Defining and classifying models of groups: The social ontology of higher-order networks

This chapter presents a series of group-based models, each built on progressively stronger ontological commitments about the nature of social groups. As a standalone paper, it includes a condensed version of the history of groups from the introduction. Readers already familiar with that material may wish to skip the section ยง1.2{reference-type="ref" reference="interface.section.review"}.

We show how the dimensions of coupling, persistence, reducibility, and alignment can be expressed through network modeling assumptions. While higher-order networks (HONs) emphasize multi-way interactions, they often rely on the annealed assumption--where group configurations shift faster than the dynamics--capturing mostly ephemeral, isolated groups. We offer one direction in which group-based models can be broadened to encompass a diversity of ontological commitments. By assuming a fixed or slowly changing (quenched) topology, we can model persistent groups and capture coupling between them. Building on this, we introduce stronger assumptions about group behavior, such as the presence of group-level states that are irreducible to individual states. These allow us to represent institutions and cultural norms as active influences in their own right. Finally, we introduce misalignment to capture cases where group-level outcomes diverge from individual preferences. Together, these modeling choices offer a conceptually grounded typology for HONs that connects network science with broader theories of group behavior.

The models here set the stage for the following chapters. Chapter [chapter:coevo]{reference-type="ref" reference="chapter:coevo"} presents an irreducible, coupled model of co-evolving individuals and institutions. Chapter [chapter:groupSkills]{reference-type="ref" reference="chapter:groupSkills"} examines the emergence of new skills in persistent but isolated research groups. This work builds on prior research [@hebert-dufresne_propagation_2010; @hebert-dufresne_source-sink_2022].

Abstract

In complex systems, the study of higher-order interactions has exploded in recent years. Researchers have formalized various types of group interactions, such as public goods games, biological contagion, and information broadcasting, showing how higher-order networks can capture these interactions more directly than pairwise models. However, equating hyperedges--edges involving more than two agents--with groups can be misleading, as it obscures the polysemous nature of the term "group interactions". For instance, many models of higher-order interactions focus on the internal state of the hyperedge, specifying dynamical rules at the group level. In doing so, these models often neglect how interactions with external groups can influence behaviors and dynamics within the group itself. Yet, anthropologists and philosophers remind us that external norms governing intergroup behavior, in the form of intergroup competition or cooperation, are essential to defining within-group dynamics. In this paper, we synthesize concepts from social ontology relevant to the emerging physics of higher-order networks. We propose a typology for classifying models of group interactions based on two key perspectives. The first focuses on individuals within groups engaging in collective action, where shared agency serves as the binding force. The second adopts a group-first approach, emphasizing institutional facts that extend beyond the specific individuals involved. Building on these perspectives, we introduce four dimensions to classify models of group interactions: persistence, coupling, reducibility, and alignment. For the physics of higher-order networks, we provide a hierarchy of nested mathematical models to explore the complex properties of social groups. We also highlight social interactions not yet explored in the literature on higher-order networks and propose future research avenues to foster collaboration between social ontology and the physics of complex systems.

Introduction

Broadly defined, the mathematical modeling of complex systems, and in particular network science, is the multidisciplinary study of how interactions in a system shape the emergent properties and functions of the system [@newman_networks_2010]. In recent years, there has been a large wave of research on higher-order networks where group interactions are modeled more directly as hyperedges which can be different entities based on their order (size). With hyperedges, group interactions at a higher order, such as adolescent cliques, committed minorities, information broadcasting, or contagion within households, can be shown to be different from the pairwise interactions at a lower order [@battiston_physics_2021; @ferraz_de_arruda_contagion_2024]. By modeling group interactions at higher orders, we can analyze them as more than mere correlated bundles of pairwise interactions. But human group interactions extend beyond individuals interacting in non-trivial ways. Group norms and institutions shape group behaviors just as groups influence individual dynamics, yet they remain irreducible to multi-way interactions. For instance, mask mandates and other shielding measures significantly alter contagion dynamics in workplaces, extending beyond the effects of higher-order interactions alone [@st-onge_paradoxes_2024]. These public health policies are the end result of cumulative cultural learning, shaped by thousands of years of socio-technical innovations and intergroup competition [@boyd_culture_1988; @tomasello_why_2009; @henrich_weirdest_2020]. Similarly, while an increase in the number of coauthors in papers may hint at the rise of teams in science [@wuchty_increasing_2007; @uzzi_scientific_2012], modeling the norms and practices within teams directly offers a different perspective on the role of collaboration in shaping scientific productivity.

To model networks beyond higher-order interactions, we propose a typology of group interactions that is informed by the long-standing history of group minds in the social sciences and, more broadly, the social ontology of groups. We establish two key connections. The first highlights the impact of group interactions on individuals, particularly as influenced by varying degrees of group persistence and coupling. Just as the jury's paradox and other anti-aggregation arguments show that group attitudes can contradict individual preferences, higher-order interactions display nonlinear behavior as they cannot be reduced to a simple additive function of independent individual influences.

The second connection focuses on (strong) group non-reducibility, distinguishing it from groups that are weakly emergent due to the absence of a simple aggregation scheme. We define strongly non-reducible groups as norms and institutions that do not directly follow from the states of members or group-level features. By adopting a group-level perspective that emphasizes the role of institutional facts, we argue that we can better investigate the coevolution of individuals and groups--an aspect that is difficult to capture by focusing on individuals-based dynamics. We can then show how possible misalignment between individuals and institutions can emerge by reintroducing individual preferences within the context of group dynamics based on group-level fitness. This approach enables us to take a step toward modeling group epistemology, cognition, and rationality within higher-order network models, as is necessary for modeling norms and institutions governing group interactions.

Taken together, we establish a typology of group models with the following dimensions: persistence, coupling, reducibility, and alignment. This typology helps address ongoing disagreements about the underlying ontology of models of groups. Consider this; in reviews of higher-order dynamics, complex contagion models---where dynamics depend on a nonlinear function of all network neighbors of a given node--are often ignored [@centola_complex_2007]. Why is that? Likely because groups are central to the definition, particularly in terms of group persistence. Why, then, are complex contagion dynamics based on social reinforcement not considered "higher-order", even when they occur over groups [@watts_influentials_2007; @osullivan_mathematical_2015]? The key distinction for a system to be considered a higher-order network model appears to be whether the dynamical rules are nonlinear and specified at the group level, such that a group interaction cannot be described as a collection of pairwise interactions. For example, in our typology, we distinguish between models assuming isolation and those depending on the states of agents outside the group, or their coupling. If coupled, the dynamics within a hyperedge are not fully determined by the state of that hyperedge, and such complex dynamics might not fall under the purview of the current higher-order network literature [@arruda_contagion_2024]. Interestingly, this distinction is more rooted in mathematical convenience than empirical evidence. In many evolutionary models of groups, the role of other groups is essential to understanding focal groups [@boyd_culture_1988; @smaldino_evolution_2018].

