The Wiring of Intelligence

Factor models

Spearman (1904) not only discovered the positive manifold but also gave it an elegant explanation. In his two-factor model, Spearman (1927) introduced the general factor, g, assuming the existence of an underlying common source that explains the scores on multiple cognitive tests (see Fig. 1a). Although lacking a formal explanation, Spearman primarily hypothesized it to be some source of mental energy. Note that whereas intelligence generally is viewed as an intraindividual characteristic, g stems from an interindividual observation and must be understood in that context (Borsboom, Kievit, Cervone, & Hood, 2009). Spearman’s factor-analytic approach inspired many scholars to propose models in the same tradition.

Among the most influential contributions is Thurstone’s (1938) theory of primary mental abilities. Thurstone initially argued that Spearman’s unitary trait is a statistical artifact and proposed a multifactor model that also explains the positive manifold and is thus consistent with g but consists of seven distinct latent constructs. Other theorists followed this approach, with Guilford (1967, 1988) pushing the limits by ultimately including 180 factors in his influential structure of intellect model. Thurstone, on the other hand, eventually had to climb down because verifying his model on a new empirical sample compelled him to add a second-order unitary factor to his model. This work set the stage for various hierarchical models of intelligence (Ruzgis, 1994).

The marriage between a multifactor theory and a hierarchical theory has evolved into what is currently the most widely supported factor-analytic model of intelligence: the Cattel–Horn–Carrol theory (McGrew & Flanagan, 1998). The first theory, Cattell and Horn’s g_f-g_c model (Cattell, 1963; Horn & Cattell, 1966), postulated eight or nine factors, including the well-known fluid and crystallized intelligence (derived from Hebb’s intelligence A and B; Brown, 2016; Hebb, 1949). Crystallized intelligence refers to one’s obtained knowledge and skills, whereas fluid intelligence refers to one’s capacity to analyze novel problems independent of past knowledge and skills. The second theory, Carroll’s three-stratum hierarchy (Carroll, 1993), postulated a hierarchy of three levels, or strata, consisting of a general ability, broad abilities, and narrow abilities. In Cattel–Horn–Carrol theory, the broad stratum consists of Cattell and Horn’s primary abilities.

Sampling models

In the past decade, after a century-long dominance of factor theories of general intelligence, three alternative theoretical approaches to explaining the positive manifold have been introduced or reintroduced. The first, the sampling (or bonds) model, was originally advocated by Thomson (1916, 1951) and Thorndike et al. (1926) as an alternative to Spearman’s g theory. Rather than by introducing a general factor that explains cognitive performance, the sampling model suggested that the positive correlations between test scores simply originate from insufficiently specific cognitive tests. Different tests share many of the involved underlying basic cognitive processes (or bonds), and this overlap in processes that the tests tap into will necessarily result in positive correlations between tests (see Fig. 1b).

Bartholomew et al. (2009), Bartholomew, Allerhand, and Deary (2013), and more recently Kovacs and Conway (2016) reintroduced the sampling theory of general intelligence. The former generalized Thomson’s model to account for multiple latent factors, and the latter further extended sampling theory to account for the effects of domain-general executive processes, identified primarily in research on working memory, as well as more domain-specific processes.

Gene–environment interaction models

A decidedly more recent explanatory mechanism for the positive manifold was introduced by Dickens and Flynn (2001, 2002). In aiming to solve the paradox of both high heritability estimates in IQ and large environmental influences on IQ, they hypothesized a gene–environment interaction, in which, through reciprocal causation, IQ influences one’s close environment and that environment in turn influences one’s IQ, creating a multiplier effect (see Fig. 1c). Moreover, rises in the IQ of others may also affect one’s IQ, a so-called social multiplier. Dickens (2007) extended the multiplier model to include multiple abilities. As Dickens and Flynn (2001) noted, their model is similar to Scarr’s (1992) model of intelligence and school achievement. More general theories that relate complex gene–environment interactions to the positive manifold can be traced back to, for instance, the work of Tryon (1935).

Evidence suggests that indeed such gene–environment interactions exist (e.g., Tucker-Drob & Bates, 2015), and social multipliers are hypothesized to be involved with differences in the Flynn effect—generational changes in the IQ test scores of a population—across intelligence domains (Pietschnig & Voracek, 2015). Moreover, in an effort to reconcile genetic and environmental claims on cognitive ability, Ceci et al. (2003) nicely summarized a variety of models that are built on multiplier principles. Among Dickens and Flynn’s (2001) models for intelligence, they distinguished four other areas in which models with similar dynamics have been proposed, such as in dynamical systems theory and bioecological theory. Each of these areas provide a compelling case for multiplier effects in cognitive development.

Network models

The final new explanation of the positive manifold, based on network modeling, was introduced by van der Maas et al. (2006) and has since seen various developments (van der Maas et al., 2019). Inspired by dynamical explanations of ecosystems—such as food webs—the idea of their mutualism model is that the cognitive system consists of many basic processes that are connected in a network with primarily positive interactions (see Fig. 1d). These processes, which the authors define “in a general sense, including such notions as modules, capacities, abilities, or components of the (neuro)cognitive system, . . . have mutual beneficial or facilitating relations” (pp. 844–845). One may recall that, although less formally, Ferguson (1954) too theorized that “differences in ability are the results of the complex interaction of the biological propensities of the organism” (p. 99). However, the mutualism model adds that, during development, the initially uncorrelated basic processes become correlated because of positive reinforcements, as “each process supports the development of other processes” (van der Maas et al., 2006, p. 845). And indeed, these mutual positive reinforcements too exhibit a multiplier effect.

