One body, many identities: a mathematical model of multidimensional segregation dynamics

ABSTRACT Individuals inhabit multiple social roles – citizen, neighbor, worker – but make residential decisions with a single body. Classic segregation models assume fixed, singular identities. This paper extends Schelling’s model by introducing agents with multidimensional identities and salience rules that determine which identity guides behavior. Across 1,920 simulations, we show that coordination failures produce persistent dissatisfaction and instability, while attention alignment can reduce or concentrate segregation. These findings challenge the assumption that identity complexity promotes integration. Instead, spatial outcomes depend on how identity salience is cognitively and socially managed. The model contributes to mathematical sociology and formal intersectionality by revealing how behavioral coordination mechanisms shape emergent patterns of segregation. 1. Introduction We live complex lives. Each of us moves between roles and contexts: parent, colleague, neighbor, believer, voter, consumer. In each of these domains, different aspects of our identity come to the fore – language, income, religion, gender, profession. Yet despite this fluidity, we cannot segment ourselves completely. We bring the same body, the same face, the same address into every encounter. Identity is not a menu we can toggle on and off depending on the setting; it is layered, persistent, and often socially visible (Goffman, Citation1959; Jenkins, Citation2014). As Georg Simmel (Citation1908/1950) observed, individuals derive their identity from the intersection of multiple “social circles” – overlapping affiliations based on attributes and associations. In modern life, the number and heterogeneity of these circles has only expanded: we now hold more roles, maintain more non-overlapping relationships, and move between social spaces that were once bounded by geography, kinship, or work. While modernity has enabled diverse social configurations, it has also increased the number of identity dimensions individuals carry into social life – and thus, the potential axes of difference. Classic models of residential segregation, most famously Schelling’s (Citation1971), have illustrated how even weak preferences for similarity on a single attribute can produce stark macro-level patterns of spatial separation (see also Clark, Citation1991; Fossett, Citation2006; Laurie & Jaggi, Citation2003 for key extensions). Recent empirical work has documented the persistence of residential segregation across multiple dimensions simultaneously (Iceland & Wilkes, Citation2006; Logan & Stults, Citation2011; Massey & Denton, Citation1993; Reardon & Yancey, Citation2009). But these models typically assume that individuals act based on a fixed identity dimension – often race or ethnicity. They do not account for the fact that individuals possess multiple persistent attributes, nor that the salience of these attributes can shift dynamically across contexts (Anthias, Citation2013; Roccas & Brewer, Citation2002; Verkuyten, Citation2016). This paper extends Schelling’s framework by formalizing what we term embodied multidimensionality: the mathematical constraint that agents with vector-valued identities must make scalar location decisions (i.e., individuals cannot “split” themselves across spaces despite holding multiple identities). Our central theoretical innovation is a perturbation theorem proving that any movement decision based on one identity dimension necessarily creates disruptions across all other dimensions. The theorem does not predict outcomes on its own, but establishes structural constraints whose consequences depend on coordination mechanisms – which our simulations then systematically explore. We introduce two core innovations. First, agents possess multiple categorical identity attributes that remain fixed throughout the simulation, reflecting the persistence of visible social categories. Second, they use one of six explicitly formalized salience rules to determine which identity dimension guides their residential preferences at any given time. These mechanisms capture different cognitive and social processes, from fixed focus to social influence to strategic minority salience. Our central theoretical claim is that multidimensional sorting creates coordination challenges that make spatial outcomes contingent on salience mechanisms. These coordination challenges arise because agents with shared attributes may fail to cluster if they focus attention on different identity dimensions. When coordination fails, movements based on one dimension create cross-cutting disruptions across other dimensions. When coordination succeeds, it can concentrate segregation on popular dimensions while leaving others integrated, creating dimensional inequality. This calls into question the intuitive assumption that diversity of preferences straightforwardly reduces segregation, and suggests that identity complexity itself may sustain spatial separation through ongoing displacement effects. The paper contributes to mathematical sociology in three ways. Most fundamentally, it maps how different salience coordination mechanisms generate distinct segregation regimes under identical structural conditions – showing that the same degree of identity complexity can produce radically different spatial outcomes depending on how agents process their multiple identities. Conceptually, it demonstrates how formal modeling can reveal counterintuitive relationships between individual complexity and collective patterns. Methodologically, it provides a tractable framework for analyzing multidimensional social processes while maintaining the analytical clarity that characterizes existing segregation models. Substantively, it offers new insights into how modern forms of identity – shaped by multiple, fragmented roles – affect patterns of integration and inequality in spatial contexts. 2. Theoretical framework and literature 2.1. Schelling’s model and mathematical extensions Schelling’s (Citation1971) dynamic model of segregation remains foundational in demonstrating unintended macro-level consequences of micro-level decisions. In his framework, agents evaluate satisfaction based on the presence of similar others in their neighborhood, and even relatively low thresholds for similarity drive the system toward spatial segregation. The model’s mathematical elegance lies in its simplicity: agents optimize a binary satisfaction function over a single attribute, producing complex emergent patterns from straightforward local rules. Extensions of segregation models have explored network effects (Henry et al., Citation2011), economic mechanisms (Benenson et al., Citation2002), and preference heterogeneity (Fossett, Citation2006), but few address multidimensional identity systematically. Subsequent mathematical extensions have incorporated heterogeneous preferences (Pollicott & Weiss, Citation2001), threshold distributions (Pancs & Vriend, Citation2007), and economic constraints (Zhang, Citation2004), but most retain the single-dimension assumption. This contrasts with empirical work on homophily, which demonstrates that individuals form ties along multiple dimensions simultaneously (Block & Grund, Citation2014; McPherson et al., Citation2001), suggesting that segregation models must account for multidimensional identity to capture real-world sorting processes. This simplification limits the models’ capacity to represent the intersectional nature of real-world residential sorting, where segregation reflects overlapping dimensions such as ethnicity, class, and religion, shaped by both structural inequalities and individual preferences (Peach, Citation1996; Reardon & Bischoff, Citation2011). Benard and Willer’s (Citation2007) status-based model introduces multiple identity axes and demonstrates how their correlation affects sorting outcomes. Their analysis suggests that segregation intensifies when identity markers correlate – a finding that motivates our systematic exploration of dimension interactions. While Benard and Willer (Citation2007) demonstrate how correlated identities amplify segregation, their model assumes agents optimize over all dimensions simultaneously – a computationally intensive and behaviorally implausible assumption as dimensionality grows. In contrast, our salience mechanisms reflect selective attention processes documented in cognitive psychology (Gigerenzer & Selten, Citation2001), where individuals focus on one dimension at a time rather than conducting exhaustive multi-attribute evaluations. This approach aligns with empirical evidence on situational identity activation (Oyserman, Citation2009) and cognitive load effects on identity processing (Shih et al., Citation1999). By formalizing salience as an endogenous process governed by explicit rules, our model systematically explores how assumptions about identity activation shape collective spatial outcomes. 