A Dynamic Computational Model of Social Stigma

The dynamics of social stigma are explored in the context of di usionmodels. Our focus is on exploring the dynamic process through which the behavior of individuals and the interpersonal relationships among them influence the macro-social attitude towards the stigma. We find that a norm of tolerance is best promoted when the population comprises both those whose conduct is driven by compassion for the stigmatized and those whose focus is on conforming with others in their social networks. A second finding is that less insular social networks encourage de-stigmatization when most people are compassionate, but it is instead more insularity that promotes tolerance when society is dominated by conformity.


Introduction
. In , the philosopher Jeremy Bentham wrote an article critically examining the harshness with which homosexuality was treated in England (Bentham & Crompton ). He could not find a sound basis for such unforgiving treatment. Yet, for many centuries before and the centuries therea er, gays have been stigmatized. However, in the last few decades, some parts of the world have experienced a significant change in their attitudes towards gays. In Great Britain, the fraction of adults expressing disapproval of sexual relations between adults of the same gender declined from % in to % in (Park & Rhead ). In the United States, there has been a comparable change in attitude as reflected, for example, in the growing acceptance of same-sex marriage. In , the Pew Research Center reported that % of Americans disapproved of samesex marriage with only % expressing approval. By , those numbers had almost flipped, as % approved and % disapproved. Of course, in some sub-groups of the United States and in many parts of the world, homosexuality is still a powerful stigma and associated with it is an intolerance of gays (Adamczyk ).
. The changing attitudes regarding gays highlights the endogeneity of an attribute as a stigma and naturally raises questions of what would lead to a movement from intolerance to tolerance. The objective of this paper is to investigate theoretically the social dynamics of stigma towards deriving insight into the conditions conducive to a society expressing acceptance of those previously stigmatized. What change in social conditions can disrupt a norm of intolerance? What are the conditions sustaining a norm of acceptance? .
To address these questions, we begin with the pioneering perspective of Go man ( ) on stigma, which he defines as an attribute that signals a deviation from the social norm and can be a source of harmful discrimination. In his micro-social theory of stigma, there are three types of actors. There are those endowed with the attribute which may be considered a stigma by others in society. Then there are those who are free of such an attribute, of which there is a subset, which Go man refers to as "the wise," who are sympathetic toward the stigmatized few. The emphasis in his theory is on the one-on-one relationship between these three types of actors rather than the attributes themselves: "The term stigma, then, will be used to refer to an attribute that is deeply discrediting, but it should be seen that a language of relationships, not attributes, is really needed." (Go man , p. ) .
Second, there is the voluminous literature modelling social di usion which generally assumes that an individual's decision to adopt or reject the object of interest depends on how widely it has been adopted among a relevant subset of agents such as friends, colleagues, and neighbors (Rogers ; Valente ). The object of interest may be a technological innovation such as a new farming practice (Ryan & Gross ), a management practice (Strang & Soule ), a cultural fad such as what clothes to wear (Crane ), a residential choice such as where to live (Schelling ), or an antisocial act such as participating in a riot or a strike (Bohstedt & Williams ; Conell & Cohn ). The situation is fundamentally di erent, however, when the "object of interest" is a person rather than an idea or a practice, and the decision is whether to approve or disapprove of them. That the object is a person adds two additional agent types to the usual presence of conformists in a model of social di usion. First, there are those individuals whose decision to accept or reject the stigmatized is driven not by conformity but by the value they attach to stigmatized agents' well-being. In other words, they care about them and that is determinative. Second, those stigmatized agents decide whether to reveal their stigma, and it is only through disclosure can the compassionate agents be induced to act. The presence of the three types of agents introduces a new triadic dynamic to social di usion models. One of the contributions of this paper is to begin to understand the implications of this dynamic, how it depends on the structure of the social network, and what can cause a shi in social norms.

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Our work also fits into the sociological and social psychological literature on social approval and conformity. Within this enormous literature, our model is most closely related to those that address the behavior of individuals who, driven by the desire to conform, act against their own belief. This literature includes research on preference falsification (Kuran ), pluralistic ignorance (Prentice & Miller ), spiral of silence (Noelle-Neumann ), and false enforcement of unpopular norms (Bicchieri ; Centola et al. ; Willer et al. ). Of particular interest is the agent-based computational model in Centola et al. ( ), in which the population of agents make compliance and enforcement decisions over a given norm. Embedded in their population is a small group of "true believers" whose conviction is so strong that they always comply with the norm regardless of the compliance by other agents. The remaining population consists of believers and disbelievers whose compliance and enforcement of the norm depend on the observed actions of others, similar to our conformists. The focus of their study is on the "false enforcement" by false disbelievers (who privately oppose but publicly support the norm), whereby they engage in pressuring others to comply with the norm "to avoid exposure as an opportunistic imposter." While there are some similarities between their true believers and our compassionists, and between their disbelievers and our conformists, the two models diverge in two important aspects. First, we focus on the acceptance/opposition (compliance) behavior of the agents without modeling their enforcement behavior. In our model the observed compliance by other agents is the sole determinant of the conformists' acceptance decisions -there is no pressuring of others for signaling purposes in our model. Second, the stigmatized agents in our model, through their decision to reveal or hide their trait, take the central role in the social dynamics of the stigmatizing behavior between the compassionists and conformists. It is the exploration of this triadic dynamic between the three agent types which makes our contribution unique. .
Finally, our work is closely aligned with the line of research on "status construction theory" as first presented in Ridgeway ( ), then refined and formalized in Ridgeway & Balkwell ( ) and Mark et al. ( ). The central question in this line of research is how to explain the presence of status di erentiation between social groups that are observationally distinct in terms of their nominal characteristics. Ridgeway ( ) presented an intuitive explanation for the di erential status values based on di erences in the resource ownership levels of the individuals. This verbal argument was formalized in Ridgeway & Balkwell ( ), which o ered an explicit model of the belief di usion process. These earlier works explained the di erential status values on the basis of resource ownership by assuming a pre-existing correlation between the nominal characteristic and the resource characteristic of individuals. Mark et al. ( ), using two micro-level mechanisms --status belief di usion and status belief loss -were able to extend the theory by showing that consensual status beliefs can emerge even in the absence of any explicit status-relevant variable such as resources. The earlier model of status construction theory was most recently extended by Grow et al. ( ). By developing a computational model that integrates the status construction theory with the network theory, they showed that a structural characteristic of social networks such as local clustering can have a significant impact on the di usion of status beliefs, resulting in noticeable regional variation in status beliefs.

