# Curiosity as filling, compressing, and reconfiguring knowledge networks

Shubhankar P. Patankar  Department of Bioengineering, School of Engineering
and Applied Science, University of Pennsylvania, Philadelphia, PA 19104 USA
Dale Zhou  Neuroscience Graduate Group, Perelman School of Medicine,
University of Pennsylvania, PA 19104 USA  Christopher W. Lynn  Initiative for
the Theoretical Sciences, Graduate Center, City University of New York, New
York, NY 10016 USA  Joseph Henry Laboratories of Physics, Princeton
University, Princeton, NJ 08544 USA  Jason Z. Kim  Department of
Bioengineering, School of Engineering and Applied Science, University of
Pennsylvania, Philadelphia, PA 19104 USA  Mathieu Ouellet  Department of
Electrical and Systems Engineering, School of Engineering and Applied Science,
University of Pennsylvania, Philadelphia, PA 19104 USA  Harang Ju
Neuroscience Graduate Group, Perelman School of Medicine, University of
Pennsylvania, PA 19104 USA  Perry Zurn  Department of Philosophy, American
University, Washington, DC 20016 USA  David M. Lydon-Staley  Department of
Bioengineering, School of Engineering and Applied Science, University of
Pennsylvania, Philadelphia, PA 19104 USA  Annenberg School for Communication,
University of Pennsylvania, Philadelphia, PA 19104 USA  Leonard Davis
Institute of Health Economics, University of Pennsylvania, Philadelphia, PA
19104 USA  Dani S. Bassett  Department of Bioengineering, School of
Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA
19104 USA  Department of Electrical and Systems Engineering, School of
Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA
19104 USA  Department of Psychiatry, Perelman School of Medicine, University
of Pennsylvania, Philadelphia, PA 19104 USA  Department of Neurology, Perelman
School of Medicine, University of Pennsylvania, Philadelphia, PA 19104 USA
Department of Physics and Astronomy, College of Arts and Sciences, University
of Pennsylvania, Philadelphia, PA 19104 USA  Santa Fe Institute, Santa Fe, NM
87501 USA  To whom correspondence should be addressed<EMAIL_ADDRESS>
###### Abstract

Curiosity is an internally motivated search for information. It is enduring
and open-ended, and may have evolved to help us build accurate mental
representations of our ever-changing environments. Due to the significant role
that curiosity plays in our lives, several theoretical constructs, such as the
information gap theory and compression progress theory, have sought to explain
how we engage in its practice. According to the former, curiosity is the drive
to acquire information that is missing from our understanding of the world.
According to the latter, curiosity is the drive to construct an increasingly
parsimonious mental model of the world. To complement the densification
processes inherent to these two theories, we propose the conformational change
theory, wherein we posit that the practice of curiosity results in mental
models with marked conceptual flexibility. To validate these three theories,
we must overcome the fundamental challenge of constructing formal models of
mental representations of knowledge. Here, we address that challenge by
formalizing curiosity as the process of building a growing knowledge network.
We then quantitatively investigate information gap theory, compression
progress theory, and the conformational change theory of curiosity. In
knowledge networks, gaps can be identified as topological cavities,
compression progress can be quantified using network compressibility, and
flexibility can be measured as the number of conformational degrees of
freedom. We leverage data acquired from the online encyclopedia Wikipedia to
determine the degree to which each theory explains the growth of knowledge
networks built by individuals and by collectives. Our findings lend support to
a pluralistic view of curiosity, wherein intrinsically motivated information
acquisition fills knowledge gaps and simultaneously leads to increasingly
compressible and flexible knowledge networks. Across individuals and
collectives, we determine the contexts in which each theoretical account may
be explanatory, thereby clarifying their complementary and distinct
explanations of curiosity. Our findings offer a novel network theoretical
perspective on intrinsically motivated information acquisition that may
harmonize with or compel an expansion of the traditional taxonomy of
curiosity.

##  1 Introduction

Humans must manage uncertainty and embrace change to thrive in a complex and
dynamic environment [1]. To this end, we continually consume information to
construct and maintain accurate mental models of the world [2, 3].
Information-seeking behavior may be driven by a variety of intrinsic and
extrinsic factors. Arising from the latter, information acquisition is an
intermediate step towards attaining a specific goal—such as increased wealth
or social recognition—that is ultimately rewarding [4]. By contrast, the
intrinsic motivation to seek information is commonly conceptualized as
curiosity [5, 1, 6]. In silico work suggests that curiosity may have evolved
to maximize long-term evolutionary fitness in rapidly changing environments
[7]. Additionally, studies have shown that humans are driven to know, even
when information is costly to obtain [8, 9] and may have no immediate tangible
utility [10, 11]. Curiosity-driven information gathering is, therefore,
inherently rewarding.

Given the significant role that it plays in our daily behavior and decision
making, several theories have sought to explain how individuals practice
curiosity. The information gap theory views curiosity as the drive to obtain
information that is missing from a mental model of the world [5]. In this
account, perception of a gap in one’s knowledge of the world creates an
aversive state of uncertainty that, in turn, motivates a search for
information to close the gap [12]. In the complementary perspective of
compression progress theory, curiosity is the drive to obtain information that
improves the compression of a mental model and thereby lowers its cost of
representation [13, 14]. Both theories provide several important explanations
for curiosity-driven information-seeking behavior. On the one hand, curiosity
that explicitly modulates uncertainty levels by opening or closing information
gaps as needed can facilitate lifelong learning. On the other hand,
compression progress enables the extraction of essential latent structures of
knowledge, offering greater capacity for generalization [15, 16]. Low-cost
representations also allow for greater functional capacity by freeing
cognitive resources that would otherwise be dedicated to information storage
and retrieval [17]. However, theoretical constructs for curiosity, such as the
information gap theory and compression progress theory, are difficult to
validate quantitatively. A fundamental challenge is constructing formal models
of mental representations of knowledge. In the absence of such models (and the
conceptual frameworks that accompany them), it is unclear how to translate
theoretical concepts such as knowledge gaps and compression progress to well-
defined variables that can be measured in experiments.

Figure 1:  Connectional approach to curiosity. _(A)_ A participant constructs
a growing knowledge network through curiosity-driven self-directed exploration
of Wikipedia, a vast networked landscape of information. Nodes represent
unique Wikipedia pages. Edges represent hyperlinks between nodes. Nodes are
colored to denote the order in which they are visited. _(B)_ Gaps in a
knowledge network can be formalized using algebraic topology and tracked in
several topological dimensions. The green and blue gaps represent a
$0$-dimensional and $1$-dimensional cavity, respectively. _(C)_ Compression
progress aims to construct internal representations of the world that are both
storage efficient and generalizable. In a knowledge network, all concepts that
belong to the same cluster can be represented parsimoniously at a higher level
of abstraction using their cluster identity. The unclustered network has $9$
nodes and $12$ edges, while the clustered network only has $3$ nodes and $3$
edges. _(D)_ A mechanical network can possess several spatial configurations,
any of which can be arrived at from any of the others through a series of
conformational changes. We formalize and measure knowledge network flexibility
as the number of available conformational degrees of freedom.