This network approach has particularly resonated in the domain of psychopathology, resulting in a recent surge of research (see Borsboom, 2017, for a comprehensive overview). In intelligence, van der Maas, Kan, Marsman, and Stevenson (2017) extended the mutualism model, allowing for test sampling, mutualistic relations and multiplier effects, and central cognitive abilities. Lastly, an entirely different research tradition—brain modeling—also provides an important class of network models. The parieto-frontal integration theory provides a network model based on various neuroimaging studies and connects parietal and frontal brain regions to explain individual differences in intelligence (Jung & Haier, 2007).

Differences and similarities

What unites these diverse models is that they all explain the positive manifold equally well. Yet a lot sets them apart. In g factor models, the correlations are due to a common source of cognitive performance in many domains. The g factor is understood as a so-called reflective latent variable. That is, in theorizing the nature of intelligence, the general factor is understood as a causal entity. Spearman’s notion of mental energy is an example of that. Note that this notion of a psychological g, reflective and causal in nature, is a hypothesized one and not uncontroversial.

In the mutualism model, there is no such common source. Rather, the positive manifold emerges from the network structure. The nevertheless apparent statistical g factor is interpreted as formative variable because an index variable of the general quality of the cognitive system (Kovacs & Conway, 2019; van der Maas, Kan, & Borsboom, 2014), akin to economical indices such as the Dow Jones Industrial Average. Contrary to psychological g, this psychometric g is well established and noncontroversial (e.g., Carroll, 1993).

In sampling theory, the statistical g factor should also be interpreted as a formative variable. But that is not to say that sampling theory and the mutualism model are very similar. In sampling theory, the positive manifold is essentially a measurement problem. If we would be able to construct very specific tests targeted at the fundamental processes, the overlap in measurement would disappear, and so will the correlations between tests. In the mutualism model, on the other hand, the correlations are real, created during development, and will not disappear when IQ tests become more specific.

In both the multiplier effect model and mutualism model, the positive manifold emerges from positive reciprocal reinforcements. However, the two models differ in several key respects. Most important, the mutualism model proposes an internal developmental process, whereas the multiplier model depicts development through an interaction with the external environment.

Finally, and importantly, as, for instance, Bartholomew et al. (2009) and Kruis and Maris (2016) noted, g theory and sampling theory and factor models and network models cannot be statistically distinguished on the basis of correlation indexes alone, nor do they necessarily contradict one another. Van der Maas et al. (2017) illustrated this in their unified model of general intelligence. However, this is not to say that these models are equivalent with respect to their explanatory power. Each conception might tap into a different granularity of general intelligence, ultimately aiding us in our understanding of the construct. In addition, it is not to say that the models cannot be distinguished. Time-series data and experimental interventions may very well distinguish between the models (e.g., Ferrer & McArdle, 2004; Kievit et al., 2017; McArdle, Ferrer-Caja, Hamagami, & Woodcock, 2002). Marsman et al. (2018) described these issues in more detail.

One dominant model

A first reason for the developmental blind spot is the dominance of g theory. Although g does not necessarily nail down its origin, be it genetic, environmental, or both, it does not naturally capture development. Ackerman and Lohman (2003) concisely summarized this by explaining that

one of the most intractable problems in evaluating the relationship between education and g is the problem of development and age. As near as we can tell, g theories have failed to provide any account of development across the lifespan. (p. 278)

Note that Nisbett et al. (2012) observed that “the high heritability of cognitive ability led many to believe that finding specific genes that are responsible for normal variation would be easy and fruitful” (p. 135). Although arguably a reification fallacy, a better understanding of the—relative—contribution of genes to intelligence is of obvious importance. Nisbett et al.’s conclusion, directly following the previous quote, summarized this obtained understanding: “So far, progress in finding the genetic locus for complex human traits has been limited” (p. 135). After several attempts in the past decade, the generally shared insight is that many genes may be involved with small effects (e.g., Hill et al., 2018; Lee et al., 2018) and that environmental factors require a better understanding. This polygenic view, or possibly even omnigenic view (Boyle, Li, & Pritchard, 2017), supports a both more complex and more realistic theory of intelligence.

The closest factor models have come to providing an account of development across the life span is in Cattell’s (1987) investment theory. In his landmark book, Cattell hypothesized that one develops a pool of crystallized intelligence by the “investment” of fluid intelligence in conjunction with the “combined result of the form of the school curriculum, and of the social, familial, and personal influences which create interest and time for learning simultaneously in any and all forms of intellectual learning” (p. 140). This idea, derived from Hebb’s (1949) two intelligences (intelligence A, “an innate potential, the capacity for development,” and intelligence B, “the functioning of a brain in which development has gone on”; Brown, 2016), was never formalized but is argued to explain the Matthew effect (Schalke-Mandoux, 2016)—a key developmental phenomenon discussed in the next section.