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There is a clear conceptual linkage between social status and social stigma --the target of our investigation -to the extent that a display of the stigmatized trait confers a lower social status to the person with such trait. In this context, stigma can be viewed as a realization of particular status ranking as agreed upon by the members of the society. The status construction theory addresses the process by which such ranking gets established, while, in our model of social stigma, we take the existence of such ranking as given but explore the process by which it can be neutralized or reinforced through individual choices to accept or reject those with the stigmatized trait. Nevertheless, the di usion mechanism driving the changes in attitude toward stigma in our model is very much in line with the mechanism driving the di usion of status beliefs on social groups. Grow et al. ( ) is particularly relevant in this regard, as the "clustered" network structure they consider is similar to that in our model where interactions among individuals are partially constrained by the community structure of the social system. Their contribution is in showing that the presence and the extent of belief inertia (in the form of consensus threshold) and the local clustering of interaction networks significantly influence the di usion of social status beliefs. They, however, assumed homogeneity among individuals in that the level of belief inertia was specified to be uniform across the population. In addition, they considered the degree of clustering in the networks only as an either-or proposition as their analysis was limited to the networks being either clustered by a fixed degree or not clustered at all. Our model diverges from theirs in two important aspects. First, given that our study of social stigma considers the interaction between compassion and conformity as well as the disclosure behavior of the stigmatized individuals, we explicitly introduce heterogeneous and type-dependent thresholds for individuals' acceptance of stigma. Second, we consider a wide variation in the local specificity (local clustering) of networks. These extensions allow us to examine in detail the impact network structure has on the interactive dynamic between the di erent types and the resulting level of acceptance in the population. We believe these extensions and the results are relevant for the study of status construction process as it pertains to a particular form of social status as implied by the stigmatizing behavior.

The Model
Spatial environment . There is a fixed population of agents, M ≡ {1, . . . , m}, who are distributed over a two-dimensional X × Y lattice. One agent is located at each point (or node) on the lattice -so m = |X| * |Y | -where a point (x,y) represents an agent's location. The lattice wraps around from right to le and from top to bottom, forming a torus; see Figure a. The use of a torus ensures symmetry in the physical environment for all agents, hence avoiding a potential edge e ect. Each agent i has a "neighborhood," denoted N (i), that consists of all other agents within the Moore neighborhood of range n. This means that each agent has (2n + 1) 2 − 1 neighbors. As a patch from the torus in Figure a, Figure b gives an example of a Moore neighborhood of range for the agent represented by the hollow square at the center. Agent i has a social network, denoted L(i), that consists of l links with a subset of other agents in the population: A network is naturally thought of as family and friends, and links are assumed to be symmetric: j ∈ L(i) ⇐⇒ i ∈ L(j). Most, though not necessarily all, of these links are with the agent's neighbors. The construction of the network is described in Section .

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Some agents may be endowed with a trait that could be the basis for a stigma. We refer to them as stigmatized (though recognizing that whether the attribute is a stigma is endogenous). The stigma considered here is of the discreditable type, such as homosexuality and atheism. The decision faced by a stigmatized agent is whether to reveal the attribute. The remainder of the population comprises agents lacking the attribute, who are referred to as normal. Their decision is whether to accept those with the attribute; that is, whether or not to stigmatize them. (It should be noted that the term "normal" is Go man's parlance in his book. We use the term purely for the sake of retaining consistency with the original work that our model derives from.) .
A stigmatized type, which we also refer to as a type S, would like to reveal that he has the attribute but is concerned with being ostracized. It is assumed that he will choose to "reveal" if and only if the fraction of agents in his social network who are accepting of those with this trait equals or exceeds a critical threshold τ S ∈ (0, 1). It is assumed that when a stigmatized agent reveals his trait that he also expresses acceptance of others like him. .
A normal agent decides whether or not to express his acceptance of those with the stigmatized trait. There are two types of normal agents, and they di er in the basis upon which they make the decision to accept the stigmatized. The acceptance decision of a conformist (CNF) is driven by a desire to conform. He will accept the stigmatized if and only if the fraction of agents in his network that have expressed their acceptance of the stigmatized is at least some critical value τ CN F ∈ (0, 1). The conformist type is common in di usion models.
There is a second normal type whose decision of whether to accept the stigmatized is based on caring about those in her network. A compassionist (CMP) will accept the stigmatized if and only if the fraction of agents in her network with the stigmatizing trait (and who have revealed it) is at least some critical value τ CM P ∈ (0, 1). Thus, in making their acceptance decisions, conformists rely on what all other agents are doing, while compassionists respond to the stigmatized. The compassionist type is unique to di usion models but is natural given the "object" of di usion involves people, as opposed to ideas or practices. .
In sum, agent i can be one of three possible types: z(i) ∈ {S, CN F, CM P }. The critical thresholds are common within each agent type, but di er between the three types, and stay fixed over time. The higher is an agent's threshold, the more resistant she is to changing her status as it takes a bigger fraction of her network links to induce the change. For a normal agent, the threshold represents the degree of her "intolerance" toward the stigmatized, while for a stigmatized agent it represents his "reluctance" to reveal himself for fear that he may be ostracized by those in his network. Finally, it is assumed that a proportion s of the population are stigmatized and the remaining proportion (1 − s) are normal. Of the normal population, a proportion w are specified as conformists and (1 − w) as compassionists. Hence, given the population of m agents, there are s · m agents who are stigmatized, (1 − s) · m · w conformists, and (1 − s) · m · (1 − w) compassionists.