One such model that has shown promise is a network model where knowledge is
composed of discrete and yet interconnected concepts [18, 19, 20, 21, 22, 23,
24]. In graph learning studies, volunteers are shown sequences of images on a
screen, where, unbeknownst to the volunteers, each image corresponds to a node
in an underlying network [25]. Based solely on observed transitions, and
despite being unaware of the underlying network’s structure, participants
successfully infer statistical regularities from the temporal order in which
images appear [26, 20, 27, 28, 29]. Crucially, the structure of the pre-
defined experimental graph can be recovered from neural activity by decoding
simultaneously acquired functional magnetic resonance imaging (fMRI) data [27,
30]. The sequential manner in which stimuli are presented in graph learning
tasks can be conceived of as a walk prescribed by the experimenter in a
limited knowledge space of objects, images, concepts, or movements. Curiosity,
too, can be conceived of as a walk, but one that is largely self-directed and
purposeful across the vast landscape of knowledge. To evaluate curious walks,
recent work gathered browsing histories from individuals who freely explored
the online encyclopedia Wikipedia. Structural features of the knowledge
networks that participants walked upon (Fig. 1A) were found to be associated
with curiosity, as measured by an independent index of participants’
sensitivity to information deprivation [31].

Here, we leverage this framework and cast curiosity as a network building
process. This approach allows us to take qualitative explanations for
information-seeking behavior, such as the information gap theory and
compression progress theory, and operationalize them in quantitative
statistics. Information gap theory posits that humans add information to
regulate uncertainty by filling gaps [5, 1, 6]. This theory can be
operationalized by treating gaps in networks as topological cavities, and by
tracking their evolution using techniques from applied algebraic topology [32,
33, 34] (Fig. 1B). In contrast, compression progress theory posits that humans
subtract or discard information [13, 14, 35] due to limited cognitive capacity
[36, 14, 37]. Compressing a network while maintaining meaningful latent
structure requires that we discard some irrelevant information while
maintaining important information about past experiences and present
priorities [35, 38, 14, 39]. This theory can be operationalized by measuring
the compressibility of a network, an information-theoretic quantity that
captures the ability of a network to be compressed [35] (Fig. 1C). Via the
network operationalization of these two theories, we come to see that
curiosity is marked as a process by which networks of knowledge densify and
simplify, raising the question of what alternative process might drive them to
sprawl and become complex.

To address this question, we expand beyond historical accounts to
operationalize our own conformational change theory of curiosity. The
conformational change theory suggests that information-seeking behavior
results in the creation of expansive knowledge networks [40] embedded in a
conceptual geometry (Fig. 1D). The notion of a conceptual geometry is
motivated by prior studies of neural population geometry and the fact that
information can be embedded and processed in locally Euclidean geometric
representations to solve complex tasks [41]. The geometry provides a key
affordance for curiosity—conceptual flexibility—as the knowledge network can
mechanically conform into different shapes. While some concepts are separated
from other concepts by fixed distances of shared-versus-unshared meaning,
other concept pairs can move closer together or farther apart as inter-concept
relations shorten or lengthen depending on time and context [42]. This
flexibility allows us to draw from past experience, cohere the past with newly
learned information, monitor conflict, and respond appropriately in different
contexts [15, 43, 44]; it may also subserve the unexpected conceptual
combinations that accompany imaginative thought and support serendipitous
discoveries [45, 46]. Mechanically akin to conformational change in proteins,
the flexible reshaping of the knowledge network can only occur if concepts are
sparsely connected; densely connected linkage networks embedded in a Euclidean
geometry are rigid. Hence, the conformational change theory of curiosity
posits a drive for conceptual flexibility that leads networks to sprawl and
become complex.

As is now evident, each of the three theories is motivated by a distinct and
uniquely important psychological drive: to reduce uncertainty by learning a
missing piece of information, to discover latent patterns by distilling
fundamental epistemic elements, and to reshape information by flexibly
reconfiguring knowledge networks. Here, we test each theory through parallel
analyses of the growth of individual and collective knowledge networks derived
from Wikipedia. At the individual scale, we construct knowledge networks for
$149$ individuals using their Wikipedia browsing histories [31] (Fig. 1A). At
the collective scale, we extract Wikipedia networks to assess knowledge growth
in $30$ disciplines such as calculus, economics, and linguistics [47]. We
treat Wikipedia pages as nodes in both sets of networks and add edges between
them according to the presence of hyperlinks between pages. For the data on
individuals, we specify network growth using the order in which individuals
visit pages; for the data on collectives, we use the years in which different
concepts originate. To model the random growth of knowledge in both data sets,
we create $25$ degree-preserving edge-rewired versions of each network. We
test the predictions of the three theories by comparing measurements of
relevant features from empirically observed knowledge networks to those from
the related null networks. First, considering the information gap theory, we
expect to find fewer-than-chance topological cavities in growing empirical
knowledge networks due to people’s hypothesized drive to close knowledge gaps
when they are perceived. Considering compression progress theory, we
hypothesize that growing knowledge networks will exhibit greater-than-chance
compressibility due to people’s hypothesized drive to distill fundamental
epistemic elements [14]. Third, considering conformational change theory [40],
we hypothesize that knowledge networks will possess greater-than-chance
capacity for conformational changes due to people’s hypothesized drive for
conceptual flexibility. In testing these hypotheses, we demonstrate the
utility of the network approach in quantitatively validating existing
theoretical constructs of curiosity as well as in formulating new ones.

##  2 Results

###  2.1 Network growth formalism

Before testing the predictions of the three theories, we clarify the network
formalism upon which they are operationalized. Consider a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ with node set $\mathcal{V}$ and edge
set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$. We define a growing
knowledge network with the tuple $(\mathcal{G},\;s)$, where $s$ denotes a map
$s:\mathcal{V}\rightarrow\mathbb{N}$ that specifies the rank order in which
nodes are added to the network. For networks built by individuals, $s$ is
determined by the order in which Wikipedia pages are first visited. For
collective networks, $s$ is determined by the years in which different
concepts originate. With $N$ nodes in a network, we construct a sequence of
graphs

$\mathcal{G}_{0}\subset\mathcal{G}_{1}\subset\cdots\subset\mathcal{G}_{N}=\mathcal{G},$
(1)

where $\mathcal{G}_{p}$ is a subgraph of $\mathcal{G}$ comprised of the first
$p$ nodes in $s$ and all $q$ connections between them that exist in
$\mathcal{E}$. Such a sequence—in which each element is a subset of the
next—is an example of a filtration. We index each subgraph in a filtration by
the number of nodes in the network at that stage. We identify topological
cavities, measure network compressibility, and compute conformational
flexibility for all subgraphs in filtrations of individual and collective
knowledge networks as well as in filtrations of related null model networks.
We perform non-parametric permutation tests to examine differences between
feature curves for empirical and null model data (see Sec. 5.5 for details).
Since networks in a given data set may have different sizes, we normalize
filtration indices to span the range [$0$, $1$], and align values of interest
to be defined on the same points before computing the mean for a feature-of-
interest across all individuals or topics [48]. For completeness, we also
report results with unnormalized values in the Supplement.