The rest of the paper proceeds as follows. We first provide a brief history of group minds, highlighting the debate surrounding group realism in the sciences (sec. 1.2{reference-type="ref" reference="interface.section.review"}). By examining how beliefs about the existence of groups have fundamentally shaped social science methodologies, we can better identify blind spots in contemporary models of group interactions. Next, we develop a hierarchy of nested mathematical models to explore increasingly complex and detailed properties of social groups (sec. 1.3{reference-type="ref" reference="typology.intro"}). We begin by presenting models of group persistence and coupling mapped to group interactions derived from aggregation-based arguments. We then introduce models of group irreducibility, incorporating the concept of individual-group misalignment. In its most detailed form, our framework allows us to mathematically consider important questions such as collective beliefs, shared intentions, and emerging institutions.

A short history of group minds

Understanding how group behavior emerges from individuals--and when it cannot be reduced to them--has long challenged both social scientists and modelers. We briefly review key positions in the debate over group minds, with a focus on ideas relevant to modeling group interactions in social systems.

The early debate

A longstanding question in the social sciences concerns whether groups are real entities with their own dynamics or just convenient labels for aggregates of individuals. This debate is best understood as one between group realism and methodological individualism (MI), shaped by early theoretical developments in sociology and economics [@schumpeter_methodological_1907; @weber_categories_1913]. Group realists like Durkheim emphasized the emergent properties of society, arguing that social norms and institutions regulate behavior and exist independently of individual intentions. While this may seem like a strong ontological commitment in retrospect, consider how professions or legal codes can hardly be reduced to any single individual [@durkheim_rules_1895 p.51]. And yet, paradoxically, they are enacted by individuals all the same. MI, as championed by Weber and Schumpeter, argued that only individuals have desires and beliefs; thus, social phenomena must ultimately be explained in terms of individual motives and actions. Schumpeter coined the term "methodological individualism" to clarify that treating groups as agents is not merely misleading--it's a category error [@schumpeter_concept_1909]. Yet, the issue persisted in different forms throughout the twentieth century [@popper_open_1945; @homans_bringing_1964; @arrow_methodological_1994][^1].

Social network analysis

Emerging in the mid-20th century, social network analysis (SNA) shifted focus to patterns of interaction among individuals, using sociograms to map relationships. Inspired by early sociologists such as Georg Simmel, SNA pioneers developed bipartite graphs with two types of nodes--individuals and the groups or events to which they belonged--to capture affiliation patterns [@simmel_conflict_1908; @feld_focused_1981; @mcpherson_hypernetwork_1982; @breiger_duality_1974]. This duality enabled researchers to study overlapping memberships and community structure indirectly, through individual-level ties. However, despite tracking group affiliations, SNA largely retained MI's emphasis on individuals as the primary units of analysis [@wellman_structural_1988; @neal_duality_2023]. Groups appeared only as aggregations of pairwise ties or node attributes, without independent dynamics of their own.

As a result, bipartite models struggled to represent genuine group-level phenomena, such as norms, institutional memory, or collective decision-making processes that exhibit inherently nonlinear dynamics. The assumption was that all group effects could be derived linearly from individual interactions, missing essential feedback loops or emergent group states. In contrast, more recent higher-order network models, based on hypergraphs or simplicial complexes, allow direct modeling of group interactions as irreducible entities with nonlinear dynamics, providing tools that are better aligned with group realist perspectives.

Groups are real

In response to MI's growing influence, a number of scholars mounted a defense of groups as legitimate units of analysis [@campbell_common_1956; @warriner_groups_1956; @horowitz_concept_1953; @wynne-edwards_animal_1962]. In Groups Are Real: A Reaffirmation, sociologist C.K. Warriner challenges methodological individualism by framing the debate as one between nominalists--who view groups as mere aggregations of individuals (with all reality vested in the individual)--and realists, who regard groups as ontologically distinct entities. He also critiques interactionists, who try to balance individual and group perspectives but tend to default to individual-level explanations, since groups lack the clear boundaries and subjective experiences of individuals. But just as individuals are not reducible to their biology, he insists that groups are not reducible to the individuals that compose them. The question is how.

At this point, scholars sought to ground group realism in observable mechanisms or formal models. Donald Campbell (1956) argued that groups are real, but the organism-like analogy is misleading [@campbell_common_1956]. To make his argument, he drew from the common fate principle--the gestalt idea that elements moving together in unison share a degree of reality. This principle helps explain why groups are perceived as real even if they lack physical unity, like a body. Think of a flock of birds or a marching band; no physical boundary binds them, but their coordinated movement signals a degree of we-ness (a property Campbell liked to call entitativity). Meanwhile, Herbert Simon (1964) reluctantly made the argument that organizational goals cannot be simply reduced to individual goals [@simon_concept_1964]. Instead, organizations operate through structured roles that shape behavior at every level. Simon argued that organizational goals are emergent; they do not simply reflect the preferences of CEOs, boards, or stakeholders but instead arise from the constraints imposed by institutional roles at every level of the organization [@simon_concept_1964]. He contended that explaining organizational behaviors in terms of individual motives is fraught with problems since organizational goals do not simply match those of the CEOs, the board, or the stakeholders; large organizations are modified in practice by employees at all levels of the organization. Simon's work bridges individualist and group-level modeling, showing how organizational goals arise from constraints on individual roles---not from the aggregation of preferences alone.

Following up on Simon's work, organizational science further embraced the idea that small working groups were the building blocks of organizations, rather than individuals. Building on this insight, organizational theorists shifted from abstract ontological debates to empirical studies of how groups function in practice. This view is discussed at length in H. J. Leavitt's paper Suppose we took groups seriously... (1976), laying the foundation for a systems-level approach to small group performance, known today as team science [@leavitt_suppose_1974; @hackman_design_1987; @katzenbach_wisdom_1992; @mathieu_evolution_2018; @wuchty_increasing_2007; @goodwin_science_2018; @hall_science_2018]. Like Schumpeter before him, Leavitt steered away from metaphysical debates. Instead, he championed empirical research on how communication structures shape group outcomes---drawing on the experimental work of Bavelas and others [@bavelas_communication_1950; @leavitt_effects_1951].