Agent states and state transition rules
. At any moment of time, the state of a stigmatized agent is either "revealed" (in which case he has also expressed acceptance of the stigmatized) or "hidden" (in which case he has not expressed acceptance with regards to the stigmatized). The state of a normal agent is either she has expressed acceptance of the stigmatized or not. With a conformist, we refer to the "not acceptance" state as "opposition," while for a compassionist (and a stigmatized agent) we refer to it as "neutral." While this semantic distinction is not important for the ensuing analysis, as the focus is on how many are accepting of the stigmatized, it seems natural to think of the "non-accepting" state as meaning opposition (otherwise, there is no harm from being stigmatized) with the exception of the compassionists who do not express either acceptance or opposition until they feel a compulsion to support those in their social network. .
Although it will be assumed that most of the stigmatized agents in the population start out by hiding their trait, it is essential for there to be a small seed group who have revealed so that the di usion process can be initiated. One could imagine that they are the brave or principled few or that the possession of the trait was inadvertently revealed. We may also allow some of the normal agents to start in the state of acceptance, though most (and sometimes all) will start in the state of non-acceptance. Notationally, a proportion p r of the stigmatized start out as "revealed" (and, therefore, "accepting"), and a proportion p a of the normals start out as "accepting." Note that the population begins in a state for which the social norm is intolerance (p a is low) and the stigmatized are hidden (p r is low).
. From these initial conditions for the population, let us describe how the state of the population evolves. For this purpose, we denote by α t i ∈ {N, A, O} the expression of attitude state of agent i in period t, where N denotes "neutral," A denotes "acceptance," and O denotes "opposition." Likewise, we denote by β t i ∈ {R, H} the disclosure state of (stigmatized) agent i in period t, where R denotes "revealed" and H denotes "hidden." The state of an individual agent's network in period t is summarized by r t i and a t i , where r t i is the proportion of revealed type-S agents in i's network and a t i is the proportion of accepting agents in i's network: For all t ≥ 1 an individual agent's states are updated based on the following rules: • For all i with z(i) = S: ( ) An individual with the stigma will switch from the state of "hidden" to that of "revealed" if the proportion of agents in his network who are accepting of the stigma is at least as great as τ S . Otherwise, he remains "hidden." If he discloses his stigmatizing trait, he expresses his acceptance of other agents with the stigma. For the stigmatized agents, (A, R) is an absorbing state such that once they are revealed, they remain in that state.
• For all i with z(i) = CN F : ( ) A conformist is "opposing" if the proportion of other agents in his network who have expressed themselves to be "accepting" of the stigma is below τ CN F . Otherwise, he is "accepting." At any point in time, a conformist can switch his state according to this rule.
• For all i with z(i) = CM P : ( ) A compassionist who is currently in the state of "neutral" will stay in that state as long as the proportion of agents in his network who are "revealed" to have the stigma remains below τ CM P . Once the proportion of revealed agents in his network is at least τ CM P , a compassionist will switch to the state of "accepting." Note that the compassionists, once "accepting," will never switch back to "neutral," as the stigmatized individuals, "once revealed," never go back to "hidden." .
It should be noted that all agents' attitudes are updated synchronously at the beginning of each time step, rather than consecutively, so that an agent's updating in a given period does not a ect another agent's updating in the same period.

Setting Up the Computational Experiments
Parameter specifications . We create a population of , agents (m = 10, 000) and distribute them on a x grid which forms the outer surface of a torus. Each agent is assigned its type at the outset based on the two parameters, s and w; we set s = . so that % of the population are stigmatized. The , stigmatized agents and , normal agents are randomly allocated over the torus. Out of those , normal agents, a proportion w are randomly selected to be conformists (CNFs) with the rest as compassionists (CMPs).

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The first issue of interest is how the proclivity for compassion and conformity in the population a ects the rate of social acceptance. To explore this issue, we consider w ∈ {0, 0.1, 0.2, . . . , 1}. The normal population consists only of compassionists when w = 0 and only of conformists when w = 1. For 0 < w < 1, the population has a mixture of the two types. .
The second issue is how the structure of the social networks a ects the di usion process and ultimately the steady-state rate of acceptance. The focus is on the degree of insularity of networks. One can think of agent i's neighborhood N (i) as defined by proximity in terms of geography (those in the same town) or education or income or some other trait. The issue is to what extent agent i's social network L(i) is largely drawn from that community or instead has links with those in other regions, educational levels, or income levels. The more that L(i) is drawn from N (i), the more insular are networks. .
More formally, we construct the network for each individual, L(i), by creating links from di erent regions of the space. For each agent i, most of her network connections will come from her own neighborhood, N (i), while the remaining connections come from outside, The range of the Moore neighborhood, n, is assumed to be in our experiments. Hence, each agent has 48(= (2×3+1) 2 −1) neighbors. A fraction (1−q) of his network connections comes from these agents in her neighborhood, while the remaining fraction q come from the other , agents. We consider q ∈ {0, 0.05, 0.1, 0.15, 0.2, 0.3, . . . , 0.9, 1}, where q = 0 is the benchmark case for our presentation. When q = 0, the networks are close to being regular lattice networks and are highly insular. As q rises above zero, an increasing fraction of an agent's network connections come from outside of his neighborhood. Our network model is a version of the well-known "stochastic block models" which explicitly introduce the concept of "local communities" into the generation of social networks; see Abbe ( ) for a recent and comprehensive survey of this group of generative network models. .
The size of an agent's network (i.e., the number of links) is specified to be l = 20. However, given that the networks are bidirectional, assigning a fixed network size for every agent in the population may not be feasible. Even when it is feasible, it is computationally intensive to construct the bidirectional networks of equal size across the entire population. Instead, we impose the condition that the networks for the population achieve a mean size of , which reduces the computational intensity considerably. Subject to the networks having mean size of , we then construct individual networks through a random matching algorithm assuming symmetry; if j is in i's network, then i is also in j's network. The random matching is done sequentially for each agent in the population until the average network size reaches . Figure shows the distribution of network sizes across agents from this procedure. . The initial conditions for the population are given by p r (the proportion of stigmatized agents who are revealed) and p a (the proportion of the normal agents who are accepting). We consider p r ∈ {0.15, 0.25} and p a ∈ {0, 0.025}. p a is kept low because if many normal agents were initially accepting then preliminary simulations showed that acceptance prevails almost irrespective of the other parameters. For our parameterizations, we want the obtaining of a norm of social acceptance to be challenging but feasible so that we can assess the conditions that promote it. Given that there are , stigmatized agents and , normal agents, when (p r , p a ) = (0.25, 0.025), the number of initially revealed stigmatized agents is 250(= 1000 * 0.25), which is of the same order of magnitude as the number of initially accepting normal agents, 225 = (9000 * 0.025). The baseline values and the set of all parameter values used in the simulations are provided in Table . parameter description baseline all values m population size , , n range of the Moore neighborhood, N (i) s proportion of the population with stigma . .
proportion of the normals who are conformists . . Given the initial conditions and the set of parameter values, we perform independent replications using a fresh set of random numbers for each run; specifically, the type and the network of each agent are re-randomized each time. In each replication, the acceptance/disclosure status of all agents is tracked as they respond to the changing state of their social networks. This is done for the first periods as that time horizon proved more than su icient for the social system to reach a steady state where the mean values of the endogenous variables remain constant over time. .
The two primary endogenous variables whose movements we follow for each replication k ∈ {1, . . . , 64} are the rate of acceptance by the normals, {RA t k } 300 t=0 , and the rate of disclosure of the stigmatized, {RD t k } 300 t=0 : number of all normal agents with α t i = A number of all normal agents ( ) number of all stigmatized agents with β t i = R number of all stigmatized agents ( ) .
For much of the analysis, we report their average values over the replications: Both endogenous variables reach their steady-states well before the terminal period. As such, we report their values at t = 300 as the steady state.