###  2.2 Information gap theory

Figure 2:  Probing information gaps as topological cavities in growing
knowledge networks. We operationalize information gaps as topological cavities
(also referred to as cycles) and track their evolution in growing individual
and collective knowledge networks. We plot the number of cycles as a function
of time. _(A)_ Topological cavities in dimension $0$, or $0$-cycles, represent
disconnected network components. _(B, C)_ Individual and collective knowledge
networks tend to possess fewer disconnected components than expected in edge-
rewired null model networks. _(D)_ In dimension $1$, a topological cavity
represents an enclosed loop formed by edges. _(E, F)_ Growing individual and
collective knowledge networks tend to possess fewer loops than expected in
edge-rewired null model networks. _(G)_ A topological cavity in dimension $2$
constitutes a void enclosed by $3$-cliques, or triangles of interconnected
nodes. _(H, I)_ Growing individual and collective knowledge networks tend to
possess more $2$-dimensional cavities than expected in edge-rewired null model
networks. Shaded regions in panels _B, C_ , _E, F_ , and _H, I_ represent
standard error. Purple curves denote the average number of cavities in edge-
rewired null model networks.

The information gap theory posits that curiosity is the drive to collect units
of knowledge that fill gaps in one’s internal representation of the world [5].
When we model internal representations as networks, the missing information
can be usefully operationalized as topological cavities, which can be tracked
in a principled manner using tools from applied algebraic topology (see Sec.
5.2 for methodological details) [32, 33]. This operationalization follows
prior work demonstrating that domains as diverse as language development in
toddlers [49], the introduction of characters in Dostoyevsky’s novels [50],
and the presentation of concepts in linear algebra textbooks [48] exhibit a
systematic creation and closing of such cavities. By employing this approach,
we can determine whether curiosity-driven exploration is motivated by a
preference for gap closure. In a network, except for dimension $0$, a
$k$-dimensional cavity, also known as a $k$-cycle, is identified as an empty
enclosure formed from $(k+1)$-cliques, where cliques are defined as all-to-all
connected subgraphs of $k+1$ nodes. The $k$-th Betti curve records the number
of $k$-cycles present at each stage of a network’s growth. Cycles of dimension
$0$ represent disconnected network components (Fig. 2A), whereas those of
dimensions $1$ and $2$ represent loop-like holes (Fig. 2D) and pocket-like
voids (Fig. 2G), respectively.

Considering information gap theory, we hypothesized that empirical knowledge
networks would contain fewer cavities than topologically similar edge-rewired
null model networks. To test this hypothesis, we compute persistent homology
for filtrations of individual and collective knowledge networks in dimensions
$0$, $1$, and $2$. We find that the number of $0$-cycles, or disconnected
network components, increases as individual knowledge networks grow, and does
so at a steeper rate in null networks than in empirical networks (Fig. 2B).
For collective knowledge networks, we find that the number of disconnected
components first increases and then decreases both in the empirical and in the
null model networks, albeit with significantly different peak values (Fig.
2C). In both data sets, for a significant duration of growth, Betti curves for
observed networks are lower than those for null model networks. In dimensions
$1$ and $2$, we find that the number of cycles increases as individual and
collective knowledge networks grow (Fig. 2D-I). This temporal trajectory could
arise from the fact that filling gaps by forging new connections can open new
gaps, making it prohibitively difficult to track (and fill) gaps among an
increasingly large number of items. In support of information gap theory, the
rate at which $1$-cycles increase is lower for the empirical networks than for
the null networks (Fig. 2E,F). In contrast to information gap theory, the rate
at which $2$-cycles increase is higher in the empirical networks than in the
null networks (Fig. 2H,I). The marked growth of $2$-dimensional cavities could
reflect an alternative drive to expand and complexify knowledge networks. All
empirical Betti curves are significantly different from the Betti curves for
the null model data ($p_{perm}<0.001$) as determined via permutation testing.
In summary, across both individual and collective knowledge networks, our
findings suggest that information gap theory explains how separate areas of
interest ($0$-cycles) grow and then are subsequently linked together, and how
loop-like holes ($1$-cycles) within specific areas of interest grow and are
subsequently filled. However, the extent and longevity of larger pocket-like
voids ($2$-cycles) remains unexplained by the information gap theory,
motivating an assessment of alternative psychological drives.

###  2.3 Compression progress theory

Figure 3:  Quantifying compression progress using network compressibility.
_(A)_ A random walk $\bm{x}$ on a network is a sequence of nodes constructed
by transitioning from a node $x_{t}$ to one of its neighbors uniformly at
random. Such a sequence generates information at a rate given by its entropy
$H(\bm{x})$. Now suppose that we group the nodes into different clusters; the
number of clusters defines the scale at which the network is described. The
random walk $\bm{x}$ is compressed into a new sequence $\bm{y}$, where $y_{t}$
is the cluster that contains node $x_{t}$. The clustered sequence $\bm{y}$
generates information about the original sequence $\bm{x}$ at a rate given by
the mutual information $I(\bm{x},\bm{y})=H(\bm{y})-H(\bm{y}|\bm{x})$. Mutual
information $I(\bm{x},\bm{y})$ is greatest—and equal to the entropy
$H(\bm{x})$—when each node is assigned independently to its own cluster. By
contrast, in the limit where the entire network is viewed as one large
cluster, the mutual information vanishes. _(B)_ At each intermediate scale
between these two extremes, we can find an optimal clustering that maximally
lowers the information rate. Network compressibility is then defined as the
maximal reduction in the information rate, averaged across all scales. _(C)_
Growing individual knowledge networks are markedly more compressible than
expected considering related edge-rewired null model networks. _(D)_ Growing
collective knowledge networks show only a slight tendency for greater-than-
expected compressibility. Shaded regions in panels _C_ and _D_ represent
standard error. Purple curves denote average compressibility values for edge-
rewired null model networks.

Originally proposed as a general algorithmic framework for reinforcement
learning, compression progress theory posits that curiosity is the drive to
continually improve the compression of a learner’s mental model of the world
[13]. By conceptualizing mental models as knowledge networks, we can measure
compressibility using recent advances at the intersection of information
theory and network science [35]. To compute the compressibility of a network,
we begin by considering a random walk $\bm{x}=(x_{1},x_{2},\cdots)$, where
$x_{t}$ is the node that appears at step $t$ (Fig. 3A). The rate at which the
sequence $\bm{x}$ generates information is given by its entropy $H(\bm{x})$.
If we group the network’s nodes into clusters, we can re-write
$\bm{x}=(x_{1},x_{2},\cdots)$ as $\bm{y}=(y_{1},y_{2},\cdots)$ by replacing
each node $x_{t}$ with its cluster identity $y_{t}$. The rate at which the
clustered sequence $\bm{y}$ generates information about the original sequence
$\bm{x}$ is given by the mutual information
$I(\bm{x},\bm{y})=H(\bm{y})-H(\bm{y}|\bm{x})$. The number of clusters that we
use to compress the network defines a scale of its description. As we decrease
the number of clusters—that is, as we increase the scale of description—the
information rate $I(\bm{x},\bm{y})$ decreases. When each node belongs to its
own cluster, the information rate $I(\bm{x},\bm{y})$ equals the original rate
$H(\bm{x})$ (Fig. 3A). By contrast, when all nodes are grouped together into
one cluster, the information rate is zero (Fig. 3A). At all scales in between,
we can find the optimal clustering of nodes that minimizes the information
rate (Fig. 3B). We then define the compressibility of the network as the
maximal reduction in the information rate, averaged across all scales of its
description (Fig. 3B) [35].