In his investment hypothesis, Cattell (1987) thus explicitly sketched an evident role for the environment in which genes and the environment are united to explain individual differences in intelligence. More recent insights, however, demand a further integration of the two. In discussing the puzzling heritability increase, Plomin and Deary (2015) explained that

Genotype-environment correlation seems the most likely explanation in which small genetic differences are magnified as children select, modify and create environments correlated with their genetic propensities. This active model of selected environments—in contrast to the traditional model of imposed environments—offers a general paradigm for thinking about how genotypes become phenotypes. (p. 98)

This developmental notion of a gene–environment interaction (Tabery, 2007) suggests a causal mechanism between the two that may give rise to the phenotype IQ.

Dickens and Flynn’s (2001) novel formal multiplier model capitalized on such a developmental gene–environment interaction, giving the high heritability of IQ a convincing explanation. According to their model (see Fig. 1c), children not only actively select their environment in accordance with their genetic endowment, but also this environment influences their IQ, creating reciprocal causal relations between the phenotype and the environment. This way, Dickens and Flynn arrived at a truly developmental model of intelligence.

Even more recently, and following a decidedly different track, van der Maas et al. (2006) showed how interactions between cognitive processes are capable of explaining high heritability. Whereas Dickens and Flynn (2001) broke g into genetic and environmental factors, van der Maas et al. (2006) proved that a single underlying factor is no intrinsic requirement for explaining some of the most important phenomena in intelligence.

One dominant phenomenon

A second reason for the developmental blind spot is the primary focus on the positive manifold. Thanks to the work of particularly Spearman (1904) and Carroll (1993), the positive manifold is an undisputed phenomenon. In turn, static factor models have provided an elegant parsimonious explanation of this phenomenon. Yet, the positive manifold lacks a similarly strong developmental companion that can function as a yardstick for the proposed models. Cattell (1987) beautifully stressed the importance of such a phenomenon:

The theorist who wants to proceed to developmental laws about abilities—who wants to be “dynamic” in his explanations of the origin, growth, and nature of intelligence—must be patient to make and record observations first. He can no more focus meaningful movement without this “description of a given moment” than a movie director can get intelligible movement in a film without the individual “static” frames themselves presenting each a clearly focused “still.” (p. 4)

One phenomenon, the Matthew effect, results from exactly those descriptions of given moments: static frames that have been put in chronological order to give a description of the development of cognitive abilities. The Matthew effect is characterized by initially diverging yet increasingly stable patterns of development, as illustrated in Figure 2a, and may serve as a primary candidate for the role of developmental companion to the positive manifold.

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Fig. 2.

Pólya urn demonstrations of (a) the Matthew effect and (b) the compensation effect. Starting with an urn that contains a white marble and a black marble, in each trial the drawn marble is replaced with two or three marbles of the same color, depending on the desired effect. The figures show the development of the proportion of marbles for 50 independent urns.

Originally coined by Merton (1968) to describe the widening gap in credit that scientists receive during their career, the term Matthew effect refers to the popular catchphrase “the rich get richer and the poor get poorer” and is named after the biblical figure Matthew. Although the Matthew effect does not necessarily involve the poor getting poorer, it does involve a widening gap between the rich and the poor in which the rich and the poor can be metaphors (e.g., for the skilled and the unskilled).

The Matthew effect is in no way an isolated phenomenon. Although the preferred term varies between (and within) disciplines, including cumulative advantage, preferential attachment, and dynamic complementarity, the intended process is essentially the same. Other related terms, on the other hand, such as the fan-spread effect and power law, may refer to the observed effect rather than the underlying process (Bast & Reitsma, 1998; Perc, 2014). Perc (2014) provided a comprehensive overview of the Matthew effect in empirical data.

Stanovich (1986) was probably the first to link the Matthew effect to education in an attempt to conceptualize the development of individual differences in reading. In this field, he argued, initial difficulties with reading acquisition can steadily propagate through reciprocal relationships with related skills, ultimately creating more generalized deficits.

Yet, the effect is not undisputed. For instance, Shaywitz et al. (1995) found a Matthew effect for IQ but not for reading when controlling for regression to the mean. Moreover, there is also evidence for the opposite developmental trajectory, the so-called compensation effect. This effect, for instance found by Schroeders, Schipolowski, Zettler, Golle, and Wilhelm (2016), describes a closing rather than widening gap.

Complicating things further, it is often hypothesized that both the factors driving and combating the gap influence development. This is, for instance, clearly explained by Schroeders et al. (2016): “It seems that the compensation effect of a formalized learning environment counteracts the effect of cumulative advantages that is present in a non-formalized setting” (p. 92). This idea at least provides an explanation for the more ambiguous status of these two developmental phenomena, especially compared with the positive manifold.

However, and importantly, evidence for a “general factor of cognitive aging” is consistent with the Matthew effect. In an exciting meta-analysis, Tucker-Drob, Brandmaier, and Lindenberger (2019) showed that “individual differences in longitudinal changes in different cognitive abilities are moderately to strongly correlated with one another” (p. 21). Such a developmental notion of a positive manifold strongly supports a Matthew effect because it implies that a future cognitive state positively depends on a past cognitive state. Moreover, the authors were right to warn that the observed general factor of cognitive aging does not necessarily imply a single cause. Rather, “an equally logical possibility is that the common factor of change represents an emergent property of dynamical systems processes that occur more directly between etiological factors and ability domains” (Tucker-Drob et al., 2019).