The Triadic Social Dynamic
. We start with the baseline set of parameters as specified in Table . Recall that % of the population (= 1, 000) are endowed with the stigmatizing trait. The initial seed population is specified at (p r , p a ) = (0.15, 0) so, at the outset of the process, stigmatized agents have revealed themselves (and are accepting), stigmatized agents are hidden (and are not accepting), and all , normal agents are not accepting.

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The thresholds for the three types of agents are set at: When an agent's network has links, a stigmatized agent discloses the attribute when at least (= 20 * 0.4) agents in his network have expressed themselves to be accepting of the stigma. A conformist is accepting of the stigma if (= 20 * 0.3) or more agents in his network are accepting of the stigma. Finally, a compassionist is accepting when one or more agents in her network have revealed themselves to have the stigmatizing trait.
. The thresholds have been chosen according to two criteria. First, we want thresholds which make the obtaining of social acceptance sensitive to the model's parameters so that insight can be acquired into the factors that promote or discourage acceptance of those who are distinctive. Having explored a range of thresholds, these particular values meet that criterion. Second, we want thresholds that are sensible. Our starting point is that people intrinsically care about those in their social networks, but pressure to conform can create a tension. A conformist is someone who is highly sensitive to those pressures. If he did not care at all about the person then one might imagine a conformist doing whatever the majority of those in his network are doing. But we suppose he does care about the stigmatized in his network and so we set the threshold for acceptance below . . τ CN F = 0.3 seems a reasonable value to capture that trade-o . In contrast, a compassionist is viewed as being highly independent and is largely driven by caring for others. With around links in one's network, τ CM P = 0.05 means that a compassionist just needs to know someone (or two) stigmatized people for them to support tolerance. It is di icult to determine a compelling value for a stigmatized person to reveal but requiring % of one's network to be accepting seems plausible. In any case, simulations have been conducted with other thresholds and, while the output does change, the qualitative insight is largely una ected. .
The mix of the conformists and compassionists is specified at w = 0.3 for the baseline analysis presented in this section. Hence, there are , conformists and , compassionists in the population of normal agents. Finally, the social networks for the individuals are highly insular in that q = 0, so all network connections are from an agent's neighborhood. .
The two endogenous variables, RA t k and RD t k , capture the aggregate behavior of the population at time t in replication k. The mean behavior of the population can then be summarized by the time series of the simulation outputs when they are averaged over the replications. Figure captures the time paths of the rates of acceptance and disclosure for: ) a single randomly chosen replication (le plots); and ) the average of the replications (right plots). Figure a shows the rate of acceptance among all normal agents, RA t , over the horizon of periods. The rate starts out at zero but rises quickly to approach the steady-state rate of over % by t = 70. Figure b reports the rate of disclosure by the stigmatized agents, which starts out at the seed rate of %, but quickly rises to stabilize at the rate of almost % by t = 70. For the two figures on the right showing the averages, the dashed curves above and below the solid curve (mean) capture the upper and lower limits of the % confidence interval around the mean. . To better understand the population dynamics, we decompose the rate of acceptance into acceptance by the conformists and by the compassionists. In Figure c, the dashed curve is the fraction of the conformists who accept (= number of conformists who accept/number of all conformists), and the solid curve is the fraction of the compassionists who accept (= number of compassionists who accept/number of all compassionists). With both fractions starting at zero, one can see that compassionists are initially accepting at a higher rate than conformists. However, the rate of increase in acceptance is soon higher for conformists and, eventually, conformists are more accepting than compassionists. In the steady state, the conformists achieve an acceptance rate of about %, while the compassionists achieve an acceptance rate of about %.
. The triadic dynamic between the three types can be seen by tracing the number of agents of each type who accept/reveal over time, especially during the transient stage prior to reaching the steady state. For a randomly chosen replication (run # ),  .
At t = 0, the dynamic is initiated by the stigmatized agents who start out in the revealed state. While none of the normals are initially accepting of the stigma, those compassionists who are connected to at least one (or two) of the revealed stigmatized agents are accepting as of t = 1 since their acceptance decisions are based on observing disclosures by the stigmatized in their networks. For this particular run, compassionists switched to accepting in t = 1 because of the revealed stigmatized agents. None of the conformists are accepting in t = 1. Their acceptance decisions are based on the observed acceptances in their networks and the only acceptances are from the stigmatized agents in t = 0, which evidently is not enough to cause any of the conformists to switch from opposition to acceptance. That changes come t = 2, for conformists are now accepting. This conversion is due to those compassionists who switched to acceptance in t = 1, which resulted in conformists finding enough agents in their networks accepting so that they now are accepting. The acceptances by the compassionists also induced fi een more stigmatized agents to reveal in t = 2. These additional disclosures induce more compassionists to accept in t = 3, which raises the number of accepting compassionists from to , . More significant is the increase in the number of conformists who are now accepting, which has risen from to . This is partly due to the additional stigmatized and compassionists who are accepting but is primarily due to the conformists who became accepting in t = 2. Recall that their acceptance decisions are based solely on acceptances by others in their networks. These conformity-based acceptances then, come t = 4, induce more stigmatized agents to reveal, which then causes more compassionists to accept, and significantly more conformists to accept. At that point, the triadic reinforcement process takes o . By t = 45, there are stigmatized agents who have revealed their attribute and , conformists who are accepting, which are their respective steady-state values. Given the steady-state number of disclosures by the stigmatized population, the compassionists achieve their steady-state in t = 46. Note that the steady state has , compassionists who are accepting, which means there are still , compassionists who are not accepting. These compassionists are not accepting because there are not enough revealed stigmatized agents in their networks. That could be due to there being no stigmatized agents or those that are in their networks remain hidden. .
As each agent in our model is assigned a specific location on the torus with a neighborhood-based social net-work, the triadic dynamic driving the di usion process has a spatial component. In Figure , we o er a series of snapshots taken from a single replication at various points in time over the horizon. At each time period t ∈ {1, 2, 6, 12, 24, 48}, the surface of the torus over which the agents are distributed is visualized as a x grid. The black dots in the plots in the le column are the positions of the conformists who are accepting, the plots in the middle column are the positions of the compassionists who are accepting, and the plots in the right column are for the agents with the stigmatizing trait where dots are black if they have revealed themselves and gray if they are hidden. Figure : Evolving attitudes toward stigma.

