# What makes Individual I’s a Collective We;
Coordination mechanisms & costs

Jisung Yoon  Kellogg School of Management, Northwestern University, Evanston,
IL  Northwestern Institute on Complex Systems, Evanston, IL  Chris Kempes
Santa Fe Institute, Santa Fe, NM  Vicky Chuqiao Yang  MIT Sloan School of
Management, Massachusetts Institute of Technology, Cambridge, MA  Seoul Lee
Kellogg School of Management, Northwestern University, Evanston, IL
Northwestern Institute on Complex Systems, Evanston, IL  Geoffrey West  Santa
Fe Institute, Santa Fe, NM  Hyejin Youn  Kellogg School of Management,
Northwestern University, Evanston, IL  Northwestern Institute on Complex
Systems, Evanston, IL  Santa Fe Institute, Santa Fe, NM  The order of authors
has not yet been determined.

###### Abstract

Collectives must coordinate and regulate individual endeavors and inputs in
order to achieve outputs that are more than the sum of individual abilities.
This principle holds true even for self-organizing systems where participation
is open and voluntary. Despite its importance, general perspectives on this
regulatory aspect are often overlooked, largely because coordination is not
easily observed in diverse organizations. Here, we use Wikipedia as a
transparent system for observing coordination and other types of regulatory
functions. We identify and quantify two primary types of coordination: one-way
coordination, characterized by rules and oversight, and two-way coordination,
marked by mutual interaction. We use scaling analysis to uncover a remarkably
consistent set of exponents, suggesting universal underlying dynamics in the
way contributors organize themselves. We find a key difference between two-way
coordination and one-way coordination. Mutual interaction through peer
interactions (measured by content reversals and related discussion pages)
increases superlinearly with an exponent of $1.3$ for both quantities. One-way
coordination increases sublinearly with an exponent of $0.9$ for
administrative activities and $0.7$ for automated bot activities. Analyzing
the interaction network between contributors, admins, and bots suggests an
inherent hierarchical structure that explains the differential (dis-)economy
of scales in coordination costs. Finally, we consider the growth of projects
and find that the influx of contributors initially causes rapid mutual
interactions, but then gradually transitions to structured supervision within
a decade. This result is a paradigmatic example of Weber’s evolution of the
organization formalism.

## Introduction

In complex systems, ranging from biological organisms to human organizations,
coordination mechanisms provide the key to understanding how numerous
individual components coalesce to form a cohesive and operational whole across
a wide range of scales [1, 2, 3]. Connecting isolated individual I’s to create
and maintain a cohesive collective We requires significant effort and
resources [4, 5, 6]. Yet, in complex systems, coordination processes and their
associated costs are often not immediately apparent, but only become clear
when applied or tested in real-world implementations.

Consider a bustling bee colony, a remarkable superorganism that might seem to
operate seamlessly. Beyond their genetic predetermination, these insect
societies require constant coordination of activities to adapt to complex
environments [7], including frequent role adjustments, communication through
pheromones [8], dances [9], and vibrations [10] in addition to resolving
occasional conflicts [11]. Currently in both corporate and academic settings
there are constant complaints about the toll of coordination. In the business
world, over the past two decades, time devoted to coordination by managers and
employees has expanded by 50% or more [12]. A recent survey indicates that
university faculty spent about 45% of their weekly work hours on meetings,
emails, scheduling, planning, and administrative tasks, which are not
immediately associated with the core functions of academia [13]. As our
society evolves into greater complexity, requiring bigger teams with experts
in specialized skills, we can expect this trend only to increase [14, 15].
However, there are fundamental questions yet to be answered about how much of
this regulatory function is optimal or efficient, and what underlying
mechanisms and processes generate the amount of regulation that we observe.

Socioeconomic systems, across their diverse manifestations, are primarily
coordinated and regulated by rules and feedback mechanisms [16, 17, 18, 19].
Akin to the genetically predetermined roles of bees, rules within these
systems can be either inherently embedded, like natural laws, or constructed
as abstract representations of formal strategies, blueprints, or cultural
norms, all aimed at meeting organizational goals. However, rules without
feedback mechanisms are often powerless. Much like the additional adaptations
bees make, the feedback mechanisms not only support the proper execution of
rules but also complement them by providing flexibility and adaptability, at
the level of interpersonal interactions. For example, authorities implement,
interpret, and execute rules or agendas through directional feedback.
Alongside this top-down governance, there is a perpetual need for open
dialogue and mutual interactions among members to minimize ambiguity and
foster adaptability, from which societal norms take shape and rules get
formulated.

What are the determining factors behind the cost of the coordination mechanism
within a collective? Several elements come into play, including the size of
the organisation [20], its structure [21, 22], and the intricacy of the
problems it grapples with [23, 24]. These variables are intertwined and often
hard to isolate from each other. For a collective to successfully navigate a
complex objective, the size, structure, and diversity of the organization need
to correspond to the required complexity of the problem at hand. This often
leads to a co-evolution between different coordination mechanisms, influenced
by a multitude of factors. For instance, large organizational units lean
towards impersonal modes of coordination mechanisms such as formalization,
standardization, and hierarchical control to ensure efficiency [25] while
complex tasks involving considerable interdependence are often better handled
through personal modes via horizontal communication channels within large
groups. Additionally, economies of scale are another consideration when it
comes to coordination costs. As an organization grows, the relative proportion
of administrators may decrease, thereby potentially reducing costs [25, 26].
Again, the economies of scale in coordination costs have been contested in
both theoretical accounts and empirical evidence [27, 28]. Therefore, it
remains unclear whether there exist endogenous mechanisms generating the
various types of coordination as more individual I’s join the collective We
[29].

To unpack the underlying mechanism and empirically assess the associated
costs, we have chosen Wikipedia as our focal point. Wikipedia is an ideal
system for studying these dynamics because of its vast collection of
individual activity records and myriad small communities associated with
individual projects, continually evolving through contributions from a diverse
array of intellectual minds [30, 31]. Individual projects are defined by
groups of varying sizes, giving us a perfect window into various scales. Yet,
each project shares approximately the same task of constructing comprehensive
knowledge structure on a specific topic. Within this community, individuals
contribute their knowledge not in isolation but in a concerted effort to
construct a cohesive knowledge structure for each project [32]. Consequently,
it requires organizational regulatory mechanisms to coordinate the diverse
efforts [1], including communication on talk pages [33, 34, 35], decisions
about other’s edits [30, 36, 37, 38, 39], the action of authorities [40, 41,
42], emergence of norms[43], and even execution of norms by automated bots
[44, 45, 46]. Notably, what sets Wikipedia apart is that it is entirely
authored and maintained by a decentralized community of volunteers, embodying
the essence of collective intelligence. For this reason, Wikipedia is
considered more like a community or ecosystem than a traditional organization
with no formal organization structure in the classification spectrum. However,
as we will demonstrate later, Wikipedia exhibits a _nascent_ hierarchical
structure [47, 31] which can greatly improve the effectiveness of coordination
within its operations.