2.2. Multidimensional social space and identity salience Peter Blau’s (Citation1977) macrostructural theory provides the theoretical foundation for understanding multidimensional identity in spatial contexts. Blau conceptualized society as a multidimensional social space where each axis represents a distinct identity or structural feature, and the likelihood of social interaction decreases with distance in this space. While Blau’s original formulation was structural and probabilistic, our model adds a behavioral layer by allowing agents to actively select which dimensions to emphasize in local decision-making. This behavioral dimension addresses a key limitation in existing multidimensional models: the assumption that all identity attributes remain equally salient across contexts. Research in social psychology demonstrates that identity salience is fluid, influenced by social context, cognitive framing, and group dynamics (Brewer, Citation1991; Oyserman, Citation2009). This aligns with extensive research on identity switching (Hong et al., Citation2000; LaFromboise et al., Citation1993), contextual identity salience (Reed et al., Citation2010; Verkuyten, Citation2016), and cognitive load effects on identity processing (Shih et al., Citation1999). Intersectionality theory similarly emphasizes that while individuals occupy multiple social positions simultaneously, the relative importance of these positions varies across situations and interactions (Cho et al., Citation2013; Crenshaw, Citation1989). Research in analytical sociology has begun formalizing these insights. Baldassarri and Bearman (Citation2007) demonstrate how political identities can be situationally activated, while Hedström and Ylikoski (Citation2010) emphasize the importance of situational mechanisms in explaining social phenomena. Our approach builds on this tradition by treating identity salience as an endogenous process governed by explicit mathematical rules. We specify six distinct salience mechanisms representing different cognitive and social processes. This allows systematic exploration of how assumptions about identity activation shape collective spatial outcomes. 2.3. The coordination challenge in multidimensional space We use the term “coordination” in two distinct senses throughout this paper, and it is important to distinguish them. Intra-personal coordination refers to how an individual agent decides which identity dimension is active at a given time – this is what the Fixed, Random, Weighted Composite, and Minority Focus rules formalize, each representing a different individual cognitive strategy. Inter-personal (social) coordination refers to how agents’ salience patterns become aligned across the population through interaction – this is what Neighbor Influence and Success Imitation rules produce, where local social environments pull agents toward shared focal dimensions. Both forms matter for spatial outcomes, but through different mechanisms. In single-dimension systems, agents with shared attributes can readily identify and cluster with similar others. Multidimensional identity fundamentally alters this dynamic because agents must decide which dimension guides their residential preferences. Even agents who share multiple attributes may fail to recognize each other as similar if they focus on different dimensions. Consider agents A and B, both with identity vector (Black, Working-class, Catholic). If A focuses on race while B focuses on class, A may move toward racial homogeneity in an area that happens to be class-heterogeneous, while B moves toward class homogeneity in a racially diverse area. Their movements, though individually rational, fail to reinforce each other and may actually disrupt clustering patterns for both race and class. This coordination challenge creates what we formalize as systemic perturbations: mathematically inevitable disruptions that arise when movement decisions based on one dimension necessarily affect spatial distributions across all other dimensions. These perturbations do not universally amplify segregation. Instead, they make spatial outcomes contingent on coordination mechanisms that determine how agents navigate their multiple identities. The implications extend beyond individual decision-making to collective dynamics. When agents coordinate their salience (through social influence or institutional arrangements), perturbations can be limited, while segregation concentrates on one dimension, leaving others mixed. When agents fail to coordinate (through attention fragmentation or chaotic switching), perturbations create cross-cutting disruptions that can increase segregation across all dimensions. 2.4. Decision rules and behavioral mechanisms Mathematical sociology has increasingly recognized the importance of explicit behavioral assumptions in formal models (Coleman, Citation1990; Hedström, Citation2005; Young, Citation1998). The choice of decision rules fundamentally shapes model outcomes, yet many models treat these choices as technical details rather than substantive theoretical commitments (Macy & Willer, Citation2002). Our approach foregrounds this issue by systematically comparing alternative salience mechanisms. The six rules we examine reflect different traditions in behavioral modeling. Fixed salience corresponds to classical rational choice assumptions of stable preferences. Random selection captures bounded rationality and cognitive limitations emphasized in behavioral economics (Gigerenzer & Selten Citation2001; Kahneman, Citation2011). Social influence mechanisms reflect insights from social learning theory and conform to extensive evidence of peer effects in residential decisions (Clark & Fossett, Citation2008), where local context dynamically shapes preferences and mobility patterns. Spatial peer effects in residential choice are well-documented (Card et al., Citation2008; Cutler et al., Citation2008; Krysan & Crowder, Citation2017), while social learning in neighborhood preferences has been demonstrated through both survey (Krysan, Citation2002) and experimental research (Bobo & Zubrinsky, Citation1996). Strategic rules like minority focus incorporate insights from social identity theory about distinctiveness motivation and identity threat responses (Brewer, Citation1991; Tajfel & Turner, Citation1979). By formalizing these alternatives mathematically, we can isolate their effects on system-level outcomes and test theoretical predictions about how different behavioral assumptions generate different segregation patterns. 2.5. Identity fluidity vs. coordination: a theoretical tension A fundamental tension runs through contemporary thinking about identity and spatial outcomes. On one side, identity fluidity – the freedom to switch between and express multiple identity dimensions – is celebrated as a marker of personal autonomy and social progress (Giddens, Citation1991; Sen, Citation2006). On the other, liberal and multicultural frameworks assume that recognizing intersecting identities promotes tolerance and integration (Crenshaw, Citation1989). Our model formalizes a coordination tension that cuts across both positions: fluidity without coordination may produce fragmentation and instability, while coordination may reduce instability but at the cost of concentrating segregation on whichever dimension agents collectively attend to. Neither fluidity nor coordination is straightforwardly desirable. Our theoretical framework challenges these assumptions by highlighting the coordination challenges that identity complexity creates. When individuals can freely switch between identity dimensions without social coordination, their movements may create cross-cutting disruptions that prevent stable community formation around any identity. Conversely, when identity salience is coordinated through social influence or institutional arrangements, it can amplify segregation by concentrating attention on particular dimensions. This creates what we term the coordination tension: stable spatial configurations in diverse societies may depend on aligning rather than diversifying identity activation. The policy implications are significant, suggesting that interventions must address not only individual preferences but also the social and institutional mechanisms through which identity salience is coordinated. 