The evolutionary dynamics of group-level features

Anthropologists and cultural evolutionists have emphasized the coevolution of human psychology and group dynamics to explain our unique capacity for large-scale cooperation [@boyd_cultural_1982; @richerson_cultural_2016]. Human cognition is uniquely attuned to social learning---especially through conformity, prestige bias, and intragroup imitation---while deviant behavior is regulated through norms and reputational sanctions. These mechanisms scaffold both individual and group-level cultural knowledge. A striking feature of human psychology is our tendency to over-imitate: children across cultures faithfully reproduce unnecessary steps in tasks. From a modeling perspective, these learning biases promote greater similarity within groups (low within-group variance) while preserving variation between them.

Such dynamics lend themselves to formal modeling via multilevel selection, or cultural group selection. When traits benefit the group more than the individual, group-level selection can dominate---even if the trait is individually costly. As Darwin himself proposed, traits like patriotism or self-sacrifice---especially in the context of intergroup warfare---may evolve because groups that promote them outcompete more selfish ones [@darwin_descent_1871; @bowles_did_2009]. Recent studies show that group extinctions due to violent intergroup competition occur on timescales of 500-1,000 years [@soltis_can_1995]. Crucially, this logic applies not only to tribes or ethnic groups but also to voluntary organizations such as churches, universities, and firms---where selection acts more rapidly and competition is often nonviolent [@henrich_weirdest_2020].

In parallel, institutional theorists such as Nelson and Winter extended evolutionary models to firms, reframing Schumpeter's notion of "creative destruction" as a process of organizational selection. Firms compete through varying strategies, technologies, and routines---some of which prove more adaptive than others [@nelson_schumpeterian_1982]. Unlike neoclassical economics, this approach emphasizes bounded rationality: agents operate with limited information and adopt heuristics, leading to persistent diversity rather than convergence to an optimum. Concepts like path dependence illustrate how early choices---such as the QWERTY keyboard---can lock in suboptimal solutions due to institutional inertia. Once embedded, organizational routines become difficult to reverse. Douglass North extended this perspective by describing institutions as emergent systems that persist beyond individual lifespans through cultural transmission: "learning embodied in individuals, groups, and societies" that is "cumulative through time" [@north_rise_1973]. For modelers, these ideas support representing institutions as evolving entities in their own right, with group-level traits such as norms, routines, and decision structures that are not reducible to individual behavior.

Cultural evolution theory and new institutional theory developed research programs around group-level traits that can evolve, stabilize, and compete in ways that go beyond individual-level dynamics. With models, they sought to avoid the earlier "Panglossian adaptationism"--the assumption that all observed traits are somewhat optimally selected. For modelers, the group-based approach raises two central challenges. First, group dynamics often cannot be captured solely through individual traits or pairwise interactions---even under assumptions of weak emergence. Second, group-level traits and individual incentives frequently diverge, especially in the context of intergroup competition or collective decision-making. This misalignment requires explicit modeling of cross-level feedback. Understanding both irreducibility and misalignment is key to modeling collective behavior in complex social systems.

Typology of groups and group-based modeling

The literature on group interactions is fragmented across communities, with ongoing disagreements about the underlying ontology of groups. We define a typology of models of groups that integrates mathematical assumptions with the long-standing history of group interactions.

We use this typology to classify broad families of models based on the kinds of higher-order interactions they can represent. While this review focuses on models rather than the empirical aspects of human groups, we highlight connections to the empirical literature where possible.

Group persistence and intergroup coupling

When discussing the existence of a group, persistence and coupling consistently emerge as key features. Early on, Georg Simmel and others studied how group size and composition influence stability. They compared smaller, tightly knit groups based on intimate connections (primary groups) with larger groups that promote formal organizations, impersonal interactions, and specialized roles (secondary groups), such as bureaucracies [@cooley_social_1909]. Broadly speaking, the idea of group persistence is intertwined with that of interactions between groups, as (ethnic) groups are believed to engage in cooperation or competition. Strongly competing groups are more tightly bound, as they may restrict interactions to maintain their advantage. In contrast, cooperative groups are more open and can facilitate the mixing of people and ideas, leading to a reduction in group differences. Although permanence and isolation are key features, their meaning varies depending on the social groups in question. Both work teams and ethnolinguistic tribes are, to some extent, permanent or relatively isolated, but somewhat in different ways.

From a modeling perspective, we distinguish groups formed through momentary interactions and those persisting in time. We define momentary group interaction as a set of individuals engaged in some shared activity within a time window that is short compared to the characteristic timescale of the dynamics. In the physics of HONs, these interactions can be modeled as hyperedges within an annealed hypergraph (see Box 1). Groups continuously form and dissolve, so the individuals with whom one interacts continuously change.

:::: spacing 1.15

::: tcolorbox []{#tcolorbox:modelbox1 label="tcolorbox:modelbox1"} The continuous formation and dissolution of groups implies the absence of dynamic correlations between the states of individuals. Therefore, knowing the expected state of a random individual is all we need to describe ephemeral groups. A simple mean-field equation for the fraction $I$ of individuals in a state--say--'active' is sufficient: $$\frac{d}{dt}I = \sum_{n=2}^\infty p_n\sum_{k=0}^n \binom{n}{k} I^{k} (1-I)^{n-k} \left[(n-k)\beta(n,k)- k\alpha(n,k)\right] ; . \label{eq:ephemeral}$$ Here, $p_n$ is the probability distribution of group sizes, while $\beta(n,k)$ ($\alpha(n,k)$) is the rate at which inactive (active) individuals become active (inactive), which is a generic function of the group size $n$ and the number $k$ of active individuals in the group. In particular, $p_n \binom{n}{k} I^{k} (1-I)^{n-k}$ is the probability that a random group has $n$ members, of which $k$ are active.