In this paper, first, we investigate how coordination mechanisms evolve when
the number of contributors expands within each project by analyzing how
various metrics scale with the number of contributors. In almost all cases, we
find good evidence of simple power-law scaling (see Eq. 2 below) whose
exponents quantitatively reveal that the nature of coordination mechanisms
systematically shifts as the number of contributors on a project increases.
Furthermore, based on these scaling exponents, we are able to quantify
comparisons between our results and other systems, including biological [48,
49], ecological [50], and urban systems [51, 52]. Our study reveals that
mutual interactions exhibit a superlinear scaling relationship with the number
of contributors (i.e., mutual interactions per contributor increases as the
size of the project increases), while supervision and coordiantion by the
rule, measured by the automated bots, increases sublinearly as contributor
numbers rise (i.e., the amount of supervision per contributor decreases as the
size of the project increases, reflecting an economy of scale). We discovered
that modularity plays a crucial role in explaining the emergence of economies
of scale in supervision and rule-based coordination. Finally, we identify a
trade-off between coordination mechanisms, primarily categorized into personal
and impersonal coordination. As time progresses, Wikipedia projects gradually
shift towards increasingly emphasizing impersonal coordination, reflecting
Weber’s theory of organizational formalism. The details of these results and
their implications are presented and discussed in the following section.

Figure 1: Schematic diagram of coordination mechanisms in Wikipedia. In
Wikipedia, discussions involve communicating with other contributors about the
content of a project (page/article). We measure the level of discussion by
looking at the length of the corresponding talk page, an accompanied page for
discussing potential improvements to the project. Conflicting opinions or
ideas among contributors can lead to disagreements and disputes, proxied by
frequent reverts. To manage conflicts among contributors, administrators take
actions known as supervision, which we measure by administrators’ activities.
Bots are automated to take action, enforcing a set of rules. [Breakpoint:
motivating scaling framework]

## Measuring Coordination Cost in Wikipedia

The goal of Wikipedia is to capture the entirety of human understanding as
manifest by over 6 million projects covering an expansive range of topics.
This grand ambition, accomplished through the collective efforts of numerous
contributors, necessitates an intricate process wherein each article’s fabric
is woven through meticulous edits to ensure accuracy, coherency, and
structural consistency. To achieve this objective, for each project (or
article/page), contributors edit segments such that the entire project is
written in a coherent and consistent structure. This often incurs disagreement
with other contributors and thus ends up with revising or reverting prior
contributors’ work. Indeed, contributors do not edit in isolation—they need to
coordinate themselves with others on the platform to resolve disagreements and
conflicts. We categorized coordination activities on Wikipedia into four
types: _mutual interactions_ , _supervision_ , and _rule enforcement_ as
illustrated in Fig 1. By measuring the volume of these coordination
activities, we can understand how Wikipedia functions as a collaborative
platform and quantify the cost of making I’s into We in a decentralized
system.

Given the vast intellectual diversity of individuals, differing views on any
given topic are inevitable. The true brilliance of Wikipedia lies in its
ability to weave these disparate threads into a cohesive narrative. However,
pursuing structural cohesion from various people comes with a price. Discord
can arise when contributors’ opinions diverge significantly. Such
disagreements become visible in the frequency of reversals — a feature that
empowers all contributors to completely annul the efforts of others, thus
reverting the article to a previous state. This process is often considered to
reveal a “norm violation” [53], and widespread practice of using reverts to
map conflicts on Wikipedia is well-established. [30, 36, 37, 38]. When simple
reversals do not resolve the discords, or reversals continue back and forth
indefinitely, a lengthy discussion with other contributors is needed to make
decisions about an project’s content and potential improvements. These
discussions are documented in the “talk page” for each project [33, 34]. All
contributors can utilize this page to deliberate on the validity of sources,
the page’s structure and organization, what content should be included, and
the conflicting opinions. In some cases, talk pages are also used to build
consensus or norms for the project in the Frequently Asked Questions section,
including documenting how previous conflicts were resolved. In our analysis,
both discussions measured by the length of the talk page and disagreements
measured by the number of reversals are forms of _mutual interactions_ toward
consensus through tools available to all contributors. We characterize mutual
interactions as _two-way coordination_ since all project participants can
utilize these tools in their interactions without requiring a hierarchical
relationship between them.

Beyond the actions that all contributors can take to affect each other,
higher-level intervention are only available to a small subset of Wikipedia
administrators, also referred to as Sysops, an abbreviation of the system
operator. Sysops comprise only 0.001% of Wikipedia users and are officially
approved through the adminship process, by a higher administrator group called
_Bureaucrats_. Sysops have powerful authority to block and unblock users,
delete and protect projects, and rename projects, often guided by the need to
deal with vandalism, enforce community policies and guidelines, and mediate
disputes. Sysops often delegate their duties and administrative privileges to
trusted Wikipedia users. We include actions taken by those who have asymmetric
privileges as _supervision_ [54, 55]. Supervision is opposed to mutual
interactions since they entail human intervention orginated from claer
distinct authority and power hierarchy within the interactions.

A final category of actions focuses on Wikibots, which in Wikipedia are
automated tools that carry out repetitive and mundane tasks required for
project maintenance. One example is anti-vandalism bots like ClueBot NG, which
are specifically designed to detect and undo instances of vandalism swiftly.
According to an empirical study, when ClueBot NG experienced a breakdown, the
number of revert edits nearly doubled [56], highlighting the bot’s
effectiveness in suppressing conflicts and upholding the quality of
Wikipedia’s project. Bots can also generate high-quality articles by
correcting evident deviations from the established template [57] and improving
verifiability of projects [58]. While Wikibots are capable of making edits at
a fast pace if they are designed or operated incorrectly, they have the
potential to disrupt the smooth functioning of Wikipedia. To address this
concern, Wikipedia established a strict bot policy, which is primarily
maintained by Wikipedia administrators. In our analysis, we consider edits by
wikibots as a proxy of _rule enforcement_ in Wikipedia [44, 45].

To provide additional context, we illustrate the case of the Phil Spector
project in Fig.1. On the associated talk pages, contributors engage in
discussions on which of Phil Spector’s musical achievements and the crime he
committed should be highlighted.Within the edit histry, a pattern of
successive reversions by contributors in similar contexts becomes evident. Due
to the absence of any inherent hierarchy among contributors regarding
discussions and reversions, conflicts among individuals can endure over
extended periods and, on occasion, escalate into acts of vandalism. In
response to these challenges, administrators intervene and protect the project
by temporarily restricting editing to mitigate ongoing conflicts and restore
order to the project. Lastly, the bots monitor and enforce established rules
by detecting abvious vandalism, incorrect, reference, and inappropriate
reverts.

In summary, in Figure 2a, we present a conceptual framework to categorize
coordination mechanisms with two principal dimensions — complexity and
authority — with a phase diagram that elucidates the interrelationship between
these two dimensions. Tasks that are straightforward occupy the low-complexity
end of the spectrum (on the left). Conversely, tasks that are ambiguous,
intricate, and require diverse inputs are situated at the high-complexity end
(on the right). The authority differentiates one-way coordination (asymmetric
interaction) from two-way coordination (mutual interactions among peers). For
instance, bots generally execute tasks that are easily codified and adhere to
organizationally agreed-upon rules, and thus, they are located in one-way but
low complexity (top left). On the other hand, content reversals and
discussions on talk pages often involve nuanced tasks without a clear
hierarchical structure, placing them in two-way but high complexity (bottom
right). Administrators, who frequently grapple with complex tasks while
wielding authority, find themselves in the quadrant characterized by both high
complexity and high authority (top right).

Figure 2: Interplay of Coordination Mechanisms Across Complexity and Authority
Dimensions. (a) Phase diagram illustrating the regulatory functions along the
complexity and authority dimensions. (b) Scaling behavior of admin activities
(c) frequency of reverts, (d) automated bot actions with contributor numbers
and (e) Talk page size. Each project is log-binned (blue) with error bars and
estimated $\beta$ in $N^{\beta}$ (the orange line), accompanied by a linear
trend (gray line) for reference. Mutual interactions, such as Talk page size
and reverts, exhibit superlinear scaling, growing faster than the number of
contributors, while supervision and rule enforcement, represented by
administrator and bot activities, show sublinear scaling, increasing more
slowly than contributor numbers. The data suggest an economy of scale
transitioning from superlinear to sublinear scaling, corresponding to a shift
from personal interactions in the lower right to more impersonal interactions
in the upper left. We highlight the project that we discussed in the paper:
George W.Bush, Phil Specter, and Quantum State.

