2.1. Latent space models of CSSs

From a modeling perspective, our work is similar to work which has aimed to identify latent structure in the perceptions of the network by considering the CSS data object in its entirety. In Kumbasar et al. (Reference Kumbasar, Rommey and Batchelder1994), the authors conduct a correspondence analysis of a $N^2 \times N$ “stacked” matrix representation of a CSS to find and compare the latent self-perceived and group-perceived social spaces of each person. In Sosa and Rodríguez (Reference Sosa and Rodríguez2021), the authors propose a three-dimensional extension of the latent space model for social networks developed by Hoff et al. (Reference Hoff, Raftery and Handcock2002). Their model identifies the self-perceived latent position of each individual, and the posterior probabilities of a Bayesian model are used to assess “whether the perception of an individual about its own position in social space agrees with the judgments of other actors.” The authors propose that this information can be used to create a weighted (single-layer) network, where the weight of an edge between persons $i$ and $j$ corresponds to how much cognitive agreement $i$ and $j$ share. Relatedly, in Sewell (Reference Sewell2019), the author proposes a latent space model which describes the deviations between individuals’ perceptions through bias and variance, explaining individuals’ tendencies to report more or less dense networks, and their confidence in the network perception, respectively. More recently, De Bacco et al. (Reference De Bacco, Contisciani, Cardoso-Silva, Safdari, Borges, Baptista, Sweet, Young, Koster, Ross, McElreath, Redhead and Power2023) proposed a latent model to uncover an unobserved underlying network from multiply reported network data (such as a CSS) by identifying latent propensities for each reporter to over or under report edges between others, as well as each reporter’s tendency to report reciprocated ties between people. These related papers which have focused on identifying latent spaces in a CSS set the precedent that we hope to continue here. In contrast to this previous work, however, the present work centers a latent space model which also describes a generative process of the varying perceptions, in order to identify who in the network shares relational schema.

2.2. Relational schema in network perceptions

From a conceptual perspective, our work is most similar to that of Janicik and Larrick (Reference Janicik and Larrick2005), Carnabuci et al. (Reference Carnabuci, Emery and Brinberg2018), Menon and Smith (Reference Menon and Smith2014), Kilduff et al. (Reference Kilduff, Crossland, Tsai and Krackhardt2008), Emery et al. (Reference Emery, Carnabuci and Brinberg2011), Brashears (Reference Brashears2013), and Brashears and Quintane (Reference Brashears and Quintane2015), which all aim to identify specific relational schema individuals use when recalling their social networks. In contrast to our proposed method, these prior works all identify specific relational schema to study the presence of in the social network, whereas we aim to identify people who share a general relational schema that can be interpreted and identified a posteriori. Janicik and Larrick (Reference Janicik and Larrick2005) specify two relational schema that influence social network perceptions: balance schema (Heider, Reference Heider1958) and linear-ordered schema (De Soto and Kuethe, Reference De Soto and Kuethe1959). Their work goes on to study how people use these schema to fill in incomplete parts of a social network. Carnabuci et al. (Reference Carnabuci, Emery and Brinberg2018) take an experimental approach to infer the relational schema that participants rely upon by measuring how learning rates change in conditions which assume different relational schema in the network. Similarly, Menon and Smith (Reference Menon and Smith2014) study how psychological priming impacts the density and characteristics of people’s perceived networks. Kilduff et al. (Reference Kilduff, Crossland, Tsai and Krackhardt2008) study multiple CSS datasets to understand how schemas like small-world assumptions (Milgram, Reference Milgram1967) and perceived popularity manifest themselves in CSSs. In Emery et al. (Reference Emery, Carnabuci and Brinberg2011), the goal is to identify how specific relational schemas impact the emergence of leadership in a social network. In Brashears (Reference Brashears2013), the author conducts experiments to understand how compression heuristics aid in the recall of large social networks, finding that people have better recall when structural assumptions of relational schema like kinship and triadic closure are accurate. In Brashears and Quintane (Reference Brashears and Quintane2015), the same experimental data is reanalyzed and the recall of triads and groups is compared to what is expected from exponential random graphs. We aim to enrich the study of these varied questions about cognitive structure. In contrast to this previous work, the operationalization we introduce identifies individuals with shared relational schema without necessitating which relational schema to identify ahead of time and enables us to contrast social structure with cognitive structure.