3. Mathematical model 3.1. Basic structure and notation Let represent a spatial grid where denotes the set of locations and defines neighborhood relationships. Each agent possesses an identity vector where each represents a categorical attribute drawn from finite set , and represents the set of all agents in the system. The fundamental constraint is that despite multidimensional identity, each agent occupies exactly one location at time . For agent at location , the neighborhood represents the set of locations within interaction distance. The neighborhood composition for attribute is defined as: This formulation captures the local similarity that drives agent satisfaction while maintaining the mathematical precision required for theoretical analysis. 3.2. The perturbation theorem The mathematical core of our argument can be stated formally: Theorem 1 (Systemic Perturbations): In multidimensional systems with , any movement decision by agent based on dimension necessarily creates perturbations across all other dimensions . Formal Statement: Let agent possess identity vector . Movement from location to based on dissatisfaction with dimension creates perturbations: where represents neighborhood composition for attribute at location and time . Proof: Agent ‘s movement simultaneously: (1) removes vector from neighborhood , altering ; (2) adds vector to neighborhood , altering ; and (3) potentially triggers subsequent relocations by agents in both neighborhoods focusing on any dimension . Corollary: The effects of these perturbations depend critically on coordination mechanisms. When agents coordinate salience focus, perturbations can be channeled to amplify segregation on popular dimensions. When coordination fails, perturbations create cross-cutting disruptions that can reduce segregation across all dimensions. Key Theoretical Insight: These perturbations do not universally amplify segregation. Instead, they make spatial outcomes contingent on coordination mechanisms that determine how agents collectively navigate their multiple identities. This insight fundamentally revises our understanding of multidimensional sorting by showing that the behavioral management of identity complexity is as important as structural diversity itself. This theorem formalizes the intuition that multidimensional identity creates unavoidable interdependencies between spatial distributions across different social categories. These interdependencies generate the systemic perturbations that distinguish multidimensional from single-dimension segregation dynamics, but their effects depend on how agents coordinate their attention across identity dimensions. 3.3. Identity salience functions The central behavioral innovation lies in formalizing how agents select which identity dimension guides residential decisions. Let represent the salience function mapping agents and time to active identity dimensions. We specify six distinct functions representing different coordination mechanisms (see ): Fixed Salience assigns each agent a permanent focus: . This represents cognitive specialization and strong identity commitment, corresponding to classical assumptions about stable preferences in choice theory. The rule aligns with Stryker’s (Citation1980) identity theory, which posits a stable hierarchy of identity salience that persists across situations, as well as with institutional constraints that emphasize specific attributes. The rule captures individuals who maintain consistent attention to particular identity dimensions throughout their residential decisions. Random Salience involves independent selection: . Random Salience is not intended as a direct model of realistic cognitive switching, but as a theoretical null case representing maximal unpredictability in identity activation. It bounds the space of possibilities and allows us to isolate what happens when no coordination mechanism whatsoever is present. The mechanism also has a partial behavioral interpretation in research on cognitive limitations in multi-attribute processing (Gigerenzer & Selten, Citation2001), where individuals switch focus based on immediate contextual cues without systematic coordination. Neighbor Influence follows local majority rule: . This mechanism reflects conformity to peer salience and social learning about which identity dimensions matter in specific contexts. The rule captures how local social environments shape which aspects of identity become focal, reflecting extensive research on peer effects in residential choice (Bruch & Mare, Citation2006) and social influence on preference formation (Card et al., Citation2008). Minority Focus emphasizes local distinctiveness: . This rule captures identity threat responses and distinctiveness motivation from social identity theory (Brewer, Citation1991; Tajfel & Turner, Citation1979). When individuals find themselves in environments where they constitute a minority on particular dimensions, they may shift attention toward those dimensions, seeking to either find more similar others or maintain distinctiveness. This rule captures a specific reactive dynamic rather than a general identity selection principle. It is most applicable to visible, rigid, or low-status identity dimensions where threat responses are most pronounced; it is not intended as a universal claim about all identity dimensions. Weighted Composite avoids discrete selection: satisfaction is defined as with . This represents comprehensive assessment with declining attention allocation across dimensions. Rather than focusing on single dimensions, agents simultaneously consider multiple attributes but weight them differentially, reflecting cognitive models of multi-attribute decision-making where attention is distributed but hierarchically organized. Success Imitation copies the salience rule of the most satisfied neighbor: . This mechanism reflects learning from successful peers and social proof dynamics (Bandura, Citation1977). Agents observe which neighbors appear most satisfied with their residential situations and adopt their salience strategies, creating potential for rapid diffusion of effective coordination patterns. Each rule represents a distinct mathematical formalization of psychological and social processes documented in the behavioral literature, allowing systematic comparison of their implications for spatial sorting. Salience is updated every time step for all agents, regardless of their satisfaction status. This reflects the continuous nature of social cognition and contextual identity activation documented in social psychology (Oyserman, Citation2009). Even satisfied agents may shift their attention between identity dimensions in response to changing neighborhood composition or social influence from neighbors. This design choice captures the dynamic interplay between identity salience and spatial patterns, allowing satisfied agents to potentially become dissatisfied as their focus shifts, thereby sustaining ongoing mobility even in seemingly stable configurations. 3.4. Satisfaction and movement rules Agent satisfaction depends on the active identity dimension. For agent with , satisfaction is binary: if , and otherwise, where represents the tolerance threshold. Note that represents the minimum proportion of similar neighbors required for satisfaction: is therefore more demanding than , not more tolerant. If multiple dimensions are equally salient – for example, in Minority Focus when two dimensions tie for minimum local representation – the agent selects uniformly at random among them. Crucially, agent satisfaction depends only on the currently active identity dimension, not on simultaneous evaluation across all dimensions (unless weighted composite is selected). This design choice reflects cognitive limitations documented in psychology (Gigerenzer & Selten, Citation2001) and ensures that our results are not artifacts of increased dissatisfaction opportunities with higher dimensionality. An agent with vector = (Black, Female, Professor) focusing on race (dimension 1) evaluates only racial composition, temporarily ignoring gender and class considerations. This approach isolates the coordination effects from simple probability increases, demonstrating that multidimensional dynamics emerge even when individual agents face constant likelihood of dissatisfaction across dimensionality conditions. Alternative definitions – such as cumulative dissatisfaction across all dimensions – would confound coordination effects with mechanical increases in unhappiness probability, obscuring the core theoretical insight. Unsatisfied agents () relocate to a random vacant location . This simple movement rule focuses attention on identity salience effects rather than housing market dynamics or search costs. While additional complexity could be introduced, the current formulation maintains analytical tractability while capturing the essential dynamics of interest. 