Considering compression progress theory, we hypothesized that growing
knowledge networks would be more compressible than topologically similar edge-
rewired null model networks. We test our hypothesis by computing network
compressibility for each subgraph in filtrations of individual and collective
knowledge networks. We find that compressibility increases monotonically as
knowledge networks grow. At all stages of growth, and in support of our
hypothesis, networks for individuals exhibit greater-than-expected
compressibility (Fig. 3C). This same trend holds, but to a much weaker extent
in the collective knowledge networks. While the early stages of growth evince
greater separation between empirical and null compressibility values, the two
curves overlap in later stages of growth (Fig. 3D). Based on non-parametric
permutation testing, compressibility curves for individual and collective
knowledge networks are significantly different from their null model
counterparts ($p_{perm}<0.001$). These data provide evidence that is critical
for an evaluation of compression progress theory. Consistent with the theory,
for individuals, a preference for greater compressibility indicates that
curiosity is driven to construct parsimonious mental representations of
knowledge. For collectives, a similar-to-expected compressibility could
reflect (i) the group’s nature as constituted by the diverse voices and
expertise that comprise it, which can enhance the relevance of details and
preclude their compression, and (ii) the fact that groups may not be
constrained by the same cognitive capacity limitations that constrain
individuals.

###  2.4 Conformational change theory

A curious learner practising curiosity solely according to information gap
theory strives for growth and completeness of knowledge. By contrast, a
learner practising curiosity solely according to compression progress theory
strives to uncover the latent organization of the world. In the process,
neither individual can keep pace with the growing complexity of the
environment; with a rapidly expanding frontier of ignorance as new unknowns
become accessible. Crucially, both theories suggest how we can usefully add or
relinquish information but neither acknowledges the worth of what we already
possess. Prior work has shown that curiosity-driven information acquisition is
not only about growing or shedding knowledge, but also about retreading and
reconsidering what one presently holds [14, 31]. Following Zurn et al. [40],
we propose that such reflection entails moving concepts flexibly in relation
to one other. Specifically, we define curiosity as the process of constructing
knowledge networks with a finely arbitrated balance between local internal
rigidity and global external flexibility. Rigidity and flexibility are
mechanical notions that require an object of interest to be embedded in
physical space. Therefore, drawing inspiration from a rich literature on
cognitive maps (see Supplement for background), we assume that knowledge
networks are embedded in Euclidean space where they possess several degrees of
freedom. We then measure flexibility as a network’s ability to undergo
conformational changes [42] and formalize our account as the conformational
change theory of curiosity.

Figure 4:  Conformational change in mechanical networks. _(A)_ In two-
dimensional space, a network with three nodes and three edges has three rigid-
body degrees of freedom: horizontal translation, vertical translation, and
rotation. _(B)_ In addition to the three rigid-body motions, a quadrilateral
frame also possesses a conformational degree of freedom, depicted here via the
angle parameter $\theta$, which allows it to change shape from a square to a
diamond. _(C)_ Rigid and flexible sub-units can be combined to construct
networks capable of undergoing large-scale conformational changes. Different
configurations of the same network can be reached by propagating
conformational changes through its structure. _(D)_ A network chain with $338$
nodes and $672$ edges folds to form a quadrifolium (panel _D_ reproduced with
permission from Kim et al. [51]).

Before measuring the conformational flexibility of growing knowledge networks,
we offer a brief introduction to mechanical networks. Consider a triangular
network in two dimensions. Each of its nodes can be located with two
coordinates (Fig. 4A). This network has three available rigid body motions:
horizontal translation, vertical translation, and rotation. Next, consider a
network comprised of $4$ nodes and $4$ edges (Fig. 4B). This network possesses
the same rigid-body motions as are available to the triangle. Additionally,
the quadrilateral possesses a conformational degree of freedom. A
conformational change in a network alters the Euclidean distance between
unconnected pairs of nodes. For instance, if a pair of adjacent nodes in the
quadrilateral is held fixed in space, the remaining nodes can be moved freely
while sweeping across an angle $\theta$ with respect to the fixed pair (Fig.
4B). Through this process, this simple network exhibits a conformational
change from a square to a diamond. Mechanical networks can exist in several
configurations, each of which can be reached through a series of
conformational changes from any of the others (Fig. 4C-D). The number of
independent conformational motions available to a network with $p$ nodes and
$q$ edges embedded in a $d$-dimensional space is $dp-q$. Among these $dp-q$
degrees of freedom are ${d(d+1)}/{2}$ rigid body motions, which include
translations and rotations. The rest, given by

$DoF_{C}=dp-q-\frac{d(d+1)}{2},$ (2)

are the available conformational degrees of freedom, or _conformational
motions_.

Figure 5:  Conformational change theory of curiosity. We propose that in the
networked space of the mind, while some concepts and their relationships have
fixed locations, others can move flexibly in a context-dependent manner. Such
flexibility affords curious humans the ability to rethink and reconfigure what
they already know in light of new information. We formalize flexibility as the
number of conformational degrees of freedom ($DoF_{C}$). In a network in
$d$-dimensional space with $p$ nodes and $q$ edges, $DoF_{C}=dp-q-d(d+1)/2$.
Assuming $d=1$ initially, we compute $DoF_{C}$ for filtrations of growing
knowledge networks. A negative value for $DoF_{C}$ indicates the presence of
self-stress, which we resolve by incrementing the $d$ by $1$. _(A)_ In the
example filtration, when nodes $3$ and $6$ are added, the network becomes
over-constrained and develops self-stress. Consequently, dimensionality first
increases from $1$ to $2$ and then from $2$ to $3$. _(B, C)_ Individual
knowledge networks require greater dimensionality and possess greater
flexibility than null model networks. _(D, E)_ Collective knowledge networks
do not exhibit greater dimensionality or conformational flexibility than null
model networks. Shaded regions in _B_ -_E_ represent the standard error.

Crucially, Eq. 2 relies on the linear independence of edges. Linear
independence entails that there are no redundant edges that over-constrain a
set of nodes beyond the formation of a rigid cluster yielding a state of self-
stress, and that the network does not exist in a rare and pathological
geometry known as a kinematic bifurcation [52, 42]. States of self-stress
imply that edges within a network bear internally balanced forces. A negative
value for the number of conformational degrees of freedom would indicate that
the network—when considered in its entirety—is over-constrained. In our
framework, we assume that such states result in a form of cognitive dissonance
whereby competing constraints between concepts cannot be resolved. We
alleviate this tension by incrementing the dimensionality by $1$ when needed.
Specifically, we increment $d$ by $1$ until $DoF_{C}$ is no longer negative.
This approach yields the minimum dimensionality required at each stage of
growth to avoid over-constraining the network. Fig. 5A depicts this process
for a representative filtration of a growing network. When node $3$ is added
to the network, which is initially embedded in a $1$-dimensional space
(green), the number of conformational degrees of freedom evaluates to
$(1\times 3)-(3)-1=-1$, indicating the presence of self-stress and requiring
that the embedding dimensionality be incremented to $2$ (orange). This process
repeats when node $6$ is added, resulting in an increase in dimensionality to
$3$ (red). To compute the number of conformational degrees of freedom for
growing knowledge networks, we assume that they are initially embedded in a
$1$-dimensional space; whenever the quantity in Eq. 2 becomes negative, we
increment dimensionality by $1$.

