simple and complex identities in similarity … porac...schelling study, agents were characterized...
TRANSCRIPT
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Simple and Complex Identities in Similarity Clustering: Using Schelling Segregation to Explore Infinite Dimensionalization in Organizational Cluster Formation
By
Christina Fang New York University
Ji-hyun Kim Yonsei University
Joe Porac New York University
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INTRODUCTION
Research in organizational theory and economic sociology suggests that organizational fields are
partly structured by collective representations that parse organizations into categorical forms or
cognitive codes (e.g., Hannan, Polos, & Carroll, 2007; Hsu, Negro, & Kocak, 2010; Vergne &
Wry, 2014). Much of this work has been focused on establishing how the socio-cognitive
category structure of organizational fields disciplines organizations by defining the
characteristics that buyers, suppliers, regulators, critics, and other audiences consider typical of
particular organizational types. Research suggests that distinctive organizations that depart from
socio-cognitive typifications are sometimes subject to legitimacy discounts in revenues (e.g., Hsu,
2006), costs (Ody-Brasier and Vermeulen, 2014), capital inflows (e.g., Pontikes, 2012), and
stock prices (e.g., Zuckerman, 1999), all of which have implications for organizational survival.
Given these effects, other research has probed how field level categories get established (e.g.,
DiMaggio, 1991; Rosa et al., 1999; Navis & Glynn, 2010), and how organizations attempt to
manipulate their categorical membership to advance their strategic interests (e.g., Santos &
Eisenhardt, 2009; Khaire & Wadhwani, 2010; Rhee 2015).
Research on the antecedents and effects of organizational categories has done much to
explicate the cognitive structure of organizational fields. At the same time, most of the existing
research takes organizational categories as givens, and tracks the usage and frequency of lexical
phrases that have become associated with particular organizational (“French restaurants,”
“grocery stores,” etc.) or product (“comedies,” “minivans,” etc.) configurations. This has left
open many questions about the nature of organizational categories themselves, particularly their
socio-cognitive underpinnings and their internal organization. Hannan, Polos, and Carroll (2007)
suggested that category nomenclatures within a field can evolve by borrowing from other
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nomenclatures and/or from recombining existing lexical phrases into new category constructions.
However, the canonical case of category formation is the creation of a new category to describe a
perceived cluster of similar organizations. In Hannan, Polos, and Carroll’s words, “Members of
audiences observe producers and products, notice similarities, try to make sense of them by
clustering similar producers/products, and possibly assign labels to clusters. These activities
comprise the first steps in the audience’s creation of what might turn into a category or even a
form” (p. 33). For Hannan and his associates, understanding this process of noticing and labeling
clusters of similar organizations “is exactly the challenge for organizational sociology” (p. 38).
In fact, a long line of research in cognitive science accords similarity among stimulus entities a
foundational role in category formation and learning (e.g., Rosch, 1978; Estes, 1994; Murphy,
2002).
However, over the years both cognitive scientists (e.g., Murphy & Medin, 1985; Sloman
& Rips, 1998) and organizational theorists interested in organizational classification (e.g.,
McKelvey, 1975) have critiqued the role of similarity in explaining category formation and
structure. At the root of these critiques is the problem of “infinite dimensionality,” or the fact
that entities and organizations can be described and compared using a very large number of
attributes and that “any two entities can be arbitrarily similar or dissimilar by changing the
criterion of what counts as a relevant attribute” (Murphy & Medin, 1985: 292). Infinitely
dimensionable stimuli have led some to limit the role of similarity in category theory by
suggesting that other cognitive processes such as causal rules (e.g., Sloman, 1995), functional
themes (e.g., Estes, Golonka & Jones, 2011), behavioral goals (e.g., Barsalou, 1983), bodily
states (e.g., Barsalou, 2008), and/or innate perceptual tendencies (Harnad, 2005) underlie
category formation and change.
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Durand and Paolella (2013) recently argued that these alternative processes “stretch”
theories of organizational categories in ways that are not accounted for by pure similarity-based
accounts. As reasonable as this suggestion might be, however, similarity assessments play too
central of a role in strategy and organizational theory to ignore similarity-based category models
completely. Indeed, much of organizational theory is an attempt to explain why organizations
are similar or different. And, in strategy research, as Farjoun and Lai (1997) noted, similarity
assessments implicitly underlie models of strategic groups, industry analysis, rivalry and
competition, diversification, and the evaluation of competitive advantage itself. In light of the
importance of similarity as a construct in strategy and organizations research, it is better to take
infinite dimensionality as given in organizational categorization and to explore its boundary
conditions and the socio-cognitive processes that are involved in overcoming the categorical
challenges it creates.
In this regard, cognitive scientists have proposed two general approaches to infinite
dimensionality, what Boster and D’Andrade (1989:132) called the “structured mind hypothesis”
and the “structured world hypothesis.” The structured mind hypothesis claims that prior
knowledge privileges some entity attributes over others in the formation of categories. Prior
knowledge includes implicit theories of the world (e.g., Murphy & Medin, 1985), motivational
interests (e.g., Barsalou, 1983), and innate perceptual tendencies (e.g., Harnad, 2005) that make
certain entity attributes more or less salient during similarity judgments. Salient attributes are
weighted more significantly in any resulting categorization. The structured mind hypothesis can
be extended to the analysis of organizational categories by recognizing the role that cultural
beliefs and routines, institutions and existing classification systems, regulatory strictures, and
group based interests can play in making certain organizational attributes more or less salient in
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category formation (Hannan, Polos & Carroll, 2007; Latour, 2005; Lounsbury & Rao, 2004; Rao,
Monin & Durand, 2005).
The structured world hypothesis, on the other hand, proposes that external stimuli are not
structured randomly, and that some entity attributes are correlated with others. In a world of
correlated attributes, it is the role of categorization processes to “carve nature at its joints”
(Rosch, 1978) by describing clusters of entity attributes that tend to occur together. Focusing on
one of the correlated attributes carries information about other correlated attributes, and certain
entities stand out and can be used as category prototypes when they are fully representative of
the correlational structure of the domain (e.g., Rosch, 1978). For example, Porac et al. (1995)
extended the structured world hypothesis to organizational categories by mapping the accepted
categories of Scottish knitwear producers. Their results suggested that these categories were
abstracted from a set of five attributes that were most correlated with other producer attributes.
In its most abstract form, the problem of infinite dimensionality reduces to the question
of how infinitely dimensionable actors combine to form clusters of actors that are similar on only
particular attributes, eventually acquiring category labels to summarize this similarity. Infinite
dimensionality is still an open question in the organizations literature on categorization, and there
have been few attempts to explore its implications explicitly (Porac et al., 1995 is one exception).
