Complexity

Volume 2018, Article ID 1918753, 19 pages

https://doi.org/10.1155/2018/1918753

## A Complex Network Framework to Model Cognition: Unveiling Correlation Structures from Connectivity

^{1}Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Spain^{2}Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain

Correspondence should be addressed to Gemma Rosell-Tarragó; moc.liamg@ogarratllesorammeg

Received 24 January 2018; Revised 20 April 2018; Accepted 3 May 2018; Published 12 July 2018

Academic Editor: Hiroki Sayama

Copyright © 2018 Gemma Rosell-Tarragó et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Several approaches to cognition and intelligence research rely on statistics-based model testing, namely, factor analysis. In the present work, we exploit the emerging dynamical system perspective putting the focus on the role of the network topology underlying the relationships between cognitive processes. We go through a couple of models of distinct cognitive phenomena and yet find the conditions for them to be mathematically equivalent. We find a nontrivial attractor of the system that corresponds to the exact definition of a well-known network centrality and hence stresses the interplay between the dynamics and the underlying network connectivity, showing that both of the two are relevant. Correlation matrices evince there must be a meaningful structure underlying real data. Nevertheless, the true architecture regarding the connectivity between cognitive processes is still a burning issue of research. Regardless of the network considered, it is always possible to recover a positive manifold of correlations. Furthermore, we show that different network topologies lead to different plausible statistical models concerning the correlation structure, ranging from one to multiple factor models and richer correlation structures.

#### 1. Introduction

Individuals differ from one another in their ability to learn from experience, to adapt to new situations and overcome challenges, to understand simple to complex ideas, to solve real-world and abstract problems, and to engage in different forms of reasoning and thinking. Such differences in performance occur even in the same person, in different domains, across time, and using distinct yardsticks [1–4].

These complex cognitive processes are intended to be clarified and put together by the concept of intelligence. Although many advances have been made, there are still open questions regarding its building blocks and nature yet to be solved [5–7].

There is a fair amount of research carried out and still going on about the theory of intelligence, and few statements have been unequivocally established. Nowadays, there are several unsolved questions which lead to prevailing discussions and active research. However, incorrect generalizations or misleading results may bring forth social and educational moves [6, 8, 9]. For this reason, there is an urgent need to understand the most important root causes, validate existing theories, and shed light to people who are responsible of educational and even social and health decision-making.

Nowadays, there are a significant number of approaches to intelligence. Developmental psychologists are often more concerned about intelligence as a subset of evolving processes throughout life, rather than about individual differences [10]. Several theorists stress the role of culture in the very conceptualization of intelligence and its influence in individuals [11], while others point to the existence of different intelligences, either measurable or not [12]. There is also an increasing interest in contributions coming from biology and neuroscience [13–17]. Yet, the most influential approach so far is based on psychometric testing [18–24].

Psychometrics has enabled successful and systematic measures of a wide range of cognitive abilities like verbal, visual-spatial, fluid reasoning, working memory, and processing speed through standardized tests [25, 26]. Even if distinct, these assessed abilities turn out to be intercorrelated rather than autonomous prowesses. That is, people who perform well in a given test tend to obtain higher scores on the others as well. This well-documented evidence concerning positive correlations between tests, regardless of its nature, is called the positive manifold. And precisely because of the existence of such complex relations, one of the main aims of this approach is to unveil the structure which best describes the relationships between a number of distinguishable factors or aptitudes that may exist. On this basis, many studies use exploratory and confirmatory factor analysis techniques, starting off from between-test correlation matrices.

Furthermore, there exists a complex correlation structure between abilities which may unveil the underlying connection between cognitive processes. Factor analysis might help clearing up such patterns and yet bring about discussion on the meaning of the outcome.

A brief historical overview since the early days of intelligence research and its development may help us understand the spectrum of existing models. Some theorists relied on the shared variance among abilities, which Charles Spearman, pioneer of factor analysis, called the factor or general intelligence [27], that is, one common factor which explains most of the variances within a population and source of improvement or decline of all other abilities, and it is still a cause for controversial.

Alternatively, hierarchical models of intelligence where each layer accounts for the variations in the correlations within the previous one were also well accepted [18, 28, 29].

Nevertheless, a fair number of scholars argued against theories of cognitive abilities or intelligence drawn upon the concept, measure, and meaning of general intelligence. Namely, Howard Gardner, stated that an individual has a number of relatively autonomous intellectual capacities, with a degree of correlation empirically yet to be determined, called multiple intelligences, among which noncognitive abilities are included [12].

