature, I encourage readers to think of agent social psychology and behavior generally.

  12. 12.

    I use the term “lover” playfully and provocatively here. Requited desire need not be romantic and is equally relevant to diverse contexts such as mutual friendship, bilateral trade and military alliances, or job search matching between potential employers and candidates.

  13. 13.

    It is also worth noting that one of the members of the requited pair must be a self-mediator, who is also the object of desire of the unrequited outsider. Self-mediators, that is, tend to become magnets for mimetic desire.

  14. 14.

    For an archetypal example of this type of psychology, consider Ivan Turgenev’s description of Anna Sergeevna in Fathers and Sons: “Personally [Anna Sergeevna] wanted nothing, although it seemed to her that she wanted everything. She had hardly been able to stand the late-lamented Odintsov… and had acquired a secret aversion to all men, whom she regarded as nothing more than untidy, ponderous and flabby, feebly importunate creatures. Once, when she’d been somewhere abroad, she’d met a handsome young Swede… He had produced a strong impression on her but that had not stopped her from returning to Russia” (Turgenev, 89).

  15. 15.

    Scenarios involving two male rivals and two female rivals competing for what they regard as the superior member of the opposite sex are familiar from comic and dramatic literature, such as Shakespeare’s A Midsummer Night’s Dream, Marivaux’s La Dispute, and Goethe’s Elective Affinities [7]. As Goethe writes: “But in that respect,” said Eduard, “chemists are much more gallant. They add a fourth party, so that nobody goes without.” “Indeed,” said the Captain, “and those cases are the most significant and the most remarkable in which the attraction and the affinity, the desertion and the uniting, can be seen, so to speak, crosswise: when four substances, united until that moment two by two, are brought into contact, desert their previous union, and unit afresh. In this letting go and seizing hold, this fleeing one thing and seeking another, one is really inclined to discern some higher prescription; one ascribes to such substances a sort of volition and power to choose and the technical term “elective affinities” seems perfectly justified” (Goethe, 89).

  16. 16.

    Romantic partners provide a limiting case of scarcity insofar as there is only of the desired object. To borrow the terminology of economics, they are both “rival” and “excludable.”.

  17. 17.

    Consider the relative infallibility of the Pope as a moral role model for devote Catholics; Steve Jobs as a professional role model for young tech entrepreneurs; Warren Buffett for aspiring value investors; Elvis for his impersonators; and so on. In each case, the psychological, social, or symbolic distance between subject and mediator contributes to a perception of infallibility, which requires disjunctions in desire to be resolved in the mediator’s favor.

  18. 18.

    As a playful but illustrative example, consider Oliver Stone’s iconic depiction of ambition and greed, Wall Street. The action of the plot is, effectively, a drama of shifting mediators. Bud Fox’s original mediator is his father, a union leader in a rust belt town whose desire is uplift the community. Bud, however, desires to become rich. Bud locates a new mediator, Gordon Gekko, who mentors him on how to achieve his desires through criminality. In the film’s climax, Bud changes his mentor back to his father, renounces his desire to becomes wealthy, and topples Gekko.

  19. 19.

    Goffman [10].

  20. 20.

    Simmel [15].

  21. 21.

    White [17].

  22. 22.

    Bourdieu [2].

  23. 23.

    Todorov’s concept of disequilibrium is adapted from Aristotle’s: “Tragedy is an imitation of an action that is complete, and whole… A whole is that which has a beginning, a middle, and an end. A beginning is that which does not itself follow anything by causal necessity, but after which something naturally is or comes to be. An end, on the contrary, is that which itself naturally follows some other thing, either by necessity, or as a rule, but has nothing following it. A middle is that which follows something as some other thing follows it. A well-constructed plot, therefore, must neither begin nor end at haphazard, but conform to these principles.” (Aristotle, Poetics, 3).

  24. 24.