2.3. Multilayer extensions of the SBM

The model we propose for analyzing CSSs builds upon a diverse set of work on SBMs for multilayer networks. Most similarly to the NNTuck work of Aguiar et al. (Reference Aguiar, Taylor and Ugander2022) is that of Schein et al. (Reference Schein, Zhou, Blei and Wallach2016) and De Bacco et al. (Reference De Bacco, Power, Larremore and Moore2017). In De Bacco et al. (Reference De Bacco, Power, Larremore and Moore2017), a multilayer SBM is proposed by estimating a separate SBM for each layer of the network, keeping the social space fixed across layers. In Schein et al. (Reference Schein, Zhou, Blei and Wallach2016), the authors propose a Bayesian Poisson Tucker Decomposition (BPTD) as a generalization of the degree-corrected, mixed-membership SBM (dc-mm-SBM) (Ball et al., Reference Ball, Karrer and Newman2011) to study multilayer networks. The model proposed in De Bacco et al. (Reference De Bacco, Power, Larremore and Moore2017) is a specific instance of the NNTuck wherein the third factor matrix in the decomposition is constrained to be the identity matrix (see Definition 1 in Section 3 for more details). Whereas Schein et al. (Reference Schein, Zhou, Blei and Wallach2016) estimate the SBM with an MCMC algorithmic approach, the NNTuck multiplicative updates procedure seeks to maximize the log-likelihood to obtain a point estimate of the same model (see Section 3.1 and Algorithm 1 for more details). See Aguiar et al. (Reference Aguiar, Taylor and Ugander2022) for more discussion on how the NNTuck relates to and generalizes other multilayer SBMs.

2.4. Layer interdependence in multilayer networks

Viewing the CSS more generally, as a multilayer network, identifying structure in the cognitive space (which we propose to do with the NNTuck) is similar to identifying layer interdependence, for which there have been a multitude of proposed methods. Beginning in the CSS literature, Krackhardt (Reference Krackhardt1987) suggested differentiating layer similarity by comparing individual layers to a consensus structure. Battiston et al. (Reference Battiston, Nicosia and Latora2014) and Kao and Porter (Reference Kao and Porter2018) introduce and use various similarity measures to identify layer communities. De Domenico et al. (Reference De Domenico, Nicosia, Arenas and Latora2015) and De Domenico and Biamonte (Reference De Domenico and Biamonte2016) cluster similar layers using information-theoretic tools. In Stanley et al. (Reference Stanley, Shai, Taylor and Mucha2016), layers are categorized into groups that were drawn from the same SBM. In De Bacco et al. (2017), the authors build multiple models using different subsets of the layers and layer interdependence is determined via each models’ performance on a link prediction task. In contrast to this previous work, the NNTuck identifies layer interdependence by identifying layer communities which have the same underlying generative model. As such, the NNTuck allows for the adjacency tensor to be low rank in the layers as well as the nodes. Because identifying layer interdependence in a CSS with the NNTuck relates the varying perceptions to one another via a shared generative process (as opposed to this other work which uses similarity measures or information-theoretic tools), it is thus a natural operationalization of existing sociological theories.