No doubt, this is partly because of the difficulty of explicating the processes of attribute
reduction empirically in organizational environments via the archival observational and/or cross-
sectional studies that have been used in prior organizational research on categories. Coding
category labels in product reviews or directory listings, for example, is not precise enough to
identity decisively the attribute clusters that labels summarize, nor how particular attributes were
selected for clustering. Cognitive scientists have typically studied attribute selection and
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clustering experimentally, but at the individual level of analysis. Organizational categorization is
a social process, however, given that attribute selection and clustering is mediated by
interorganizational communication, coordination, and/or imitation over time. This endogeneity
makes intra-individual studies of similarity and categorization less applicable to organizational
contexts, although the problem of infinite dimensionality remains.
Following from the above reasoning, our goal in the present research is to explore the
structured mind and structured world hypotheses by taking advantage of a well-known paradigm
of similarity clustering in a simulated agent-based environment. Boster and D’Andrade
acknowledged that the cognitive mechanisms underlying these two hypotheses are not mutually
exclusive, and that either or both mechanisms can be deployed to reduce the complexity and
dimensionality of stimulus entities for purposes of semantic categorization. We thus explore
each mechanism independently. Our choice for an empirical platform is Schelling’s (1971) well-
known segregation simulation. In Schelling’s model, simulated agents are arrayed within a two-
dimensional grid in such a way that some grid cells contain agents and some are empty. Agents
are able to move around to empty cells according to well-specified rules. In the original
Schelling study, agents were characterized with a single binary color dimension as either black
or white. Agents were endowed with a homophily motive such that they preferred to be near
agents who were similar in color to them. Schelling manipulated the strength of this motivation
by varying the percentage of similar neighbors surrounding a given agent that was necessary to
trigger a move to another location. Agents evaluated their surrounding neighbors for similarity
during each round of the simulation, and moved to a more homophilic location when the
percentage of similar neighbors in their current location was below the pre-defined similarity
threshold. Schelling found that marked clusters of similar agents formed after a small number of
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rounds with no more than moderate preferences toward homophily (movement thresholds of
60% neighborhood similarity).
The Schelling paradigm is useful for our purposes because it represents an instantiation
of a dynamic social clustering environment in which similarity assessments are the driving
motivation behind actor location and relocation. Schelling’s (1971) original paper stimulated a
large subsequent literature exploring various parameters of the paradigm, and the mechanics of
the Schelling model are well known. A major line of research has been to modify one or more of
the key model assumptions to determine the robustness of Schelling’s original insights. For
instance, some (e.g., Bruch & Mare, 2006; Pancs & Vriend, 2007; Van de Rijt, Siegel, & Macy,
2009) explored alternative forms of preference function than the discrete, threshold-based
function in Schelling (1971). Others varied the size and shape of neighborhoods, the number of
empty cells (e.g., Vinković & Kirman, 2006), and/or the rule and the order of migration (e.g.,
Pancs & Vriend, 2007). Most of this work showed a high degree of congruence with Schelling’s
basic predictions. A related stream of work tested Schelling’s predictions in various empirical
settings (e.g., Card, Mas, & Rothstein, 2008; Clark, 1991; Clark & Fossett, 2008), with similar
support. In addition, given the generality of Schelling’s insights, scholars from outside the social
sciences have built on Schelling’s model to explore segregation in their own domains. For
example, Vinković and Kirman (2006) reported an analogue between Schelling’s segregation
dynamics and particle dynamics found in physical worlds, and Nielsen, Gade, Juul, and
Strandkvist (2015) used Schelling segregation to model the clustering and segregation of
biological cells.
Despite the extensive literature on the Schelling model, to our knowledge all subsequent
work, like Schelling’s (1971), has characterized agents with a single binary attribute. The
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possibility that agents are infinitely dimensionable, and the clustering challenges associated with
multi-attribute agents, has not been considered in prior research. Research in the Schelling
tradition has thus assumed that some type of attribute reduction occurs prior to clustering
dynamics. In the present research, we relax this assumption and experimentally manipulate the
number of attributes that agents consider in their location decisions. We incorporate a realistic
model of multi-attribute similarity assessments derived from Tversky (1977). In Experiment 1,
we first establish the comparability of our simulation mechanics to Schelling’s (1971) by
replicating his basic finding in the one attribute case. We next introduce three-attribute and ten-
attribute extensions in Experiment 2. We show that Schelling segregation becomes increasingly
less apparent as the number of attributes gets larger. In Experiment 3, we test the structured mind
argument by differentially and exogenously weighting random attributes and show that attribute
weighting modulates this effect and re-establishes segregation even in the multi-attribute case.
In Experiment 4, we begin our exploration of the structured world argument by comparing multi-
attribute clustering when the attributes are randomly allocated to agents and when the attributes
are non-randomly structured into mutually exclusive crisp sets. Consistent with the structured
world hypothesis, we show that segregation occurs when attributes are perfectly correlated in
crisp configurations but not when they are randomly distributed among agents. Finally, in
Experiment 5, we relax the crispness of the attribute sets and explore the effects of moderately
correlated attributes in segregation dynamics. Our results suggest that correlated attributes seed
clustering and segregation around category “prototypes” that best represent the correlational
structure of the attribute space. Together, these five experiments provide suggestive evidence
that supports both mechanisms of attribute reduction. We end our paper by drawing out the
implications of our results for research on organizational categorization.
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Before proceeding with a more detailed exposition of Schelling mechanics and our
extensions, it is important to address the question of whether the Schelling paradigm is a realistic
context for studying organizational clustering, identity formation, and categorization.
Schelling’s (1971) original purpose was to show that racial segregation in city neighborhoods
could occur with only moderate preferences for homophily among neighbors. The simulation’s
two-dimensional grid represented geographic space, and the movement of actors on the grid was
considered analogous to the movement of actual individuals into and out of real neighborhood
positions. Subsequent research on the Schelling model has continued to conceptualize the two-
dimensional grid as geographic or physical space. In organizational contexts, the clearest
extension of the Schelling model is thus to identity formation and categorization in geographical
clusters of organizations (e.g., Romanelli & Kessina, 2005), or models of spatial competition
(e.g., Hotelling, 1928). Indeed, Romanelli and Kessina (2005) hint at the problem of infinite
dimensionalization in regional identity formation by suggesting that consensus around the
characteristics of the work activities within a cluster is a facilitating condition for the creation of
a regional identity categorization. This suggestion is supported by Porac et al.’s (1995) report of
geographical clustering in the Scottish knitwear industry, where each geographical cluster was
associated with particular values on a small set of consensually recognized attributes. Porac et al.
suggested that common resources, information sharing and collaboration, and imitation all
contributed to the homophilic motivations of knitwear firms in their sample.
