“4” in “Curiosity Studies”
4
A Network Science of the Practice of Curiosity
Danielle S. Bassett
Curiosity is not an accidental isolated possession; it is a necessary consequence of the fact that an experience is a moving, changing thing, involving all kinds of connections with other things. Curiosity is but the tendency to make these conditions perceptible.
In other words, knowledge is a perception of those connections of an object which determine its applicability in a given situation. . . . Thus, we get at a new event indirectly instead of immediately—by invention, ingenuity, resourcefulness. An ideally perfect knowledge would represent such a network of interconnections that any past experience would offer a point of advantage from which to get at the problem presented in a new experience.
—John Dewey, Democracy and Education
What we know today about the neural basis of curiosity has capitalized on conceptual frameworks and empirical advances across many fields of science. The disciplines that have contributed the most to this conversation in recent years include biology, psychology, neurology, and psychiatry, spanning the gamut from basic science to clinical medicine. Although an exact definition of curiosity from a neuroscience perspective has remained elusive,1 most scientists and practitioners would agree that curiosity is accompanied by some sort of information-seeking behavior.2 A particularly important characteristic of this behavior is that it appears to be internally motivated,3 meaning that no one forces a person to be curious. Naturally then, the scientific study of curiosity tends to uncover the motivations for and neural correlates of information-seeking behavior.4
One manner in which to formalize this study is to examine perturbations of curious thought, which—as it turns out—occur quite ubiquitously in the world we know. By studying how curious thought is modulated by natural or unnatural factors, one can begin to infer underlying mechanisms. A canonical example of a natural perturbation of curious thought is normative neurodevelopment: as children grow from infants to adults, the type of curious thought they produce appears to change in kind. In young children,5 information seeking can amount to heightened attention and focus on objects that are “bright, vivid, startling,”6 while in older adults, information seeking is naturally accompanied by voluntary movements (of the eye or body) to gain more knowledge. One might envision the infant obsessed with the cotton-stuffed ball, with colorful velvet patches on the outside, and a rich smattering of various sorts of ribbons attached with teeth-resistant stitching; and one might contrast this parochial vision with that of a graduate student entering the Ren Library at Trinity College, Cambridge, seeking a definitive tome on “neurons, networks, and nebulae.”7 In other words, information seeking can be distinguished based on the types of information that the subject seeks.
Likewise, information seeking can also be characterized by the manner in which the information is sought. Indeed, the practice of curiosity can differ across individuals,8 may change with age and cognitive development,9 and is likely impacted by stress and socioeconomic status,10 as well as prior experience. Intuitively, the practice of curiosity could be impatient or enduring. It could involve seeking completely unknown information or vaguely familiar information. It could involve gathering the new information and keeping it logged separately like bits of trivia, or it could involve determining the links between bits of information, fitting them into one’s existing body of knowledge. While these manners of curiosity are intuitive, it remains difficult to precisely define them, categorize them into classes, write down mathematical formulations for their nature, and form generative models for their processes. In other words, we lack a science of the practice of curiosity.
In this chapter we develop the conceptual foundations for such a science. We suggest that the practice of curiosity can be defined as knowledge network building. This proposition offers an interdisciplinary perspective on curiosity that is informed by neuroscience, psychology, linguistics, and network science. By drawing on concepts and tools across these disciplines, we suggest that knowledge can be represented mathematically as a network. While prior scholarship has focused on definitions of curiosity more akin to the force that enables us to seek knowledge, we focus on the manner of network growth in our minds and the potential to quantitatively characterize and mathematically model that growth using tools from network science. The proposal formalizes many of the intuitions that we have about the practice of curiosity, and by that formalization provides the foundations from which to construct explicit hypotheses that can be tested empirically in humans.
The chapter is organized as follows. In the first section, we define what a network is both conceptually and mathematically. In the second section, we discuss how networks can be used to represent knowledge, and we review key efforts in the study of language networks, semantic networks, and concept networks. In the third section, we outline the general idea of network building, and we review models of network growth from various fields of biology, including genetics, vasculature, and neuroscience. In the fourth section, we unpack more explicitly the bridge between the practice of curiosity and models of network growth, placing special emphasis on how this intersection can be informed by theories of learning and education. In the fifth section, we highlight several future directions in empirical science, mathematics, and their intersection that could further inform a science of the practice of curiosity, and in the last section, we conclude.