## Scaling Factor in Coordination Costs

As a first step, we delve into the concept of the scaling factor [59], a
pivotal determinant of the costs associated with coordination mechanisms. It
is clear that coordination costs rise with an increase in organizational size
[20]. However, the intriguing question here is the rate at which they grow.
For instance, one may ponder whether certain coordination costs increase at a
faster or slower pace than the organization’s size. Can we find consistent
patterns of growth in alignment with our proposed framework outlined above? To
study how coordination costs increase with the organization size, we compare
organizations of different sizes using a scaling perspective (See Methods).
Here, we regard the contributors in each project as a functioning
organization. Using the number of unique contributors on each project $N$, as
the measure of organization size, we consider power-law scaling as follow

$Y=Y_{0}N^{\beta},$ (1)

where $Y$ represents the coordination cost and $Y_{0}$ is a normalization
constant. The exponent $\beta$ quantifies the rate of increase in $Y$ with
relative increases in $N$ as being greater ($\beta>1$) or less ($\beta<1$)
than linear.

We focus our empirical analysis on Vital Articles, a subset of projects
selected by Wikipedia as covering important topics, aiming at higher quality.
Among them, we only consider projects with at least one edit in their talk
pages, one revert in their content, and one action from an administrator. This
resulting subset comprises 26,014 projects tuned to our questions. Fig. 2
shows how coordination costs scale with the number of contributors. We observe
that both talk page size and the number of reverts exhibit a superlinear
scaling relationship ($\beta_{\text{talk}},\beta_{\text{revert}}\simeq 1.3$,
Fig. 2 b-c). This implies that discussions and disagreements in Wikipedia grow
faster than the number of contributors — _descaled economy of scale_ ,
reflecting the societal characteristics of coordination costs driven by
interactions, which may follow mechanisms similar to those driving urban
features with superlinear scaling [51, 52]. On the other hand, administrator
activities and bot activities increase sublinearly,
$\beta_{\text{adm.act.}}\simeq 0.9$ and $\beta_{\text{bot.act.}}\simeq 0.7$,
respectively (Fig. 2 d-e.) These results display _economies of scale_ ,
consistent with economies of scale of infrastructure in urban systems [51] and
administrative staff in companies [25]. The sublinear scaling relationship is
robust to measuring number of unique administrators and bots (Fig.
LABEL:fig:costs_number), $\beta_{\text{adm.}}\simeq 0.7$ and
$\beta_{\text{bot}}\simeq 0.4$, respectively. This robust economy of scale
aligns with prior research on the natural development of bureaucracy [47].
Lastly, the FAQ length, which can be seen as an established norm derived from
collective discourse, exhibits sublinear scaling ($\beta_{\text{FAQ}}=0.67$,
as illustrated in Fig. LABEL:fig:faq). This provides insight into how
coordination contributes to the establishment of norms, culture, and rules.
Notably, it indicates that the increase in FAQ length occurs at a pace even
slower than the growth rate of the number of administrators.

Furthermore, we have identified a plausible explanation for how complexity and
authority influence the scaling factor of coordination costs. Let us revisit
the phase diagram (Fig. 2 a). According to the results, when the coordination
mechanism deals with complexity but lacks authority (bottom left), $\beta$ has
a high value ($\beta\simeq 1.3$). Conversely, when the coordination mechanism
exhibits similar complexity (top left) but relies on authority, $\beta$ can
decrease ($\beta\simeq 0.9$). Furthermore, when the coordination mechanism
handles tasks of low complexity (top left), $\beta$ may even decrease further
($\beta\simeq 0.7$). As suggested by March and Mintzberg, these aspects can be
classified into two coordination mechanisms: feedback and rules [60, 16].
Individuals can coordinate and adjust themselves _personally_ in response to
feedback through communication and discussion. On the other hand, according to
Weber, rules and protocols can _impersonally_ manage individuals [61]. These
two ways of coordinating individuals can complement and substitute each other
to fulfill the coordination costs that the group size demands. In this
context, our findings indicate a shift in the scale economy, transitioning
from a superlinear to a sublinear scaling pattern. This transition aligns with
a shift from personalized interactions (bottom right) to increasingly
impersonal interactions (top left) in Fig. 2a.

## From Modular Interactions to Hierarchical Structure

The superlinear relationship of mutual interactions suggests that as the
number of contributors increases, the more interact per person. Then, if every
interaction requires administration, the number of administrators might also
increase at the same rate $\approx N^{1.2}$. However, contrary to this
expectation, the number of administrators increases sublinearly by the
relationship of $\approx N^{0.7}$. This discrepancy may raise an intriguing
question: Where does this gap originate? Does the structure of the interaction
of mutual interactions reveal an economy of scale in supervision, despite the
lack of economy of scale in mutual interactions [62]?

To understand this (dis-)economy of scales, we examine mutual interaction
network (See Methods) from the mention relationship in edit history. As an
example, the mutual interaction network of the Phil Spector project is shown
in Fig. 3a. Overall, the network is modular, and administrators (red nodes)
and bots (black nodes) are broadly interacting across modules. Based on
observations, we construct a schematic representation of mutual interaction
networks, presenting two hypotheses in Fig. 3b. First, a modular structure
will naturally emerge from the interactions among contributors and the number
of modules exhibits sublinear scaling because the size of the module is
heterogeneous. Second, administrators and bots hold the responsibility of
managing these emergent modules.

Figure 3: Emergence of Formal Hierarchical Structures from Modular
Interactions. (a) Mutual interaction network of the Phil Spector project. Node
color indicates the extracted module with community detection. Red nodes
represent administrators, and black nodes represent bots. (b) Schematic
diagram of the mutual interaction network. The modular structure naturally
arises from mutual interactions. Administrators and bots manage these emergent
modules. The number of managed modules varies depending on the types of
administrators and bots involved. (c) Scaling behavior of the number of
administrators and bots compared to the number of modules. Each project is
log-binned (blue) with error bars, and estimated $\beta$ in $N^{\beta}$ (the
orange line), accompanied by a linear trend (gray line) for reference. The
number of administrators and bots shows an almost linear scaling
($\beta$=1.09). Scaling behavior of the number of the module compared to the
number of nodes (inset). The number of modules shows a sublinear scaling
relationship. (d) The extent of module coverage for each role, increasing from
delegated administrator to administrator (sysop) to Wikibots (see
Fig.LABEL:fig:adminship in SI). The average degree of each role is shown in
the inset.

We tested our hypotheses by expanding our analysis of the mutual interaction
network to encompass all vital articles. Here, we define the module as the
extracted modules from community detection (See Methods). First, modular
structures emerged across the spectrum of vital articles, with an average
modularity score of 0.79. Furthermore, as illustrated in Fig. 3c (inset), the
number of modules $N_{m}$ increases sublinearly ($\beta=0.69$) as the number
of nodes $N_{n}$ increases. As expected, the module size exhibits
heterogeneity. In Fig. LABEL:fig:mention, it becomes evident that the size of
the largest module follows a scaling pattern with $\beta=0.7$. On the
contrary, the average module size scales to a lesser degree ($\beta=0.3$), and
the median module size does not show any discernible scaling behavior. Second,
Fig. 3c shows the unique number of administrators and bots increases almost
linearly ($\beta=1.09$) with the number of modules in the mentioned network,
that is, $N_{\text{admin + bot}}\sim N_{m}$. This suggests that administrators
do not involve every individual in mutual interaction $N$, but rather its
coarse-grained module, a dense group of interactions $N_{m}$. This explanation
could help to clarify why economies of scale in supervision emphasize modules
rather than individual entities.