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The triadic dynamic can now be made more concrete by visualizing the evolving attitudes toward stigma at the individual agent level. Starting from the top row (t = 1), there are agents whose stigmatizing traits are initially revealed; these are the seed agents who are shown as the black dots in the right plot in the first row. Note that none of the conformists are accepting of the stigma at this point. The seed agents who are initially revealed initiate the social dynamic by inducing some compassionists (though no conformists) to react to their disclosure. In t = 1, those compassionists who are connected to one or more revealed agents (or two or more when there are more than links) switch to accepting as indicated by the black dots in the middle plot in the first row. Moving on to t = 2, the acceptances by the compassionists now induce some of the connected conformists to accept, while simultaneously motivating some of the stigmatized agents to reveal themselves. These additional disclosures further invite acceptances by both conformists and compassionists, which in turn induce disclosure by more stigmatized agents, and the triadic feedback mechanism continues from there. The rest of Figure shows the mutually reinforcing nature of the interactions among the three types as time goes on. For this particular run, we observe gradual di usion of disclosure by the stigmatized agents and acceptance by the normal agents. By t = 48 the population has already reached its steady state.

The E ect of the Mix of Compassion and Conformity on Social Acceptance
. In deciding how to treat those with distinctive attributes, a person can turn externally to others for guidance or internally to what she thinks is proper. Societies may di er in terms of the strength of orthodoxy and how much emphasis its members give to conforming, even when it may mean harming those who ones care about. In our model, this social heterogeneity is captured by the parameter w which is the proportion of normal agents who accept the stigmatized when such acceptance is su iciently common within their networks, and 1 − w is the proportion of normal agents who accept the stigmatized when there are at least a few in their networks. In this section, we investigate how the mix of compassion and conformity in the population a ects whether the society ends up tolerant of those who are distinct or instead ostracizes them. .
The initial exercise involved performing replications for each w ∈ {0, 0.1, 0.2, . . . , 1}, holding all other parameters at the baseline values including the network structure (at q = 0). Figure a shows the mean rates averaged over the replications. Figure b reports the distribution of the rates from all replications using a box-and-whisker chart, where the box represents the range of rates that are between % and % quantiles.
Given the baseline parameter configuration, Figure a shows that the mean rate of social acceptance of the stigmatized is maximized at w = 0.7, which means there are , conformists and , compassionists. Though not reported here, the rate of disclosure closely follows the rate of acceptance so that it also attains its maximum at w = 0.7. That it is a mixture of conformists and compassionists which maximizes acceptance was found for a wide range of parameterizations. .
Property : The rate of social acceptance is generally maximized when the population consists of both compassionists and conformists.
. Towards explaining Property , let us begin by considering the rate of acceptance at the two extreme values, w = 0 and w = 1. When w = 1, the population consists only of conformists. Recall from the previous section that, once there are enough agents who are accepting in the population, conformists convert to acceptance at a faster rate than compassionists. The problem when all normals are conformists is that, unless the initial population has many agents accepting, the absence of compassionists prevents the creation of a critical mass of acceptance to start inducing conformists to convert from opposition to acceptance. In contrast to conformists, who need many in their networks to accept before they will accept, compassionists will accept in response to only one or two stigmatized agents having revealed themselves. Hence, without compassionists, social acceptance fails to spread altogether because there is no initial acceptance by the compassionists that can subsequently induce acceptances by conformists. Replacing some of the conformists with compassionists (i.e., reducing w below ) can raise the acceptance level early on, which can then induce conformists to accept and eventually lead to the population-wide di usion of acceptance. For this reason, social acceptance is higher with a mix of compassionists and conformists than when all normal agents are conformists. . This intuition is verified by separately inspecting the acceptance behavior of the two types, which is reported in Figures and . In Figure , the time paths of the acceptance rate for conformists and for compassionists are plotted for two replications for w = 0.9. For run # in Figure a, almost % of the compassionists immediately accept in response to the stigmatized agents who are revealed in t = 0, but their acceptance rate stays at that level for the remainder of the horizon. These initial acceptances were insu icient to create the critical mass required to induce conformists to start accepting. In contrast, Figure b reports a case where the early acceptance by compassionists is su icient to induce some of the conformists to start accepting which then induces the stigmatized agents to reveal, and then there is a sequence of triadic reinforcements. Eventually, the rate of acceptance by the conformists surpasses that by the compassionists, reaching acceptance by all conformists at the steady state. Out of the replications for w = 0.9, replications failed to take o (as in  . w = 0.7; and (c) w = 0.6. It should be noted that successful takeo s were observed in out of replications when w = 0.8, and in out of replications when w = 0.7 or w = 0.6. The property to highlight in Figures and is that the acceptance rate among compassionists exceeds that of conformists early on in the di usion process, which substantiates the claim that acceptance by compassionists is a prerequisite for acceptances by conformists. w x: proportion of type-S in the networks of CMPs  .
While it has been shown and explained why acceptance is higher when there are at least some compassionists, more intriguing is why social acceptance is not maximized when all normal agents are compassionists. Note from Table that for w = 0 the median of the steady-state rates of acceptance is only .
(with the maximum value attained at . ); hence, only about % of the normals are accepting of the stigma, even though all of them are compassionists. This lack of acceptance is partly due to the small number of stigmatized individuals (only % of the population), which means that some compassionists will have no agents with the stigma in their social networks which, by itself, will prevent them from accepting. However, that rather mechanical reason for the lack of acceptance does not fully explain why many compassionists are not accepting. In Tables and , we report the rate of acceptance for compassionists (Table ) and conformists (Table ) depending on the proportion of stigmatized agents (whether revealed or hidden) in their social networks. For example, when w = 0, . % of the compassionists had networks comprised of to % of stigmatized agents. Note that the acceptance criterion for the compassionists is to accept if the proportion of revealed stigmatized types is at least %. The takeaway from Tables and is that many compassionists have ample stigmatized agents in their social networks but are still not accepting. For example, even when their social networks have -% of their links with stigmatized agents, only . % of compassionists are accepting. That some compassionists with multiple stigmatized agents in their social networks are not accepting means that those stigmatized agents are remaining hidden. The reason that the acceptance rate for compassionists is not as high as it could be is then a "coordination failure": Some compassionists are not accepting because the stigmatized agents in their social networks are remaining hidden, and those stigmatized agents are remaining hidden because there are not enough agents in their social networks who are accepting. .
By replacing some of those compassionists with conformists, some of these coordination failures can be corrected. Consider raising w from to . ; thereby replacing compassionists with an equal number of conformists. Some of those conformists will accept even when there are no stigmatized agents revealed, as long as enough agents are accepting. Those additional acceptances can induce the stigmatized agents to reveal and that can cause some compassionists to accept; in that way, conformists are disrupting the coordination failure between the stigmatized and the compassionate. To see that conformists can accept where compassionists would not, consider the acceptance rate for those normal agents with social networks for which -% of the links are with stigmatized agents (Tables and ). When there are no conformists (w = 0), compassionists accept at the rate of . %. When % of normal agents are conformists (w = 0.1), conformists are accepting at a higher rate of . %. More relevant, compassionists are now accepting at a higher rate; compare . % with . % when there are no conformists. While it is not universally the case that the optimal mix entails at least some conformists, it is very commonly true.