Summarizing, it should not come as a surprise that Protopapas, Parrila, and Simos (2014) suggested to focus on the reciprocal relations that drive Matthew effects rather than estimating the gap itself. And intriguingly, one deceptively simple mechanism—driven by such reciprocal relations—can actually explain the Matthew and compensation effects. However, before we introduce the mechanism, we first briefly discuss a second blind spot: idiography.

Pólya’s urn and the Matthew effect

Here, we are concerned with the urn’s ability to simulate typical developmental patterns of cognitive ability. We consider the unmodified version of Pólya’s urn that we described previously. Figure 2a shows the proportions of white marbles for 250 trials from 50 independent urns. As you can see, the urn compositions quickly diverge. The earlier trials have the largest effects, with gradually decreasing influence over time. Indeed, this pattern closely resembles the general Matthew effect, in which at the start, within-person variance is high, gradually decreasing over time, and between-person variance is low, gradually increasing over time.

Pólya’s urn might thus be conceived as a model for a developmental process that produces a Matthew effect. The initial configuration of the urn depicts the genetic component, whereas the trials represent the environmental experiences. The white marbles can, for instance, represent skills that reinforce advantageous experiences, whereas the black marbles might represent misconceptions that reinforce disadvantageous experiences. Via a strict random process, the urns’ configurations diverge in a similar vein as the multiplier process of in Dickens and Flynn (2001). In this Pólya process, skills and misconceptions are reinforced, ultimately growing toward a stable state.

However, one significant additional property over the multiplier mechanism must not be overlooked. Because the distribution of color of each independent urn is exactly equal at the start, it is shown that Pólya’s urn does not require initial genetic differences for the multiplier process to do its work. This illuminating effect is not necessarily an artificial property of Pólya’s urn: For example, Freund et al. (2013) showed that individual differences actually can emerge in genetically identical mice. Summarizing, Pólya’s urn shows that neither genetic nor environmental differences are necessarily required for individual differences to arise, although surely, one should not be fooled into thinking that those therefore play no role in intelligence. We briefly expand on this observation in the section following the next.

Pólya’s urn and the compensation effect

Naturally, environmental influences are less random than assumed here. One strong systematic influence is formal education, which is hypothesized to create compensatory effects that counteract the Matthew effect. Figure 2b shows what this effect could look like. To obtain the effect, we slightly adapted Pólya’s urn to allow for one possible effect of education. Rather than reinforcing an advantageous experience with one extra marble, we now reinforce it with two extra marbles, with p = .5, while keeping the rule for reinforcing disadvantageous experiences the same. An adaptation such as this can, for instance, be conceptualized as the beneficial effect of practice and instruction in education. Likewise, remediation of disadvantageous experiences such as errors may too create a compensation effect and can be modeled by not reinforcing such an experience with an extra marble.

Pólya’s urn and the third source

One more connection deserves to be discussed: Pólya’s urn and the third source of developmental differences. The third source refers to phenotypic variability that cannot be attributed to either genetic or environmental factors. To explain the third source phenomenon, two articles (Kan, Ploeger, Raijmakers, Dolan, & van der Maas, 2010; Molenaar, Boomsma, & Dolan, 1993) proposed rather complicated nonlinear models. Conveniently, the third source becomes directly apparent in Pólya’s urn. In the two examples of Pólya processes in Figure 2, both the genetics (the initial urn configurations) and the environment (the rules for drawing and replacing the marbles) are identical. Yet, the developmental trajectories vary greatly.

The third source is also clearly shown when heritability estimates are calculated for Pólya’s urn. If solely genetic and environmental factors would contribute to phenotypic variability, heritability estimates in Pólya’s urn—with identical initial urn configurations and growth mechanisms—would be 1 . In Figure 3, we show the heritability estimates over time based on identical twins with identical environments. Clearly, heritability estimates are not 1 even for twins whose initial urns already contain 10 times as many marbles as the urns used in Figure 2 and are therefore less prone to diverge.

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Fig. 3.

Heritability estimates of 200 identical twins in Pólya’s urn. Initial urn configurations are sampled such that each urn contains black and white marbles. The standard growth mechanism is used for all twins: One marble is randomly drawn from the urn and replaced by two marbles of the same color. This process is repeated until the urn contains 800 marbles. At each developmental step (x-axis), the heritability is calculated (y-axis) as the squared correlation between the proportions of white marbles in current and past developmental steps of a twin pair, averaged over all twin pairs. This process is repeated for different initial urn sizes (line type and line color).

Pólya’s urn as a network

By transforming the example of Pólya’s urn to a network representation, the benefit of networks in an idiographic science is clearly shown. Conveniently, Pólya’s urn permits a simple transformation to such a network representation. Imagine a network with two types of nodes: black nodes that represent some kind of misconception and white nodes that represent some kind of correct conception. Analogous to the urn example, the initial network may consist of a disconnected black and white node. Now, on each trial, one node is randomly attached to one of the existing nodes in the network, copying its color. This simple mechanism ensures that the probability of a new node receiving a certain color is proportional to the number of existing nodes with that color, essentially a preferential attachment mechanism.