Then, how do administrators and bots manage the modules? Are they involved in
overseeing multiple modules or positioned centrally within modules? While one
might initially assume a straightforward scenario where each admin manages
approximately one module, we find a more nuanced structure emerges upon closer
examination of the network. In Fig. 3d, we adress these questions with the
average module coverage $\langle\mathpzc{M}\rangle$ (see Methods) for each
administrative role. The $\langle\mathpzc{M}\rangle$ of delegated
administrators is 1.10, whereas the $\langle\mathpzc{M}\rangle$ of
administrators, who hold higher authority than delegated administrators, is
1.12. This suggests that administrators with greater authority span a larger
number of modules $\langle\mathpzc{M}\rangle$. Based on this observation, we
can reconstruct a hierarchical structure as depicted in Fig.3a, where peer
interactions are clustered and overseen by administrators at various levels.
Bots, characterized by their high scalability and impersonal nature, exhibit
the highest $\langle\mathpzc{M}\rangle=2.72$, engaging in broader modules
rather than being central [63]. It is noteworthy that the increasing pattern
of $\langle\mathpzc{M}\rangle$ also aligns with the increasing order of the
economy of scales, suggesting that the economy of scales originates from the
modular structure.

Then, is there any difference in the strategy of how administrators and bots
manage the multiple modules? By definition of the module coverage,
$\langle\mathpzc{M}\rangle$ has a tendency to increase when a node possesses
greater capability (or scalability). As an example, in the inset of Fig. 3d,
bots exhibit a significantly higher average degree compared to administrators.
To quantify different strategies in the network, we introduce module awareness
score $r$ (See methods). Our findings indicate that human administrators are
more inclined towards the module-aware strategy ($r_{\text{del. adm.}}=0.34$,
$r_{\text{adm.}}=0.26$), while bots tend to align with the individual-aware
strategy ($r_{\text{bot}}=0.14$). In simpler terms, the economy of scale in
supervision arises from the module-aware abilities of human administrators,
whereas the economy of scale in rule enforcement is derived from the extensive
scalability of bots. Our overarching findings remain consistent in the revert
network, another type of mutual interaction network.

## Emergence of Impersonal Coordination

The scaling relationship (Eq. 2) provides an average description of the
behavior of coordination costs. There are cases where some projects require
unusual coordination costs. As discussed above, the coordination costs are
determined not only by the size $N$ but also by the complexity of functions
$C$, which are often interrelated. For complex topics, it is often the case
that a larger group of individuals is required to produce a full perspective.
Similarly, the more complex the subject, the higher the potential for
disagreements and disputes, drawing a greater crowd into the discourse. Table
LABEL:table:costs summarizes the scaling exponents deconstructed by article
category. Overall, the findings are remarkably consistent with the previous
integrated analysis: talk page size and the number of reverts exhibit a
superlinear scaling relationship, while administrator and bot activity shows a
sublinear relationship. On the other hand, projects in the category of
Everyday Life and Mathematics—a topic generally devoid of controversy—the talk
page size increases only mildly superlinearly with the rise in contributors
($\beta\approx 1.1$), which stands in stark contrast to more contentious
topics such as History or Philosophy and religion, where the talk page size
rises at a far brisker pace ($\beta\approx 1.5$). The results indicate that
the complexity of the topic has to be further differentiated epistemological
complexity from controversy-prone topics.

| $\xi^{\text{talk}}$ | $\xi^{\text{revert}}$ | $\xi^{\text{bot act}}$ | $\xi^{\text{adm act}}$
---|---|---|---|---
$\xi^{\text{talk}}$ |  |  |  |
$\xi^{\text{revert}}$ | 0.092 |  |  |
$\xi^{\text{bot act}}$ | 0.080 | 0.088 |  |
$\xi^{\text{adm act}}$ | 0.386 | 0.229 | 0.427 |
Table 1: Correlation between residuals The residuals of talk page size,
reverts, bots’ activities and admin activities are denoted as
$\xi^{\text{talk}}$, $\xi^{\text{revert}}$, $\xi^{\text{bot act}}$ and
$\xi^{\text{adm act}}$, respectively. Note that there are no correlations
among residuals except for admin activities. The p-values for all correlations
in this table are significant ($\ll 0.001$).

To provide insight into the factor factors influencing coordination costs
beyond the average behavior predicted by the power-law scaling model, we
compute residual (See Methods) of each coordination cost. Residual is the
difference between the observed and predicted values, denoted as
$\xi_{i}\equiv\log\frac{Y_{i}}{Y(N_{i})}=\log\frac{Y_{i}}{Y_{0}N_{i}^{\beta}}$
[64]. Table 1 illustrates the correlations among the residuals, highlighting
their interrelationships after the removal of the average description of the
behavior of the coordination cost. First, it’s evident that administrative
activities consistently positively correlate with other types of the
coordination activities, while all other correlations are negligible ($<0.1$).
This observation could suggest that governance, represented by administrative
activities, plays a central role in the collective intelligence coordination
process [65, 66]. Second, we find no correlation among residuals except for
administrator activities. This finding contradicts the common assumption that
one coordination method can replace another, implying that an increase in
rules, protocols, or supervision might reduce the need for mutual
interactions. The absence of such a trade-off raises the following questions:
What might be the underlying reason for this lack of correlation? Could it be
that the strong correlation observed between administrative activities and
hidden trade-off mechanisms is blending or confounding the dynamics within
these four residuals? For instance, controversial topics concurrently drive
coordination costs with increasing administrative activities, resulting in
positive correlation with administrative activities, but not among themselves.

Figure 4: Transition from Personal coordination to Impersonal Coordination (a)
Compensation among coordination costs. Blue dots and whiskers indicate the
average value of each principal axis categorized by a specific criterion with
standard errors. The arrows in the plot indicate the loadings of each
variable, showing how each original variable contributes to each principal
component. Notably, all variables demonstrate positive loading for principal
axis 1. However, for principal axis 2, we observe positive loading (indicated
by a red arrow) and negative loading (represented by a blue arrow) for
different variables. (b) Transition from personal coordination to impersonal
coordination. Dots and whiskers indicate the average value of each principal
axis categorized by a project year with standard errors. The time series of
PC2 scores is depicted in the inset.

To address aforementioned queries, we employ principal component analysis on
the residuals (See Methods). Fig. 4 presents resulting two primary orthogonal
principle component. We name Principle Component (PC1) as the “controversy
axis” since it demonstrates a simultaneous escalation of all coordination
costs aligned with administrator activity, with a loading factor of 0.8. In
other words, administrators typically focus their attention on controversial
projects and rarely facilitate the transition of projects from high-conflict
states to low-conflict states [39], leading to a concurrent increase in all
coordination costs. In line with this alignment, we observe contentious
topics, such as History or Philos./Religion, tend to be positioned on the
right side of the figure (PC1 $>$ 0), while less contentious topics such as
Everyday life or Mathematics tend to be situated on the left side of the
figure (PC1 $<$ 0). To illustrate this with a clearer example, let’s consider
the PC1 values of two individuals in the people category. Phil Spector has a
PC1 value of 0.35, suggesting that the topic is relatively contentious. On the
other hand, Goethe, also in the people category and with a similar number of
contributors, has a PC1 value of -0.01.