Although geographical identity formation among clusters of organizations is a natural
extension of the Schelling paradigm, we also believe that the paradigm is of more general
relevance in modeling attribute reduction in organizational categorization. The two-dimensional
grid can be represented in other theoretically meaningful ways. For example, the grid may
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represent positions on easily changeable attributes, with similarity comparisons being made
using other more stable actor attributes. White(2004), for example, suggested that markets are
characterized by producer positions along quality and volume dimensions. Given this
representation of the Schelling two-dimensional grid, movement among cells in the space might
be considered producer choices to be more or less similar to other producers occupying particular
quality and quantity positions. As Hannan, Polos, and Carroll (2007) noted, there are many
reasons why organizations may seek homophily in a characteristics space, leading to clustering.
We leave it up to the reader to consider other possible organizational representations of
Schelling’s two-dimensional grid.
A MULTIATTRIBUTE EXTENSION OF THE SCHELLING PARADIGM
The Space
Following Schelling (1971), our model uses a two dimensional lattice consisting of a 20 x 20
lattice with a total of 400 cells. Cells in this lattice are populated by agents with m-dimensional
attributes, each of which is a randomly generated binary variable (i.e. either 1 or -1). Each agent
can occupy only one cell and a cell cannot be occupied by more than one agent. To allow agents’
migration to other parts of the lattice, randomly chosen cells of the lattice are specified to be
vacant. Figure 1a) illustrates a 10 x 10 grid. As seen, agents reside in cells which are marked by
their distinct binary attribute structures. For instance, one agent may be (-1,1, -1), and another
may be denoted as (1, -1, -1). Note that there are also vacant cells, marked by empty slots spread
out randomly in the grid.
---------------------------------------- Insert FIGURE 1 about here
----------------------------------------
Similarity Measure
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Each agent finds himself more or less similar to each one of his eight neighbors in his Moore
neighborhood: if an individual’s coordinates on a grid are represented as (i, j), the eight
surrounding cells((i-1, j-1), (i, j-1), (i+1, j-1), (i-1, j), (i+1, j), (i-1, j+1), (i, j+1), (i+1, j+1)) are
its neighbors. Because agents evaluate their neighborhood from their unique position, each agent
has a unique set of neighbors. Figure 1b) provides an example of such a Moore neighborhood.
The focal agent in the center is surrounded by 8 cells, only 6 of which are occupied by actual
neighbors with the remaining two cells empty.
As in Schelling (1971), an agent’s overall similarity score to all his neighbors is
computed as the average of dyadic level similarity scores. In Schelling (1971)’s original model,
an agent is characterized by a single binary number (e.g., black or white). Thus, in Schelling
(1971)’s world, a neighbor is either perfectly same or perfectly different. However, in our
context, since there are m dimensions in an agent’s characteristic, we need a similarity score that
summarizes across the m dimensions. We use Tversky (1977)’s ratio similarity judgment model
to characterize agents’ similarity calculation with multiple attributes. The similarity between two
agents a, and b, is denoted as s(a, b) and determined by:
A)-BfB)-AfB)AfB)Afbas
((((),(
⋅+⋅+∩∩
=βα !!!!!!!!!!!!!!!!!(1)
Here, A and B denote the set of features associated with agent a and b, respectively. α and β
denote weights attached to a and b, respectively, where α, β ≥ 0 and f(·) is a monotonic function
(In our case, we use a count function). We simplify by fixing α = β = 1, and s(a, b) becomes
simply the ratio of the features the two agents share in common to the total number of features
they each have. By using the ratio model of similarity, we normalize the similarity score s so that
it lies between 0 and 1. In other words, s(a, b) reduces to:
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B)AfB)Afbas
∪∩
=((),(
(2)
Consider the two agents in Figure 1c): agent a and b each has {1,-1,-1} and {1,-1,1} as their set
of features. Since the first and second features (underscored) are in common, and there are three
features in total, the similarity between a and b is the ratio of the two features shared in common
to the total number of features, 0.67 (=2/3).
Agent i’s level of similarity with all its neighbors (denoted by S(i)) is simply the average
of all dyadic similarity scores:
])([
),()( ][
jn
jisiS j∑
=!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(3)
where [j] denotes the neighbors of i and n(·) represents the number of elements of the argument
set. Figure 1d) illustrates how we compute S(i) for our focal agent: we get 0.39 by simply adding
up all dyadic similarities for the 6 neighbors (since 2 cells are empty) and dividing by 6.
Migration
At the beginning of the simulation, agents are randomly distributed on the lattice. In each period,
a single agent is randomly chosen from the lattice population. The chosen agent decides whether
he is happy or content with the current neighbors. If the agent’s current similarity is above a
certain threshold (which we parameterize as Th and ranges from 0.0 to 1.0), he is satisfied and
will remain in his current position. If not, he attempts to migrate to another vacant position in the
conceptual space, by scanning the nearby vacant slots not yet occupied. His radius of search
expands outwards gradually until he finds a slot where his overall similarity exceeds his
threshold level of happiness. Note that it is not sufficient for the new similarity to be greater the
current level of similarity if it is lower than the threshold. In the case that no empty cell is
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satisfactory, the agent remains in his current position. We continue this until the system has
reached a steady state where migration ceases: all agents either have no incentive to migrate or
cannot find any spot that would make them content.
As seen in Figure 1d), suppose the computed similarity of 0.39 is less than 0.7, the
desired threshold, the focal agent is unhappy as his similarity with his neighbors is less than his
threshold. As a result, he scans his neighborhood for vacant slots and computes his would-be
similarity if he had moved to one of the nearest neighboring slots. Suppose he first considers a
possible move to the cell immediately to his left. His new neighborhood would have been Figure
1e) and 1f), and the associated similarity is 0.78 which exceeds the threshold of 0.7. Because this
new neighborhood makes him happy, the focal agent would carry out a migration to occupy the
cell to his immediate left. If this new neighborhood does not make him happy, he would
increasingly expand his search for vacant slots until he exhausts all possible slots.