The description given by Eq. ([eq:ephemeral]{reference-type="ref" reference="eq:ephemeral"}) is equivalent to a annealed structure with an infinite number of individuals (nodes) and groups (hyperedges), where every group is possible and all groups are reshuffled on a timescale much (ideally, infinitely) shorter than the timescale (here defined by either $1/\beta$ or $1/\alpha$) at which the dynamical process unfolds. Both structural and dynamical correlations are absent in these models. ::: ::::

In contrast, group permanence generally induces dynamical correlations. For example, if an individual in a household gets sick, others in the household are likely to be infected sooner or later. Since the household persists over time--enforcing repeated interactions--the states of its members become correlated through the dynamics (here, a contagion process). We can therefore study how specific group interactions influence the local and global dynamics. In the limit of never-changing groups, these define a quenched structure, representable as a static hypergraph, where hyperedges (groups) connect to each other through shared nodes (e.g. a household and a sports team with a member in common). In this case, neglecting dynamical correlations may lead to a poor understanding of the system's behavior. One can define models that, in addition to accounting for the quenched topology, are able to preserve dynamical correlations within groups by tracking the state evolution of each group [@burgio_network_2021]. The large amount of information used by such models makes them highly accurate but also computationally costly. Moreover, all that information is often difficult to access in real-world systems.

An alternative approach is offered by approximate master equations (AMEs) models [@hebert-dufresne_propagation_2010; @st-onge_master_2021; @burgio_adaptive_2023]. Although assuming an infinite, annealed structure, such models can work as satisfactory approximations for large quenched (or slowly varying) topologies due to the dynamical correlations they account for. Groups are assumed to be constantly reshuffling, but not uniformly at random; they rather do so while respecting the current probability distribution of group states (e.g. proportion of groups with $i$ active and $j$ inactive nodes, for every $i$ and $j$), so that dynamic correlations within groups are preserved. More sophisticated formulations of AMEs can also account for some dynamic correlations between groups [@burgio_adaptive_2023] (see Box 2).

:::: spacing 1.15

::: tcolorbox []{#tcolorbox:modelbox2 label="tcolorbox:modelbox2"} To account for persistence, we must at least preserve correlations within groups, since the duration of a group interaction induces correlations among the states of its members. The latter can also change state because of their membership in other groups. As a minimal model, we might, therefore, use approximate master equations. These describe the temporal evolution of the probability distribution $G_{n,i}$ that a group of size $n$ has $i$ active individuals (between 0 and $n$). The description tracks the processes within a focal group exactly and the processes within external groups in a mean-field fashion. The distribution $G_{n,i}$ thus evolves according to $$\begin{aligned} \notag \frac{d}{dt}G_{n,i} =&~ (n-i+1)\left[\beta(n,i-1) + \rho\phi\right]G_{n,i-1} - (n-i)\left[\beta(n,i) + \rho\phi\right]G_{n,i} \ &+ (i+1)\left[\alpha(n,i+1) + \rho'\psi\right]G_{n,i+1} - i\left[\alpha(n,i) + \rho'\psi\right]G_{n,i} ; , \label{eq:box2} \end{aligned}$$ where $$\phi = \frac{\sum_{n,i}(n-i)\beta(n,i)G_{n,i}}{\sum_{n,i}(n-i)G_{n,i}} ; ,\
\psi = \frac{\sum_{n,i}i\alpha(n,i)G_{n,i}}{\sum_{n,i}iG_{n,i}} ; , \label{eq:box2_aux}$$
are the respective probabilities of being activated or deactivated in random external groups. The factors $\rho$ and $\rho'$ quantify the coupling between groups for the activation and deactivation mechanisms, respectively. The case of isolated groups corresponds to $\rho = \rho' = 0$. In the case of susceptible-infected-susceptible (SIS) type dynamics, recovery is typically modeled as an intrinsic process, independent of group composition or external coupling. This corresponds to setting $\rho'=0$, while allowing $\rho>0$ to model infection through external contact. ::: ::::

If groups persist over time, it then makes sense to ask if and how they interact with the outside social world. In Box 2, strongly coupled groups are those in which out-group states provide information about within-group dynamics. For example, children playing together can spread infections between households. Similarly, group coupling affects information flow in social contagions--polarized groups freely exchange information internally but not externally. In pairwise networks, modularity captures this, but perfectly modular networks differ from quenched, coupled ones; in the former, contagion remains a sum of independent influences. In contrast, higher-order structures introduce nonlinear reinforcement, where repeated interactions within persistent groups amplify contagion beyond independent pairwise transmissions.

The coupling is distinct from adaptive networks, where individuals can move between groups based on several mechanisms, such as leaving when dissatisfied [@gross_adaptive_2007; @marceau_adaptive_2010; @burgio_adaptive_2023] or ascribed migration [@mcelreath_mathematical_2007 ch.6]. Instead, our modeling allows for recombination, where a mix of ephemerality with persistence of group dynamics can result in something new. By modeling how individuals form, reshuffle, or stay put, we can achieve an adaptive group-structured system. In some cases, this approach might be better suited to model group processes such as intergroup competition [@wilson_intergroup_2003; @henrich_cultural_2004; @richerson_cultural_2013]. In other cases, such as with social systems exhibiting strong polarization, coupling can instead be extended by adding a polarity to the coupling parameter. In this case, antagonistic influence could simply lead groups to adopt the opposite behaviors of what adverse groups are exhibiting, without requiring people to move [@smaldino_coupled_2021].

The description of persistent group-based dynamics remains limited in that the state of a group is fully derived from that of its members. Group interactions in households impacting contagion dynamics seem reasonable enough, but what about clans within ethnolinguistic tribes competing with each other, or tacit organizational knowledge? To account for persistent cultural behaviors, we propose shifting our view from groups as sets of individuals engaged in multi-way interactions to a stronger notion of emergence based on cultural group-level traits [@smaldino_cultural_2014].

Irreducible group-level features

In many ways, the vast majority of group interactions are mediated by cultural group-level traits, as human behaviors are regulated by institutions. Consider how sports teams consist of players, staff, and management, all working together to optimize team performance. Teams succeed or fail not only due to synergistic individual performance but also due to persistent group-level traits, such as how management decides to allocate resources among its members [@turchin_ultrasociety_2016]. Cultural traits are considered to operate at the group level because they are shaped by the ratio of within-group and between-group competition, rather than by individual performance alone [^2]. For example, if management chooses to distribute resources more evenly, it reduces within-group competition for the best contracts [@tiokhin_shifting_2024]. In this context, contracts promoting equity serve as one of many norms that facilitate cooperation among team members, contributing to the group's overall organization. In Box 3, this idea is captured as an abstract group-level feature, $\ell$, which can be framed as fostering altruistic states among teams.