In contrast to PC1, which exhibits positive loadings across all dimensions,
Principal Component (PC2) unveils a trade-off mechanism among coordination
costs by breaking down administrator and bot activity with positive loadings,
in contrast to talk page length and reverts, which exhibit negative loadings.
Consequently, we refer to PC2 as the “compensation axis”. Specifically, when
focusing on coordination costs with high loadings, an increase in bot activity
is associated with a decrease in reverts. Nevertheless, most plausible
explanation is that bot activity has the potential to suppress conflicts, as
the mechanisms behind bot activity, which is the _impersonal coordination_ by
the rule, inherently mitigate conflicts and promote smoother coordination
[56].

Aligned with the idea that PC2 is primarily linked to potential compensation
driven by bot activity, the PC2 scores for each project indicate the extent of
alignment with “impersonal” coordination governed by rules. In the inset of
Fig. 4b, we present the PC2 scores grouped by the project’s age, which serves
as a proxy for the article’s maturity. During the initial stages of a project
(left), PC2 scores tend to exhibit negative values, signifying a weak degree
of impersonal coordination. However, as the articles mature (right), their
alignment with PC2 increases, indicating a smooth transition from personnel
coordination to impersonal coordination takes place.

Taking into account the scores for PC1 and PC2 together, the transition from
personal coordination to impersonal coordination is depicted in Fig. 4b. In
the initial stages of the project, the project tends to exhibit a high PC1
score but a low PC2 score, indicating a reliance on personal coordination.
Furthermore, their variance is substantial compared to older projects. As
projects mature, the average PC1 value tends to decrease or remain relatively
stable, while the average PC2 value steadily increases, which is evidence of
the emergence of impersonal coordination. This shift aligns with a theoretical
perspective in the literature on organizational design. Early-stage
organizations often employ informal coordination modes, like mutual
interactions, as they navigate various corporate strategies that are not yet
well-defined. These informal or organic ways help them to explore and build
new business models from the ground up under high uncertainty [67]. However,
as organizations evolve and mature, organization typically often shift toward
bureaucratization, adopting formal structures or rules to achieve both
stability and efficiency in coordination [68].

Similarly, contributors of nascent Wikipedia articles collaborate much like
the founding members of entrepreneurial ventures. They start from scratch to
build the initial outlines of the articles and flesh them out. When they face
disagreements regarding the fundamental structures of these articles, they are
compelled to engage in unstructured discussions on the articles’ talk pages
and manually revert others’ edits. This is because there are no established
rules to govern these unprecedented conflicts in newly created articles. These
seemingly cumbersome processes of mutual interactions help the initial
contributors to establish the basic framework of these nascent articles. Once
they have established structures, a significant portion of coordination needs
can be dealt with based on more formal ways, such as bots and adminstrators.
This transition is possible because matured articles have accumulated enough
prior cases to reference when new conflicts arise, and most conflicts
typically concern minor issues that can be addressed without changing the
entire structure of the article. For validation, we cannot observe a similar
emergence of impersonal coordination when examining the influence of the
number of contributors, as depicted in Fig. LABEL:fig:real_size_vert.

Furthermore, for a deeper understanding of the trade-off between the
coordination mechanism, we build the minimal model suggesting possible
mechanisms (See Supporting Information). Within the model, the parameter
$\sigma$ governs the necessary level of regulation for a project based on its
size. A larger $\sigma$ value indicates a more significant deviation from the
expected scaling behavior. Subsequently, each coordination mechanism allocates
a proportion of the overall required regulation following a predetermined
compensation order, which is very naïve mechainsm of the compensation. With
these two simple mechaism Fig.LABEL:fig:synthetic_combination, we successfully
reproduce our compensation results (Fig.4a). In alignment with our explanation
of PC1, as the deviation from the expected scaling behavior increases (as
$\sigma$ rises), all coordination costs simultaneously increase. In the case
of PC2, when employing the compensation order “bot-admin-talk-revert”, we are
able to replicate results that closely resemble empirical findings. This
implies that our proposed straightforward compensation mechanism could provide
an explanation for both empirical data and the underlying structure of
compensation in Wikipedia, characterized by the sequence “bot-admin-talk-
revert”. Interestingly, this order coincides with order of the increasing
pattern in the scaling factors.

## Discussion

We investigated how coordination costs scale with the size of the Wikipedia
communities and how they vary across different categories of articles. By
analyzing the coordination costs in Wikipedia, we found that mutual
interaction costs scale superlinearly with the number of contributors. On the
contrary, the cost of supervision and the enforcement of rules increase
sublinearly with the number of contributors. Our residual analysis also
highlighted that article content and the interaction between contributors
shape the coordination costs of Wikipedia pages.

Our findings contribute to the literature on coordination costs and scaling in
complex systems by providing insights into the coordination challenges online
communities face. Poeple ofthen ovelook the cost of the collective
intelligence and tend to assume the absence of formal organization structure.
Nevertheless, in line with the the nature law of hierarchy [69, 47], we reveal
the natural emergence of hierarchical structures and the impersonal
coordination in Wikipedia. The results also have practical implications for
online communities, as they demonstrate the importance of understanding the
coordination costs associated with different categories of content and the
need to develop appropriate coordination mechanisms as the community size
increases.

In our increasingly complex society, the efforts and expertise of highly
specialized individuals are distributed across the globe. This necessitates
their productive coordination spanning vast multi-level networks to create a
diverse array of complex goods and services [15]. Also, the impact of
successful collaboration is becoming increasingly important in knowledge
production, showing the significance of collaborative efforts in producing
successful outcomes and advancing scientific progress [70, 14, 71, 72]. This
reality underscores the urgent need for enhanced comprehension of coordination
mechanisms and their latent structures at many different scales.

## Materials and Methods

### Wikipedia Data

We used Wikipedia XML dump of English Wikipedia on January 9, 2022, comprising
edit history of 6,412,821 distinct pages in Wikipedia. Each edit includes a
unique identifier of the editor (username for registered contributors and IP
address for unregistered contributors), timestamp, comments, and edit size.
Additionally, we extracted contributor’s roles from the user group table of
the SQL dump. Our primary focus lies on the crucial articles in the curated
collection, Vital articles. These articles have been carefully selected to
cover a broad spectrum of subjects and are considered essential for obtaining
a comprehensive understanding of human knowledge. Encompassing a wide range of
disciplines, including history, science, arts, geography, and more, these
articles represent significant topics, events, individuals, and concepts. They
undergo regular evaluation and updating to ensure their accuracy, relevance,
and comprehensiveness, thus serving as indispensable pillars of knowledge
within Wikipedia’s vast and extensive encyclopedia.

### Scaling Analysis

To study how coordination costs increase with the number of contributors, we
employ the scaling framework [59]. Using the number of unique contributors on
each page $N$, as the measure of organization size, we consider power-law
scaling to take the form of

$Y=Y_{0}N^{\beta}.$ (2)

$Y$ denotes the coordination cost, while $Y_{0}$ serves as a normalization
constant. The parameter $\beta$ characterizes the pace at which $Y$ increases
in response to relative changes in $N$, with values of $\beta$ exceeding 1
indicating a faster increase and values below 1 indicating a slower increase
compared to expected if it were linear. Here, we use a specific strategy to
fit the data into the scaling law originated to focus on understanding growth
behavior which is the interest of this research. In our data set, the majority
of projects have a relatively small number of contributors, typically around
1,000. However, there are a few exceptional projects with a significantly
larger contributor base. If we were to fit a model using ordinary least
squares (OLS) regression, the model would naturally prioritize explaining the
majority of the data. However, in this paper, our focus is on understanding
the growth patterns of these projects, rather than modeling the majority of
the data. To address this, we begin by logarithmically binning the data and
fitting the model using the binned data. This approach ensures that every
scale, from small to large projects, is given equal importance in our
analysis.