What Is a Network?
To make the notion of knowledge network building concrete, we must first clarify what we mean by a network. A network is a representation of a complex system, which in turn is a system that is composed of many interacting parts and in which the pattern of interactions is far from homogeneous and therefore defies simpler modeling efforts.11 Specifically, a network is a representation in which the system’s components are represented by network nodes, and a relationship between two components is represented by an edge (or a link) between two nodes. Commonly, network representations are encoded in a graph G=(V,E) with vertices V representing nodes and edges E representing relationships between them (see Figure 4.1).12 Further, a common storage object for a graph is an adjacency matrix, which is an N-by-N matrix A, where N is the number of nodes in the network, whose element Aij indicates the strength of connectivity between node i and node j. Naturally then, networks are an excellent way in which to represent and probe relational data.13
Figure 4.1. A network is often encoded in a graph, in which a system’s parts are represented by nodes (circles), and in which relations between parts are represented by edges (lines). Art by J. K. Rofling, www.jkrofling.com.
Historically, network representations have been commonly exercised in the context of social groups, largely in an effort to understand patterns of social interactions,14 quantify the influence of a single individual on collective behavior,15 and predict voting patterns or political tumult.16 Across these efforts, it has proven critical to carefully consider the definition of network nodes.17 While a single person is perhaps the clear initial choice for a network node, there are arguments both to choose larger components (extraperson objects such as groups, parties, communities, or countries) or smaller components (intraperson characteristics including brains or brain areas driving social behaviors). Indeed, the choice of what constitutes a node depends on the scientific question at hand; some levels of description will be more or less sensitive to the phenomenon of interest. In other words, the term “social network” may be a misnomer for a multiscale network system that can be interrogated either in a scale-specific manner or in a cross-scale manner.
Importantly, the concept of a multiscale network is not only relevant for the historical fodder of network science (social networks) but also particularly appropriate for the organ that produces our curiosity: the human brain.18 Network representations of the brain usually begin with a subdivision of the cortical and subcortical tissue into parcels, which are thought to perform different functions, and whose boundaries are defined by anatomical19 or functional markers.20 These parcels are then connected with one another either using estimates of hardwired connections, as defined by imaging markers of white-matter tracts, or by estimates of functional connections, as defined by similarities in the time-dependent activity traces measured from pairs of parcels.21 Expanding this representation to a multiscale network enables us to capture interactions between neural units defined across both spatial and temporal scales. Intuitively, these networks then represent the patterns of interactions between functional units of the brain that enable the complex patterns of thought characteristic of humans. Indeed, individual differences in the architecture of these brain networks across people have been linked to individual differences in openness to experience,22 creativity,23 and information-seeking behaviors.24
Knowledge as a Network
The multiscale network housed inside of the human skull enables us to acquire knowledge, learn new languages, and build conceptual frameworks and theories to explain the world around us. While the Oxford English Dictionary provides a quite general account of knowledge as “the sum of what is known,”25 Webster’s 1828 Dictionary of the English Language more specifically claims that knowledge is “a clear and certain perception of that which exists, or of truth and fact; the perception of the connection and agreement, or disagreement and repugnancy of our ideas.”26 Indeed, knowledge is quite naturally thought of as a set of ideas and a pattern of connections between those ideas. That is, we do not simply hold disconnected concepts in our minds; instead, we hold concepts and their relationships.