    As Forster famously wrote in Aspects of the Novel: “Let us define a plot. We have defined a story as a narrative of events arranged in their time-sequence. A plot is also a narrative of events, the emphasis falling on causality. “The king died and then the queen died,” is a story. “The king died, and then the queen died of grief” is a plot. The time-sequence is preserved, but the sense of causality overshadows it” (Forster, 86).

  25. 25.

    As far back as the Greeks, one finds a particular version of the cascade in the form of the law of unintended consequences. Oedipus, for example, initiates an investigation that inadvertently sets off a chain reaction of events. However, in the classical model, causality is still located in individual choice. The version of the cascade present in the model of triangular desire, though, is rather different: it describes the ripple effects in an out-of-control social process—akin to the typhus epidemic in Dickens’ Bleak House that rips through the ensemble, from Jo to Charley to Esther, etc. This version of the cascade has been most extensively developed in the context of complex systems theory, where it features prominently in the models of self-organized criticality and phase transitions. The quintessential examples of the cascade are the avalanche and the epidemic—natural and social processes that that sweep through a system, radically altering their structure and in which causality cannot be localized in a single entity, but is rather a characteristic of an entire set of relations between entities. In the case of the Girardian mechanism, we are confronted by a cascade of desires. The closest analogues are models of fads, informational cascades, and belief diffusion.

  26. 26.

    This double plot closely resembles the logic of the classical Bildungsroman, such as Wilhelm Meister’s Apprenticeship (1796), which narrates the trajectory of an individual as they find and accept a particular role in society.

  27. 27.

    Structural Balance Theory (also known as ‘Social Balance Theory’ or SBT) was originated in the mid-1940s by Fritz Heider, who studied patterns of belief coherence in psychology. In the mid-1950s, Cartwright and Harary generalized Heider’s theory of coherence and applied it to social relations, representing stable and unstable configurations with basic graph theory on signed networks. The rules of Structural Balance Theory are simple and intuitive. Imagine a set of nodes representing people, nation states, or other agents. Each pair of nodes is joined by a friendship (+) or an enmity (−) tie. The unit of analysis is the triad, which can have one of four states: (+)(+)(+), indicating 3 mutual friends; (−)(−)(−), indicating 3 mutual enemies; (+)(+)(−), indicating one node is friends with two who are enemies; (+)(−)(−), indicating two nodes are allied against a mutual enemy. A triad is stable if there is no social or psychological pressure for any node to change the valence its relationships. This can be conveniently summarized by the requirement that the product of the signs be positive, ergo (+)(+)(+) or (+)(−)(−). This encapsulates several social-psychological heuristics: (1) my friend’s friend is my friend; (2) my friend’s enemy is my enemy; (3) my enemy’s friend is my enemy; (4) my enemy’s enemy is my friend. Changing any one link in an unstable triad will make it stable, while changing any one link in a stable triad will make it unstable, leading to cascade effects when triads are embedded in large signed networks.

  28. 28.

    Heider [11].

  29. 29.

    Cartwright and Harary [3].

  30. 30.

    For treatment of a dynamic network model based on SBT, see “Character Networks for Narrative Generation: Structural Balance Theory and the Emergence of Proto-Narratives” [Sack 12]. For an alternative model of plot generation, see “Simulating Plot: Towards a Generative Model of Narrative Structure” [Sack 13].

  31. 31.

    In The Strength of Weak Ties [8], Granovetter distinguished between strong social ties, which include deep relationships with frequent contact, such as family and close friendships, and weak ties, which include acquaintanceships with infrequent contact. Granovetter proposed the Strong Triadic Closure Property (STCP): if A is strongly connected to B and C, then B and C must be at least weakly connected. STCP is a version of ‘transitivity’ and captures the intuition that if I am friends with two individuals, they are likely to become acquainted with each other and may eventually become friends.


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Appendix: The Mathematics of Desire

Appendix: The Mathematics of Desire

Over the preceding pages, I developed a number of results about the dynamics governing triangular desire on a network. These results are formalized below into a set of definitions and theorems about the behavior of the model. There is not space in here for full proofs, but I will offer a sketch of the main arguments.

