# Network Models for Cognitive Development and Intelligence

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Factor Model Dominance

## 3. Alternative Explanations for the Positive Manifold of Cognitive Abilities

#### 3.1. The Sampling Model

#### 3.2. Network Models

_{i}) is modeled as a function of autonomous growth, i.e., growth that is not dependent upon other processes, and the influence of other processes. The first term of Equation (1) models the autonomous growth of each sub-process, this is a typical s-shaped logistic growth equation where a

_{i}specifies the growth rate for a particular process and K

_{i}indicates the upper limit in the growth of that particular cognitive process. The second term of Equation (1) models the mutualistic interactions between the different processes. The strength of the interactions between processes is represented in the interaction matrix

**M**, which is assumed to be equal for all people. The second equation defines K

_{i}(the process-specific limiting capacity) as a linear function of genetic (G

_{i}) and environmental (E

_{i}) factors (see [13] for details). The full model is:

_{0}, the ability within each process x

_{i}will grow over time until it reaches a state of equilibrium. Individual differences in x

_{i}, or composites of x (such as IQ or g), are due to individual differences in limiting capacities K

_{i}. In the proof and simulation of the positive manifold, the K’s are uncorrelated between the cognitive sub-processes, which means that there is no single source of individual differences.

**M**most likely differs between subjects because not all growth processes will start at the same time, and the linear model for the genetic and environmental impact ignores statistical interactions between the two. However, there are two major advantages. The first advantage is that this model can easily be investigated using both simulations and formal proofs. Van der Maas and colleagues demonstrated (analytically) that the equilibria of Equation (1) only depend on

**M**and

**K**, and that a positive covariance matrix always characterizes these stable states [13]. For example, a mutualism model with equal interactions M

_{ij}= c and a one factor model with equal factor loadings imply exactly the same covariance matrix.

#### 3.3. g, Sampling, and/or Mutualism

**K**, the person-specific limiting capacity determined by genetics (

**G**) and the environment (

**E**); this naturally explains the increased influence of heredity with age on intelligence, but the authors acknowledge that the role of an external environment is missing [13]. However, this can be accounted for with Dickens’ multiplier model, which is very similar to the mutualism model [34,35]. In this model, some of the reciprocal causal effects are routed through the environment. Strong performance in one domain (Dickens uses basketball as an example) leads to environmental support (in the form of better training), which in turn leads to even better performance. These multiplier effects can also occur across domains. In this way, the mutualism/Dickens model incorporates gene by environment interactions and possibly explains the so-called Flynn and Jensen effects [36].

_{f}

_{1}) than others, reflected by stronger connections to other nodes. Such central processes will correlate very strongly with the g-factor extracted from a factor analysis of the separate cognitive ability test scores. All of these explanations can be included into a unified network model (Figure 2) with the specification of

**M**, depicted in Figure 3.

## 4. Network Psychometrics

#### Complex Behavior in Networks

_{i}, of symptom X

_{i}being active at time t is modeled by a logistic function:

**M**is again a weight matrix of connections, c is general connectivity constant, and b

_{i}is the threshold of a symptom. A higher threshold means that a higher stressor (S

_{i}) is needed to activate the symptom.

**X**(e.g., a particle) being in some state

**x**(e.g., positive), as shown in Equation (5):

**τ**are thresholds (analogous to the thresholds b in Equation (3)) possibly influenced by an external field, β is the inverse temperature parameter and related to the general connectivity parameter in Equation (3), and Z is the integration constant.

## 5. Discussion

## Author Contributions

## Conflicts of Interest

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1 | Though factor analysis on sum scores of subtests is most common, it is not a necessity. An alternative is item factor analysis [59], for instance. The positive manifold and “statistical” g can be investigated with both continuous and dichotomous scores. We focus here on the Ising model since it is very simple and the formal relation with a psychometric model has been established. |

**Figure 1.**The reflective and formative latent variable model. In the reflective model (left) the latent variable (e.g., temperature) causes the manifest scores (e.g., thermometer values at different locations and times). In the formative model (right) the latent variable (e.g., economical index) summarizes the manifest scores (e.g., economic success of different companies).

**Figure 2.**The unified model of general intelligence allowing for test sampling, reciprocal effects (both mutualistic and multiplier), and central cognitive abilities (such as working memory, x

_{f}

_{1}). The x

_{f}and x

_{c}nodes represent separate cognitive abilities in the intelligence network. The c

_{i}and f

_{i}represent test results of crystalized and fluid cognitive abilities, respectively, the sum of which is IQ. The g-factor can be extracted using factor analysis on f (and c) tests. See Equations (1) and (2) for more details on the internal workings.

**Figure 5.**Depictions of models of the WAIS-III dataset [56] that were compared to the mutualism network model of intelligence: the WAIS-III measurement model (

**a**) and the hierarchical g-factor model (

**b**). The constrained mutualism network model (

**c**) fitted the data best. The AIC and BIC for the saturated model were 72181 and 72709, respectively.

**Figure 6.**The cusp catastrophe that underlies stage transitions in cognitive development illustrated by the dynamics of a simple business card. Vertical pressure (Fv) on the card is the splitting variable (e.g., test conditions) and horizontal pressure (Fh) is the normal variable (e.g., instruction). With low vertical pressure (path

**a**), switches in the middle point of the card are continuous, with high vertical pressure (path

**b**) they are sudden, exhibiting hysteresis, bimodality, divergence, and other typical characteristics of phase transitions.

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Van Der Maas, H.L.J.; Kan, K.-J.; Marsman, M.; Stevenson, C.E.
Network Models for Cognitive Development and Intelligence. *J. Intell.* **2017**, *5*, 16.
https://doi.org/10.3390/jintelligence5020016

**AMA Style**

Van Der Maas HLJ, Kan K-J, Marsman M, Stevenson CE.
Network Models for Cognitive Development and Intelligence. *Journal of Intelligence*. 2017; 5(2):16.
https://doi.org/10.3390/jintelligence5020016

**Chicago/Turabian Style**

Van Der Maas, Han L. J., Kees-Jan Kan, Maarten Marsman, and Claire E. Stevenson.
2017. "Network Models for Cognitive Development and Intelligence" *Journal of Intelligence* 5, no. 2: 16.
https://doi.org/10.3390/jintelligence5020016