RESULTS
Experiment 1: Replicating the Schelling Model: the Threshold Effect
Before we explore how multiple attributes influence cluster formation, we first replicate the
classic Schelling model with only one attribute. Figure 2(a) plots the average level of similarity
across all threshold levels. In the classic Schelling model, there is only one attribute: whether the
agent is black or white. As seen in Figure 2(a), the threshold effect on average similarity is not
monotonic: average similarity gradually increases up until some level of threshold and declines
as threshold level increases further. The levels of similarity are very similar for the two extreme
values of threshold (Th = 0.1 and 0.9). When threshold is low (e.g., <0.2), the average level of
similarity initially is likely to be much higher than the threshold. Recall that agents are initially
distributed randomly in the grid. As such, on average, the level of similarity is about 0.5, a value
that far exceeds the low threshold levels such as 0.1 or 0.2. Virtually all the agents are content
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with where they are. Thus, hardly any agent has the incentive to migrate. Thus the system
reaches a steady state within a short period. On the other hand, when threshold is very high, there
is strong incentive to be with highly similar neighbors yet there are not many available places to
move to that satisfy the high thresholds. After a few agents find their perfect spots, no further
improvement is possible. At high threshold levels above 0.7, it is increasingly difficult to find
similar others above thresholds and as a result, agents remain ‘unhappy’. Thus at both extremes,
the average similarity level of the system hovers around 0.5, which is exactly the initial level of
similarity.
---------------------------------------- Insert FIGURE 2 about here
----------------------------------------
These results are consistent with the basic results of the Schelling model: even at
moderate levels of threshold (e.g. 0.6), a clear pattern of segregation emerges where groups of
similarity seeking agents are located close together in geographical clusters.
To see this emergence of ‘segregation’ visually, we track in Figure 2(b) the evolution of
agents’ spatial location which is marked by different colors. Since there is only one attribute,
black and grey represents two possible types. The first and the second panel of this ‘heat map’
represent the map at the first and the last period, respectively. Clearly, when the threshold level
is around 0.6 or 0.7, strong segregation occurs as indicated by the presence of large clusters of
similar colors (i.e. either black or grey). Agents of the same type tend to be found with similar
others, and we see the emergence of clusters. More segregation implies a higher similarity level.
This visual pattern indicates that there is a link between the levels of similarity and the levels of
segregation. In other words, the higher the average level of similarity, the more agents tend to be
clustered.
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Experiment 2: Effects of the Number of Attributes
How does adding more attributes complicate the basic story? We plot in Figure 3(a),
average similarity across different threshold levels for the 1, 3 and 10 attribute cases
respectively. As seen, the number of attributes has a clear flattening impact on average
similarity. While the general inverted U shaped relationship is preserved when we introduce
more attributes, average similarity is gradually lowered at every threshold level as the number of
attributes increases. In results not reported here, we also try the number of attributes = 20, and
the resulting curve is almost entirely flat. In other words, as the number of attributes increases, it
is increasingly difficult to achieve clustering of similar others. This indicates that category
formation is a small numbers phenomenon under the condition of random attribute assignment.
---------------------------------------- Insert FIGURE 3 about here
----------------------------------------
In Figure 3(b), we again visualize cluster formation by using a ‘heat map’. With multiple
attributes, we use different colors to indicate how high or low the focal agent’s level of similarity
with its neighbors. Cells with darker color (i.e., close to black) represent higher level of
similarity of the focal agent with its neighbors. A highly dark region represents a cluster of agent
with similar types. With one attribute, highly dark regions emerge when threshold is between 0.6
and 0.7. This observation is consistent with the upper panel of Figure 2. Comparing across
different number of attributes, we observe the lightening pattern as the number of attributes
increases. This further suggests that high levels of similarity are indicative of high levels of
clusters formation.
To understand the flattening result from multiple attribute cases, it is helpful to note that
cluster formation is a result of migration, which is in turn driven by two forces. First, to migrate,
agents must be discontent, which means that their neighborhood similarity must be lower than
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their threshold level of happiness. Second, for migration to take place successfully, these
discontented agents must have a place to move to. In Figure 4, we plot the contrast between 1)
number of agents that are discontent and 2) the number of available slots that would theoretically
make them happy (normalized by 100), at the beginning of the migration process. Two effects
are clear. First, for threshold levels greater than 0.5, the greater the number of attributes, the
more agents who are discontent. However, for threshold levels less than 0.5, the higher the
number of attributes, the less number of agents who are discontent. As the number of attributes
increases, the number of discontent agents begins to rise at increasingly higher threshold levels
(0.1, 0.3 and 0.4 for 1, 3, and 10 attributes accordingly). We label this latter one ‘slow rising
discontent’ effect. As the number of attributes increases, it is easier to satisfy any given
individual: similarity along more dimensions may be increasingly easier to attain theoretically.1
Hence, as the number of attributes increases, the number of discontent agents who are motivated
to migrate is increasingly slower to rise. Less number of individuals would be motivated to move
and migrate in search of similar others, as the number of attributes increases.
However this ‘slower rising discontent effect’ is only half of the story. Motivation to
move (initially) does not necessarily mean that agents succeed in finding an ‘ideal’ spot in a
neighborhood of sufficiently similar others. The second effect is related to the availability of
spots for the discontent agents to move to. For any given individual who is motivated to move
(i.e., their current similarity level being lower than the threshold), he can move to the next
nearest vacant spot in a new neighborhood where similarity with new neighbors exceeds !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1!The!number!of!attributes!changes!the!theoretical!distribution!of!similarity!before!migration!sets!in.!when!the!agents!have!only!1!attribute/dimension,!i.e.,!either!1!or!0,!the!baseline!distribution!of!similarity!is!spread!out!across!all!similarity!levels,!with!the!most!frequent!scores!being!0.5.!When!we!increase!the!number!of!attributes!to!3!and!10,!the!average!level!of!similarity!remains!the!same!at!0.5.!However,!the!distribution!of!the!similarity!scores!is!now!more!concentrated,!and!extreme!values!of!similarity!(e.g.!those!at!around!0!or!1)!become!much!less!likely.!This!implies!that!at!low!threshold!levels,!the!higher!the!number!of!attributes,!the!more!concentrated!the!initial!distribution!of!similarity,!and!the!smaller!the!number!of!discontent!agents.!In!other!words,!the!discontent!curve!is!slow!to!rise,!for!low!threshold!levels.!
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threshold. A spot is available if it enables a focal agent to successfully migrate. We simply
divided the number of agents who are motivated to move and can find a satisfactory spot by the
number of agents who are motivated to move. Thus the availability curves in Figure 4 represent
the percentage of discontent agents who can in fact move to satisfactory locations. As seen, the
number of qualified slots starts out high but declines dramatically at higher threshold levels.