Each of the Pólya’s urn networks in Figure 4—graphs created with the qgraph package (Version 1.6; Epskamp, Cramer, Waldorp, Schmittmann, & Borsboom, 2012) for the R software environment (Version 3.6.1; R Core Team, 2019)—represents a different individual and already illustrate their benefit in an idiographic science: For each individual, the values of the nodes and presence of the links are shown. However, the unidimensional structure of this simple urn example is too simplistic for the ultimate objective to describe a multidimensional intelligence. While preserving the network perspective, in the next part, we introduce a new, formal, and multidimensional model of intelligence—a theory that explains both stationary and developmental phenomena, an abstraction that concretely describes an individual’s skills and knowledge on the level of specific educational items, and an avenue for separating the role of genetics and the environment.

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Fig. 4.

Two instances of a Pólya’s urn network. Both networks started with a single black node and a single white node (t = 0). In addition, the networks share an identical growth mechanism: A new node is randomly connected to one of the existing nodes and copies its color. The numbers show the time points at which the nodes were added.

Fortuin–Kasteleyn

The definitions of nodes and edges give us a minimal description of the wired intelligence network. Now, the model aims to describe the probabilities with which skills are either obtained or unobtained and how they are related. It is this description of probabilities that enables us to explain the established stationary phenomena.

The following model, proposed by Fortuin and Kasteleyn (1972, hereafter referred to as the Fortuin–Kasteleyn model) in the statistical physics literature, forms the basis of this approach,

p ( x , ω ) = 1 Z F ∏ i = 1 p − 1 ∏ j = i + 1 p { θ ω 〈 i , j 〉 x i x j + 1 2 + ( 1 − θ ) ( 1 − ω 〈 i , j 〉 ) } ,

(1)

in which θ is a parameter of the model that describes the probability that any two skills become connected, the factor ( x i x j + 1 ) / 2 is equal to zero whenever x i ≠ x j and equal to one whenever x i = x j , and Z F is a normalizing constant. To illustrate the model, Table 1 shows the various possible states of a network with two nodes and gives the probabilities for those states given three different values of θ .

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Table 1.

Evaluation of the Fortuin–Kasteleyn Model With Two Nodes

State	p ( x , ω ) ∝	p ( x , ω )	θ = 1	θ = 1 / 2	θ = 0
x 1 = + 1 ; x 2 = + 1 ; ω = 1	θ	θ / Z F	1 / 2	1 / 6	0
x 1 = − 1 ; x 2 = + 1 ; ω = 1	0	0	0	0	0
x 1 = + 1 ; x 2 = − 1 ; ω = 1	0	0	0	0	0
x 1 = − 1 ; x 2 = − 1 ; ω = 1	θ	θ / Z F	1 / 2	1 / 6	0
x 1 = + 1 ; x 2 = + 1 ; ω = 0	1 − θ	( 1 − θ ) / Z F	0	1 / 6	1 / 4
x 1 = − 1 ; x 2 = + 1 ; ω = 0	1 − θ	( 1 − θ ) / Z F	0	1 / 6	1 / 4
x 1 = + 1 ; x 2 = − 1 ; ω = 0	1 − θ	( 1 − θ ) / Z F	0	1 / 6	1 / 4
x 1 = − 1 ; x 2 = − 1 ; ω = 0	1 − θ	( 1 − θ ) / Z F	0	1 / 6	1 / 4
	Z F = 4 − 2 θ	1	1	1	1

Note: The eight possible states of a network with two nodes (Column 1), evaluated under the Fortuin–Kasteleyn model (Columns 2 and 3), for three different values of θ (Columns 4, 5, and 6). States differ with respect to whether the nodes ( x 1 and x 2 ) are obtained ( + 1 ) or unobtained ( − 1 ) and connected ( ω = 1 ) or not connected ( ω = 0 ). The values in the final three columns represent the probabilities of observing a certain state of the network (Column 1) given the value of θ (Columns 4, 5, or 6) and are calculated from the model (Column 3, i.e., Column 2 proportional to its sum). The bottom row sums over all states. Fortuin–Kasteleyn = Fortuin and Kasteleyn (1972).

An important property of the model is that whenever two skills are connected to one another, they are necessarily in the same state, that is, they are either both present or both absent. Consequently, whenever two skills are in different states, that is, one skill is present while the other is absent, then these two skills cannot be connected to one another. With this simple rule and the single parameter θ , the model can describe the joint probability distribution of both the nodes (i.e., skills or knowledge) and their relations.

Idiography

This unaltered Fortuin–Kasteleyn model already has highly beneficial properties for the study of intelligence, making it a convenient point of departure. Figure 5 illustrates a few of the properties thus far discussed. To begin with, we can ensure idiography, the first modeling principle. Because the model is characterized by both random nodes and edges, both skills and their relations may vary across individuals. This is clearly seen in the differences between the individuals in the figure, Cornelius and Pete, which are instances of the exact same model. In Pete’s network, considerably more knowledge is obtained than in Cornelius’s network, while at the same time it is less densely connected (both within domains and between domains). In addition, interesting differences between domains exist; Pete clearly performs differently on two of the four domains. The careful eye spots that the connected nodes form clusters only with either obtained knowledge or unobtained knowledge, a homogeneity that is dictated by the model.