We hypothesized that knowledge networks would possess greater conformational
flexibility than corresponding null model networks. We test this hypothesis by
computing the number of conformational degrees of freedom in filtrations of
individual and collective knowledge networks (Fig. 5C, E). In parallel, we
track the minimum embedding dimensionality required to prevent self-stress
from developing in the growing networks (Fig. 5B, D). We find that individual
knowledge networks need greater dimensionality and possess greater
conformational flexibility than null model networks (Fig. 5B, C). By contrast,
measurements of dimensionality and flexibility for collective networks cannot
be as easily distinguished from their corresponding null model data (Fig. 5D,
E). However, for both data sets, the empirical curves for dimensionality and
conformational flexibility are significantly different from the corresponding
curves from the null model data ($p_{perm}<0.001$). Our findings suggest that
individuals value the ability to reconsider what they already know in light of
newly acquired information. On the other hand, collective knowledge displays
less capacity for global reconfiguration over the long time scales evaluated
in this study; future work could investigate the existence and dynamics of
internal sectors that change shape over different time scales or during
paradigm shifts.

##  3 Discussion

In this work, we formalize curiosity as the process of constructing a growing
knowledge network. We leverage tools from network science to quantitatively
examine several theoretical constructs for curiosity such as the information
gap theory and compression progress theory. Information gap theory suggests
that curiosity is the drive to obtain units of knowledge that fill gaps in
understanding [5]. Compression progress theory posits that curiosity is the
drive to uncover the latent organization of the world [13]. We probe
information gaps as topological cavities in growing knowledge networks and
quantify compression progress using network compressibility. The two theories
offer complementary perspectives on curiosity; the information gap theory
suggests that new information is acquired to fill knowledge gaps, whereas the
compression progress theory suggests that new information is used to distill
the essential epistemic elements of knowledge. While these perspectives
describe how knowledge networks become denser and simpler through information
acquisition, an alternative formulation is needed to explain how they become
expansive and more complex. Therefore, we build upon a recently proposed
conceptual framework [40] to develop the conformational change theory of
curiosity. We posit that knowledge networks are embedded in a Euclidean
geometry, which allows concepts to move flexibly in relation to one another.
We then view curiosity as the practice of constructing mechanically flexible
knowledge networks. Formally, we measure conceptual flexibility as the number
of conformational degrees of freedom available to a growing knowledge network.
Throughout our investigations, we take a multi-scale view and probe evidence
for each theory in individuals and in collectives. Across the two scales of
granularity, we determine the precise contexts in which each theoretical
account is explanatory, thereby clarifying their complementary and specific
affordances.

Information gap theory and topological cavities in knowledge networks.
Information gap theory suggests that humans tolerate a finite amount of
uncertainty in their knowledge of the world [5]. Exposure to a small amount of
previously unknown information brings into focus the presence of a knowledge
gap, pushing the level of uncertainty past an acceptable threshold. This
increased uncertainty then prompts a search for information to fill the
knowledge gap and resolve the unknown. In this work, we formalize gaps as
topological cavities in growing knowledge networks and track their evolution
in dimensions $0$, $1$, and $2$ [49, 47, 34]. Each dimension is characterized
by a different kind of topological gap: $0$-dimensional gaps correspond to
disconnected network components, $1$-dimensional gaps correspond to loop-like
holes, and $2$-dimensional gaps correspond to pocket-like voids. Across all
dimensions, we find that the number of cavities increases as individual
knowledge networks grow. Stated differently, associations between familiar
concepts remain undiscovered even as we acquire more information. Hence, in
addition to the common view of an expanding frontier of ignorance, knowledge
growth is accompanied by an ever-expanding interior of ignorance [47]. Except
for the $0$-th dimension, we report similar results for knowledge networks
built collectively. Filling a $0$-dimensional cavity entails adding an edge
between two disconnected network components. Such edges may be easier for
collectives to add than for individuals since interdisciplinary sub-fields
within scientific domains are motivated to link disparate sub-areas of
knowledge [53]. Importantly, and in support of the information gap theory, the
number of $0$\- and $1$-dimensional cavities is lower in observed individual
and collective knowledge networks than in the corresponding null model data,
reflecting a downward pressure on the number of gaps created, consistent with
a gap-filling drive. Therefore, from a networks perspective, gaps—as
envisioned by information gap theory, those that are prioritized for
filling—may best correspond to topological cavities of dimensions $0$ and $1$.
Stated differently, information gap theory provides an explanation for the
markedly damped growth of lower dimensional cavities; however, a different
account is needed to explain the contrasting proliferation of higher
dimensional cavities, both in individuals and in collectives.

Compression progress theory and efficient network representations of
knowledge. To gain a deeper intuition, we turn to compression progress theory,
which derives inspiration from resource limitations that underpin brain
function [13]. We represent knowledge as a network of concepts and their
inter-relationships, and we compute network compressibility [35] to determine
whether curiosity drives compression. We find that growing individual
knowledge networks consistently exhibit greater-than-expected compressibility,
consistent with the theory. This finding can be contextualized by considering
the fact that as we interact with the world, we encounter and consume large
quantities of information. Constructing perfectly accurate mental models would
entail storing each unit of acquired knowledge separately. However, finite
resources constrain us to build compressed or efficient abstractions of
observed data that can generalize across contexts [15]. According to
compression progress theory, information that—when acquired—facilitates such
abstraction is more valuable [13]. Our results support this proposition and
suggest that individuals preferentially seek such information. By contrast,
the compressibility curve for collective knowledge networks tends to align
with the curve for the corresponding null model data in later stages of
growth. This finding can be contextualized by considering the fact that
collectives can store vast quantities of detailed information in a distributed
manner and, hence, do not face the same resource limitations that individuals
do. In summary, while compression progress theory is supported by our data
from individual knowledge networks, the building of collective knowledge
networks appears to require a different account.

Conformational change theory and the mechanical flexibility of knowledge
networks. The conformational change theory of curiosity is an alternative
account that is built on two assumptions. First, we assume that humans encode
conceptual knowledge in cognitive networks. Second, we assume that knowledge
networks are embedded in Euclidean space, where they possess several degrees
of freedom. Both assumptions are predicated on how humans encode spatial and
abstract knowledge [27, 54, 23, 24]. Evidence from spatial navigation studies
demonstrates that mental representations of space take the form of labeled
cognitive graphs. Each node represents a physical location and is accompanied
by local metric information such as angles and Euclidean distances to its
immediate neighbors [18, 54, 23]. Furthermore, hexadirectional modulation,
which is the telltale signature associated with an underlying map-like neural
code, is observed in the neural signals when individuals navigate discrete and
continuous abstract concept spaces [55, 56] (see Supplement for details on
mental representations of spatial and non-spatial knowledge). Building on
Euclidean cognitive graphs, we operationalize conceptual flexibility in
knowledge networks as the number of conformational degrees of freedom. We find
that growing individual knowledge networks have greater-than-expected
embedding dimensionality and conformational flexibility. According to
conformational change theory, embedding dimensionality increments when growing
knowledge networks become over-constrained and develop self-stress. We find
that such stress arises more frequently in individual knowledge networks than
in null model data. This observation is consistent with the conformational
change theory of curiosity, and suggests that individuals’ idiosyncratic
acquisition of information leads to a frequent reshaping of concept relations
based on context. By contrast, in knowledge networks built collectively we
find that the evolution of mechanical features-of-interest cannot be
distinguished from their evolution in null model data. Collective networks
grow through a dynamic interplay of consensus and dissensus between large
groups of individuals. Therefore, it is possible that due to the long time
scales that we focus on in this study, dynamic events associated with
collective knowledge growth, such as paradigm shifts, are simply concealed
from view in local sectors of each field.