3.1. The nonnegative Tucker decomposition

The degree-corrected, mixed-membership SBM (dc-mm-SBM) (Ball et al., Reference Ball, Karrer and Newman2011) is a generative model of a single-layer network that assumes each of the $N$ nodes in the network have soft (“mixed”) membership in each of $K$ different communities, and that an affinity matrix captures the Poisson rates at which nodes in each community form an edge with one another. More specifically, for a network of $N$ nodes represented by an adjacency matrix $A \in \mathbb{Z}_{0+}^{N \times N}$ , the dc-mm-SBM assumes that each node $i$ has a nonnegative outgoing membership vector, $\mathbf{u}_i \in{\mathbb{R}}_+^K$ , and a nonnegative incoming membership vector, $\mathbf{v}_i \in{\mathbb{R}}_+^K$ . Each membership vector represents the node’s soft assignment to $K\leq N$ groups when forming outgoing and incoming edges, respectively. The $K\times K$ nonnegative affinity matrix $\boldsymbol{{G}}$ describes the rates of connections between nodes in each communitiy. Then, for $N \times K$ matrices $\boldsymbol{{U}}$ and $\boldsymbol{{V}}$ with rows defined by $\mathbf{u}_i$ and $\mathbf{v}_i$ , the NNTuck assumes that $\boldsymbol{{A}}\sim \text{Poisson}({\boldsymbol{{U}}} \boldsymbol{{G}}{\boldsymbol{{V}}}^\top )$ . While the model allows for adjacency matrices $\boldsymbol{{A}}$ consisting of nonnegative count data, it is common practice to employ the dc-mm-SBM for model-based analyses of binary network data. Note that constraining all the elements of $\mathbf{u}_i, \mathbf{v}_i,$ and $\mathbf{G}$ to be nonnegative allows for these parameters to be interpretable as membership weights and affinities.

The nonnegative Tucker decomposition (NNTuck) as a generative model is a multilayer network extension of the dc-mm-SBM. dc-mm-SBM. It builds on related work on multilayer SBMs (e.g, Schein et al., Reference Schein, Zhou, Blei and Wallach2016; De Bacco et al., Reference De Bacco, Power, Larremore and Moore2017; Tarrés-Deulofeu et al., Reference Tarrés-Deulofeu, Godoy-Lorite, Guimera and Sales-Pardo2019) by generalizing the SBM to a multilayer setting and using tensor decompositions to jointly model the layers, much like how matrix decompositions underlie traditional (single-layer) SBMs.

Consider a multilayer network with $N$ nodes and $L$ layers represented by adjacency tensor ${\boldsymbol{\mathcal{A}}} \in \mathbb{Z}_{0+}^{N \times N \times L}$ . The NNTuck multilayer SBM again assumes that each node $i$ has nonnegative membership vectors $\mathbf{u}_i \in{\mathbb{R}}_+^K$ and $\mathbf{v}_i \in{\mathbb{R}}_+^K$ , representing the node’s soft assignment to $K\leq N$ groups when forming outgoing and incoming edges, respectively (note that for an undirected network, $\mathbf{u}_i = \mathbf{v}_i$ ). Furthermore, the NNTuck assumes that each layer $\ell$ has a nonnegative vector $\mathbf{y}_\ell \in{\mathbb{R}}_+^C$ describing the layer’s soft membership to each of $C\leq L$ communities. Just as matrices $\boldsymbol{{U}}$ and $\boldsymbol{{V}}$ in the dc-mm-SBM describe latent community structure in the nodes of single-layer networks, the factor matrix $\boldsymbol{{Y}}$ in the NNTuck describes latent structure in the layers of a multilayer network. Finally, the NNTuck assumes a nonnegative core tensor $\boldsymbol{\mathcal{G}} \in \mathbb{R}_{+}^{K \times K \times C}$ whose frontal slices are $C$ different affinity matrices. Let $\mathbf{u}_i, \mathbf{v}_i, \mathbf{y}_\ell$ be the rows of nonnegative matrices ${\boldsymbol{{U}}},{\boldsymbol{{V}}},$ and $\boldsymbol{{Y}}$ , respectively. Then the NNTuck multilayer SBM assumes that

(1) \begin{equation}{\boldsymbol{\mathcal{A}}} \sim \text{Poisson}(\boldsymbol{\mathcal{G}} \times _1{\boldsymbol{{U}}} \times _2{\boldsymbol{{V}}} \times _3 \boldsymbol{{Y}}). \end{equation}