There is simply no neighborhood that would satisfy the higher thresholds of these unhappy
agents. They remain unhappy, depressing the average similarity levels for the system. We label
this second effect the ‘insufficient destination effect’. This effect means that even though agents
are motivated to move, they succeed less and less in finding similar others as the threshold levels
increases. As a result, average similarity of the system remains low regardless of the number of
attributes. Despite the motivation to migrate, average similarity of the whole system remains
‘depressed’ at the initial level of 0.5.---------------------------------------
Insert FIGURE 4 about here ---------------------------------------
These two effects together explain the flattening effect of attributes. At lower threshold
levels, more attributes means that agents are less motivated to migrate. Thus as the number of
attributes increases, similarity scores are lower because migration simply is slower to set in. At
very high threshold levels, regardless of attributes, migration does not take place successfully as
there are simply not enough vacant slots to move to. Again, similarity remains low because
migration fails. At moderate thresholds, migration operates smoothly lifting the average
similarity of the system to a level above the baseline. The resulting pattern is therefore what we
have observed in Figure 2: as the number of attributes increases, the relationship between
similarity and threshold is increasingly less pronounced. In other words, it is increasingly
difficult to observe category formation as we increase the number of attributes.
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One important question is whether the visual patterns we observe in these ‘heat maps’
indeed represent stable, meaningful clusters of agents of different types. For category structure to
emerge, clusters need to be stable and not temporary. To ensure that clusters we observed are
stable, we run each simulation until the 10,000th period, well past the equilibrium defined as the
last time period when migration occurs. The longest time taken to reach a steady state was 6,102
time steps, whereas the average time to steady state was 3,336. We then use the agent locations
as of the 10,000th period to produce the ‘heat maps’ in Figure 3b).
To summarize, clustering or identity formation becomes more difficult as agents evaluate
more attributes. Identity clustering is a small number phenomenon: identity satisfaction or
clustering depends on not comparing oneself on many, let alone infinite, attributes or dimensions.
Experiment 3: Effects of Unequal Attribute Weighting
Next, we explore how the structured mind approach works to solve the problem of
infinite dimensionalization. Recall that in the structured mind approach, certain attributes are
made more salient in determining similarity by our common prior knowledge. So far we assume
that, in multi-attribute cases, different attributes receive the same weight. Next, we weigh
selectively certain attributes. To examine the structured mind approach, we reformulate the
similarity function between a and b by adding a weighting parameter ωi which represents how
much weight is given to the ith attribute as follows.
! !, ! = ! !! ∙ !!!!!
!!!!!!
where
1≔ 1!!!!!!if#ith!attribute(in(a(is(equal(to(ith!attribute(in(b(0!!!!!!otherwise)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
and
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!! = !!
!!!
---------------------------------------- Insert FIGURE 5 about here
----------------------------------------
Figure 5 examines the effect of unequal weighting using four illustrative cases based on
three attributes. First, there are two extreme cases: 1) each attribute is weighed equally where (ω1,
ω2, ω3 ) = (1, 1, 1); 2) there is an favorite attribute receiving all weights where (ω1, ω2, ω3 ) = (3, 0,
0). Note that case 2) reduces three attributes to one, and should produce results identical to the
one attribute case. Second, we include two intermediate cases where one or two attributes are
given more weight than others: 3) (ω1, ω2, ω3 ) = (2.5, 0.25, 0.25); and 4) (ω1, ω2, ω3 ) = (1.375,
1.375, 0.25).
As seen in Figure 5, the curves corresponding to the two intermediate cases are in
between the two extreme baselines (i.e. equal weighting and extreme dominance). As we move
from the equal weighting baseline (at the bottom) to the extreme weighting case (at the very top),
the system attains increasingly higher level of similarity, and thus higher degree of clustering. In
other words, the unequal weighting of attributes facilitates clustering and increases the average
level of similarity from the equal weighting scheme case. To understand this, it is important to
note an equivalence between smaller number of attributes and unequal weighting: giving
disproportionately more weights to some select attributes should have the same effect as
considering a smaller number of attributes (only those deemed important). A three attribute case
where (ω1, ω2, ω3 ) = (3, 0, 0) is identical to a single attribute case. Unequal weighting in effect
reduces the number of attributes that are included in the calculation of similarity. Given that our
baseline results that clusters are more easily formed in the single attribute case, it is therefore not
! 20!
surprising that unequal weighting is associated with higher similarity than equal weighting with
the same number of attributes. Clustering is not as easy as in the single attribute case, but easier
than the equal weight, multi-attribute case. These results indicate that one possible way to
facilitate the formation of clusters is to compare on many attributes but to weight the attributes
unequally.
Experiment 4: Effects of Fuzzy vs. Crisp Sets
An alternative approach to infinite dimensionalization is the structured world hypothesis.
In contrast to the structured mind approach which assumes that different actors attend to similar
salient attributes, the structured world approach argues that different attributes are correlated in
the real world and this correlation facilitates category formation. In this section, we manipulate
the degree of correlation among attributes.
In our baseline case, we randomly assigned 0’s or 1’s to each attribute. This creates a
random distribution of attributes in the population: knowing a value in an attribute dimension
does not allow any prediction about other attribute dimensions. Since there is no correlation
among attributes, we can call this baseline attribute structure ‘fuzzy’. For any randomly chosen
pair of individuals, it is unlikely that the two are different on every dimension. Very few agents
are either 100% similar or 0% similar, since the average similarity is 0.5.
A different attribute structure is the opposite of ‘fuzzy sets’, where agents’ attributes are
specified to either 100% or 0% similar, i.e., ‘crisp’. For instance, we can assign some agents
(0,0,0,0,0,1,1,1,1,1) and the rest (1,1,1,1,1,0,0,0,0,0). If an individual holds (0,0,0,0,0,1,1,1,1,1)
and the neighbor also holds (0,0,0,0,0,1,1,1,1,1), then the level of similarity is 1. If the neighbor
holds (1,1,1,1,1,0,0,0,0,0), the similarity level is 0. Thus, individuals fall into two “crisp” sets,
and there is perfect correlation between attributes. Knowing a value in an attribute dimension
! 21!
does allow a prediction about other attribute dimensions. Note that in a single attribute case, the
attribute structure is by definition crisp: there are only two types of agents.
In the example given above, two randomly chosen individuals are either 100% similar or
0% similar. A more generalizable specification is to include other intermediate levels of
crispness in agents’ attribute structures. First, we divide the entire population into two groups.