Using computational measures to operationalize conceptual theories. Our study
of curiosity theories stands on the backdrop of significant work in psychology
that delineates different types of information acquisition. Classical
perspectives from psychology separate notions of curiosity into two broad
categories [57]. Perceptual curiosity is the desire for increased perception
of specific sensory stimuli and is reduced via continued exposure to such
stimuli. By contrast, epistemic curiosity is a more general desire for non-
perceptual knowledge. Based on the breadth of knowledge sought, epistemic
curiosity is further classified as specific or diversive [58]. Specific
curiosity is associated with the desire to reduce uncertainty about an
ambiguous stimulus and leads to exploration in search of a particular piece of
information. Diversive curiosity is less restrictive and refers to the desire
to obtain wide-ranging knowledge. Each category represents a different
manifestation of self-driven information-seeking behavior. Across these sub-
types, many scales have been developed to quantify curiosity in individuals
[59, 60, 61, 62, 63]. However, such efforts rely mainly on questionnaires to
measure curiosity at discrete points in time and are unable to record its
dynamic nature. By formulating curiosity as a process of knowledge network
building, we offer calculable, theory-based measures such as topological gaps,
compressibility, and conformational degrees of freedom that can be used to
characterize the longitudinal process of information-seeking, whether in
laboratory experiments or in the wild. Important future directions include
mapping each network measure to the most suitable sub-type of curiosity and
expanding the current taxonomy of sub-types to accommodate novel network
theoretical perspectives.

Implications for the study of reinforcement learning. The computational
metrics we examine are relevant not only for the study of human curiosity, but
also potentially for that of artificial intelligence. Compressibility, for
instance, was originally proposed as an intrinsic learning signal to guide
reinforcement learning [13]. Our work provides several candidate metrics—such
as the number of topological cavities, network compressibility, and
conformational flexibility—that can act as suitable curiosity-based signals
for task settings where the environment can be modeled as a network. Out of
the three measures, while it is currently computationally intensive to compute
persistent homology and network compressibility, the number of conformational
degrees of freedom can be determined inexpensively. Information acquisition in
reinforcement learning is a means to an end, where the end is a reward
associated with the successful completion of a specific task [1, 64]. An agent
seeking to collect high total reward during interactions with its environment
must strike a balance between exploitation and exploration. The agent must
exploit, or productively use, those actions that are currently known to yield
high reward but must also occasionally explore untested actions that may
eventually turn out to be better. In many real-world settings, external
rewards are highly infrequent or even completely absent and, thus, cannot
reliably guide behavior. In such sparse reward environments, curiosity-like
intrinsic motivations can lead to improved exploration and, by extension,
improved task performance [65, 66]. The design of intrinsic (or curiosity-
based) reward signals for reinforcement learning is an increasingly important
area for further research [67], and may benefit from computational insights
into human behavior, such as those derived from our analyses here.

Methodological considerations. Several methodological considerations are
pertinent to our work. First, in the applied mathematics literature, a network
filtration is typically specified based on rank-ordered edges as opposed to
rank-ordered nodes [68, 69, 70]. Edges are first ordered by decreasing
weights. The filtration then proceeds by sequentially incorporating higher-
ranked edges into the network. Here instead, we assign ranks to nodes and
construct node-ordered filtrations of knowledge networks. At each growth
stage, we add a node and all edges that exist between the node and other nodes
already present in the network. It is possible that future efforts could gain
greater granularity in the constructed filtrations by assigning ranks to both
nodes and edges. For instance, edges (hyperlinks) that individuals click may
be ranked higher than edges that individuals do not click. Second, we
construct knowledge networks for individuals based on Wikipedia browsing data
acquired for fifteen minutes each day for three weeks. However, in reality
daily knowledge acquisition is continuous and not limited to Wikipedia as a
source. Furthermore, participants may act on their curiosity about a specific
concept during un-tracked sessions and continue to explore related ideas
during tracked sessions. The KNOT study, where the data for individuals are
sourced from, implicitly mitigates such discontinuities by linking the last
page of each browsing session to the first page of the following session.
Third, in the conformational change theory of curiosity, we resolve states of
self-stress by incrementing $d$ by one until $DoF_{C}$ is no longer negative.
Importantly, we only consider self-stress in the entire ensemble of nodes and
edges. It is possible, however, that certain subsets of the network may be
over-constrained, even if the entire network, as a whole, may not appear to
be. Such over-constrained subgraphs can be identified using algorithms such as
the pebble game [71]. Self-stress may also occur due to pathological
geometries in networks referred to as kinematic bifurcations [42]. To address
such geometries, we require a precise embedding of concepts in Euclidean
space, which remains an important area for future work.

##  4 Conclusion

We conceptualize curiosity as a process of knowledge network building in order
to examine three theoretical accounts: information gap theory, compression
progress theory, and conformational change theory. Formalizing curiosity in
terms of networks helps us to _quantitatively_ operationalize predominantly
_qualitative_ theoretical constructs. Information gaps can be identified as
topological cavities, compression progress can be quantified using network
compressibility, and flexibility—as premised on the conformational change
theory—can be quantified as the number of conformational degrees of freedom.
We use data acquired from Wikipedia to construct growing knowledge networks
for individuals and for collectives. We find that as networks grow, knowledge
gaps increase in number, suggesting an expanding interior of ignorance. Yet,
in support of an aversion to gaps predicted by information gap theory, we also
find fewer-than-expected disconnected network components (or $0$-dimensional
topological cavities) and fewer-than-expected loops of edges (or
$1$-dimensional topological cavities) in growing knowledge networks. This set
of findings suggests that knowledge “gaps” as conceptualized by information
gap theory may best translate, in a network theoretical sense, to $0$ and
$1$-dimensional cavities. We also find that growing individual knowledge
networks possess greater-than-expected compressibility, indicating that
information acquisition is driven to construct parsimonious mental world
models. In addition, we find that knowledge networks built by individuals
become increasingly flexible with growth, foregrounding the longstanding
relevance of conformational change in the mind. Our results lend support to a
pluralistic view of curiosity, wherein intrinsically motivated information
acquisition fills knowledge gaps and builds increasingly compressible and
flexible mental representations of the world. Our findings offer a novel
network theoretical perspective on intrinsically motivated information
acquisition that may harmonize with or compel an expansion of the classical
taxonomy of curiosity.

##  5 Methods

###  5.1 Data

####  5.1.1 Knowledge networks built by individuals

Knowledge networks for individuals are constructed with data obtained from the
“Knowledge Networks Over Time” (KNOT) study [72, 73, 31]. These data are
comprised of Wikipedia browsing histories of 149 individuals (121 women, 26
men, 2 other) collected between October 2017 and July 2018. At the time of
data acquisition, participants were aged between $18.21$ and $65.24$ years
($\mu=25.05$, $\sigma=6.99)$; $6.71$% identified as African American/Black,
$25.50$% identified as Asian, $5.37$% identified as Hispanic/Latino, $49.66$%
identified as White, $5.37$% identified as Multiracial, $5.37$% identified as
Other, and $2.01$% provided no racial or ethnic information. Every evening for
$21$ days, participants were prompted to open a browser and navigate to
wikipedia.org. They were then instructed to engage in $15$ minutes of self-
directed information search with no restrictions placed on how they could
traverse from one page to another. At the end of each session, participants
used tracking software (Timing), pre-installed on their personal computers or
laptops, to export and upload their browsing histories.