The exponents are robust to the entire English Wikipedia articles beyond vital
articles with $\beta_{\text{talk}}=1.379$, $\beta_{\text{revert}}=1.309$,
$\beta_{\text{adm. act.}}=0.910$, and $\beta_{\text{bot act.}}=0.679$ Also,
among the 48,864 vital pages on Wikipedia, 22,680 pages have not undergone any
actions by administrators. To delve deeper into this matter, we performed a
robust check by re-conducting a scaling analysis on talk page size, number of
reverts, and bots’ activity, including pages that have not received any
intervention from administrators. We again confirm that our results are
consistent (Fig.LABEL:fig:costs_robust), affirming our earlier observations.

### Mutual Interaction Network

To understand this (dis-)economy of scales, we construct the _mutual
interaction network_ from the edit history when contributors mention other
contributors explicitly (e.g., Undid revision by @Jisung). In this network,
each node is the contributor who has mentioned another person or was mentioned
by another person at least once. Nodes are linked when there exist mentions.
For simplicity, we ignore directionality in this network. As almost 92% of
mentions are revert actions, the constructed mention network wouldn’t be too
different from the revert network. For the analysis, we remove components (or
sub-graph) with two nodes, which is a dyadic relationship. We extracted the
modular structure with the community detection [73]. Overall, the constructed
network is modular (average modularity is 0.79).

### Module coverage and module-awareness

To quantify the extent of module coverage of each node, we employ module
coverage denoted as $\mathpzc{M}_{i}$, which is quantified by the unique
number of modules managed by a given node $i$, which serves as a proxy for its
role as a broker or structural hole [74]. As an illustration, in Fig. 3b, the
leftmost administrator has two connections, but both connections are within a
single module, resulting in $\mathpzc{M}_{i}=1$. On the other hand, the top
administrator has four connections spanning three unique modules, yielding
$\mathpzc{M}_{i}=3$. We calculate the average module coverage capacity for
delegated administrators, administrators, and bots, yielding
$\langle\mathpzc{M}\rangle$ for each type. When we consider the concept of
module coverage, the value $\langle\mathpzc{M}\rangle$ tends to increase as
the average degree is high. For example, bots often exhibit a high
$\langle\mathpzc{M}\rangle$ due to their elevated average degree. To measure
module coverage behavior while normalizing for degree size, we utilize the
module awareness ratio $r=\frac{\langle\mathpzc{M}\rangle}{\langle k\rangle}$
for delegated administrators, administrators, and bots. The value of $r$ can
vary depending on the node’s strategy for managing connections within the
network.

For better understanding, we propose two extreme cases. The first extreme
scenario is the perfect module-aware strategy, wherein nodes establish
connections solely with the module and not with individual nodes. In this
case, multiple connections within the same module are not allowed, resulting
in $\mathpzc{M}_{i}\simeq k_{i}$ and a value of $r\simeq 1$. Conversely, the
other extreme scenario is the perfect individual-aware strategy, where nodes
establish connections with individual nodes without considering the modular
structure. In many large networks, it is theoretically possible for the size
of the largest module to increase proportionally to the size of the entire
network [75], while the sizes of smaller modules remain constant. If nodes
adopt the individual-aware strategy, connections are more likely to be
distributed in the larger module. Consequently, $\mathpzc{M}_{i}$ will remain
constant as $k_{i}$ increases, resulting in a value of $r\simeq 0$. Likewise,
we can map the node’s strategy in the network along a continuous spectrum
ranging from the individual-aware strategy ($r=0$) to the module-aware
strategy ($r=1$)

### Residual

As the scaling relationship (Eq. 2) only provides an average description of
the behavior of coordination costs, differences between observed and predicted
values provides insights into the factors that affect coordination costs
beyond the mean behavior predicted by the scaling law. To investigate this, we
calculate the residual, $\xi_{i}$, [64] for each coordination cost as follows:

$\xi_{i}\equiv\log\frac{Y_{i}}{Y(N_{i})}=\log\frac{Y_{i}}{Y_{0}N_{i}^{\beta}},$
(3)

where $Y_{i}$ is the observed value of coordination costs and $Y(N_{i})$ is
the expected value of coordination costs given the number of contributors
$N_{i}$. For the comparison, Table LABEL:table:raw, present the Pearson
correlation between coordination costs, which exhibits strong correlations
primarily influenced by the number of contributors. On the other hand, Table 1
shows the correlation among residuals, highlighting their relationships after
removing the average description of the behavior of coordination costs.

### Principal component analysis

Principal Component Analysis [76], commonly known as PCA, is a powerful
statistical technique used for dimensionality reduction and data compression.
PCA allows us to extract the most significant information from a dataset by
capturing the underlying structure and reducing redundancy. PCA provides a
streamlined and more manageable representation of data, facilitating better
insights and analysis. In our analysis, we apply PCA to the residuals of each
project$\xi$ are represented as a four-dimensional vector,
[$\xi^{\text{talk}}$, $\xi^{\text{revert}}$, $\xi^{\text{admin act}}$,
$\xi^{\text{bot act}}$], and extracted two principal components (PC1 and PC2).
Explained variance ratio for Principal Component 1 (PC1) is 0.43, and for
Principal Component 2 (PC2) is 0.23. In PCA, loadings are the correlation
between original variables and principle components. A substantial loading on
a specific principal component indicates pronounced influence on that
particular component.

## Acknowledgements

The authors would like to acknowledge the support of the National Science
Foundation Grant Award Number 2133863. J.Y and H.Y thank Eric Rupley for the
generous support and interesting discussions.

## Author Contributions

JY collected and performed analysis. JY and HY conceptualized, designed, and
drafted the study. All authors interpreted the analyses, wrote the manuscript,
and approved the final version of the manuscript.

## Additional Information

Supporting Information is available for this paper. Correspondence and
requests for materials should be addressed to Dr. Youn.

## Data availability

Wikipedia edit history data are available at wiki-dumps,
https://dumps.wikimedia.org/. We collect the list of vital articles from
https://en.wikipedia.org/wiki/Wikipedia\%3AVital_articles.

## Code Availability

The code for this analysis is available upon request.

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into a polar electric field $\vec{E}$ and axial magnetic field $\vec{H}$.

This formalism also handles group transformations in a natural way. For
example, the Hodge duality between a spatial plane and an orthogonal axis in
relative 3D space defines the Gibbs 3-vector _cross product_ ,

$\displaystyle\vec{\sigma}_{i}\times\vec{\sigma}_{j}$
$\displaystyle\equiv-I(\vec{\sigma}_{i}\wedge\vec{\sigma}_{j})=\epsilon_{ijk}\vec{\sigma}_{k},$
(166)

that is closed for grade-1 3D vectors at the expense of breaking the
associativity of the wedge product, but this definition is identical to the
spin-(1/2) rotation group commutator relations for the Pauli spin-matrices,

$\displaystyle\vec{\sigma}_{i}\vec{\sigma}_{j}-\vec{\sigma}_{j}\vec{\sigma}_{i}=\epsilon_{ijk}\,2I\vec{\sigma}_{k},$
(167)

if the pseudoscalar $I$ is identified with the scalar imaginary $i$ when
restricted to 3D space.