Second, for one group, when assigning actor attributes, we set the probability of receiving 1’s (vs.
0’s) equal to a parameter “c” (for the degree of crispness). For the other group, they receive 1’s
with the probability of (1 – c). When the degree of crispness is 0.5, all the actors receive 1’s with
the probability of 0.5. If the degree of crispness is 0.7, half of the population receives 1’s with
probability of 0.7. The other half receives 1’s with probability of 0.3 (=1 – 0.7). In this way, by
varying the degree of crispness, some members are more or less likely to have 1’s in their beliefs
and other members tend to have 0’s. This degree parameter has a maximum value of 1.0,
corresponding to the case where there are only two groups: one having only 1’s, and the other
having only 0’s. This is the specification underlying Figure 6. Since cases between 0 to 0.5 are
symmetric to the cases between 0.5 and 1, we present only results based on the latter intervals.
----------------------------------------
Insert FIGURE 6 about here
----------------------------------------
As seen from Figure 6, clustering becomes clearer as c increases (i.e., the attribute
structure becomes more crisp: at each threshold level, average similarity increases monotonically
as degree of crispness increases. This means that higher degree of crispness in effect facilitate the
emergence of clusters even as the number of attributes has the opposite effect. It is important to
note that the case of c = 1.0 (i.e., perfectly crisp case) exactly overlaps with the single attribute
! 22!
case. So long as sets are crisp, the number of attributes matters much less. The small number
problem is only a concern when the sets are fuzzy.2
Experiment 5: Prototypicality and Cluster Formation
As discussed earlier, scholars have argued that categories are formed around prototypical
members (e.g., Rosch and Mervis, 1975). According to Rosch and Mervis (1975), prototypical
members have the properties of other category members and tend not to have properties of
category non-members. In our set up, because each attribute can take one of two possible values
(i.e. 1 or 0), there are two fuzzy cluster configurations: one in which members tend to have more
1’s versus 0’s and vice versa. Members with all 1’s and all 0’s are, by definition, the most
prototypical members of the first and the second cluster configuration, respectively.
To understand the role of prototypical members in cluster formation, we take the
following steps. We first create members with randomly distributed attributes (i.e., a completely
“fuzzy set”). Then, we replace a fixed percentage of members (we call this the “prototypical
ratio”) with prototypical members, half of which have all 1’s and the other half all 0’s. For
example, when the prototypical ratio is 0.2, we replace 20% of the initial randomly created
population with 10% members having all 1’s and 10% with all 0’s.
Next, to see if prototypical members are ‘centers’ of their respective clusters, we need a
measure to capture, for each individual, how prototypical he or she is. This new parameter,
‘typicality index’, measures how close an individual is to the most prototypical members, and is
calculated as follows:
Typicality Index = Number of 1’s – Number of 0’s
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2 The inverted U shaped relationship between average similarity and thresholds holds across all degrees of crispness.
! 23!
The greater the difference between this index and 0 (either positive or negative), the more
prototypical the member. A larger difference implies that there are either more 1’s or more 0’s.
For example, in a four attribute case, members with either all 1’s or all 0’s will have 4 and -4 as
their typicality index, the maximum distance from 0. On the other hand, members with two 1’s
and two 0’s will have 0 for their typicality index.
Figure 7 shows the effect of prototypical members using one typical simulation run.
Note that these color heat maps are constructed using typicality indexes of individuals located in
the grid rather than the types or similarity levels of individuals as in Figure 2b) and 3b). Colors
located at the top and the bottom of the color bar indicate more prototypical members (that is,
those with the brightest and darkest colors). In the 1 attribute case (Figure 7a), there are only two
colors as there are only two types: 0’s and 1’s. Comparing heat maps across different
prototypicality ratios, it seems that having prototypes does not have a pronounced effect on
cluster formation. It is perhaps not surprising because all members are initially by definition
prototypical. Replacing a portion of the initial population with artificially created prototypes
does not enhance clustering any further than the initial condition.
The role of prototypical members becomes more pronounced as the number of attributes
increases. Consider the 10 attribute case (Figure 7c). When the prototypicality ratio is zero, there
is virtually no change in the visual pattern between the initial stage and t=10,000. However, as
the ratio of prototypical members increases to 0.2 or 0.4, more clusters emerge in the final stage.
In addition, prototypical members (as indicated by darker/lighter colors) are clustered together
surrounded by less typical members. As everyone becomes a prototype (i.e. at prototypicality
ratio = 1.0), the role of prototypical members becomes less pronounced.
! 24!
---------------------------------------- Insert FIGURE 7 about here
----------------------------------------
Do prototypical members form the center of the emergent clusters?’ If indeed they
facilitate the formation of clusters, we would expect that clusters will be formed around
prototypical members in their neighborhood. In other words, in the case of 10 attribute case,
clusters will be formed around members with the typicality index of 10 (or -10 in the other
category) and they are surrounded with members with 8 (or -8). As the boundary of a local
neighborhood increases, the expected tendency of members with either a high or low typicality
index being clustered together would be weakened. While Figure 7 illustrates visually the effect
of prototypicality, its value is limited because we can only use one typical simulation run.
To conclusively demonstrate the role of prototypical members in cluster formation, we
need a more robust and generalizable way to capture the typicality indexes of individuals located
at the center of clusters and the typicality indexes of members as the clusters expand spatially.
For the latter, we calculate the average typicality index for each expanding neighborhood degree.
The 1st degree neighbors are the eight surrounding spots around the focal member. The 2nd
degree neighbors are the sixteen neighbors around the 1st degree neighbors. Using this rule, we
can define the expanding neighborhood of each actor. We expect the following patterns: 1)
prototypical members (i.e., members with either very high or low typicality index) are expected
to be surrounded by similar prototypical members and this tendency becomes weaker as
neighborhood boundaries increase; 2) non-prototypical members (i.e., members with mid-range
typicality index) are not expected to be surrounded by neighbors with any particular patterns.
Figure 8 is based on 500 independent runs, each of which consists of 300 members on a
20 x 20 lattice. It plots the average typicality index as a function of the neighborhood degree.
Given that it is a one attribute case, there are only two types of individuals – those with 1’s
! 25!
(typicality index = 1) and those with 0’s (typicality index = -1). Seen from Figure 8, as the
neighborhood degree increases (i.e. as the neighborhood expands), the average typicality index
declines from 1 to 0 for prototypical members with 1’s (and increases from -1 to 0 for
prototypical members with 0’s).3 In other words, members with typicality index of 1 are
surrounded by immediate neighbors with similarly high typicality index. As we move outwards
from this center, neighbors become much less prototypical (as their typicality index approaches
zero). The reverse pattern is observed for the members with typicality index of -1.