:::: spacing 1.15

::: tcolorbox []{#tcolorbox:modelbox3 label="tcolorbox:modelbox3"} In irreducible groups, the dynamics affect both the state of the members of a group as well as group features $\ell$ that do not directly follow from the state of the members. We therefore split the dynamics into two sets of transitions, one governing members and one governing emergent group features, like this $$\frac{d}{dt}G_{n,i}^\ell = \frac{d}{dt}M_{n,i}^\ell + \frac{d}{dt}E_{n,i}^\ell ; .$$ []{#eq:box3 label="eq:box3"}
The first set of transition rates, $\frac{d}{dt}M_{n,i}^\ell$, governs transitions related to the members of the group and is equivalent to Eq. ([eq:box2]{reference-type="ref" reference="eq:box2"}) but with dynamical parameters $\alpha$ and $\beta$ now being a function of the group feature $\ell$.

The second set of transition rates, $\frac{d}{dt}E_{n,i}^\ell$, governs the transitions for the group features $\ell$. We can imagine that the group mind might want to promote or hinder activation and that $\ell$ might capture the level of group activity dedicated to this function (e.g., the creation of group norms). We could then assign a perceived fitness $Z^\ell$ to the levels $\ell$ (potentially also a function of the state of its members, i.e., $Z^{\ell}{n,i}$). In a very simple form, group minds might then perform a biased random walk over the fitness landscape of group features $\ell$: $$\label{eq:box3_groups} \frac{d}{dt}E{n,i}^\ell = h(n,i,\ell-1) \dfrac{Z^{\ell}{n,i}}{Z^{\ell-1}{n,i}} G_{n,i}^{\ell-1} + h(n,i,\ell+1) \dfrac{Z^{\ell}{n,i}}{Z^{\ell+1}{n,i}} G_{n,i}^{\ell+1} - h(n,i,\ell) \dfrac{Z^{\ell-1}{n,i}+Z^{\ell+1}{n,i}}{Z^{\ell}{n,i}} G{n,i}^{\ell} ; ,$$ where the transition rate for group features, $h(n,i,\ell)$, accounts for potential dependencies on the current state of the group (resources, costs, etc.). ::: ::::

By assuming group-level independence, we can directly model how groups adopt a series of discrete levels of institutional strength to enhance their capacity to promote individually costly behaviors. In the current form of the model, adopting stronger norms results in a faster adoption rate of costly behaviors among members, but this can be further elaborated (as we do in sec. 1.3.3{reference-type="ref" reference="secion:alignment"}). In our team example, modern teams could engage in increasingly sophisticated strategies to promote team equity. Just as coupling involves knowing the state of out-group members to predict the state of individuals within groups, irreducibility can be understood as the amount of information about the group required to predict the state of its members. When group state is perfectly correlated with that of its members, it is fully reducible.

Without group-level traits, groups are indistinguishable if their members find themselves sharing the same states. Once we account for group states, these may vary or remain constant while shaping the dynamics within groups. For example, a team might change its communication platform, influencing group dynamics. But practically, we are interested in how those changes may lead to groups being more or less successful, exhibiting group-level fitness. In Box 3, group-level fitness influences how groups decide to scale institutional strength up or down, based on a trade-off between achieving success and balancing other factors, such as the potential costs of scaling up group norms. With group-level features, we can begin to inquire about the impact of perceived success, or perhaps prestige, of neighboring groups on within-group dynamics [@richerson_cultural_2016; @hebert-dufresne_source-sink_2022].

We highlight some key findings based on the dynamics of models from Box 3. First, there are critical thresholds in terms of collective costs, benefits, and the individual adoption rate $\beta$ that determine the widespread adoption of costly behaviors, as expected. Then, the phenomenon of institutional localization has been identified--that is, a specific institutional level dominates the fitness landscape within certain parameter ranges [@st-onge_paradoxes_2024; @hebert-dufresne_source-sink_2022]. This has raised questions about the conditions under which groups invest in scaling up their policies versus relying on other groups to bear the burden of increased institutional efforts while reaping the benefits within their own groups--what has been dubbed as "institutional free-riding" [@st-onge_paradoxes_2024]. Lastly, it was found that when individual behaviors are perceived as excessively costly (perhaps adopting equitable contracts by sports players in our running example is seen as such) institutions may respond by intensifying their efforts to promote those behaviors [@st-onge_paradoxes_2024]. This "call to action" suggests that worst-case scenarios can drive institutional change, whereas more tolerable situations may be accepted, leading to lower overall adoption rates.

Irreducible features help distinguish between groups as mere containers of interactions and groups as entities with higher-level traits subject to selection. The group independence assumption is a useful simplification in that we can directly model institutional dynamics without having to explain their emergence. Yet, organized groups rarely emerge in a mathematical vacuum. Increased institutional strength is fundamentally correlated with changes in resource management and social hierarchy, which we both include as components of misalignment.

Alignment of individuals and groups

We define alignment as the degree to which individual-level states--such as preferences, beliefs, or actions--are congruent with group-level dynamics. Alignment arises when group behavior reinforces individual goals, and vice versa. Misalignment, by contrast, occurs when groups produce outcomes that persist despite contradicting or neglecting the goals of their members. Alignment interacts with previous dimensions to capture a diversity of group phenomena in nature.

In animal behavior, alignment can be as simple as bird flocks or fish schools emerging from individuals obeying local interaction rules---such as repulsion, attraction, or predator avoidance [@reynolds_flocks_1987; @ioannou_predatory_2012]. These decentralized dynamics generate group-level benefits (e.g., anti-predator shielding) that align with individual fitness. Similar effects arise in human systems; for example, the presence of more cyclists on the road improves safety for all, even without explicit coordination. In such cases, group-level properties emerge from simple local rules that reinforce individual goals.

In contrast, misalignment arises when group-level patterns feed back in ways that contradict individual incentives. A simple example is that of Roger's Paradox, where overreliance on social learning degrades decision quality in changing environments [@rogers_does_1988]. When individuals copy others rather than directly engaging with the world, groups risk becoming unresponsive or locked into outdated norms [@torney_social_2015]. A similar dynamic appears in Condorcet's jury paradox, where individually rational votes aggregate into irrational majorities. In both cases, individuals act sensibly based on local information, yet the group produces outcomes no one intended. Real-world examples abound: alarm calls become noisy under excessive imitation [@brown_social_2003], and speculative bubbles form when investors herd without reevaluating fundamentals [@lo_wisdom_2022]. In such models, misalignment arises within persistent but reducible groups--structures where individual behavior correlates over time, but the group exerts no independent influence.