Positive manifold

The next property of this model we turn to is the positive manifold it produces. As discussed in the introduction, the uncontested significance of this phenomenon renders a plausible explanation a burden of proof for any serious theory of intelligence. Intuitively, this property of the Fortuin–Kasteleyn model is shown by the fact that the correlation between any two nodes x i and x j in the network is positive. This is the case whenever the probability that the two nodes are in the same state, p ( x i = x j ) , is larger than the probability that the two nodes are in a different state, p ( x i ≠ x j ) . The following equation by Grimmett (2006, p. 11) confirms this,

p ( x i = x j ) = 1 2 + 1 2 p ( i ↔ j ) ≥ 1 2 ,

in which p ( i ↔ j ) is the probability that nodes i and j are connected by an open path (i.e., are in the same cluster). Because p ( i ↔ j ) is nonzero, p ( x i = x j ) is strictly larger than 0 . 5 , and the positive manifold emerges.

Figure 6 shows the positive manifold using a heat map. Each of the patches in the figure represents a correlation between two nodes. The positive manifold can be deduced from the fact that all patches indicate a positive correlation. For this figure, we considered 1,000 idiographic networks, construed from the Fortuin–Kasteleyn model, with some important extensions laid out in the following sections.

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Fig. 6.

Heat map of the correlational structure of the nodes of the Fortuin and Kasteleyn (1972) model. As analogues to Spearman’s very first observation of the positive manifold in the correlational structure of his cognitive tests, the exclusively positive patches illustrate the positive manifold as a constraining property of the Fortuin–Kasteleyn model. In addition, the hierarchical structure of intelligence is clearly reflected in the block structure. Networks were generated with θ W = . 07 , θ B = . 005 , and µ = . 03 . Corr = correlation.

Hierarchical structure

Arguably the second most important stationary phenomenon in intelligence is its hierarchical structure. Although the debate on whether g is organized in a bifactor or higher-order structure continues to keep some intelligence researchers occupied, the fact that some cognitive domains form clusters, with higher correlations within clusters than between clusters, is uncontested (e.g., Carroll, 1993; Spearman, 1904). This phenomenon is reflected in the typical block structure seen in correlation matrices of intelligence tests (see Fig. 6).

Crucially, although the block structure clearly is present, the blocks are not fully isolated. Indeed, the small but meaningful correlations outside the blocks indicate interactions between the blocks. Simon (1962) termed this property near decomposability and demonstrated its ubiquitous presence across a multitude of complex hierarchical systems. In his words, “intra-component linkages are generally stronger than intercomponent linkages. This fact has the effect of separating the high-frequency dynamics of a hierarchy—involving the internal structure of the components—from the low frequency dynamics—involving interaction among components” (p. 477). The presence of both a general factor and a hierarchical structure can be seen to reflect this.

The human brain serves as a convenient illustration. The functional specialization of our brain can cause different cognitive tasks to tap into structurally dispersed brain areas (e.g., Fodor, 1983; Johnson, 2011; Spunt & Adolphs, 2017), making within-community connectivity more likely and between-community connectivity less likely. An example is the (increasing) functional specialization of arithmetic (e.g., Dehaene, 1999; Rivera, Reiss, Eckert, & Menon, 2005). Note that the hierarchical structure seems consistent with brain networks of fluid intelligence (Santarnecchi, Emmendorfer, & Pascual-Leone, 2017). Naturally, this does not exclude the possible existence of processes that play a more general, or maybe central, role in cognitive functioning, such as mitochondrial functioning (Geary, 2018).

In network models, the block pattern is generally referred to as a community structure. In the model proposed here, we impose such a structure by creating communities of nodes that have a higher probability of connecting with nodes within their community than with nodes in other communities. Note that the model is ignorant with respect to the exact substance, or content, of a component because for the theoretical purpose of the model, the levels of the hierarchy are irrelevant. Yet, for illustrative purposes, in the specification of the model hereafter, we assume that the communities are known.

Suppose that there are two communities: skills that are related to mathematics and skills that are related to language. We partition the set of nodes V into two groups, V = (V_M, V_L), one associated to each community. Likewise, we partition the set of edges E into three parts, E = (E_M, E_L, E_ML), in which E_M are the relations between different mathematics skills, E_L the relations between different language skills, and E_ML are all the relations between a mathematics-related skill and a language-related skill. In principle, we may associate to each community of skills c a unique probability θ _c to connecting its members and associate to each pair of communities c and d a unique probability π _cd to connect members from the community c to members of the community d. However, for now, it is sufficient to have one probability θ _W to connect the skills within a community and one probability θ _B to connect skills between two different communities. If θ _W > θ _B , it follows that p ( i ↔ j ) is larger for any two skills i and j within the same community than for two skills i and j that are not a member of the same community. As a result, a hierarchical pattern of correlations emerges from the model, with higher correlations between skills within a community than between skills from different communities.

Figure 5 shows two networks that arose from our extended model, of which a formal expression is given in the Appendix. The four communities in the networks are denoted by the various node shapes. Note that communities should not be confused with clusters—the nodes, or groups of nodes, that are isolated from the rest of the network. In these networks, the within-community connectivity was set to θ W = . 07 and between-community connectivity to θ B = . 005 . Of course, these probabilities can be seen as an empirical estimation problem, which we do not consider here. Figure 6 shows how the community structure is reflected in the correlational structure of 96 nodes across 1,000 extended Fortuin–Kasteleyn models.