4. Simulation analysis 4.1. Experimental design We conduct comprehensive simulation analysis using a factorial design that systematically varies key parameters while maintaining mathematical precision. The experimental setup involves 1,920 simulation runs across four factors: identity dimensions , salience rules (all six mechanisms), tolerance thresholds , and 20 replications per condition to ensure statistical reliability. Each simulation runs on a 40 × 40 toroidal grid with 1,280 agents, maintaining 80% occupancy to ensure meaningful neighborhood interactions while preserving sufficient movement opportunities. The toroidal topology eliminates edge effects while preserving spatial structure. Binary identity attributes are assigned randomly with equal probability across categories, creating orthogonal dimensions that isolate the effects of coordination mechanisms from correlation-based clustering. Agents act in sequence (i.e., asynchronously): at each time step, a single randomly selected unsatisfied agent moves, and salience is updated for all agents before the next agent is evaluated. Unsatisfied agents relocate to a randomly chosen vacant location on the grid. The Moore neighborhood (radius-1, 8 surrounding cells) determines local composition. The model employs tolerance thresholds that map directly to empirical neighborhood preferences observed in residential choice studies. The tolerance parameter captures varying degrees of similarity preference, where represents highly tolerant agents requiring only 20% similar neighbors, reflects moderate tolerance with a 30% threshold, indicates selective preferences, and represents demanding similarity requirements. These parameter values align with empirical studies demonstrating that most individuals prefer neighborhoods with 20–50% similar neighbors (Bobo & Zubrinsky, Citation1996; Charles, Citation2006; Krysan, Citation2002). 4.1.1. Measurement innovation: capturing multidimensional dynamics We employ a single segregation measure to capture the core dynamics of multidimensional sorting: Dimension-Specific Segregation computes mean homogeneity within each dimension: where represents the proportion of agent ’s neighbors who share attribute at time . This measure captures the average level of local similarity that agents experience for identity dimension within their neighborhoods. Segregation Spread measures dimensional inequality within each run: For each run, this captures the difference between the most and least segregated dimensions. When aggregated across runs, this reveals whether coordination mechanisms create uneven sorting patterns where some dimensions become highly segregated while others remain integrated, or whether they produce uniform effects across all dimensions. This measurement approach directly captures the effects of our perturbation theorem. Crucially, low segregation spread indicates that all dimensions have similar segregation levels, but this similarity can occur at either high or low absolute values. We therefore always interpret spread alongside mean segregation (). Because spread is defined as the difference between the maximum and minimum dimension-specific segregation, it directly captures the range of outcomes across dimensions. We therefore do not report maximum and minimum values separately, but note that all interpretations are consistent with the underlying extrema. For Fixed Salience, low spread coincides with declining mean segregation – hence attention fragmentation reduces segregation across dimensions relatively uniformly. For Neighbor Influence, high spread accompanies stable mean segregation, indicating that coordination concentrates segregation on popular dimensions while leaving others more integrated. System convergence is defined operationally as achieving fewer than 1% unhappy agents or reaching the 10,000-tick computational timeout. This threshold balances computational tractability with meaningful stability detection, as preliminary analysis revealed that genuine equilibria typically stabilize within this range. The convergence criterion ensures that observed segregation patterns represent stable residential configurations rather than transient sorting dynamics. For analyses of overall segregation levels, we average across dimensions; for analyses of dimensional inequality, we report spread (max – min). Both are reported in respectively, and should always be interpreted together. 4.2. Key empirical findings The simulation results provide compelling empirical support for our theoretical framework while revealing the critical importance of coordination mechanisms in shaping multidimensional sorting dynamics. Our analysis demonstrates that the relationship between identity complexity and spatial outcomes cannot be understood without explicit attention to how agents navigate their multiple identities in residential decisions. 4.2.1. The breakdown of stable equilibria The convergence analysis (see ) reveals that dimensional effects on system stability are far more selective than one might intuitively expect. Contrary to the intuition that more identity dimensions necessarily destabilize sorting, we observe that specific salience mechanisms drive instability while others maintain robust convergence properties even in high-dimensional systems. Three salience rules – Fixed, Neighbor Influence, and Success Imitation – demonstrate remarkable resilience to dimensional complexity. These mechanisms converge reliably across all dimensionalities tested, typically within reasonable time frames that increase only modestly with additional identity dimensions. This finding suggests that certain approaches to managing multidimensional identity can effectively insulate residential sorting processes from the perturbation effects we theorized. The dramatic instability effects emerge primarily from two specific mechanisms: Random and Minority Focus salience rules. These exhibit the exponential convergence delays and eventual breakdown that our perturbation theorem predicts. Random salience creates instability through continuous cognitive switching between identity dimensions, preventing agents from sustaining focus long enough to achieve stable residential configurations. Minority Focus generates instability through a different mechanism – by systematically directing attention toward locally underrepresented attributes, it creates ongoing displacement pressures that prevent any neighborhood composition from achieving equilibrium. Weighted Composite occupies an intermediate position, showing modest increases in convergence time with dimensionality but maintaining overall stability. This suggests that distributing attention across multiple dimensions simultaneously can partially buffer against perturbation effects, though not as effectively as the focused attention strategies employed by the most stable rules. This pattern fundamentally revises our theoretical understanding of multidimensional sorting dynamics. Rather than dimensional complexity inevitably creating system instability, our results suggest that the behavioral mechanisms agents use to navigate multiple identities determine whether dimensional complexity becomes problematic. Some cognitive and social processes appear inherently compatible with multidimensional residential decision-making, while others amplify the perturbation effects that our mathematical framework identifies. 4.2.2. Three pathways: fragmentation, coordination, and volatility The segregation outcomes in show the overall level of segregation at convergence/time-out averaged over dimensions. shows the spread of of segregation over dimensions. Results reveal that multidimensional identity effects cannot be captured by any single theoretical relationship. Instead, different salience mechanisms generate entirely distinct pathways that reflect the underlying cognitive and social processes they represent. This finding validates our decision to formalize multiple behavioral alternatives rather than assuming a universal response to dimensional complexity. 4.2.2.1. Pathway 1: attention fragmentation (Fixed salience) Fixed Salience produces the most counterintuitive results: segregation decreases systematically with additional identity dimensions across all dimensions (see ). In single-dimension systems, Fixed Salience achieves segregation levels comparable to other mechanisms (~0.72). However, by four dimensions, segregation drops to ~0.55 across all dimensions, with minimal segregation spread (~0.02, see ). This pattern reflects attention fragmentation: when agents focus on different dimensions, their movements create cross-cutting disruptions that prevent any dimension from achieving high segregation. An agent focused on race moves toward racial homogeneity but may disrupt class-based clustering; an agent focused on religion moves toward religious similarity but disrupts ethnic patterns. The mathematical mechanism is straightforward: if agents are distributed equally across n dimensions, then on average only 1/n of agents pay attention to any given dimension, reducing the coordination necessary for strong segregation on that dimension. It is worth noting that from the individual agent’s perspective, Fixed Salience represents concentration of attention on one dimension; from the system’s perspective, however, diversity in which dimension each agent focuses on produces fragmentation of collective attention across all dimensions. The policy implications are profound: promoting individual choice in identity salience – encouraging people to “be themselves” and focus on personally meaningful dimensions – may actually reduce segregation through this fragmentation mechanism. However, it comes at the cost of preventing meaningful identity-based community formation around any particular dimension. 