We treat all pages visited by an individual as nodes in a knowledge network.
Edges between nodes are specified based on the presence of hyperlinks. Prior
work has found that pairs of pages connected by hyperlinks are significantly
more similar to each other compared to pairs that are not connected by
hyperlinks [31]. Thus, we add an undirected and unweighted edge between Page 1
and Page 2 if either Page 1 links to Page 2 or Page 2 links to Page 1.
Hyperlinks are not required to exist bidirectionally for an edge to exist
between two nodes. We determine the presence of hyperlinks based on how
Wikipedia appeared on August 1, 2019 and adapt code from the wikinet package
for network construction [47]. Each node (or page) in the browsing data is
accompanied by an index that denotes the temporal order in which it was
visited. For every individual, the nodes and edges as well as the order of
node visitation is used to specify a growing knowledge network.

####  5.1.2 Knowledge networks built collectively

In its role as an encyclopedia, Wikipedia represents a large repository of
knowledge acquired over thousands of years through collective human effort.
Building on prior work, we construct domain-specific collective knowledge
networks by taking subgraphs of the larger Wikipedia network [47]. Information
in Wikipedia is organized in a hierarchical manner, which makes it possible to
identify articles that pertain to a particular domain of interest. We
capitalize on this structure to construct knowledge networks for thirty
topics: abstract algebra, accounting, biophysics, Boolean algebra, calculus,
cognitive science, commutative algebra, dynamical systems and differential
equations, dynamical systems, earth science, economics, education, energy,
evolutionary biology, geology, geometry, group theory, immunology, linear
algebra, linguistics, meteorology, molecular biology, number theory, optics,
philosophy of language, philosophy of law, philosophy of mind, philosophy of
science, sociology, and software engineering. All pages listed under a topic
are treated as nodes in the topic’s network. For instance, the network for
molecular biology contains pages for ‘allele’, ‘lymphocyte’, and ‘antibody’ as
nodes. Similar to knowledge networks for individuals, edges between nodes are
added on the basis of hyperlinks. Typically, articles also contain information
about the year in which the concept they describe first became known; the year
attribute is used as an index to specify node order in a growing graph. More
details on the network construction process (such as the procedure followed
when a page has no year attribute) are available from Ref. Ju et al. [47].

###  5.2 Detecting topological cavities

In order to identify cavities of various dimensions in a network, we construct
a higher-order relational object known as a simplicial complex. While a graph
is comprised of a set of nodes and a set of edges, a simplicial complex
consists of simplices. A simplex represents a polyadic relationship among a
finite set of $k$ nodes. Geometrically, a $k$-simplex is realized as the
convex hull (enclosure) of $k+1$ generally placed vertices. For $0\leq k\leq
2$, a node is a $0$-simplex, an edge is a $1$-simplex, and a filled triangle
is a $2$-simplex. Simplices follow the downward closure principle, which
requires that any subset of vertices, known as a face, within a simplex also
form a simplex. For instance, a $2$-simplex (filled triangle) has three
$1$-simplices (edges) as its faces, each of which in turn is comprised of two
$0$-simplices (nodes). In graph theoretical terms, a $k$-simplex corresponds
to a $(k+1)$-clique, which is an all-to-all connected subgraph of $k+1$ nodes.
We can construct a simplicial complex by assigning a $k$-simplex to each
$(k+1)$-clique in a binary graph. Thus, the resulting combinatorial object is
sometimes also referred to as the clique complex of the graph. We denote the
clique complex of the graph $\mathcal{G}_{p}$ as
$\mathcal{X}(\mathcal{G}_{p})$.

In a clique complex, a $k$-dimensional topological cavity is identified as an
empty enclosure formed by $k$-simplices. Whether a collection of simplices
encloses a cavity is determined in part by its boundary. The boundary of a
$k$-simplex $\sigma$ is defined as the set $\partial\sigma$ of its
$(k-1)$-faces. The boundary of a set of simplices
$K=\\{\sigma_{1},\;\sigma_{2},\;\cdots,\;\sigma_{m}\\}$ is obtained by taking
the symmetric difference $\Delta$ of the boundaries of its constituents,

$\partial
K=\partial\\{\sigma_{1},\;\sigma_{2},\;\cdots,\;\sigma_{m}\\}=\partial\sigma_{1}\;\Delta\;\partial\sigma_{2}\;\Delta\;\cdots\;\Delta\;\partial\sigma_{m}.$

The symmetric difference is an associative operation that returns the union of
two sets without their intersection. A set of $k$-simplices with an empty
boundary is called a k-cycle. At first glance, it may seem adequate to
identify cycles of various dimensions in a simplicial complex and treat them
as topological cavities. However, note that any collection of
$(k+1)$-simplices has a $k$-cycle as its boundary. For example, the boundary
of a $2$-simplex is a $1$-cycle that is “filled in” by the $2$-simplex. Thus,
it is necessary to distinguish non-trivial cycles that constitute true
cavities from those that trivially belong to the boundaries of higher-
dimensional simplices. Finally, we introduce the notion of equivalence. Two
$k$-cycles $K_{1}$ and $K_{2}$ are equivalent if $K_{1}\;\Delta\;K_{2}$ is the
boundary of a collection of $(k+1)$-simplices. Homology refers to the counting
of non-equivalent cycles of various dimensions in a clique complex. It is
customary to refer to non-equivalent cycles simply as _cycles_ for brevity.

The graph filtration from Eq. 1 induces a related filtration of clique
complexes

$\mathcal{X}(\mathcal{G}_{0})\subset\mathcal{X}(\mathcal{G}_{1})\subset\cdots\subset\mathcal{X}(\mathcal{G}_{N})=\mathcal{X}(\mathcal{G}).$
(3)

At each stage in the filtration, we add a node and replace all cliques that
may result from its addition with relevant simplices. While some newly added
simplices create cavities, others close older ones. Equivalence allows us to
compute persistent homology wherein we track the evolution of each cavity from
the moment it is first born to the moment it is completely filled in by higher
simplices. At any index $p$ of the filtration, the $k$-th Betti number
$\beta_{k}(p)$ records the number of active cavities of dimension $k$. We then
define the $k$-th Betti curve as the sequence of numbers
$\\{\beta_{k}(p)\\}_{p=0}^{N}$. We compute persistent homology for all
knowledge networks using the Ripser.py package in Python [74]. Visualizations
of persistent homology results are generated with code adapted from [48].

For a more comprehensive treatment of topological data analysis, we direct the
interested reader to Refs. [75, 33, 32, 76, 77, 34].

###  5.3 Computing network compressibility

In order to estimate the compressibility of a network, we consider a binary
graph $\mathcal{G}_{p}$ with $p$ nodes and $q$ edges, which can be represented
by a symmetric adjacency matrix $M\in\mathbb{R}^{p\times p}$. A message
containing information about the network’s structure can be conveyed to an
arbitrary receiver by encoding it in the form of a random walk
$\bm{x}=\left(x_{1},\;x_{2},\;\ldots\;\right)$. The walk sequence is generated
by transitioning from a node to one of its neighbors uniformly at random.
Thus, for a random walk on $\mathcal{G}_{p}$, the probability of transitioning
from node $i$ to node $j$ is $P_{ij}=M_{ij}/\sum_{j}M_{ij}$. Since the random
walk is Markovian, the rate at which such a message transmits information (or
its entropy) is given by

$H(\bm{x})=-\sum_{i}\pi_{i}\sum_{j}P_{ij}\log P_{ij}.$ (4)

Here, $\pi_{i}$ is the stationary distribution representing the long-time
probability that the walk arrives at node $i$, which is given by
$\pi_{i}=\sum_{j}M_{ij}/2q$.