General ability

Note that the Fortuin–Kasteleyn model has no preference toward obtained or unobtained pieces of knowledge. However, in the proposed model, we assume that there actually is a preference and indeed toward general ability. Reminded of the previously mentioned 6-year-old named Cornelius, this preference reflects the fact that, for instance, education usually is at the level of the student and aims at attainable goals (i.e., learning generally takes place in the zone of proximal development; Vygotsky, 1978), and interactions with the environment fit the individual to a large extent (i.e., there is a significant degree of reciprocity in people’s intelligence and how they create and select their environments and experiences; Scarr & McCartney, 1983). Moreover, anticipating the growth perspective that is introduced in the next section, it is evident that individuals tend to become more able rather than less able (i.e., if we ignore cognitive decline due to aging and degenerative diseases).

To account for this bias toward aptitude, we impose a so-called external field µ that is minimally positive. External fields are used in physics to represent some outside force that acts on variables in a network, and to understand this idea, magnetism provides a clarifying illustration. In the study of magnetism, variables in the network may represent the electrons in a piece of iron that either have an upward spin (i.e., x i = 1 ) or a downward spin ( x i = − 1 ). When the spins align, the iron is magnetic. One way to magnetize a piece of iron is by introducing an external field (i.e., holding a magnet close to the object) that pulls the electrons in a particular direction.

By applying a minimally positive external field, we thus ensure that the nodes in the network have a slight preference toward general ability. Consequently, on average, knowledge is more often obtained than unobtained in the population. For an individual network, this fact implies that it is more likely that the knowledge in a particular cluster is all obtained rather than all unobtained. Our model extension that includes the external field µ is given in the Appendix. In the networks used to create Figures 5 and 6, we set µ = . 03 .

Moreover, as is shown in the Appendix, the probability that a cluster consists of obtained knowledge is proportional to the size of the cluster. This means that the larger the cluster is, the more likely the nodes in the cluster reflect obtained knowledge. This idea ensures—with high probability—that the giant component (the largest cluster) consists of pieces of knowledge that are obtained rather than unobtained. Note that the external field could be negative too, in the rare situation that the environment elicits misconceptions.

Scenario 1: development toward equilibrium

Al-though we do not know the exact causal mechanisms that drive development, we can observe the model during development. In the first scenario, we started the network in an undeveloped state, with solely unobtained pieces of knowledge and no edges. We then used a Gibbs sampling procedure to grow the network toward its equilibrium state: the extended Fortuin–Kasteleyn model with its desirable properties. The positive manifold and hierarchical structure displayed in Figure 6, discussed in the previous section, necessarily follow from this approach because they are properties of the Fortuin–Kasteleyn model.

Here, we are concerned with how the model develops over time. As the Gibbs sampler rapidly converge the networks to an equilibrium state, we slowed down the process by updating the nodes and edges in each iteration with p = . 15 . This way, we aimed to get insight into how the model behaves toward its equilibrium state. In Figure 7, we illustrate, for 1,000 networks, the growth in the number of obtained pieces of knowledge across the first 30 iterations of the decelerated Gibbs sampler. We considered 96 nodes across four domains and set the external field to .03, the within-community connectivity to .07, and the between-community connectivity to .005.

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Fig. 7.

The Matthew effect when the Gibbs sampler—used to sample a network from the model—is decelerated. The graph in (a) shows the fan spread that characterizes the Matthew effect. The graph in (b) shows the variance in obtained pieces of knowledge across networks. The graph in (c) shows the average variance in obtained pieces of knowledge across subsequent states of individual networks. The dashed line indicates the smoothed variance; the solid line indicates the actual variance.

Figure 7 provides a clear indication of the Matthew effect. First, Figure 7a shows the fan-spread effect that characterizes the Matthew effect. Although all simulated persons were conceived with the exact same cognitive networks, early differences in the number of obtained pieces of knowledge become more pronounced over time, until they stabilize. Figure 7b shows that the variance in obtained pieces of knowledge across networks indeed increases and ultimately stabilizes.

Moreover, Figure 7c shows that the variance in obtained pieces of knowledge across subsequent states of individual networks decreases over time, an effect that is also observed in the example of Pólya’s urn that is displayed in Figure 2c. However, as opposed to the Matthew effect in the Pólya’s urn example, the positive external field in the wired intelligence model ensures that a general ability arises.

This scenario, in which a cognitive network grows from an undeveloped state into an equilibrium state, was also used by van der Maas et al. (2006) in their description of the mutualism model. A criticism of this approach is that from the onset of development until the moment it reaches its equilibrium state, the exact properties of the network are unknown. In the next scenario, we avoided this issue by inspecting the networks solely in equilibrium but across different sizes.

Scenario 2: development in equilibrium

Rather than observing the model during the sampling dynamics, in the second scenario, we investigated the development of the model across equilibrium states of networks of different sizes like—in the previously cited words of Cattell (1987)—the movie director that inspects the stills. To this end, we sampled networks ranging in size from 20 to 300 nodes, in steps of four nodes. We set the external field to .005, the within-community connectivity to .07, and the between-community connectivity to .01.

