The latent cognitive structures of social networks

TL;DR

This work proposes a method for identifying cognitive structure across multiple network perceptions, analogous to how community detection aims to identify social structure in a network, and proposes a specific model instance and related statistical test for testing when there is social-cognitive agreement in a network.

Abstract

When people are asked to recall their social networks, theoretical and empirical work tells us that they rely on shortcuts, or heuristics. Cognitive Social Structures (CSS) are multilayer social networks where each layer corresponds to an individual's perception of the network. With multiple perceptions of the same network, CSSs contain rich information about how these heuristics manifest, motivating the question, Can we identify people who share the same heuristics? In this work, we propose a method for identifying cognitive structure across multiple network perceptions, analogous to how community detection aims to identify social structure in a network. To simultaneously model the joint latent social and cognitive structure, we study CSSs as three-dimensional tensors, employing low-rank nonnegative Tucker decompositions (NNTuck) to approximate the CSS--a procedure closely related to estimating a multilayer stochastic block model (SBM) from such data. We propose the resulting latent cognitive space as an operationalization of the sociological theory of social cognition by identifying individuals who share relational schema. In addition to modeling cognitively independent, dependent, and redundant networks, we propose a specific model instance and related statistical test for testing when there is social-cognitive agreement in a network: when the social and cognitive structures are equivalent. We use our approach to analyze four different CSSs and give insights into the latent cognitive structures of those networks.

Figures (13)

• Figure 1: In this work we analyze csspl as multilayer networks represented by an $N \times N \times N$ adjacency tensor (left). The frontal slices of the adjacency tensor are visualized in blue, yellow, and green. The $N$ frontal slices of the css are adjacency matrices representing the perception each person has of their network. We use the ntd to model the css with a multilayer sbm, which decomposes the adjacency tensor into latent social spaces and a latent cognitive space (right).
• Figure 2: A visualization and example showing the connection between the sbm, the ntd, and relational schema. (A) We assume that each person generates their perception of the network according to a stochastic block model (SBM). A person’s perception of the existence of an edge is drawn according to the rate specified by the affinity matrix in each person’s SBM. Previous empirical and theoretical work on how people store and recall large social networks suggests that a coarse model of relationship, like the SBM, well represents this cognitive process. (B) We could estimate a separate affinity matrix to describe each person’s network perception. (C) However, the NNTuck allows us to identify when people share the same generative process for their perceptions, interpretable as sharing the same relational schema.
• Figure 3: The test-AUC averaged across a tubular fivefold xval task for the Krackhardt advice (left) and friendship (right) css datasets. The pink and black lines correspond to the ntd model assumptions of cognitive redundancy and independence, respectively. Each other colored line corresponds to a different value of $C$ in assuming cognitive dependence in the css, and the x-axis corresponds to different choices of the social latent space parameter $K$. Based on this xval task, we choose to examine the social and cognitive factor matrices of the advice and friendship css datasets corresponding to the cognitively dependent ntd with $K=C=3$, and $K = 3,C= 5$, respectively.
• Figure 4: The latent social and cognitive spaces in the high tech firm from krackhardt1987, identified by estimating a cognitively dependent ntd of the advice css with $K=C=3$. The plotted network is of the network's consensus structure, with an edge shown if at least 50% of the network perceived its existence. Each node's position is determined by the departmental affiliation and hierarchy structure of the firm, where the person in the middle is the president, persons 1, 3, 17, and 20 are vice presidents, and the rest are supervisors. Each node is colored according to its proportional membership to each group, where a darker color denotes more proportional membership. We see that persons 5 and 16 belong mostly to the same cognitive space as the president, persons 0 and 13 belong mostly to the same cognitive space as person 14, and everyone else belongs to the third cognitive space.
• Figure 5: The latent cognitive space of the krackhardt1987 advice css, rewritten relative to the relational schema of the president of the company, the supervisor we refer to as person 14, and person 10. Each node is colored according to its proportional membership to each cognitive group, where dark pink denotes more membership. Note that, because this plot shows the cognitive membership of each node relative to persons 6, 14, and 10, person 6 (the president) has his entire membership in the first cognitive group, and persons 14 and 10 have their entire membership in the second and third cognitive groups, respectively.

Theorems & Definitions (8)

• Definition 1: Cognitively independent ntd
• Definition 2: Cognitively dependent ntd
• Definition 3: Cognitively redundant ntd
• Definition 4: Social-cognitive agreement ntd
• Definition 5: Cognitive independence
• Definition 6: Cognitive dependence
• Definition 7: Cognitive redundance
• Definition 8: Social-cognitive agreement