---------------------------------------- Insert FIGURE 8 about here
----------------------------------------
In the 3 attribute case (Figure 8b), there are six groups of individuals depending on their
typicality indexes (3,2,1,-1,-2, and -3). Six lines thus describe how average typicality changes as
a function of neighborhood degree. We find that the members with Typicality = 3 are, on average,
surrounded by members with Typicality = 2 in its immediate neighborhood and average
typicality of neighbors gradually decreases as the neighborhood expands. The pattern is
symmetric for members with Typicality = -3, who are surrounded by members with similarly
high typicality values and average typicality increases as the neighborhood expands. In other
words, the prototypical members (Typicality = 3 or -3) are surrounded by similarly prototypical
members. Actors with lower typicality values (e.g., Typicality = 1 or -1) are surrounded by
similarly low level prototypicality members. This is why the curve for Typicality = 1 (Typicality
= -1) is located lower (higher) than Typicality = 3 (Typicality = -3) and decreases (increases)
more gradually. Therefore, prior results from Figure 7 are confirmed in a more general way:
prototypical members are more likely to be surrounded by other prototypical members and they
tend to be at the center of clusters. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3!Recall that there are only two types of members in a single attribute case.!
! 26!
Our analyses so far have revealed two potentially competing drivers for cluster formation.
In the previous section we find that a crisp attribute structure facilitates category formation. In
the current section, prototypical members also are found to facilitate emergence of structure. The
question is - which mechanism is more important? In the initial formulation of a crisp structure,
we considered two attribute types: (0,0,0,0,0,1,1,1,1,1) and (1,1,1,1,1,0,0,0,0,0). By comparison,
when we explored the role of prototypicality, we considered two related attribute types:
(1,1,1,1,1,1,1,1,1,1), and (0,0,0,0,0,0,0,0,0,0). Note that the later, prototypical types are also crisp,
but the former crisp types are NOT prototypical. In other words, prototypicality is theoretical
subset of crispness. This means that crisp attributes that are not prototypical may not enhance
cluster formation. In results not reported here, we find this to be true. To summarize, we
systematically injected two types: (1,1,1,1,1,0,0,0,0,0), and (0,0,0,0,0,1,1,1,1,1) into the
population and found that there was little change in cluster formation.
Complementary and Sensitivity Analysis
So far, we assumed that members migrate to the nearest satisfactory spot, an assumption of local
migration that is consistent with the classic Schelling model. In this section, we relax this
constraint and allow members to find the most satisfactory spot anywhere in the grid without
local constraints. With this global migration rule, the thresholds we previously applied in the
baseline Schelling settings do not apply any more as individual do not merely attempt to meet
their threshold level of happiness. Rather, they attempt to maximize their happiness. In results
not reported here, we compare the clustering patterns under these migration rules, based on
typicality indexes (other parameters are identical to the baseline 10 attribute case). We find that
clusters are larger and seems to be more salient under global migration. In the previous analysis
with fuzzy attributes (Figure 2), we reported the highest level of similarity under the 10 attribute
! 27!
case to be 0.67 when threshold is 0.6. Under the global migration rule, the highest level of
similarity reaches 0.78. This seems to be a clear improvement from the baseline case.
Clustering seems more likely for two reasons. First, under global migration, members can
realize any improvement in similarity scores even if the improvement is small. In contrast, under
local migration, potential improvement may be foregone when it does not result in the individual
meeting his or her threshold. This possibility of realizing incremental improvement is akin to the
availability argument we made earlier: when the amount of improvement is no longer
constrained to be above a certain threshold, more neighborhoods becomes candidate or ideal
spots to migrate towards. Second, while members under the local migration rule may stop
searching upon finding the first spot that gives a similarity level above the threshold (no matter
how small the gap is), members in global migration keep searching till finding the best fit overall.
Combining these two, migration is more likely (due to the first mechanism) and once migration
happens, the distance is farther (due to the second mechanism). This results in more cluster
formation.
DISCUSSION
There are strong experiential reasons to assume that categories of organizational forms are
induced from clusters of similar organizations. However, one of the underlying theoretical
conundrums of category models based on similarity clustering is the fact that organizations, and
all external entities for that matter, are infinitely dimensionable. This well-known problem has
led cognitive scientists to propose two general arguments regarding how infinite dimensionality
is resolved during the formation of categories in any knowledge domain. Domain knowledge
and belief systems can privilege certain attributes over others, and thus lead to clustering and
categories formed around those attributes that are most salient and weighted more heavily in a
! 28!
particular context. Or, the external entities themselves may be structured in a way that
configurations of correlated attributes occur together, implying that attributes that are highly
correlated with others have special informational value and thus are privileged during category
formation.
We have used the robust paradigm of Schelling (1971) segregation to examine some of
the details of each of these mechanisms, and have found both to be effective in mitigating the
challenges that infinite dimensionality presents. First, our results suggest that pure Schelling
segregation with randomly assigned attributes is a small numbers phenomenon in that it only
occurs when the number of actor attributes is low. However, weighting attributes differentially
in similarity judgments mitigates this effect and reinstates segregation, albeit at a reduced level.
This supports the “structured mind” hypothesis by suggesting that attribute reduction through
knowledge filtering is at least theoretically plausible. We also found evidence supporting the
“structured world” hypothesis by showing that organizing actor attributes into perfectly crisp and
non-overlapping sets completely reinstates Schelling clustering regardless of the number of
attributes considered. However, as many have noted in the cognitive and organizations literature,
rarely do we find perfectly crisp and non-overlapping sets of attributes in the environment.
Given this, our results also suggest that imperfectly correlated attribute configurations can
reinstate clustering as well, with prototypical attribute actors moving to the center of similarity
clusters and other less prototypical actors positioned around them.
Our results thus lend support to similarity-based models of organizational category
formation, but with two important provisos. First, although it is easy to see how beliefs, cultures,
standard operating procedures, existing classification systems, institutional priorities and the like
can act to privilege certain attributes over others, the structured mind hypothesis essentially
! 29!
requires that any explanation for cluster and category formation must first identify, and perhaps
explain, how various sources of knowledge act together to make some attributes more salient
than others. As Murphy and Medin (1985) acknowledged many years ago, this means that a
theory of categories must be rooted in a theory of knowledge. In organizational contexts, it is
one thing to map the structure of categories-in-use. It is quite another to explain why such
categories are put into use. For the latter, the structured mind hypothesis suggests that a theory
of organizational categories must include a theory about how sources of knowledge come
together during category formation for purposes of attribute reduction.