.
In sum, both those who conform -which may mean promoting tolerance or intolerance -and those who intrinsically care about people irrespective of social norms have a role to play in promoting tolerance. Without those who intrinsically care, it is di icult to break away from a norm of not accepting those who are distinct. How far acceptance (and also disclosure by the stigmatized) goes is limited however when society lacks conformists. Those who seek to conform help push acceptance further and, in particular, help break coordination failure in some networks with the stigmatized hiding their traits and compassionists not expressing support because there is no one in their networks who have revealed themselves to have the stigma.
. While some conformists are needed for maximal acceptance, it is noteworthy that the rate of social acceptance can drop precipitously in response to adding conformists. Examining Figure a and Table , note the sudden drop in the rate of acceptance when the fraction of conformists is increased from % to % to %. While w = 0.7 leads to maximal acceptance, w = 0.9 results in a rate of acceptance close to zero for % of the replications. In comparison, the acceptance rate only gradually declines as the fraction of conformists is lowered from % to % to %. The sensitivity of social acceptance to having too many conformists is a general (though not universal) property. It is then better for society to err on being overly compassionate than being overly conformist.
. Property : While both types are generally needed to achieve a high level of social acceptance, there can be a critical value of w such that the rate of acceptance drops sharply when the fraction of conformists exceeds that critical value. .
That the rate of social acceptance is highly sensitive to reducing the fraction of compassionists is due to the role they play. As previously explained, a critical mass of compassionists is crucial to induce conformists to start accepting. Short of that critical mass will prevent the triadic reinforcment dynamic from taking o . In contrast, the role of conformists is in spreading acceptance and breaking coordination failures between compassionists and stigmatized agents. That e ect is more linear so we observe that the rate of social acceptance is smoothly declining as the fraction of conformists is reduced below the value that maximizes tolerance. .
In concluding, Figure