In Figure 8, we illustrate the growth in obtained pieces of knowledge across the increasing number of nodes in the respective networks. Naturally, the number of obtained pieces of knowledge cannot be higher than the total number of nodes in a network, hence the fact that all observations are in the lower right triangle. Figure 8a again shows the Matthew effect. This time, all observations are completely independent from one another, so no lines are shown.

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Fig. 8.

The Matthew effect and bifurcation in developing networks. Points represent the number of obtained pieces of knowledge (y-axis) across networks of different sizes (x-axis). Each point represents an independent observation. The gray rectangles show the parts of the figures that overlap.

Note that when the networks continue to develop, they start to bifurcate. In Figure 8b, it is shown that this is the case for networks that contain roughly 100 or more nodes. This pattern bears similarities to the Matthew effect found in science funding (Bol, de Vaan, & van de Rijt, 2018), in which a similar divergence is seen for scholars that are just below and just above the funding threshold. In the case of science funding, the effect is partly attributed to a participation effect: Scholars just below the funding threshold may stop applying for further funding. In education, this participation effect is institutionalized through stratification: Students just below an ability threshold will receive education on a different level.

Here, the bifurcation arises from the preference of the Fortuin–Kasteleyn model to form clusters and from the fact that only nodes that are in the same state can become connected. As soon as all nodes are connected in a giant component, the networks contain either obtained or unobtained pieces of knowledge. In addition, the positive external field ensures that the larger the cluster, the higher the probability that it contains obtained pieces of knowledge, and hence the larger the number of observations near the diagonal. Finally, the low connectivity in the simulated networks creates some observations in between the two forks of the bifurcation.

The growth of cognitive networks can also shed a new light on the positive manifold. In Figure 9, the positive manifold is shown again but this time across four different network sizes. We considered networks with 40, 80, 120, and 160 nodes and show that the positive manifold steadily increases with more nodes. This property of the model reflects a much discussed phenomenon in intelligence: the age dedifferentiation hypothesis. Dedifferentiation is the gradual increase of the factor g or, put differently, the increasingly common structure in intelligence across individuals. The hypothesis states that such dedifferentiation takes place from adulthood to old age. Age dedifferentiation’s antagonist, the age-differentiation hypothesis, posits that differentiation takes place from birth to early maturity.

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Fig. 9.

Heat maps of the correlational structure of cognitive networks with 40, 80, 120, or 160 nodes (note the scale of each heat map) across four different domains. Each heat map is based on 1,000 networks. The exclusively positive patches illustrate the positive manifold, and the block structures illustrate the hierarchical structure. With networks increasing in size, the positive manifold increases too. Corr = correlation.

Evidence for the age differentiation and dedifferentiation hypotheses is both poor and problematic. Many scholars have tried to summarize the evidence, but all come to the conclusion that the evidence for either of the hypotheses is inconclusive. Methodological problems such as selection effects and measurement bias are commonly mentioned to account for the inconclusive evidence (e.g., van der Maas et al., 2006, which also discusses similar problems with ability differentiation). In the model proposed here, the strength of the positive manifold is primarily determined by the size of the giant component, which increases with the size of the network.

Scenario 3: a growth mechanism

Ultimately, the formulation of a formal developmental theory of intelligence requires the identification of mechanisms of growth and possibly decline. Although we are unaware of growth mechanisms that keep the Fortuin–Kasteleyn structure intact, we nonetheless end this section with a third approach. We first describe a simple growth mechanism, and because it may force the network out of equilibrium, we then briefly discuss an additional method that repairs the network to ensure that the discussed stationary phenomena remain guaranteed during development.

In this scenario, we conceptualize growth as the addition of previously absent nodes and edges to the cognitive network in which those new nodes may represent obtained as well as unobtained pieces of knowledge. ³ Effectively, in the growth model, edges are sampled from the finite set of possible edges that constitute the full network. Nodes connected by a sampled edge—if previously absent—are added to the network. Then, the communities of the nodes connected by the edge are determined, and the nodes are actually connected in the cognitive network with a probability respective to the determined communities. We further explicate this growth mechanism below.

Growth mechanism

Let us start with the sampling mechanism for the edges. In our approach, we focus on growing the network topology—the wiring of skills and knowledge in a cognitive network—and let the states of the skills follow this process. To do so, we make use of the following factorization of the model:

f ( x , w ) = f ( x | w ) f ( w ) ,

in which  f ( w )  is the model for the topology, known as the random cluster model (Fortuin & Kasteleyn, 1972; Grimmett, 2006), that describes the wiring of the cognitive network.

This idea can be summarized as follows. Suppose that there is a full theoretical network G = ( V , E ) that includes all potential skills, knowledge, and their relations. At conception, an individual may start with an empty network or a small initial subset of the network that may represent his or her genetic endowment. As time proceeds, skills, knowledge, and their relations are added to the network by sampling edges and their attached skills and knowledge. An edge e = 〈 i , j 〉 between skills i and j is included in the individual’s cognitive network, with simply the previously discussed probability θ W to connect the nodes of the same community and probability θ B to connect the nodes of two different communities.

This very basic growth model allows for substantial variation across individual networks, thus satisfying the idiography principle. Edges may or may not be sampled, and once sampled, the attached nodes may become connected or disconnected, directly or indirectly via paths. The nodes may become obtained or unobtained pieces of knowledge and may end up isolated or connected with nodes of the same or other communities in small or large clusters. All of this process may all vary across development.