More generally, the six unit planes of spacetime form a closed subalgebra
under the wedge product, which is then precisely the _Lie bracket_ for the
_Lorentz group_ of spacetime rotations generated by the unit planes directly.
That is, the three space-time planes $\vec{\sigma}_{k}\sim-i\mathbf{K}_{k}$
with positive signature generate hyperbolic boost rotations,

$\displaystyle\exp(\theta\,\vec{\sigma}_{k})=\cosh\theta+\vec{\sigma}_{k}\,\sinh\theta,$
(168)

while the three purely spatial planes $I\vec{\sigma}_{k}\sim-i\mathbf{J}_{k}$
with negative signature generate elliptical spatial rotations,

$\displaystyle\exp(\theta\,I\vec{\sigma}_{k})=\cos\theta+I\vec{\sigma}_{k}\,\sin\theta.$
(169)

Thus, Eq. (166) is also equivalent to the Lorentz group Lie bracket relations
$[\mathbf{K}_{i},\mathbf{K}_{j}]=-i\epsilon_{ijk}\mathbf{J}_{k}$, with the
other relations,
$[\mathbf{J}_{i},\mathbf{J}_{j}]=i\epsilon_{ijk}\mathbf{J}_{k}$ and
$[\mathbf{J}_{i},\mathbf{K}_{j}]=i\epsilon_{ijk}\mathbf{K}_{k}$, obtained as
variations with extra factors of the pseudoscalar $I$. It follows that any
Lorentz transformation of an element $A$ of the algebra has the form of a
group inner automorphism with half-angle rotors generated by a unit spacetime
plane $B$,

$\displaystyle A$ $\displaystyle\mapsto R(\theta)AR(\theta)^{-1},$
$\displaystyle R(\theta)$ $\displaystyle\equiv\exp(\theta\,B/2),$ (170)

where the rotor inverse is $R(\theta)^{-1}=R(\theta)^{\sim}=R(-\theta)$. In
particular, rotations in purely spatial planes
$R(\theta)=\exp(I\vec{n}\,\theta/2)$ around a 3D spatial unit vector
$\vec{n}=\sum_{k}n_{k}\vec{\sigma}_{k}$ have precisely the same form used for
Pauli spin-rotations.

###  A.2 Spacetime Clifford bundle

The preceding treatment holds for a particular spacetime volume containing all
geometric content in the spacetime Clifford algebra. However, each point in
spacetime has an infinitesimal volume around it that can contain local
geometric objects at that point, forming a _Clifford bundle_ over spacetime in
which _fields_ $\Phi(x)$ can be defined of arbitrary geometric grade. The
dynamics of these spacetime fields both enact local geometric transformations
at each point $x$ and connect the geometric content of nearby points by
_parallel transporting_ elements of their tangent algebras along curves
connecting those points.

More precisely, given a flat spacetime manifold $\mathcal{M}$, each point
$x\in\mathcal{M}$ has a tangent space $\mathcal{T}\mathcal{M}(x)$ spanned by a
local tangent vector basis $\\{\gamma_{\mu}(x)\\}_{\mu=0}^{3}$. Products of
these tangent vectors construct a spacetime algebra
$\text{Cl}_{1,3}[\mathbb{R}](x)$ that is local to the point $x$. Each tangent
space formally has a dual cotangent space of functions
$\mathcal{T}^{*}\mathcal{M}(x)=\mathcal{T}\mathcal{M}(x)\to\mathbb{R}$ that is
spanned by a basis of one-forms $\\{\omega^{\mu}(x)\\}_{\mu=0}^{3}$ defined
from the tangent vectors $\gamma_{\mu}(x)$ to jointly satisfy the Euclidean
metric $\omega^{\mu}(x)[\gamma_{\nu}(x)]=\delta^{\mu}_{\nu}$. The one-form
basis is in one-to-one correspondence with a _reciprocal vector basis_
$\\{\gamma^{\mu}(x)=(\gamma_{\mu}(x))^{-1}\\}_{\mu=0}^{3}$ in the tangent
space, defined such that
$\gamma^{\mu}(x)\cdot\gamma_{\nu}(x)=\delta^{\mu}_{\nu}$, according to the
identity $\omega^{\mu}(x)[v(x)]\equiv\gamma^{\mu}(x)\cdot v(x)$ for any
tangent vector $v(x)\in\mathcal{T}\mathcal{M}(x)$. This correspondence means
that any one-form
$\alpha(x)[v(x)]=\sum_{\mu}\alpha_{\mu}\omega^{\mu}(x)[v(x)]=(\sum_{\mu}\alpha_{\mu}\gamma^{\mu}(x))\cdot
v(x)\in\mathcal{T}^{*}\mathcal{M}(x)$ can be identified with a tangent vector
$a(x)=\sum_{\mu}\alpha_{\mu}\gamma^{\mu}(x)\in\mathcal{T}\mathcal{M}(x)$ with
the same components $\alpha_{\mu}$ in the reciprocal basis. This corresponding
vector also has the component expansion
$a(x)=\sum_{\mu}a^{\mu}\gamma_{\mu}(x)$, so $\gamma_{\mu}(x)\cdot
a(x)=a_{\mu}=a^{\mu}\eta_{\mu\mu}$ and the two component expansions of the
same geometric object are related by the metric as expected from tensor
component analysis. In a similar way, the entire Grassmann algebra of forms in
the cotangent space can be identified with geometric objects in the graded
structure of the tangent space, which is a useful simplification.

These independent tangent Clifford algebras and associated cotangent spaces of
forms at each point $x$ must then be connected by linear translations between
nearby points. The _4-vector derivative_ (or _Dirac operator_ , familiar from
the Dirac equation for the quantum electron),

$\displaystyle\nabla_{x}$
$\displaystyle\equiv\sum_{\mu=0}^{3}\gamma^{\mu}(x)\frac{\partial}{\partial
x^{\mu}(x)}$ (171)

is the appropriate generator of translations along the spacetime manifold,
potentially modified by adding an appropriate _connection_
$D_{x}=\nabla_{x}+A(x)$ that enacts the parallel transport of relevant
geometric content during translation. Note the natural expansion of
$\nabla_{x}$ into reciprocal (dual) basis vectors, which will be clarified in
the next section. Each $\partial/\partial x^{\mu}(x)$ is the partial
derivative along a particular coordinate function $x^{\mu}(x)$ evaluated at
the point $x$. Also note that the point
$r=\sum_{\mu=0}^{3}x^{\mu}(x)\gamma_{\mu}(x)$ can itself be expressed in terms
of these same coordinate functions as a 4-vector displacement $r$ from the
origin, assuming a flat manifold with parallelized tangent vector directions
for simplicity, which conveniently lifts a vector representation of the
manifold points into the same tangent Clifford algebra for ease of
calculation.

It is customary to omit the point $x$ for brevity in $\nabla_{x}$ and write
the partial derivatives more compactly as $\partial_{\mu}$, yielding the
simpler form $\nabla=\sum_{\mu}\gamma^{\mu}\partial_{\mu}$. Further noting
that $\gamma^{k}=(\gamma_{k})^{-1}=-\gamma_{k}$ for $k=1,2,3$ and
$\gamma^{0}=\gamma_{0}$ then yields convenient relative frame expansions of
the (covariant) vector derivative,

$\displaystyle\nabla$
$\displaystyle=\gamma_{0}(\partial_{ct}+\vec{\nabla})=(\partial_{ct}-\vec{\nabla})\gamma_{0},$
(172)

which differ in sign from similar expansions of (contravariant) 4-vectors like
the displacement $r=\gamma_{0}(ct-\vec{x})=(ct+\vec{x})\gamma_{0}$ in the
tangent space.