With irreducible groups---those possessing persistent states like norms or policies--misalignment takes on a more institutional character. These group-level structures can persist even when they diverge from the preferences or goals of individual members. Consider researchers choosing between slower, reproducible methods and faster, less rigorous ones. When success metrics reward publication volume, individuals are incentivized to cut corners---even if the group would benefit from quality-focused norms [@dawson_role_2022; @tiokhin_shifting_2024]. This isn't necessarily a case of bad actors engaging in p-hacking; rather, individuals may continue practices they have learned and seen rewarded, despite growing evidence of their detrimental effects [@smaldino_natural_2016]. Groups may attempt to enforce better norms through institutional policies, but unless those norms are strong, visible, and valued, individuals may simply defect through apathy. In this context, misalignment stems not from resistance but from indifference.

This mismatch between individual incentives and collective structures isn't just a product of faulty aggregation. Institutions evolve on different timescales and are often sustained by mechanisms like prestige [@way_gender_2016], norm entrenchment [@richerson_cultural_2016], or institutional lock-in [@nelson_search_1993]. They may persist even when they no longer serve their members, creating structural inertia that deepens misalignment. Conversely, institutions can promote better practices---such as transparency or methodological rigor---while individuals may exhibit inertia on shorter timescales, failing to adapt even when norms shift in that direction.

We explore this dynamic more formally in Box 4, which presents a minimal model of institutional misalignment. Individuals belong to groups (e.g., cliques), each with an institutional quality level $\ell$, determining the perceived value of public goods $s(\ell)$. Strategy updates---such as choosing to cooperate or defect---depend on peer behavior and institutional satisfaction. The probability of adopting cooperation follows a sigmoid function $f(x)$, where $f(s(\ell)-1)$ captures responsiveness to institutional strength. Here, individuals behave more cooperatively when institutions are perceived as strong, and vice versa.

:::: spacing 1.0

::: tcolorbox []{#tcolorbox:modelbox4 label="tcolorbox:modelbox4"}

Alignment in irreducible groups takes the same basic form as in Box 3 but with the additional component that individuals have opinions about how their institution manages their public goods. A minimal implementation of the latter amounts to adding an institution-dependent term of spontaneous transition in the dynamic equation for the state of the individuals. We thus write $$\begin{aligned} \label{eq:box4_ind} \notag \frac{d}{dt}M_{i}^\ell =&~ (n-i+1)\left[\gamma f(s(\ell) - 1) + \beta(i-1) + \rho \beta \phi \right]G_{i-1,\ell} \ \notag &- (n-i)\left[ \gamma f(s(\ell) - 1) + \beta i + \rho \beta \phi \right] G_{i,\ell} \ \notag &+ (i+1) \left[ \gamma f(1-s(\ell)) + \alpha(n-i-1) + \rho \alpha \psi \right] G_{i+1,\ell} \ &- i\left[\gamma f(1-s(\ell)) + \alpha(n-i) + \rho \alpha \psi \right] G_{i,\ell} ; , \end{aligned}$$

where, for simplicity, we removed dependence on group size $n$ by assuming all groups are equally sized (more generally, we would have an index $n$ to account for distributed size). As in Box 3, after summing over $\ell$, the terms $\phi$ and $\psi$ represent the influence of other groups on individual transitions. The function $f(x)$ gives the probability that a defecting individual switches to cooperation (activates), and depends on the perceived quality $s(\ell)$ of the public good, where $ds(\ell)/d\ell > 0$. The parameter $\gamma$ governs the rate of spontaneous behavioral switching. Without loss of generality, we set $\ell = 1$ to represent the threshold at which individuals are indifferent to institutional performance, and fix $s(1) = 1$. We choose $f(x)$ to be symmetric about zero (i.e., $f(x) + f(-x) = 1$), so that $f(s(1)-1=0)=1/2$ represents indifference, and $f(-x) = 1 - f(x)$ gives the probability for a cooperator to switch to defection.

The second set of transition rates $\frac{d}{dt} E_{i}^\ell$ is similar to Eq. [eq:box3_groups]{reference-type="ref" reference="eq:box3_groups"}, but--mirroring Eq. [eq:box4_ind]{reference-type="ref" reference="eq:box4_ind"}--we give institutions more agency by including a term of spontaneous, fitness-independent change ($\propto \mu$). Then, as in Eq. [eq:box3_groups]{reference-type="ref" reference="eq:box3_groups"}, the transition rates be any function of $i$ and $\ell$ (and $n$). Supposing that upgrading and maintaining stronger institutions entails some cost, we write $$\begin{aligned} \notag \frac{d}{dt}E_{i}^\ell =&~ g(bi - c\ell) \left[\mu + \rho \dfrac{Z^{\ell}{n,i}}{Z^{\ell-1}{n,i}}\right]G_{n,i}^{\ell-1} - g(bi - c(\ell+1)) \left[\mu + \rho \dfrac{Z^{\ell+1}{n,i}}{Z^{\ell}{n,i}}\right]G_{n,i}^{\ell} \ \notag &+ \left[ \mu g(c(\ell+1) - bi) + \rho g(bi - c\ell) \dfrac{Z^{\ell}{n,i}}{Z^{\ell+1}{n,i}}\right] G_{n,i}^{\ell+1} \ &- \left[ \mu g(c\ell - bi) + \rho g(bi - c(\ell-1)) \dfrac{Z^{\ell-1}{n,i}}{Z^{\ell}{n,i}} \right] G_{n,i}^{\ell} ; , \end{aligned}$$

where the function $g$ accounts for the balance between the available resources ($\propto bi$) and the cost to sustain ($\propto c\ell$). The overall dynamics allows us to explore how individual- and group-level traits coevolve driven by alignment. ::: ::::

Crucially, the model allows institutional misalignment to become self-reinforcing. If a group underinvests in norms (low $\ell$), cooperation declines, leading to weaker outcomes and even lower investment---a "race to the bottom". Conversely, well-maintained institutions can stabilize cooperation, even under noise or temptation to defect. This dynamic reflects a feedback loop: individuals are sensitive to norms only when the group visibly invests in collective goods. Otherwise, they default to inaction, and group performance collapses.