Perhaps the simplest illustration of this networked nature of knowledge is evident when considering language. Language is composed of units defined over different temporal scales, including phonemes, syllables, and supra-syllabic objects, each of which is represented and processed in different areas of the brain.27 At the finest level of phonemes, pairs of phonemes are found beside one another with some specific probability, and the set of probabilities defines a network architecture for the language. The rules by which these probabilities are defined are the topic of a large body of work in artificial grammars.28 At the coarser level of phonological word-forms, or lexemes, one can similarly construct a network representation by linking lexemes, if they are phonological neighbors of each other in the adult lexicon.29 The structure of this network has specific implications for the process of retrieving word-forms from the mental lexicon, and also motivates questions regarding the mechanisms that might lead to certain network structures.30
While phonological neighborhood is a natural metric by which to link units of language, its relationship to knowledge per se is arguably rather tenuous. Closer to our focus are semantic networks, which represent semantic relations between concepts: for example, mammal (node) is (edge) an animal (node).31 Early efforts argued that these networks are tree-like structures with connections determined by class-inclusion relations,32 while later work argued that such strict hierarchical structures may not be relevant for a large majority of concepts.33 Since those early efforts, data continue to mount supporting the notion that semantic networks are not particularly tree-like,34 but instead have a small-world organization (local clustering accompanied by a few long-distance connections), and a scale-free organization (most nodes having relatively few connections and a few nodes having many connections).35 Both this clustering and heterogeneity are thought to impact the formation and search of semantic memory.36 One can expand on the simple semantic network by adding causal links between concepts; this extension takes concepts organized into categories and enables causal inference, causal reasoning, and causal perception from them.37
The number of possible semantic relations between concepts is massive. This fact increases the potential complexity of semantic networks and can hamper simple interpretations of that complexity. A more tractable place to start is to choose a single relation, or a set of similar relations, and study the network architectures that emerge. For example, one might wish to study the network of concepts in which words are represented as network nodes, and two words are connected with one another if they share similar meaning. Technically speaking this approach moves us from networks of semantic relatedness (broadly defined) to networks of semantic similarity.38 A strict way to construct this network is to use information regarding synonyms in a dictionary or thesaurus; alternatively, one can perform laboratory experiments in which one asks a human participant to list a sequence of words in which each word is related to the next by meaning.39
Indeed, this discussion naturally raises the question of how to measure the knowledge network of a single individual, with the eventual goal of understanding how their knowledge network might be built through the practice of curiosity. Intuitively, individual semantic networks can be observed and measured through either verbal or written form. Verbal assessments include free association tasks,40 or asking participants to produce narratives or stories, with or without visual prompts such as pictures. The structure of these narratives can provide insights not only into healthy cognition but also into the minds of those with cognitive impairments or mental health disorders.41 An alternative to verbal measurements is to use written forms, such as stories, blogs, articles, or books. Common network representations of these data include word co-occurrence networks,42 where words are represented as network nodes and two words are connected with one another if they are less than x words away from one another in the text; here, x is a threshold that is often chosen in the range of two to ten. Other measures of word-to-word relations beyond co-occurrence have also proven useful, and many computational algorithms have been devised to build such networks from large corpora.
Network Building: Models of Network Growth
While the previous section described the existence of knowledge networks, both in the population at large and in individual humans, it did not address the question of how those networks are built. Indeed, how does one build any sort of network? Intuitively, one might imagine that to build or grow a network, one must have a rule for choosing nodes to add to the network and a rule for choosing how to link those new nodes to existing nodes in the network (see Figure 4.2). In addition, one might consider whether or not older nodes or older edges die out after a certain time has elapsed, or after a certain amount of growth has occurred. These ideas lead to questions of conservation: are there energetic, spatial, or other constraints on the system that inform the rules of network growth?
Figure 4.2. Knowledge networks can be built in a purposeful manner, with explicit rules for choosing nodes to add to the network and for choosing how to link those new nodes to existing nodes in the network. This figure is an artist’s rendition of the knowledge network created by a daydream, where the meandering nature of thought produces a network with loopy structure. Art by J. K. Rofling, www.jkrofling.com.