4.2.2.2. Pathway 2: coordinated amplification (social learning mechanisms) Neighbor Influence and Success Imitation demonstrate how coordination mechanisms can maintain segregation despite multidimensional complexity. These rules achieve moderate average segregation levels (~0.57 by four dimensions, see ) that remain stable across increasing dimensionality, indicating that social coordination prevents the attention fragmentation that reduces segregation in other mechanisms. While average segregation differs little from Fixed Salience (~0.55), the key distinction lies in the spread (): Neighbor Influence generates dramatically higher dimensional inequality, indicating that coordination creates qualitatively different outcomes by concentrating segregation on popular dimensions while leaving others relatively mixed. The coordination occurs through social influence: when agents observe neighbors focusing on specific dimensions, they adopt similar salience patterns, creating collective attention toward particular identity attributes. This coordination enables effective clustering on popular dimensions but leaves unpopular dimensions relatively mixed, generating the dimensional inequality that characterizes this pathway (see ). Neighbor Influence operates through local conformity: agents adopt the most common salience focus in their immediate neighborhood, creating spatial clusters of attention that reinforce segregation on particular dimensions. Success Imitation operates through performance-based learning: agents copy the salience strategies of their most satisfied neighbors, potentially creating rapid diffusion of effective coordination patterns across the system. The implications suggest that social coordination mechanisms, while promoting stability, risk amplifying segregation by concentrating attention on particular identity dimensions. Communities with strong social influence networks may achieve integration on some dimensions while experiencing extreme segregation on others, creating complex patterns of inclusion and exclusion. 4.2.2.3. Pathway 3: volatile instability (Random and Minority focus) Random and Minority Focus create persistent instability through uncoordinated salience switching, often achieving higher segregation levels than single-dimension systems while preventing convergence (see ). These mechanisms sustain high segregation across multiple dimensions (see ) while creating ongoing dissatisfaction that drives continued mobility. Random salience creates instability through continuous cognitive switching between identity dimensions, preventing agents from sustaining focus long enough to achieve stable residential configurations. The mathematical effect is to create ongoing perturbations as agents continuously shift their evaluation criteria, disrupting emerging spatial patterns before they can stabilize. Minority Focus generates instability through systematic attention toward locally underrepresented attributes, creating ongoing displacement pressures as agents seek to either find similar others or flee areas where they constitute minorities. This creates cascading relocations as movements based on minority status on one dimension affect minority status on other dimensions, generating the persistent mobility that characterizes this pathway. Importantly, Minority Focus is not fully uncoordinated in a local sense – agents do respond systematically to their immediate environment – but it is globally uncoordinated, because agents react independently to their own local minority status without aligning on any shared focal dimension. Both mechanisms demonstrate how uncoordinated identity processing can amplify the perturbation effects we theorized, creating systems that sustain higher segregation than single-dimension equivalents while preventing stable equilibrium. The result is ongoing residential instability that may characterize diverse societies lacking effective coordination mechanisms. Weighted Composite produces intermediate outcomes that do not cleanly fit any of the three pathways. It achieves moderate average segregation and moderate spread, with modest but not dramatic increases in convergence time. This suggests that simultaneously weighting multiple identity dimensions partially buffers against the perturbation effects that destabilize Random and Minority Focus, without fully coordinating attention in the manner of Neighbor Influence. The hierarchical weighting structure provides some directionality while preserving multi-attribute sensitivity, placing it between focused and volatile strategies. 4.2.3. The persistence of dissatisfaction The temporal dynamics (see ) reveal how systemic perturbations manifest as persistent agent mobility rather than simply delayed equilibrium. Single-dimension systems exhibit the smooth exponential decay characteristic of stable dynamical systems, with unhappiness dropping rapidly from initial random levels to near zero within hundreds of time steps. This pattern confirms that classical Schelling dynamics successfully resolve residential dissatisfaction when agents optimize over single attributes. However, the introduction of additional identity dimensions fundamentally alters these dynamics in ways that directly reflect our perturbation predictions. The transition from rapid convergence to persistent oscillation occurs gradually but unmistakably as dimensionality increases. Two-dimension systems maintain the basic convergence pattern but require longer time horizons, reflecting the increased coordination challenges we theorized. Three-dimension systems begin showing the irregular convergence curves that suggest ongoing perturbations are disrupting residential satisfaction even as the system approaches apparent equilibrium. Four-dimension systems provide the clearest evidence for our theoretical claims about systemic instability. Rather than eventual convergence to zero unhappiness, these systems sustain prolonged dissatisfaction throughout the simulation duration. This persistent unhappiness cannot be explained by slow convergence or temporary disequilibrium – instead, it reflects the mathematical reality that movements based on one identity dimension continue to disrupt satisfaction on other dimensions, creating the cascading relocations that define multidimensional perturbation dynamics. The salience rule differences in unhappiness evolution further validate our emphasis on coordination mechanisms. Random and Minority Focus rules often achieve apparent convergence but then exhibit renewed mobility as coordination breaks down, while other rules show more consistent but slower convergence patterns. These differences demonstrate that the temporal dynamics of multidimensional sorting depend critically on how agents process their complex identities. 4.2.4. Robustness across tolerance levels The threshold analysis in demonstrates that our core theoretical predictions hold across different levels of agent selectivity, confirming that multidimensional coordination effects reflect fundamental behavioral differences rather than artifacts of specific parameter choices. The figure reveals several important patterns about how tolerance levels interact with coordination mechanisms. At low tolerance thresholds (), when agents are easily satisfied with minimal similarity requirements, segregation levels are relatively compressed across all salience rules, with most mechanisms achieving moderate segregation levels around 0.52–0.62. The key pathway differences are still visible but muted: Fixed Salience shows slight declines with dimensionality, while coordination mechanisms maintain stable levels. At intermediate tolerance levels (), the characteristic patterns of our three pathways become most pronounced. Fixed Salience shows clear declining segregation with increasing dimensionality, coordination mechanisms (Neighbor Influence, Imitate Satisfied) maintain stable moderate levels, and volatile mechanisms (Random, Minority Focus) show their distinctive patterns with some decline in higher dimensions. At high tolerance levels (), when agents require substantial similarity for satisfaction, the dimensional effects become most dramatic. While most mechanisms initially achieve high segregation levels (0.85–0.9) in 2–3 dimensional systems, significant declines occur in 4-dimensional systems across multiple mechanisms. Random and Minority Focus show particularly sharp drops from dimensions 3 to 4, declining from ~0.87 to ~0.75 and ~0.90 to ~0.65 respectively. Fixed Salience and Weighted Composite show even more dramatic declines, indicating that both attention fragmentation and volatile coordination mechanisms struggle under the combined pressure of high dimensionality and demanding similarity requirements. The consistency of pathway rankings across tolerance levels validates our theoretical framework, while the pronounced dimensional effects at high tolerance levels suggest that coordination challenges become most severe when agents have demanding similarity requirements in complex identity landscapes. 