The model also incorporates bounded rationality via a sensitivity parameter $\alpha$. When $\xi$ is low, behavior becomes noisy and less strategic; when $\xi$ is high, behavioral thresholds sharpen, approximating rational adaptation. This flexibility allows us to interpolate between stochastic imitation and goal-oriented decision-making.

Importantly, the institution itself evolves. Group-level quality $\ell$ rises or falls based on internal dynamics (e.g., number of cooperators) and comparisons to other groups. Institutions that succeed in managing public goods attract more cooperation, which in turn reinforces their strength---linking micro-level behavior to macro-level evolution.

If research groups exist in a cultural context that strongly values individual achievement (e.g., the myth of the heroic inventor), efforts to shift the unit of selection from individuals to groups may be resisted [@henrich_weirdest_2020]. Our model captures this interplay: cooperation spreads through peer imitation and group identity but is moderated by the institution's perceived quality. A group may invest heavily in promoting cooperative norms---such as mentoring, code review, or rigorous methodology---but if members remain indifferent, these norms may fail to take hold. Irreducible institutions do not merely shape strategy---they also define roles, responsibilities, and prestige. In this way, misalignment interacts with social differentiation: institutions like academia offer stable roles (e.g., professor, advisor) that come with behavioral expectations, but if those roles become disconnected from individuals' lived experiences or values, misalignment arises not just in behavior, but in identity and purpose.

By treating institutions as persistent group-level states that both shape and respond to individual behavior, we can model the coevolutionary feedback between people and structure. Misalignment, in this view, is not a statistical quirk---it is a dynamic process through which group norms and individual actions gradually diverge, potentially undermining collective goals. Conversely, individuals can also diverge from long-term, sustainable goals promoted by groups. Capturing this process is essential for understanding real-world systems where alignment must be earned, not assumed.

Discussion

We outlined four dimensions of group models, integrating insights from higher-order interactions and the long-standing history of group interactions. We found that models of higher-order interactions mainly address weakly emergent group behaviors, whether ephemeral or persistent. By incorporating group-level features, we improve our ability to represent the coevolution of individuals and institutions. Before concluding, we highlight some challenges and future directions in applying our framework to empirical research.

Measuring group-level features

Measuring co-authorships is easier than measuring the norms and institutions shared by research groups. Individual interactions often seem easier to measure than group interactions. Yet, this does not mean that social networks are more real than group interactions. Individual interactions are not as easy to measure as they seem; social networks are subject to assumptions and are often error-prone [@young_robust_2020]. What we define as nodes and edges depends on the research question; for example, friendship can be defined reciprocally or not, leading to different networks [@butts_revisiting_2009]. Then, it is good to remind ourselves that any kind of social interaction occurs within specific sociotechnical contexts. Individual productivity is easier to measure because we have made it easy, and co-authorship is particularly prominent because Western scientists have long attributed ideas to individuals [@henrich_weirdest_2020]. Ultimately, the key is to use the right tools for the work at hand, and researchers studying teams in science should model changing norms and policies, as the meaning of co-authorship varies over time and across communities. The availability of group interactions does not exempt us from clear assumptions about their meaning.

That being said, how would we start to measure group-level features? Anthropologists, among other fields, made tremendous efforts to measure and characterize cultural groups throughout the world. In recent times, researchers have begun to take on the challenges of coding norms and policies and making them accessible through open databases [@slingerland_database_2024; @turchin_seshat_2015; @kirby_d-place_2016]. Another example is the Oxford COVID-19 Government Response Tracker data, which was a tremendous effort by public health researchers to track policies adopted by governments in response to the COVID-19 pandemic. A key part of their effort was ensuring intercoder reliability in how a team of experts quantified the strength of government responses. In general, this ability to organize scientific teams on a larger scale to agree on relevant norms has been shown to be key to modeling group-level features [@slingerland_coding_2020].

Measuring misalignment

Strongly emergent misalignment, perhaps due to being a mouthful, is also the most challenging dimension to estimate because it involves variability in individual preferences and group dynamics. How do we know if sports team management or a research group is misaligned with the interests of its members? Worse still, what about misalignment in organizations such as a "Mafia" or a "Church"? We must rely on evolutionary models; if the group-level feature is assumed to promote group success, in terms of higher fitness, then misalignment occurs whenever individual preferences are not aligned with that (functional) goal. Sometimes, it is obvious, as with suicide bombers willing to die for their group to prosper. Most of the time, this is more nuanced, as with religious acts that entail serious risks and significant investments of time, leading to increased performance of the group relative to other religious groups [@power_social_2017]. Modeling functional group behaviors has been controversial, but the combination of contemporary theoretical modeling and fieldwork can help us identify the necessary ingredients to determine when group-beneficial traits are favored, as one of many potential alternative stable states [@boyd_group_1990; @boyd_culture_1988; @boyd_group_2002; @soltis_can_1995]. Once again, a key aspect of quantifying misalignment is intercoder reliability among domain experts when identifying latent norms and policies, as well as individual variability, which can then be used to make predictions.

Group minds redux

When early social scientists discussed the idea of "group minds", they made logical (verbal) arguments about society exhibiting organism-like characteristics. This has led to a kind of Panglossian evolutionism, where group adaptation is everywhere while motivating a Victorian laissez-faire [@campbell_how_1994], which was later dismissed. In our typology, we are assuming that group-level features, such as norms and institutions, do influence group-level fitness. Importantly, fitness is defined generally in our model, but it must be empirically specified. In the case of interfirm competition, for instance, empirical research continues to specify how exactly organizational breakthroughs in management and innovation improve overall firm success, whether through differential survival or proliferation [@richerson_cultural_2016]. It is related but not reducible to the idea of group minds as collective intelligence, or weak emergence. In this view, there is a similar ongoing effort to empirically assess how teams perform better than the sum of individual performance, most notably through complementary skills, communication strategies, or diversity of expertise [@page_diversity_2010]. As such, both the question of whether particular group-level features and weak emergence entail the existence of group minds is open and subject to empirical inquiry, though it has now been influenced by extensive modeling work.

It is worth mentioning that philosophers have argued at length in recent years about the meaning of groups exhibiting cognitive-like properties [@french_collective_1984; @pettit_groups_2001]. As with collective intelligence, some aspects of it are generally accepted, such as ants exhibiting memory-like properties through pheromone trails [@gordon_ant_2010]. However, as with the organism-like analogy, what is meant exactly by a corporation lying about the fuel efficiency of its vehicles is still an ongoing ontological debate [@lackey_epistemology_2021]. In its strongest form, philosophers argue that groups capable of believing, knowing, asserting, or even lying can be held accountable for their actions, effectively becoming moral entities--an issue that falls beyond the scope of this review [@lackey_epistemology_2021].