The structured world hypothesis brings with it different explanatory challenges. We have
shown that the problem of infinite dimensionality can be mitigated by non-randomly structuring
attributes into configurations of correlated attributes. Even imperfect correlations can be used to
privilege certain attributes, those with most informational value, over others in a bottom up
fashion, without prior knowledge acting as a filter. This was the explanation advanced by Porac
et al. (1995), for example, in their study of competitive clustering and category structure in the
Scottish knitwear industry. However, Porac et al. was a cross-sectional study, and the authors
freely admitted that an explanation for why the industry was ordered into a particular category
structure had to be based in a historical analysis of the forces that encouraged and selected out
certain configurations of attributes over others. Again, it is one thing to map the existence of
categories at any given time, and quite another to explain the historical evolution of the attribute
clusters on which that structure is based.
! 30!
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! 33!
FIGURE 1 The Fundamentals of Schelling (1971) Model
Attri
bute
1 At
tribu
te
2 Attri
bute
3
:!A!
:!B!
S(A,B)!=!2/3!
Average!Similarity!=!0.39!(<0.7)!
2/3 1/3
0/3
0/32/32/3
Focal Actor
3/3
3/3
2/3
2/3 1/3
0/3
0/32/32/3
Focal Actor
3/3
2/32/32/3
Focal Actor
3/3
2/3
Average!Similarity!=!0.78!(>0.7)!
� �� ��
� �� �
(a)
(b) (c)
(d) (e) (f)
! 34!
FIGURE 2 Replicating the Schelling Model
(a) The Effect of Threshold in Schelling’s Original Model
!Note) 300 agents on 20x20 grid; average of 100 independent runs; error bars indicate 95% confidence intervals
(b) Visualizing the Effect of Threshold
Th 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Initial State
! ! ! ! ! ! ! ! !
Steady State
! ! ! ! ! ! ! ! !Note) 300 agents on 20 x 20 grid; black(�) and grey(�) cell represents each type; white cells (�) are empty cells.
0.0 0.2 0.4 0.6 0.8 1.00
20
40
60
80
100
Ave
rage'Sim
ilarity'(%)
T hres hold '(T h)
! 35!
FIGURE 3 Effects of the Number of Attributes
(a) The Effect of Threshold under Multiple Attributes
Note) 300 agents on 20x20 grid; average of 100 independent runs; error bars indicate 95% confidence intervals
(b) Visualizing the Effect of Threshold under Multiple Attributes
Th� 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 attribute
! ! ! ! ! ! ! ! !
3 attributes
! ! ! ! ! ! ! ! !
10 attributes
! ! ! ! ! ! ! ! !Note) White cells (�) are empty cells. Darker cells indicate individuals surrounded by similar individuals that lighter cells are.
0.0 0.2 0.4 0.6 0.8 1.00
20
40
60
80
100
10(a ttributes
2(a ttributes
Ave
rage'Sim
ilarity'(%)
T hres hold '(T h)
1(a ttribute
! 36!
FIGURE 4 Discontent and Availability (Initial Period)
!
- 20x20, 300 agents, 100 independent runs - Availability = 1: chosen agent wants to migrate and is able to find a happy spot / 0: the chosen agent wants to migrate but no available
spot (values with no data point = the chosen agent is happy) !
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
100
1 1 1
1
1
1
1
11
1 1
3 3 33
3
3
3
33 3 3
10(a tt(a va ilability3 (a tt(a va ilability
10
10
%
T hres ho ld -(T h)
10
1(a tt(a va ilability
D is content(
! 37!
FIGURE 5 Effects of Unequal Attribute Weighting
Note:
- 3 attributes / 20 x 20 grid size / results at t=5,000 / 100 independent runs - Equal Weight: (�1, �2, �3 ) = (1, 1, 1) - Unequal Weight 1 (privileging 1 attribute): (�1, �2, �3 ) = (2.5, 0.25, 0.25) - Unequal Weight 2 (privileging 2 attributes): (�1, �2, �3 ) = (2.75/2, 2.75/2, 0.25)
0.0 0.2 0.4 0.6 0.8 1.0
50
60
70
80
90
100
Ave
rage1Sim
ilarity1(%)
T hres hold 1(T h)
!1 !A ttribute!Unequa l!W eig ht!1!Unequa l!W eig ht!2!E qua l!W eig ht
! 38!
FIGURE 6 Degree of Crispness and Average Similarity!
- !- Note: “c” denotes degree of crispness / number of attributes is 10 unless indicated otherwise / 20 x 20 grid size / 300 actors /
results at t=5,000 / 100 independent runs.!!
0.0
0.2
0.4
0.60.8
1.0
50
60
70
80
90
100
1(A ttributec=1.0
c=0.9
c=0.8
c=0.7
c=0.6
(
Ave
rage.Sim
ilarity.(%)
T h res ho ld .(T h )
c=0.5
! 39!
- FIGURE 7!
Varying Prototype Ratio across Different Numbers of Attributes a) 1 attribute
Prototype Ratio 0.0 0.2 0.4 0.6 0.8 1.0
t = 0
t = 1
0,00
0
b) 3 attributes Prototype Ratio 0.0 0.2 0.4 0.6 0.8 1.0
t = 0
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-2
-1.5
-1
-0.5
0
0.5
1
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
! 40!
t = 1
0,00
0
c) 10 attributes
Prototype Ratio 0.0 0.2 0.4 0.6 0.8 1.0
t = 0
t = 1
0,00
0
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20-4
-3
-2
-1
0
1
2
3
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -10
-8
-6
-4
-2
0
2
4
6
8
10
! 41!
FIGURE 8 Average Typicality Index across Expanding Neighborhood
(a) 1 Attribute (b) 3 Attributes (c) 10 Attributes
!4
!3
!2
!1
0
1
2
3
4
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
Typicality)Inde
x)of)Neighbo
rs
Neighborhood)Degree
Typicality=1Typicality=!1
!4
!3
!2
!1
0
1
2
3
4
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10thTypicality)Inde
x)of)Neighbo
rsNeighborhood)Degree
Typicality=:3Typicality=:1Typicality=:!1Typicality=:!3
!4
!3
!2
!1
0
1
2
3
4
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
Typicality)Inde
x)of)Neighbo
rs
Neighborhood)Degree
Typicality=10Typicality=6Typicality=0Typicality=!6Typicality=!10