As one might imagine, the answers to these questions may vary from system to system; rules may depend on the purposes of the network, on persistent pressures from evolution, and on transient demands from the environment. Some of the simplest growth models, however, ignore physical constraints and simply follow a set of abstract topological rules. For example, the Barabási-Albert model (also known as the preferential attachment model) begins with a single edge connecting two nodes, and then iteratively adds a single node to the network by linking the new node to m existing nodes, with a preference for nodes of high degree.43 This growth model tends to form networks with highly skewed distributions of degree (the number of edges each node has): a scale-free distribution, in fact, whereby most nodes have relatively few connections and a few nodes have many connections. A recent extension of the Barabási-Albert model is the affinity model, which is designed to create more explicit hierarchical structure in the network by assigning each node an affinity parameter, and then linking nodes with similar affinity parameters.44
While these growing models create networks with architectural features that are commonly observed in concept networks—such as local clustering and skewed degree distributions—they do not explicitly account for realistic constraints in the environment or in the mind of the thinker. Two particularly salient examples of network growth models, which do account for realistic constraints, address the development of neural and vasculature systems. In neural systems a common observation is that neurons are more likely to connect to one another if they are close in physical space.45 A natural model for this process is the distance drop-off growth model, which distributes nodes uniformly at random in a physical space, and then connects nodes to one another according to a probability that is a function of node–node distance.46 This model creates degree distributions that are consistent with empirical data, and also recapitulates observed patterns of assortative mixing:47 nodes with high degree (also called hubs) tend to connect to other nodes of high degree. While this and similar models focus solely on the growth of nodes and edges in a single system, models of vasculature have coupled models of underlying tissue growth, with the overlaid vasculature network growth. Notably, recent work demonstrates that the growth of the underlying tissue, coupled to the dynamical equations for network development, can explain the emergence of highly optimized transport networks in animal and leaf vasculature.48
Thus a variety of network growth models span those that evolve by abstract topological rules and those that evolve by physically or biologically motivated rules. Do these notions help us in constructing growth models for knowledge networks? This question has been most actively addressed in the context of semantic networks. For example, Steyvers and Tenebaum suggest a simple model for semantic growth in which new words or concepts are added to the network in such a way as to differentiate the connectivity pattern of an existing node,49 generating both small-world architecture and scale-free degree distributions. Interestingly, this model suggests a mechanism for the effects of learning history variables (age of acquisition, usage frequency) on behavioral performance in semantic processing tasks. In a similar spirit, Hills et al. consider the learning of word association networks and demonstrate that a preferential attachment model incorporating word frequency, number of phonological neighbors, and connectedness of the new word to words in the learning environment offered a reasonable fit to a data set of noun acquisition from children under thirty months of age.50 While both models are described as generically applicable across individuals in the broader population, it is also of interest to examine if and how individual differences in semantic networks are correlated with individual differences in the participant’s personality or creativity,51 under the assumption that the manner in which an individual interacts with the world may impact their network’s growth.
Bridging the Practice of Curiosity with Models of Network Growth
While growth models exist for semantic networks, no current efforts address the active growth of knowledge networks through the practice of curiosity. And what is the practice of curiosity? Can one practice curiosity? Many studies operate under the idea that curiosity is an innate or default state: a capacity that is best characterized as a trait of a person, or the common mode in which the person operates.52 This notion is similar to the notion that a person has a natural level of self-assurance, irritableness, or self-referential processing, also referred to as mindfulness. Yet, in truth, self-assurance and irritability can vary over short timescales, and mindfulness is far from fixed in a single person. In fact, mindfulness training can fundamentally alter a person’s patterns of thought, leading to a change in their decision-making,53 working memory, spatial memory, verbal fluency,54 and cognitive flexibility,55 by altering the activity of specific areas of the brain.56 Similarly, curiosity can be argued to be far from fixed in a person, but instead can wax and wane naturally from moment to moment.57 Furthermore, curiosity can be modulated by external factors including those present in learning environments.58 The fact that curiosity can vary and be varied opens the possibility of practicing curiosity with the aim of self-betterment.
Here we define the practice of curiosity as the performance of mental tasks characteristic of curious thought. Just as in mindfulness training where one practices a certain set of mental states and transitions (or lack of transitions) between them, so in curiosity training one practices mental states of curiosity, and mental state transitions following a line of inquiry. Moreover, one might practice choosing the objects of curiosity, following patterns of curious search, and making time and space in one’s life to act on one’s information-seeking proclivities. Metaphorically, one walks along one’s network of knowledge and seeks to build new webs, add new edges, add new nodes, or leap into the black (or blank) space beyond one’s knowledge in the hopes of landing on some deliciously unexpected idea (see Figure 4.3). The manner in which we walk, build, and leap may be informed by our personalities, our educational experiences and learning capacities, and other characteristics that differ from person to person.
Building network growth models for the practice of curiosity then entails several distinct ingredients. First, one must determine the type of node one seeks; while semantic networks are arguably the most common knowledge network studied in the current literature, one could argue that the ideas one tends to search for in curious acts are often larger than a single concept. Second, one must determine the type and distance of links one is willing to make: How distinct may two ideas be for one to still acknowledge their relationship? Third, one must determine the manner of incorporating the new node and edge into the existing knowledge network (if at all): What sort of architecture does one wish to build? Is it dense or sparse? Ordered or disordered? Low-dimensional or high-dimensional? The answers to these questions require some explicit notions of distance, geometry, and space: the distance between ideas, the geometry of the network, and the space in which the network exists.59 Building on the work of Peter Gärdenfors, one can ask: “What is the geometry of curious thought?60 And how does it relate to one’s own conceptual space?”61 Such a space may even be poetic.62
Figure 4.3. One can walk along one’s network of knowledge, deciding to build a steady path or to leap to a new space in the hopes of landing on an unexpected idea. Art by J. K. Rofling, www.jkrofling.com.