###  A.3 Spacetime differential forms

Given a scalar field $\phi(x)$ on the manifold, the _exterior derivative_ $d$
creates a _differential one-form_ ,

$\displaystyle d\phi(x)[v(x)]=v(x)\cdot\nabla\phi(x),$ (173)

that takes a tangent vector $v(x)$ and returns a linear approximation to a
change in $\phi(x)$, in the form of a _directional (partial) derivative_ along
$v(x)$. Note that the differential is entirely characterized by the total
vector derivative $\nabla\phi(x)$ of the field. This identification between
the exterior derivative $d$ acting in the cotangent space
$\mathcal{T}^{*}\mathcal{M}$ and the vector derivative $\nabla$ acting in the
tangent space $\mathcal{T}\mathcal{M}$ is convenient, and is why the partial
derivative components in Eq. (171) are naturally expressed in the reciprocal
basis.

Taking the exterior derivative $d$ of a one-form field such as
$\omega(x)[v(x)]=a(x)\cdot v(x)$ similarly raises its grade,

$\displaystyle d\omega(x)[v(x),w(x)]$ $\displaystyle=[[\nabla\wedge a(x)]\cdot
v(x)]\cdot w(x),$ (174)

and can be identified with the 4-curl $\nabla\wedge a(x)$ of the object $a(x)$
in the tangent space that corresponds to $\omega(x)$ in the cotangent space.
This identification between the exterior derivative $d$ and the 4-curl
$\nabla\wedge$ holds generally for differential forms of any grade in the
cotangent bundle, including grade-0 scalar fields where
$\nabla\wedge\phi(x)=\nabla\phi(x)$ is identified with the gradient. The
exterior derivative satisfies the useful identity $d^{2}(\cdot)=0$ (equivalent
to a _Bianchi identity_), which is also satisfied by the corresponding four-
curl, $\nabla\wedge(\nabla\wedge(\cdot))=0$.

Given the metric and the Hodge star operation $\star$, the _codifferential_
$\delta\equiv\star^{-1}d\star$ can also be defined on differential forms,
which lowers the grade by one rather than raising the grade like the exterior
derivative. Using the correspondence between $\star$ in the cotangent space
and $I$ in the tangent space, this definition yields

$\displaystyle\delta\omega(x)$
$\displaystyle=\star^{-1}d\star\omega(x)=I^{-1}\nabla\wedge(Ia(x))=\nabla\cdot
a(x),$ (175)

so is equivalent to the grade-lowering 4-divergence $\nabla\cdot{}$. Moreover,
the codifferential satisfies another useful (_Bianchi_) identity
$\delta^{2}(\cdot)=\star^{-1}d^{2}\star(\cdot)=0$, which is also satisfied by
the corresponding four-divergence, $\nabla\cdot(\nabla\cdot(\cdot))=0$.

It then follows that the vector derivative under the Clifford product
precisely unifies both the grade-raising exterior derivative and the grade-
lowering codifferential of the cotangent space of forms, but is expressed in
the tangent algebra,

$\displaystyle(\delta+d)\omega(x)$ $\displaystyle\sim\nabla a(x)=\nabla\cdot
a(x)+\nabla\wedge a(x).$ (176)

It then naturally follows that the square of the vector derivative gives the
correct 4-Laplacian, or scalar wave operator,

$\displaystyle(\delta+d)^{2}\omega(x)$ $\displaystyle=(\delta
d+d\delta)\omega(x)$ (177)
$\displaystyle\sim\nabla^{2}a(x)=[\partial_{ct}^{2}-|\vec{\nabla}|^{2}]a(x).$

By working with the vector derivative in the tangent Clifford bundle, all the
convenience and clarity provided by differential forms is preserved, but
without the complication of excess function arguments as is necessary with
forms. This simplification enables much closer correspondence to standard
notations and vector calculus methods, as used in the main text. In fact, one
can argue that the unification of the exterior derivative and codifferential
encourages more structurally transparent calculations than with differential
forms alone.

For example, the Maxwell electromagnetic equations of motion can be written as
the single Clifford algebra equation $\nabla F=j$, where $F$ is a bivector and
$j=j_{e}+(j_{m}/c)I$ is a complex 4-vector containing both electric and
magnetic sources. This contrasts with the pair of equations commonly seen in
treatments that use differential forms, $\delta F=\mu\,j_{e}$ and $\delta
G=\mu\,j_{m}/c$, where $F$ is the Faraday 2-form and $G=\star F$ is the dual
Maxwell 2-form treated separately, with the currents $j_{e}$ and $j_{m}$ both
treated as one-forms despite the magnetic current $j_{m}$ properly having the
character of a pseudo-4-vector geometrically. One can fix this standard
differential forms treatment by writing instead,
$(\delta+d)F=\mu\,(j_{e}+\star j_{m}/c)$, which then becomes identical in the
cotangent bundle to the content of the spacetime Clifford algebra formulation
in the tangent bundle, $\nabla F=\mu\,j$, albeit using less compact notation.

###  A.4 Lagrangian densities

Recall that a Lagrangian density properly is a _4-volume integration measure_
that is a function of the dynamical fields; extremizing the action as the
integrated Lagrangian density with respect to variations of the dynamical
fields then produce the classical equations of motion. Thus, the Lagrangian
density should geometrically correspond to a volume 4-form on spacetime that
is invariant under the correct physical symmetries, including those of
spacetime itself.

To make explicit how this idea connects to the tangent Clifford bundle used
here, a 4-form is a local linear map
$d^{4}S(x)[v_{0}(x),v_{1}(x),v_{2}(x),v_{3}(x)]=\mathcal{L}(x)\cdot(v_{0}(x)\wedge
v_{1}(x)\wedge v_{2}(x)\wedge v_{3}(x))$ in the cotangent space at a point $x$
that takes four 4-vector arguments in the tangent space at $x$, constructs an
invariant pseudoscalar 4-volume in the tangent space out of them, then
contracts that volume with another local 4-volume $\mathcal{L}(x)$ to produce
a scalar measured value. The 4-volume $\mathcal{L}(x)$ is a grade-4 object in
the tangent Clifford algebra, so is proportional to the local unit
pseudoscalar $I(x)$ at the point $x$. A spacetime Riemann sum,
$\sum_{k}d^{4}S(x)[\delta V_{k}(x)]$, implicitly passes in small vector
increments $(\delta x)^{\mu}\gamma_{\mu}(x)$ along independent vector
directions to construct small invariant 4-volumes $\delta
V_{k}(x)=(\prod_{\mu}(\delta x)^{\mu})I(x)$ that are proportional to the local
pseudoscalar $I(x)$, then evaluates the 4-form measure $d^{4}S(x)[\delta
V_{k}(x)]$ on each such local volume, and sums those measured values over a
grid of small volumes that partition the spacetime volume being integrated.
Taking the limit as the small volumes $\delta V_{k}(x)$ become infinitesimal
in magnitude yields a spacetime action integral $S=\int
d^{4}S(x)=\int\mathcal{L}(x)\cdot d^{4}V(x)$ as the limiting value of the
Riemann sum, with resulting directed volume measure $d^{4}V(x)=|d^{4}V|I(x)$.

The important takeaway is that such a integration measure 4-form is fully
characterized by a grade-4 _pseudoscalar field_ $\mathcal{L}(x)$ in the
tangent Clifford bundle, which we can identify as the geometric content of the
Lagrangian density. For the action $S$ to yield an invariant scalar value, as
usually assumed, the density $\mathcal{L}(x)$ must be proportional to the unit
4-volume field $I(x)$. The traditional scalar Lagrangians in Eqs. (23) and
(32) leave out this volume factor as implicit.

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