Big groups and networked norms

Our typology is meant to represent group interactions, both small and large. However, some group interactions are too abstract and hierarchical to currently fit into our modeling work; for example, a federal government issuing an executive order like a mask mandate. As such, the key level of analysis for our typology is the level at which policies are implemented, for instance, local health departments. At this level, groups can be thought of as copying and learning from each other. There is also an additional layer of global top-down influence, referred to as "higher-level" political-economic institutions, which interact with both "lower-level" institutions, such as those related to kinship, marriage, religion, and people's cultural psychology. Only by developing models that can capture the joint influence of individual variability along with that of lower-level institutions can we begin exploring the larger, more diverse "pluralistic" political institutions [@henrich_big_2015; @acemoglu_why_2013].

Our family of models focused on models with binary states at the individual level and a single abstract group-level state that influences those states. Nothing prevents our work from being extended to more complex states at all levels. One more challenging generalization is to consider overlapping institutions and identity as part of the connection between groups. People have overlapping memberships, which are known to significantly impact the dynamics of systems. Similarly, norms and institutions often come in bundles, with specific networks of practices.

Conclusion

As the modeling of group dynamics and higher-order networks flourishes, we will continue to see new mathematical approaches to describe groups. By using our typology to compare models, we hope to not only clarify and classify existing efforts, but also identify types of groups and group interactions that are currently ignored by these efforts.

Appendix

In our short history of group minds, we noted at least one important feature missing from our typology; social differentiation. Differentiation refers to the diversity among members in roles, responsibilities, or rewards---and is key to understanding how coordination, hierarchy, and inequality arise within groups [@redhead_social_2022; @smith_leadership_2016]. In biological and social systems alike, differentiation enables synergy: the performance of a group can exceed the sum of its parts when members specialize and coordinate their actions [@araujo_team_2016; @guimera_team_2005]. We left this category out of the main text because our current typology does not yet include a group-based model with differentiated individuals. But here we provide a short overview of how differentiation is addressed across relevant modeling traditions---from network science to cultural evolution and higher-order interactions---to draw attention to novel modeling opportunities in that area.

Differentiation as heterogeneity

Differentiation in networks can be as simple as node metadata---fixed attributes like class, gender, or rank. These are often treated as exogenous labels that may (or may not) correlate with community structure [@newman_structure_2016]. When such attributes shape interaction patterns---for instance, assortativity by class or profession---we enter the realm of heterogeneous mixing. This introduces structure into interaction but still assumes randomness conditioned on classes (e.g. node features or degree-based). As such, it cannot capture dynamical correlations---state dependencies that build up over time through repeated interactions. In contrast, persistent groups preserve local structure, enabling individual states to co-evolve through reinforcement, norm enforcement, or coordinated roles. Differentiation in this sense is structuring group interactions---not just a label but a product of sustained interaction.

Differentiation improving group coordination

In evolutionary models, social differentiation enhances group performance by enabling more sophisticated forms of coordination. In correlative coordination games, any shared behavior---regardless of content---yields higher payoffs than uncoordinated actions. These models explain how arbitrary conventions (e.g., driving on the right) can emerge and stabilize [@smaldino_modeling_2023]. In asymmetric coordination games, one strategy leads to better group outcomes but imposes initial costs on individuals---such as contributing to a public good that only succeeds if enough others do the same [@boyd_group_2002]. Differentiation means that individuals adopt specialized strategies, like moralists who both cooperate and punish defectors to help the group reach and maintain beneficial equilibria [@boyd_group_1990]. A third form, complementary coordination, involves interlocking roles---"I cook, you clean" kind of scenarios. These games produce surplus only when roles are matched. If rewards are balanced, roles can rotate; if not, differentiation may harden into hierarchy, with some roles becoming persistently higher status or more burdensome. Over time, such roles may align with social markers like gender, age, or skill, embedding stratification within the group structure [@smaldino_modeling_2023]. In this view, differentiation is not just functional---it is formative, shaping how groups coordinate, specialize, and reproduce internal inequalities.

Differentiation as synergy in HONs

Whereas differentiation in cultural diffusion and coordination games emphasizes how roles and strategies contribute to group success, higher-order network models highlight how group structure itself---rather than internal role composition---can generate synergy. In these models, individuals interact through hyperedges: explicit group-level interactions where the composition and size of each group shape outcomes. Recent work on order-heterogeneous hypergraphs shows that even without role differentiation, cooperation can be amplified if the synergy factor scales super-linearly with group order ($R(g) \propto g^\beta$)---meaning larger groups yield disproportionately larger payoffs [@alvarez-rodriguez_evolutionary_2020; @burgio_evolution_2020]. This structural differentiation---determined by who participates with whom, and in which group sizes---shifts the critical thresholds for cooperation and alters relaxation times. Such models complement cultural diffusion approaches [@boyd_group_2002], but go further by assigning high-order effects to the topology of participation itself. In this view, differentiation is not just about roles or strategies, but about the topology of participation---how group size and overlap encode evolutionary leverage.

Differentiation sustaining group-level features

In all models we've seen so far, groups are reassembled each round, and synergy arises from transient structure or strategy---not from persistent group identity or evolving institutions. Irreducible group structures, such as the churches, ethnicities, or unions, do more: they create, stabilize, and legitimize roles over time. The moralist, for example, is not merely a strategic role but can become a codified identity embedded within a system of beliefs, or a priest. Institutions interact with differentiation to determine who does what, for whom, and at what cost. From the perspective of higher-order network models, this kind of institutional differentiation remains fundamentally understudied---most work still focuses on ephemeral, isolated groups. Yet differentiation is more than heterogeneous mixing; it shapes not only immediate coordination but also collective performance across evolutionary timescales.

[^1]: See [@udehn_changing_2002] for a review of the changing face of methodological individualism. Group realism is entangled with that of functionalism, where groups are real because they occupy functional roles that promote the common good, as in the organism-like analogy of groups.

[^2]: This argument draws on the Price Equation, which shows that the ratio of within-group to between-group variance determines the conditions under which group selection can outweigh individual selection [@richerson_cultural_2016]. Here we avoid introducing yet another formalism to the paper and keep the text focused on the formalism discussed in the mathematical boxes.