The E ect of the Network Structure on Social Acceptance
. The next task is to investigate how social acceptance of the stigmatized is influenced by the structure of agents' social networks. To lay the groundwork, we begin by reviewing some previous findings on network e ects and di usion, and then relate our model to past models.
. It is well-accepted that network structure is a key determinant of di usion patterns. Granovetter ( ) identified the strength of weak links whereby agents may benefit from the paucity of mutual friends in their job search as otherwise distant nodes in the network can provide new information that improves the rate of diffusion: "[W]hatever is to be di used can reach a larger number of people, and traverse greater social distance (i.e., path length), when passed through weak ties rather than strong." (p. ) The "strength of weak ties" notion was given further support when Watts & Strogatz ( ) -WS from here on -discovered the "smallworld" networks in which the rate of di usion of information significantly increased with a small number of long random ties. However, Centola & Macy ( ), herea er CM , provided an important qualification to that finding. Previous work, including WS , considered settings in which an agent's exposure to one other agent exhibiting some conduct was su icient for that conduct to then be adopted. Referred to as a "simple contagion," CM re-examined the model of WS with a "complex contagion," which means that adoption of some conduct requires exposure to two or more agents exhibiting that conduct. Of particular note, CM found that the proportion of random (non-local) ties has a non-monotonic e ect when the contagion is complex. While a few randomized ties can improve propagation, more than that can significantly harm di usion. (This result is explained below.) .
Our model belongs to this class of social di usion models. The structural parameter for the social networks, q, is equivalent to the proportion of random ties in WS and CM . When q = 0, all the connections in an individual's network come from her own neighborhood N (i) (though within the neighborhood they are randomly chosen). More generally, a proportion q of an agent's network connections is drawn (randomly) from the population-at-large outside of the agent's immediate neighborhood. This is equivalent to the method of "random re-wiring of links" as implemented in WS and CM . Figure o ers a visualization of the global social network in our model for randomly constructed networks for q ∈ {0, 0.05, 0.1, 0.15}, where nodes are placed on the basis of their network position. When q = 0, the network is a long circular chain connecting the individual members of the population through their strictly local networks. As q rises, some of the links to one's own neighbors are replaced with random links to those external to the neighborhood. As a result, the tight circular property of the chained network gradually weakens and we observe an increasing number of direct links between agents positioned far apart from one another. .
In line with WS , there are two distinct channels through which the randomness parameter q influences di usion via social networks in our model. First, the extent of random ties in the social networks a ects the average length of the shortest paths between any two agents in the population. Let d ij denote the length of the shortest path between agent i and another agent j, where the "length of a path" is defined as the number of edges that the path contains. By taking an average of the path lengths between all pairs of agents i, j ∈ M , i = j, we compute the mean path length, P L(q): Second, the random ties a ect the local clustering coe icient, which is a measure of the extent to which one's friends are also friends of each other. More specifically, for a given agent i, the local clustering coe icient is: LCC i (q) = number of pairs of i's friends, (j, k), such that j ∈ L(k) and k ∈ L(j) number of pairs of i's friends ( ) .
Averaging over the population, we obtain the mean local clustering coe icient: . Based on the networks generated from the replications, P L(q) and LCC(q) were computed for all q ∈ {0, 0.05, 0.1, 0.15, 0.2, 0.3, . . . , 1}. Consistent with the results reported in WS , P L(q) and LCC(q) are monotonically decreasing in q. In Figure , P L(q)/P L(0) and LCC(q)/LCC(0) are plotted with respect to q, where, for purposes of comparison, the measures are normalized by their values at q = 0. The intuition behind the declining shape of the two curves is as follows. First, when q = 0, an individual's network consists of connections drawn strictly from within her local neighborhood. While any two agents in the population can typically be connected through a chain of local networks, the path length tends to be high on average. As q is raised, the social networks become more cross-cutting due to the random links; agents positioned spatially far apart from each another can be connected without going through a long chain of intermediate agents. Hence, mean path length is reduced. Second, the rise in distant links from an increase in q makes it less likely that an agent's direct links are also directly connected; that is, it becomes less likely that members of an agent's network are also members of each other's networks. With strictly local networks (q = 0), an agent is likely to share many links with her neighbors as their neighborhoods will extensively overlap; in other words, their friends are likely to know each other. As the network becomes increasingly random and global, the mutual ties become weaker and this is reflected in LCC(q) declining in q.
Figure : E ect of q on path length and local clustering coe icient. .
WS used the rapidly declining average path length as an explanation for their "small-world" network result: It only takes a few randomly connected (weak) links to substantially reduce the path length, and that speeds up the di usion process. Note that P L(q)/P L(0) drops sharply with respect to q in Figure . However, CM showed that this result only holds for simple contagions. When the contagion is complex, an agent needs to have multiple exposures to the conduct before it is adopted. If there is a high degree of local clustering, it becomes more likely that when an agent's link has the conduct then so does another link for that agent because those two links are likely to be connected and thereby influence each other. An increase in q then has two countervailing e ects on the di usion process when the contagion is complex: ) it reduces the mean path length, which speeds up the di usion ("small-world" e ect) required for adoption; and ) it reduces the local clustering coe icient, which weakens the extent of exposure necessary to exceed the threshold for adoption ("shared-friends" e ect). .
Figure shows that the two countervailing forces have a di erential impact on the di usion process as q is raised. Note that the path length declines steeply in the beginning and then very slowly a er that. In contrast, the local clustering coe icient declines much more gradually. The result of CM on complex contagions may be understood as the result of the small-world e ect dominating when q is low, and the shared-friends e ect dominating when q is su iciently high. When q is low, the steep drop in P L(q) (which promotes di usion) dominates the mild decline in LCC(q) (which hinders di usion); the marginal gain from the strengthened small-world e ect exceeds the marginal loss from the weakened shared-friends e ect. However, the steep drop in P L(q) is restricted to low values of q. When q rises further, the additional decline in P L(q) is very small, almost becoming negligible for high values of q. In that case, the shared-friends e ect is increasingly dominant.

.
Our model enriches the preceding models by allowing for heterogeneous contagions. CM assumed all agents have the same threshold, while the agents in our model have di erent thresholds depending on their types. The compassionists make their acceptance decisions purely on the observation of stigmatized agents in their networks. For the average network size of , exposure to a single revealed stigmatized agent is sufficient for a compassionist to accept; hence, the contagion for compassionists is simple. Conformists, on the other hand, base their acceptance on observing multiple acceptances by others in their networks. For an average network of links, a conformist will adopt acceptance only when exposed to at least six agents who are accepting of the stigmatized; hence, the contagion is complex. As w is increased, the fraction of conformists rises which means di usion is more dependent on the complex contagion. Also note that the contagion for stigmatized agents is complex as they require at least eight agents (for a network with links) to have adopted acceptance before they reveal. .
To see the impact of q on the rate of acceptance of the stigmatized, Table and Figure report the mean rate of acceptance for a range of values for q and w, given the initial conditions of (p r , p a ) = (0.25, 0.025). All other parameters are at their baseline values. The next property can be inferred from these results.  .
Property : Suppose the initial fraction of random ties, q, is low. Given enough number of compassionists in the population (i.e., w is low), a rise in the fraction of random ties in social networks tends to increase the rate of social acceptance. Alternatively, when there are few compassionists (i.e., w is high), a rise in the fraction of random ties lowers the rate of social acceptance. .
As an example to illustrate the property, compare w = 0 (all compassionists) and w = 0.9 (almost all conformists) as q is raised from to . so that some weak ties are introduced into networks. The rate of acceptance rises from . to . for w = 0, but falls from . to . for w = 0.9. When there are mostly conformists, the di usion in our model is largely driven by a complex contagion, for which the shared-friends e ect is dominant. In that case, moving to a network structure with weak ties reduces the extent of multiple exposure to acceptance and thereby leads to less acceptance by conformists. When w is instead low, the di usion mechanism is dominated by the simple contagion of compassionists. Now, the small-world e ect is crucial and, as a result, some weak ties (q is positive but low) promotes more exposure to revealed stigmatized agents throughout the population and thus encourages acceptance among compassionists. .
It is also worth highlighting Property in Figure . Note the steep drop in acceptance in response to a small increase in the fraction of conformists.
. Summing up, the insularity of social networks tends to promote tolerance in a society that is dominated by orthodoxy. It is more likely that the insularity will generate pockets of conformists who switch to acceptance because there are many common connections among them who are accepting. (Though it is important to remember from Section that having some people motivated by compassion, rather than orthodoxy, is crucial for initiating that process.) If instead society is full of people who are more driven by caring for those in their networks then tolerance is more widespread when there is less insularity of networks.