4.3. Comparison with empirical segregation studies While our model is not directly testable against existing data, some empirical patterns are consistent with the dynamics we theorize. Ellen (Citation2000) documents that racially integrated neighborhoods often experience ongoing demographic change rather than stable equilibrium, while Friedman (Citation2008) shows how gentrification creates simultaneous shifts along multiple dimensions (race, class, education) – the kind of cascading cross-dimensional change our perturbation theorem formalizes. Hwang and Sampson (Citation2014) document how changes along one dimension (economic) frequently trigger changes along others (racial, cultural) in Chicago neighborhoods, generating sustained residential turnover consistent with our perturbation predictions. Their finding that neighborhoods experiencing economic upgrading are substantially more likely to subsequently experience racial turnover illustrates the kind of cross-dimensional cascading our model formalizes. Similarly, Delmelle’s (Citation2017) analysis of neighborhood change trajectories in 50 US metropolitan areas is consistent with the persistent instability our model predicts. His finding that only 23% of neighborhoods maintained stable demographic profiles over three decades is compatible with our theoretical argument that multidimensional perturbations can prevent stable equilibria, though of course many other factors contribute to neighborhood instability in practice. The persistence of such multidimensional sorting, despite substantial changes in discriminatory practices and individual attitudes, supports our theoretical emphasis on coordination mechanisms rather than preference-based explanations for segregation persistence. Recent work by Iceland and Sharp (Citation2013) documents continuing segregation across multiple dimensions simultaneously, suggesting that the coordination challenges we identify may be more fundamental than policy interventions targeting single dimensions. These empirical patterns are consistent with our model, but we caution against treating them as direct validation. Our model makes theoretical predictions about mechanism, not empirically testable forecasts calibrated to real data. Future work should design empirical tests specifically targeting salience coordination effects, for instance by using panel data on neighborhood composition combined with measures of identity salience at the individual level. 5. Discussion 5.1. Multidimensionality and coordination mechanisms A key contribution of this study is demonstrating that dimensionality alone does not destabilize sorting. Rather, instability arises when identity salience fluctuates in ways that introduce systemic inconsistency. Our perturbation theorem explains this in formal terms: any movement decision based on one identity dimension perturbs all others. However, our simulations demonstrate that the degree of resulting instability depends critically on how agents coordinate their salience switching. Random and Minority Focus salience rules generate persistent dissatisfaction and delayed convergence, as agents fail to align their movements around shared focal points. This echoes findings in social psychology showing that unstable identity salience increases cognitive dissonance and disrupts coordination (Roccas & Brewer, Citation2002; Swann et al., Citation2009). Meanwhile, Fixed, Neighbor Influence, and Success Imitation facilitate smoother convergence and, under higher dimensionality, can even reduce final segregation levels through different mechanisms. This pattern challenges the assumption – common in both popular discourse and some strands of intersectional theory – that simply recognizing identity complexity promotes tolerance and diversity. In our model, such recognition often leads to disruption unless it is channeled through mechanisms that coordinate salience across agents or stabilize individual attention patterns. At the same time, persistent instability is not inherently undesirable. Systems that fail to converge may prevent permanent segregation along any single dimension, sustaining cross-group exposure and ongoing mixing. In this light, the “failure to converge” we observe under Random and Minority Focus salience could be reinterpreted as resilience against rigid sorting – a form of dynamic integration that, while restless, avoids locking in stable segregation. 5.2. Integration, stability, and attention alignment Our results reveal a fundamental coordination challenge: achieving stable spatial configurations in multidimensional societies may depend on aligning attention mechanisms rather than simply celebrating identity diversity. When agents maintain coordinated salience focus – through social influence or institutional arrangements – systems achieve stable configurations. However, these come with trade-offs: attention fragmentation reduces segregation but prevents meaningful identity-based community formation, while coordination can amplify segregation through dimensional inequality. This finding suggests that the relationship between identity complexity and spatial outcomes depends critically on behavioral coordination mechanisms. Rather than structural diversity alone determining spatial patterns, how this complexity is cognitively and socially managed becomes crucial. The key insight is that multidimensional integration may require coordination mechanisms – shared focal points that channel salience toward common dimensions or institutional arrangements that provide predictable attention patterns. 5.2.1 Policy Implications and Trade-offs This creates analytical challenges for multicultural policy evaluation. Simply promoting awareness of identity complexity, without coordination mechanisms, may reduce segregation through attention fragmentation but prevent stable community formation. More problematically, coordination mechanisms that stabilize salience can amplify segregation through dimensional inequality, creating societies where integration on some dimensions occurs alongside extreme segregation on others. Potentially effective approaches might focus on institutional arrangements that provide salience coordination while preserving meaningful diversity recognition. This could involve creating contexts where different identity dimensions become temporarily focal while maintaining overall system coordination. However, salience coordination carries risks. Fixing attention on single dimensions (nationality, religion, class) can foster majoritarianism and suppress legitimate intersectional claims. The challenge is developing coordination mechanisms that stabilize salience without eliminating meaningful diversity recognition. This might involve creating institutional contexts where different identity dimensions become sequentially focal while maintaining overall system coordination. Recent empirical work supports this coordination framework. Successful integration programs often provide structured interaction contexts that temporarily channel attention toward shared goals (Allport, Citation1954; Pettigrew & Tropp, Citation2006). Similarly, neighborhood stabilization efforts that emphasize local rather than demographic identity (Sampson, Citation2012; Small, Citation2004) may succeed by providing alternative coordination mechanisms that override demographic salience switching. 