Definition 1—Mimetic Network: A set of nodes N = {n1, n2, …, nk}, a set of directed object links O = { (ni, nj), …}, and a set of directed mediation links M = { (nk, nl), …} such that each node has exactly one out-directed object link and exactly one out-directed mediation link.

Definition 2—Equilibrium: A mimetic network is in equilibrium if, for every node ni contained in N, the object of ni and the object of ni’s mediator are identical.

Definition 3—Mutual Mediator Set: A subset N’ of nodes in the mimetic network N such that the mediator of each node in N’ is also a member of N’. That is, the nodes in N’ only have mediation relations amongst each other, not with any nodes outside of N’.

Definition 4—Minimum Mutual Mediator Set: A mutual mediator set that cannot be partitioned into any non-empty disjoint mutual mediator sets. That is, a mutual mediator set is “minimum” if we cannot subdivide it into smaller mutual mediator sets.

Theorem 1—Convergence of Desire: In equilibrium, all nodes in a minimum mutual mediator set must have the same object of desire.

Sketch of proof: By definition, in equilibrium, each node must have the same object as its mediator. That is, the nodes at either end of a directed-mediation link must have the same object, regardless of the direction of the link. In a minimum mutual mediation set, a path along mediation-links must exist from any node in the set to any other: otherwise, the set would contain disjoint subsets and therefore it would not be minimum. By the transitive property (O1 = O2 = O3, etc.), all nodes must therefore have the same object of desire.

Theorem 2—Non-Identity of Desire: If object self-links are not permitted, all nodes in a minimum mutual mediator must desire an object that is not a member of that mutual mediator set. (If object self-links are permitted, this does not hold).

Sketch of proof: Proceed by contradiction. Assume that a minimum mutual mediator set is in equilibrium and also that the object of desire is a member of the set. By the “Convergence of Desire” theorem, all nodes must have the same object of desire. Therefore, there must exist a node within the set that is its own object of desire. If object self-links are not permitted, however, then we have a contradiction and therefore one of the initial assumption must be false: either the mutual mediator set is not actually in equilibrium or the nodes must desire an object that is not a member of the set.

Theorem 3—Existence of Equilibrium without Narcissism: If object self-links are not permitted, an equilibrium outcome will exist if and only if N can be partitioned into at least two disjoint mutual mediator sets.

Sketch of proof: Proceed by contradiction. Assume N is in equilibrium and cannot be partitioned. Therefore the network consists of just one minimum mutual mediator set. Then by the “Non-Identity of Desire” theorem, the members of the set must desire an object that is outside of the set. But the set is equal to the network. Therefore the nodes must desire an object outside of the network. But this is not possible, so we have a contradiction. Therefore, N must be partitionable.

Theorem 4—Existence of Equilibrium with Narcissism: If object self-links are permitted, an equilibrium solution always exists.

Sketch of proof: Consider the case that all nodes in the network desire the same object. Then, by construction, every node must have the same object as its mediator, therefore the network is in equilibrium.

Theorem 5—Number of Objects of Desire in Equilibrium: Let A be the number of minimum mutual mediator sets into which N can be partitioned. Let B be the number of distinct objects of desire in equilibrium. Then B ≤ A. That is, the number of minimum mutual mediator sets into which we can partition N sets an upper bound on the number of distinct objects of desire there can be.

Sketch of proof: Let A be the number of minimum mediator sets into which a network can be partitioned. By the “Convergence of Desire” theorem, all member of a mutual mediator set must have the same object of desire. Therefore, there can be at most A distinct objects of desire.

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Sack, G.A. (2021). Geometries of Desire: Simulating René Girard’s Mimetic Theory. In: Yang, Z., von Briesen, E. (eds) Proceedings of the 2020 Conference of The Computational Social Science Society of the Americas. Springer Proceedings in Complexity. Springer, Cham.

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