Conclusion
. The first contribution of this paper is providing a model that formalizes the conceptual framework of Go man ( ) within a larger social structure in which di erent actors interact through their social networks. This approach allows one to explore the micro-to-macro link that has been lacking in the stigma literature and do so in an explicit and systematic way. The second contribution is using this model to develop new insight into the determinants of a norm of tolerance with regards to those who possess an attribute that may be a source of stigma. .
The model o ered two innovations to previous models of di usion, both of which are motivated by the object of di usion being acceptance of a person as opposed to believing an idea or adopting a practice. As the object is a person, we allowed the adoption decision of some agents (compassionists) to be driven by sympathy for the stigmatized; they are accepting of them when their social networks include them. However, that acceptance requires knowing that the stigmatized are in their social networks. That leads us to the second innovation which is to allow those with the attribute to decide whether to reveal it or keep it hidden. Adding in the conforming type of agent typically present in di usion models -who are accepting of the stigmatized when enough others in their social networks are accepting -the model embodies a triadic dynamic among the stigmatized agents, the compassionists, and the conformists. .
Exploring this framework, the paper o ers two new findings regarding when a norm of tolerance prevails. One finding is that the maximal rate of acceptance is achieved when the population includes both those driven by sympathy for the stigmatized and those driven by the desire to conform. Some compassionists are needed to achieve a critical mass of acceptance that can then induce conformists to begin to accept. At the same time, some conformists are needed to break the coordination failure that can occur between compassionists -who are not accepting because the stigmatized agents in their social networks are hidden -and stigmatized agents -who are remaining hidden because compassionists (as well as conformists) are not accepting. .
A second finding is that the relationship between the insularity of social networks and a norm for tolerance depends on the disposition of agents in society. If compassion for the stigmatized is dominant then less insular networks contributes to the promotion of a norm of acceptance. That result is driven by the "small-worlds" e ect of Watts & Strogatz ( ). However, if conformity is dominant then more insularity is desirable. That result comes from the complexity of the contagion for conformists, as defined in Centola & Macy ( ), and the importance of the "shared-friends" e ect in di usion.

.
The framework presented and explored here was rather sparse in its structure. That was intentional in order to identify some basic insight into the triadic dynamic. It is to be emphasized that the framework is highly flexible and can encompass richer social networks and more diverse agent types. Consider, for example, the stigma of homosexuality. As is well documented, religion is strongly correlated with attitudes to gays, including on issues such as same-sex marriage. One could then build social networks by first locating churches and places of employment on the torus. With those in place, the social networks would be constructed by attaching people to churches and employers and making links more likely with those who attend the same church and work for the same employer. There could be di erential rates of church attendance in the space so as to capture, for example, regional di erences in the United States. While we had some agents driven solely by sympathy and others by conformity, richer forms of heterogeneity could be encompassed. An agent could adopt acceptance when a weighted average of the fraction of revealed stigmatized agents and the fraction of accepting agents in her social network exceeds some threshold. With such a rich structure, one could make predictive statements about the change in acceptance of gays over time and space. What can we say about the geographic spread of tolerance? How will it be correlated with agents' traits? What would happen in a world without churches? What types of events will tend to disrupt a norm of intolerance? Can we explain the recent rapid acceptance of same-sex marriage in the United States? The framework is flexible enough to take on many relevant questions related to the social dynamics of stigma.
tends to have a rather moderate to small sized networks. The stochastic block model we use generates a distribution which diverges from this type of distribution. Nevertheless, it does display a fairly wide variation in the network sizes of individuals.
For each parameter configuration considered in this paper, we performed sixty-four independent replications in parallel using cores at the Wharton HPCC (High Performance Computing Cluster). In this paper, we report only those runs for the sake of computational e iciency and analytical consistency. However, a much larger number of runs were carried out for a representative subset of parameter configurations. All the results reported in this paper are found to be robust to increasing the number of runs.
That periods were su icient is confirmed by running many replications for longer horizons.
Recall that the mean number of links is . So, these thresholds in terms of the number of agents can be slightly higher or lower than as described. Of particular note, for those compassionists who have more than links (but no more than links, which is always the case), it will take at least two revealed stigmatized agents in their networks to induce them to accept.
See Appendix B.
The grid lines are removed from these figures to improve the visual representation of the agents' states as they evolve within this space. As noted earlier, a torus is formed by extending the le edge to the right edge and the top edge to the bottom edge.
The horizontal line in the box represents the median, while the diamond represents the % confidence interval about the mean. The lines at the top and bottom are, respectively, the maximum and the minimum.
In Appendix A, it is reported that there is a very high positive correlation between the rate of acceptance and the rate of disclosure.
Note that compassionists with a low threshold of acceptance are needed to initiate the cascade of acceptance, while conformists with a high threshold (but with the potential for strong positive feedback) are needed to sustain the cascade. This is reminiscent of and consistent with the theory of collective action as presented in Oliver & Marwell ( ): "The problem of collective action [. . . ] is whether there is an organization or social network that has a subset of individuals who are interested and resourceful enough to provide the good when they act in concert [. . . ] What matters for successful mobilization is that there be enough people who are willing to participate and who are also reachable through social-influence networks. [. . . ] the theory of collective action explains why most action comes from a relatively small number of participants who make such big contributions to the cause that they know (or think they know) they can 'make a di erence.' " (pp. -) This insight was also anticipated in the earlier works by Olson ( ) and Hardin ( ) in the context of asymmetries in collective action.
The path may entail going through several individuals. For instance, if i and j know each other (i.e., they are in each other's network), then d ij = 1. If i and j do not know each other but they both know another agent k, then d ij = 2, so i knows k and k knows j. Typically, there are many di erent paths that can be taken to connect i and j. The "path length" is the shortest of all feasible paths.
It should be noted that the analysis carried out in CM (as well as WS ) is restricted to those cases where the entire population reaches full adoption over the relevant horizon. Their interest is in measuring the "time to saturation" of the population. In contrast, we focus on the rate of acceptance (or adoption), allowing for the possibility that the steady-state may not involve saturation.
When a compassionist has more than links then she requires exposure to at least two revealed stigmatized agents. Though that is then a complex contagion, the more general point is that the contagion associated with conformists is more complex than that associated with compassionists.
That the results are robust to initial conditions is shown in Appendix B for other values of p r and p a .