5.3. Implications for intersectionality and behavioral modelling From a methodological standpoint, this work contributes to formal intersectionality modeling (Cho et al., Citation2013; Leung, Citation2020; Núñez, Citation2021) by demonstrating how the joint effects of multiple identity dimensions – and the rules governing their activation – produce emergent, nonlinear outcomes. The divergence in outcomes across salience rules under identical dimensional conditions illustrates the importance of explicitly modeling behavioral assumptions rather than treating them as technical details (Hedström, Citation2005; Macy & Willer, Citation2002). Our simulation offers a behavioral companion to Blau’s (Citation1977) structural theory of multidimensional social space. Whereas Blau assumed fixed orthogonal traits and declining interaction probabilities, our model introduces agent-level cognition into the system, showing how different processing rules for identity salience mediate the structural consequences of heterogeneity. This demonstrates how individual-level cognitive constraints can generate system-level properties that are not reducible to simple aggregation of preferences. The measurement approach – focusing on dimension-specific segregation and segregation spread – provides tools for analyzing complex systems where coordination mechanisms produce fundamentally different patterns. Average segregation across dimensions captures the overall effects of attention fragmentation versus coordination, while segregation spread reveals whether coordination creates dimensional inequality or uniform effects. This methodological contribution extends beyond segregation studies to other domains where multidimensional outcomes resist simple characterization. More broadly, the results support recent arguments in mathematical sociology that complex outcomes often hinge on micro-level processing bottlenecks (Manzo, Citation2013; Young, Citation1998). The salience rules in our model function as such bottlenecks: seemingly minor cognitive differences in how agents process multidimensional identity produce qualitatively different macro-level equilibria. Fixed Salience generates attention fragmentation with declining average segregation and low spread, while Neighbor Influence creates coordination with stable average segregation and high spread. These patterns demonstrate how individual-level behavioral mechanisms can generate emergent system properties that fundamentally alter spatial sorting outcomes. This connects to broader theoretical debates about emergence and reduction in social science (Coleman, Citation1990; Demeulenaere, Citation2011), demonstrating how coordination mechanisms – not just individual preferences or structural diversity – determine collective outcomes in multidimensional systems. 5.4. The paradox of diversity recognition Our findings reveal a troubling paradox for contemporary diversity policy. Interventions designed to promote recognition of identity complexity – encouraging people to embrace their multifaceted selves and switch flexibly between different aspects of identity – may inadvertently undermine the coordination necessary for stable integration. When agents pursue individual identity exploration without coordination mechanisms, the result can be either persistent instability (Random, Minority Focus) or integration through fragmentation that prevents meaningful community formation (Fixed Salience). This suggests that the relationship between diversity recognition and integration outcomes is far more complex than current policy frameworks acknowledge. Simply increasing awareness of intersectionality or promoting flexible identity expression may not produce the tolerant, integrated communities that advocates envision. Instead, such interventions may require careful attention to the coordination mechanisms through which multidimensional identities are socially managed. The policy implications are profound but complex. On one hand, our results suggest that some forms of identity-based community formation may require constraining rather than celebrating identity fluidity. Stable ethnic, religious, or cultural communities may depend on shared salience focus that our Fixed or Neighbor Influence mechanisms represent. Policies that encourage constant identity switching may undermine these communities even when motivated by inclusive ideals. On the other hand, coordination mechanisms that stabilize salience can amplify segregation through dimensional inequality, creating societies where some identity dimensions become highly segregated while others achieve integration. This raises questions about which identity dimensions should receive coordination priority and how to manage the trade-offs between different forms of integration and segregation. 5.5. Limitations and methodological considerations Several constraints limit our analysis and suggest extensions for future research. First, the model assumes exogenous, orthogonal identity categories, while real dimensions often correlate and evolve through social interaction. Correlated identity dimensions might reduce perturbation effects by aligning movement incentives, but systematic investigation is required. Preliminary analysis suggests that when identity dimensions correlate positively (e.g., race and class), coordination challenges may diminish as agents with similar attributes on one dimension are likely to share attributes on others, reducing cross-cutting disruptions. Second, our spatial framework abstracts from economic constraints, housing market dynamics, and institutional factors shaping real mobility patterns. Economic segregation often reinforces identity-based sorting, while housing market constraints can prevent agents from acting on residential preferences. Incorporating these elements could reveal when market constraints amplify or dampen identity-based coordination challenges. Third, the model lacks endogenous tie formation – agents cannot develop relationships that alter preferences or coordinate salience rules. Real neighborhood integration often depends on cross-cutting social ties that might provide natural coordination mechanisms (Blau, Citation1977; Feld, Citation1981). Extensions incorporating network formation could illuminate how social capital emerges in multidimensional contexts and whether successful coordination mechanisms can spread through social influence. Fourth, we focus on residential contexts while multidimensional sorting occurs across multiple spatial domains (workplaces, schools, social venues) that could create feedback loops with neighborhood composition. Multi-context extensions might reveal how segregation in one domain affects integration possibilities in others, potentially identifying intervention points where policy can leverage cross-domain dynamics. Fifth, salience rules remain fixed rather than evolving through learning or social influence, missing potential adaptive mechanisms that might emerge in real communities. Models with endogenous salience learning could explore whether successful coordination mechanisms can emerge spontaneously or whether institutional intervention is necessary for stable coordination. Sixth, the Minority Focus rule is likely most applicable to visible, rigid, or stigmatized identity dimensions – such as race or disability – where distinctiveness motivation and identity threat responses are most pronounced. Social identity theory does not predict minority focus as a universal response; for dimensions where high-status or majority membership is preferred, agents might instead seek majority affiliation. Future refinements should specify scope conditions for each salience rule more precisely. Seventh, we assume fixed categorical identity attributes throughout. For fluid dimensions – such as political opinion or religious affiliation – identity attributes themselves may change through social interaction, potentially altering the coordination dynamics we document. Extensions incorporating endogenous identity change could explore how attribute fluidity interacts with salience coordination (Della Posta et al., Citation2015). Eighth, we assume binary identity categories. Real social dimensions often involve many categories, which would increase coordination challenges further and may alter the scaling relationships we observe with dimensionality. Similarly, introducing heterophily – preference for contact with dissimilar others on some dimensions – might stabilize diverse neighborhoods in ways our current model cannot capture. Ninth, the model also does not capture localism or neighborhood attachment. The tolerance parameter τ reflects preference intensity, not place affinity; incorporating a tenure-based staying premium or co-resident familiarity would formalize this and likely dampen the perturbation dynamics we document. 6. Conclusion This paper presents a mathematical framework for understanding how identity complexity and behavioral coordination jointly shape spatial segregation. By extending Schelling’s model to include vector-valued identity and formalized salience mechanisms, we reveal that the relationship between dimensional complexity and segregation is fundamentally mediated by coordination processes rather than determined by structural diversity alone. 