Two different approaches with reference to the relationship between observable variables and attributes or constructs prevail in present research and theorizing not only in psychology but also in clinical psychology, sociology, and business research amongst others: formative and reflective models [30]. In the first of this conceptualization, observed scores define the attribute, whereas in the latter, the attribute is considered as the common cause of all observables. As an example, the classic definition of general intelligence could fall into a reflective model. But also, in clinical psychology, a mental disorder may be thought to be a reflective construct that brings about its observable symptoms [31]. Possible correlation between observables might be therefore due to its underlying common cause. Conversely, the aggregate outcome of education, job, neighbourhood, and salary leads to socioeconomic status (SES), a standard example of formative model.

A more recent approach aims to combine distinct possible factor models by only using the information about the factorial structure found by each study [32].

Both formative and reflective models, along with similar alternatives, may elicit discussion regarding two different issues: one first source of debate is rooted in the meaning and interpretation of such models, while a second cause stems from disregarding the role of time, that is, the dynamics of the system is not explicitly considered.

The abovementioned problems can potentially be overcome if we consider that variables, that is, observables, scores, or indicators, are the characteristics of nodes in a network. These latter are directly connected through edges, which reflect the coupling between variables. Dynamical systems theory is therefore the proper framework to formalize and study the behaviour of such systems [33]. Starting from an initial state, the system evolves in time according to a system of coupled differential equations and eventually reaches an attractor state of the system.

Examples of previous works on a dynamic systems approach to social and developmental psychology are the modeling of language acquisition and growth, the dynamics of scaffolding, dyadic interaction in children, teaching and learning processes, and coconstruction of scientific understanding among many others [34–43].

Noteworthy, a substantive piece is prevalently missing: the topology of the network on where the process is taking place, which may be a determinant fact that enables nodes to communicate between each other and brings about correlations not explicitly enforced in the model. Therefore, the objective and contribution of the present work is exploring the significant role of the network topology or connectivity structure between the variables deemed meaningful to the case of cognitive abilities or intelligence models.

In this work, we evince the tight connection between a centrality measure of the network and the stable solution of the studied models. Moreover, we show that distinct network topologies may explain different correlation structures.

The paper is organized as follows: Section 2 introduces basic notions of networks and explored topologies. Section 3 describes and formalizes the two studied models of cognition. Section 4 and Section 5 go through the main results, concerning dynamics and correlations, respectively. The final discussion and the conclusions are presented in last section. Further mathematical methods can be found in the appendix.

#### 2. Network Topology

A network, , is a collection of vertices or nodes, , linked by edges, , which are given meaning and attributes. Networks can describe complex interconnected systems such as social relationships, transportation maps, and economic, biological, and ecological systems. We consider networks that have neither self-edges nor multiedges, called simple networks [44].

The adjacency matrix of , written , is the -by- matrix which entries equal 1 if node is linked to node and 0 otherwise. Networks can be directed or undirected, although we stay on the latter case.

The topology of a network characterizes its shape or structure and the distribution of connections between nodes. Besides the attributes of nodes and edges, the topology of a network determines its main properties and makes it distinguishable from others. One main property is the degree of a node , , which is the number of edges connected to it. Although networks may describe particular real systems, regardless of its nature, they can be classified to one of the most well-known families of networks. Right after, we briefly describe the four network models explored in the present work. (a)Complete network (Figure 1): within the family of deterministic networks, a complete network is characterized by its nodes being fully connected, that is, each node is connected to the others, such that all off-diagonal elements of the adjacency matrix are equal to 1, .(b)Erdös-Rényi network (Figure 2): one of the most renowned random networks is generated by the Erdös-Rényi (ER) model [45]. Given the number of nodes, , and the probability of an edge, , this model, , chooses each of the possible edges with probability . However, generally, real networks are better described by heterogeneous rather than ER networks. Therefore, ER networks are often used as null hypothesis to reject or accept models concerning more complex situations.(c)Heterogeneous network (Figure 3): there is a wide range of networks coming from real systems (either found in nature or human driven) which topology is far from being homogeneous, but it rather entails degree distributions which are characterized by a power law, also called scale-free when the networks are large enough [46]. The Internet network, protein regulatory network, research collaboration, online social network, airline system, cellular metabolism, company, and industry interlinks are few examples of them [47, 48].(d)Newman modular network (Figure 4): in addition to the degree distribution, another important feature is the presence of communities or modules within a network, mainly in social but also in metabolic or economic networks [49, 50]. A module or community can be defined as a subset of nodes which is more densely linked within it than with other subsets of nodes. One particular method to generate such modules within a network is the Newman model, which distributes the nodes in a number, , of modules not necessarily isolated from the others [51–53]. Similarly as ER networks, with a probability , an edge between pairs of nodes belonging to the same community is created, whereas pairs belonging to different communities are linked with probability . In the model, the number of nodes, ; the total average degree, ; and , which stands for the average degree within a community, are fixed. Hence, and are given by where and . As grows, the network modularity increases [54], that is, the communities become easier to identify.