The geometry of the network that one builds may depend on processes of implicit learning that occur as we watch others perform acts of curiosity, either by visual or auditory observation or by reading their written work. One of the most commonly studied forms of implicit learning is known as statistical learning,63 whereby we acquire knowledge about statistical regularities in our environment. The neural computations supporting this type of learning can facilitate either the encoding of pairwise relationships between objects or concepts or higher order relational patterns between them.64 In the context of learning the practice of curiosity from others,65 this human capacity could manifest in acquiring knowledge about the types of ideas others search for, how they connect them, and, over time, how these small steps lead to the growth of knowledge networks. Learning the practice of curiosity can be strengthened further through so-called reinforcement learning processes, where one is explicitly told by another that one has responded correctly.66 This confirmation of accuracy can reinforce the learned behavior; one’s nature or idea may be validated as “curious” by another person.
The notion of learning the practice of curiosity from others naturally motivates a discussion of education and educational forms. In the common forms of education, is the practice of curiosity taught? Does one learn what category of nodes to look for, what types of edges to draw, and what sort of networks to build? Of course, it is entirely possible that a teacher or professor lecturing in front of a class for a semester may be able to impart some knowledge about the practice of curiosity, as a byproduct of demonstrating their own. But perhaps the more natural means of transferring a mode of knowledge network building is by mentorship or apprenticeship. Here, the one-on-one nature of the interaction can denoise the mentee’s observed statistical regularities, perhaps leading to a swifter and more accurate acquisition of the knowledge offered.
Future Directions
While previous sections have laid out a framework for a network science of the practice of curiosity, many questions remain that directly motivate ongoing and future empirical research. The first and most natural place to begin is to empirically characterize the objects of curiosity that individuals seek—whether they be concepts or causal relations, ideas or principles, and whether they be crystalized or hazy, simple or complex. The second objective is to empirically measure the relationships that humans seek to make between those objects of curiosity, and the third is to characterize the evolution of the participant’s knowledge network as they build. It seems natural to tackle these challenges both over short time scales in living subjects (in the course of a traditional laboratory experiment) and over long time scales in deceased (or merely absent) subjects, by examining the evolution of their written work. Collectively, such studies would provide insights into the manner of network evolution that accompanies the practice of curiosity.
To complement these empirical measurements, one needs to build fundamental theories and mathematical models to explain the observations. Specifically, one needs to use these empirical measurements to determine the parameters and rules of network growth models characteristic of a cohort of participants, as well as variations in those parameters or rules that are representative of individual people. It is possible that the rules by which we build semantic networks are similar to the rules by which we build knowledge networks through acts of curiosity. However, it is also possible that curiosity enables us to reach farther for ideas than we would naturally—to seek, search, and track with greater fervor and with greater dedication than would otherwise be our want, leading us to stretch out tendrils into the knowledge network space that would otherwise remain tight local neighborhoods. A careful blend of theory and experiment could prove or disprove this inkling.
In this chapter we have described a formalism embedded in the natural sciences in which to study the thoughts and acts of curiosity. We described the mathematical notion of a network or graph and how it can be used to represent information about different sorts of knowledge, from grammars to semantics. We then described models of network growth and their relation to the evolution of semantic networks, which parsimoniously capture complex patterns of relationships between concepts. We then defined the practice of curiosity and described how it can be characterized as a purposeful growth of one’s knowledge network, which can be influenced by one’s personality, learning capacity, and educational experiences. We suggested that further developing, empirically testing, and validating a network science of the practice of curiosity could inform not only an individual’s internal practice but also educational practices at large. We look forward to future efforts clarifying these ideas.
Notes
The author is grateful to (i) Perry Zurn for inspirational discussions that were instrumental in the formulation of these ideas, (ii) Arjun Shankar and Perry Zurn for helpful comments on earlier versions of this manuscript, (iii) the Center for Curiosity at the University of Pennsylvania, which motivated the author to devote time to developing the ideas put forth in this chapter, and (iv) Brennan Klein for illustrations.
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