6.1. Summary of key findings Our systematic analysis of 1,920 simulations demonstrates that coordination mechanisms produce three distinct pathways for multidimensional sorting. Attention Fragmentation (Fixed Salience) reduces segregation across all dimensions through cross-cutting disruptions but prevents meaningful identity-based community formation. Coordinated Amplification (Neighbor Influence, Success Imitation) can achieve stability and integration on some dimensions while amplifying segregation on others through dimensional inequality. Volatile Instability (Random, Minority Focus) sustains high segregation across multiple dimensions while preventing system convergence through ongoing perturbations. The perturbation theorem establishes that multidimensional identity creates inevitable interdependencies, but coordination mechanisms determine whether these generate fragmentation, amplification, or instability. The measurement innovation – distinguishing average, maximum, and spread – reveals how different aspects of multidimensional sorting can move in opposite directions, providing tools for analyzing complex systems where intuitive predictions fail. The temporal analysis demonstrates that multidimensional systems sustain ongoing dissatisfaction rather than simply experiencing delayed convergence, reflecting the mathematical reality that movements based on one identity dimension continue to disrupt satisfaction on other dimensions. 6.2. Theoretical implications for mathematical Sociology For mathematical sociology, the model illustrates how formalization can surface hidden trade-offs and counterintuitive dynamics in multidimensional systems. The divergence in outcomes across salience rules under identical structural conditions validates the importance of explicitly modeling behavioral mechanisms rather than treating them as technical details. This contributes to ongoing debates about micro-macro linkages by demonstrating how individual-level cognitive constraints generate emergent system properties that cannot be reduced to simple preference aggregation. Our findings also contribute to formal intersectionality research by providing a tractable framework for analyzing how multiple identity dimensions interact through behavioral processes. Unlike approaches that assume simultaneous optimization across all dimensions, our selective attention mechanisms offer a more psychologically plausible account of how individuals navigate complex identity landscapes while maintaining analytical tractability. The perturbation theorem establishes that multidimensional identity creates inevitable interdependencies between spatial distributions across different social categories, but the effects depend on coordination mechanisms rather than universally amplifying segregation. This insight extends beyond residential segregation to other domains where multidimensional sorting occurs, suggesting that coordination challenges may be fundamental features of complex social systems. 6.3. Policy relevance and normative tensions This yields a provocative theoretical implication: to achieve stable spatial configurations in multidimensional societies, attention alignment mechanisms may be as important as structural diversity itself. Coordinating salience – whether via cultural scripts, institutional norms, or social learning – appears to be necessary for stable outcomes, but comes with risks of amplifying segregation through dimensional inequality. These findings create significant analytical challenges for multicultural policy evaluation. Simply promoting awareness of identity complexity, without coordination mechanisms, may reduce segregation through attention fragmentation but at the cost of preventing meaningful identity-based community formation. More problematically, coordination mechanisms that stabilize salience can amplify segregation through dimensional inequality, creating societies where integration on some dimensions occurs alongside extreme segregation on others. This suggests that effective integration policy must address not only individual preferences but also the coordination mechanisms through which multiple identities are activated and managed. The challenge is developing institutional arrangements that promote stability without eliminating meaningful diversity recognition – what we might term “coordination without domination.” We emphasize that our model maps these trade-offs rather than prescribing optima: we do not assume convergence is normatively desirable. Stable segregation along a single salient dimension may well be worse for social welfare than persistent mixing with ongoing dissatisfaction. Potential approaches might include institutional salience coordination (creating contexts where different identity dimensions become sequentially focal), managed attention fragmentation (deliberately designing policies that encourage attention dispersion), or temporal coordination (establishing predictable “salience schedules” that allow for identity expression while preventing ongoing perturbations). 6.4. Broader implications and future research The core insight extends beyond residential segregation to other domains where multidimensional sorting occurs: educational tracking, occupational segregation, political coalition formation, and online community organization. In each context, understanding how identity salience is activated and coordinated may be crucial for predicting emergent patterns of integration and separation. For contemporary debates about diversity and inclusion, our findings suggest that interventions focused solely on recognition and representation may be insufficient without attention to coordination mechanisms. The challenge is not simply acknowledging complexity but managing it in ways that promote rather than undermine social cohesion. Future research might extend this framework to endogenize salience learning, correlate identity dimensions, or incorporate nonresidential social spaces. Extensions could also explore how coordination mechanisms emerge and evolve in real communities, investigate the role of institutional arrangements in managing salience coordination, or examine how multidimensional sorting in different domains creates feedback loops that reinforce or undermine integration efforts. 6.5. Final reflection The fundamental insight remains: in a world of multidimensional identity, spatial outcomes depend as much on how people coordinate their attention as on what identities they possess. The paper maps how different salience coordination mechanisms generate distinct segregation regimes under identical structural conditions – a finding that calls into question the intuition that diversity of identity necessarily promotes integration, and that suggests salience coordination deserves as much analytical attention as structural diversity itself. Understanding this coordination challenge – and developing institutional mechanisms to address it constructively – represents a crucial frontier for both mathematical sociology and social policy in increasingly diverse societies. Our analysis reveals that the relationship between identity complexity and spatial outcomes is far more nuanced than either assimilationist or multiculturalist frameworks suggest. Neither suppressing identity differences nor simply celebrating them may be sufficient for achieving stable spatial configurations. Instead, the key may lie in developing sophisticated coordination mechanisms that allow for meaningful identity expression while maintaining the social cohesion necessary for collective life. This places new demands on both theoretical understanding and policy design. Theoretically, we need more sophisticated models of how coordination emerges, evolves, and can be influenced through institutional design. Practically, we need policy frameworks that can navigate the trade-offs between different forms of integration and segregation while preserving both individual autonomy and collective welfare. The stakes are considerable. As societies become increasingly diverse across multiple dimensions, the coordination challenges we identify may become more pronounced. Understanding how to manage these challenges successfully – enabling diversity without fragmentation, coordination without domination – may be essential for the future of democratic pluralism in complex societies. Disclosure statement No potential conflict of interest was reported by the author(s). References - Allport, G. W. (1954). The nature of prejudice. Addison-Wesley. - Anthias, F. (2013). Intersectional what? Social divisions, intersectionality and levels of analysis. Ethnicities, 13(1), 3–19. https://doi.org/10.1177/1468796812463547 - Baldassarri, D., & Bearman, P. (2007). Dynamics of political polarization. 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