Assigning clusters to nodes leads to a coarse-grained sequence
$\bm{y}=\left(y_{1},\;y_{2},\;\ldots\;\right)$, where $y_{t}$ is the cluster
containing node $x_{t}$. The number of clusters $n$ can be used to define a
scale of the network’s description $S=1-\frac{n-1}{p}$. For example, when
$n=p$, the network is described at a fine-grained scale $S=1/p$; by contrast,
when $n=1$, the network is described at the largest possible scale $S=1$. In
general, the distorted sequence $\bm{y}$ is non-Markovian. However, we can
still use Eq. 4 to find an upper-bound on its information rate. At every scale
of description, it is possible to identify a clustering of nodes that
minimizes this upper bound. After computing these optimal clusterings across
all scales, we arrive at a rate-distortion curve $R(S)$, which represents the
minimal upper bound on the information rate as a function of the scale $S$.
The compressibility $C$ of the network is then given as the average reduction
in $R(S)$ across all scales [35],

$C=H(\bm{x})-\frac{1}{p}\sum_{S}R(S).$ (5)

Visually, this quantity represents the total area above the rate-distortion
curve and below the entropy of the original random walk $H(\bm{x})$ (Fig. 5B).
For a graph filtration such as in Eq. 1, we abuse indexing notation and define
the compressibility curve as the sequence $\\{C(p)\\}_{p=0}^{N}$, where $p$
denotes the number of nodes in subgraph $\mathcal{G}_{p}$.

###  5.4 Computing mechanical network features

Consider a set of nodes $\mathcal{V}=\\{1,\dotsm,N\\}$ embedded in $d$
dimensions. Each node $i\in\mathcal{V}$ is located at a particular coordinate
in space $\bm{x}_{i}\in\mathbb{R}^{d}$. On its own, this system possesses $dN$
degrees of freedom, as each node is able to move independently in space. If we
connect these nodes with edges in the set
$\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$, then each edge
$e_{ij}\in\mathcal{E}$ between node $i$ and node $j$ removes one degree of
freedom along the direction of edge extension. Each edge generates a
constraint that keeps the distance between the nodes constant, such that

$\displaystyle(\bm{x}_{i}-\bm{x}_{j})^{\top}(\bm{x}_{i}-\bm{x}_{j})=\mathrm{constant}.$

To linear order, this constraint can be modified by taking the total
derivative of both sides and dividing by 2 to yield

$\displaystyle(\bm{x}_{i}-\bm{x}_{j})^{\top}(\mathrm{d}\bm{x}_{i}-\mathrm{d}\bm{x}_{j})=0$
(6)

where $\mathrm{d}$ is the differential operator. Intuitively, Eq. 6 is simply
a dot product between the vector pointing from $\bm{x}_{j}$ to $\bm{x}_{i}$,
and the node motions. Hence, Eq. 6 implies that the nodes must move
perpendicular to the edge such that the edge does not change length. If we
compile all such constraints for every edge in $\mathcal{E}$ then we obtain
$E=|\mathcal{E}|$ constraints on the node motions. If these constraints are
independent, then the total number of degrees of freedom is reduced to $dN-E$.
Among these degrees of freedom, $d(d+1)/2$ are rigid body motions that do not
change the distance between any pair of nodes. Hence, the number of
conformational degrees of freedom is given by

$\displaystyle DoF_{C}=dN-E-\frac{d(d+1)}{2}.$

This line of reasoning was first put forth by Maxwell [78] in 1864.

Many important extensions of this idea exist. One important extension
considers the violation of independent constraints. This violation can occur
in several ways. One such way is over-constraining a network. For example, a
network of $N=4$ nodes embedded in $d=2$ dimensions is over-constrained if we
place edges between all node pairs, such that
$\mathcal{E}=\\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\\}$. Here, there are
$|\mathcal{E}|=6$ edges, such that $DoF_{C}=[(2\times 4)-6]-3=-1$. For a
network to possess negative degrees of freedom, there must exist patterns of
edge compressions and tensions that are load-bearing, such that there are
balanced internal forces held within the edges and experienced by the nodes
[52]. In the conformational change theory of curiosity, we treat such states
of self-stress as aversive. Practically, whenever $DoF_{C}$ becomes negative,
we resolve competing constraints by incrementing the embedding dimensionality
by $1$.

###  5.5 Statistical testing

We use non-parametric permutation testing to determine whether feature curves,
such as those for compressibility and conformational flexibility, for
empirical knowledge networks differ significantly from those for corresponding
null model networks [79]. For a given feature, we first compute the area $A$
between the average curve for the observed data and the average curve for the
null model data using numerical integration in Python. We then pool all data
together and randomly re-assign each data point to either the empirical data
group or the null model data group. Each group results in a pseudo-curve of
values for a given feature of interest. We compute the area $A^{\prime}$
between the pseudo-curves for the two groups and repeat this process for
$I=1000$ iterations. For the group difference between empirical and null model
data, we define the p-value $p_{perm}$ as the number of times $A^{\prime}$ is
greater than $A$ divided by the number of iterations $I$.

###  5.6 Data and code availability

All data used in the manuscript are available upon request from the
corresponding author. All code used is available at
https://github.com/spatank/Curiosity.

##  6 Citation diversity statement

Recent work in a number of scientific fields has identified a bias in citation
practices such that papers by women and other minority scholars are under-
cited relative to the number of such papers in the field [80, 81, 82, 83, 84,
85, 86, 87, 88]. Here, we sought to proactively choose references that reflect
the diversity of the field in thought, form of contribution, gender, race,
ethnicity, and other factors. First, we predicted the gender of the first and
last authors of each reference using databases that store the probability of a
first name being carried by a woman [84, 89]. By this measure (and excluding
self-citations to the first and last authors of our current paper), our
references contain 16.27% woman(first)/woman(last), 11.80% man/woman, 19.30%
woman/man, 52.63% man/man citation categorizations. This method is limited in
that a) names, pronouns, and social media profiles used to construct the
databases may not, in every case, be indicative of gender identity and b) it
cannot account for intersex, non-binary, or transgender people. Second, we
obtained predicted racial/ethnic category of the first and last author of each
reference using databases that store the probability of a first and last name
being carried by an author of color [90, 91]. By this measure (and excluding
self-citations), our references contain 4.67% author of color/author of color,
9.86% white author/author of color, 20.34% author of color/white author, and
65.12% white author/white author citation categorizations. This method is
limited in that a) names, Census entries, and Wikipedia profiles used to make
predictions about gender may not be indicative of racial/ethnic identity, and
b) it cannot account for Indigenous and mixed-race authors, or those who may
face differential biases due to the ambiguous racialization or ethnicization
of their names. We look forward to future work that could help us to better
understand how to support equitable practices in science.

##  7 Acknowledgments

The authors gratefully acknowledge helpful discussions with Drs. Lorenzo
Caciagli, Erin G. Teich, and Kieran Murphy. This work was supported by the
Center for Curiosity. The authors would also like to acknowledge additional
support from the Army Research Office (Grafton-W911NF-16-1-0474,
Falk-W911NF-18-1-0244, DCIST-W911NF-17-2-0181) and the National Institute of
Mental Health (1-R21-MH-124121-01). The content is solely the responsibility
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