When estimating this model from multilayer network data, maximizing the log-likelihood of observing $\boldsymbol{\mathcal{A}}$ under the model given by Eq. (1) is equivalent to minimizing the KL-divergence between $\boldsymbol{\mathcal{A}}$ and ${\boldsymbol{\mathcal{\hat{A}}}} = \boldsymbol{\mathcal{G}} \times _1{\boldsymbol{{U}}} \times _2{\boldsymbol{{V}}} \times _3 \boldsymbol{{Y}}$ . This equivalence is essential to why the NNTuck estimated by minimizing the KL-divergence is synonymous with the multilayer SBM.

3.2. Interpretation of the NNTuck of a CSS

Through its multiple factor matrices, the NNTuck multilayer SBM models both the latent social structure—sometimes referred to as the Blau space (Blau, Reference Blau1977; McPherson, Reference McPherson1983; McPherson and Ranger-Moore, Reference McPherson and Ranger-Moore1991)—and the latent layer structure of a multilayer network. When viewing a CSS as a multilayer network, each layer corresponds to a different individual’s perception of the social network. Therefore, the latent layer structure captured by the NNTuck of a CSS is precisely the structure identified among the varied perceptions of the network, what we hereafter refer to as the latent cognitive structure of the CSS. In the setting of CSSs, then, the NNTuck identifies three latent spaces in a network: the outgoing and incoming social groups in the $\boldsymbol{{U}}$ and $\boldsymbol{{V}}$ factor matrices, respectively, and the $\boldsymbol{{Y}}$ factor matrix, which identifies $C$ cognitive groups and each individual’s membership to each of these groups.

3.2.1. Relationship to social cognition and relational schema

Aiming to interpret this factor model through sociological theories, we will not review all relevant or adjacent sociological literature surrounding cognition, social representations, schema, or culture’s various impact on the three. While tempting, it is far beyond the scope of the present work. Instead, we focus on connecting the NNTuck to Baldwin’s (Reference Baldwin1992) theory of relational schema and Howard’s (Reference Howard1994) theory of social cognition.

A succinct summary of Baldwin’s theory of relational schema (emphasis our own) is that,

“[relational schema] describe expectations about the nature of relationships between people, defining what aspects of social interactions individuals will pay attention to, as well as the attributes of others that are meaningful within those interactions”(Brands, Reference Brands2013).

Relational schema describe the generalizations or patterns we rely on when making assumptions about relationships or when making new social connections. In data collection for CSSs, each person is asked to report on their perceptions of the relationships between every pair of people in their network. It has been empirically shown that people rely on compression heuristics when recalling large networks (Brashears, Reference Brashears2013), and it is reasonable to assume that the perceptions reported in CSSs are the product of some such compression heuristics. In Section 4.3, we use the NNTuck to understand the cognitive latent space of a longitudinal CSS collected repeatedly over several weeks, allowing us to investigate the heuristics involved (and not involved) in that early formation of the CSS for that population.

About social cognition, Howard writes,

“Social cognition articulates explicitly how social structures are carried in individuals’ mental systems …The content of the group dimensions on which people are prone to categorize reveals the connections between cognitive and social structures.”

CSS datasets provide empirical access to “how social structures are carried in individuals’ mental systems,” and we propose that the NNTuck, through the $\boldsymbol{{Y}}$ factor matrix, can identify specific ways in which any differences in these perceptions can be attributed to differences in such mental systems.

Because the NNTuck is an extension of the SBM to multilayer networks, it serves as a generative model of multilayer networks, where the role of the $\boldsymbol{{Y}}$ factor matrix is to identify layers which can be modeled using the same generative SBM. In the context of the CSS, the $\boldsymbol{{Y}}$ factor matrix identifies individuals in the network who rely on the same generative process when recalling their perception of the network. As such, we propose that individuals in the same cognitive group can be thought of as having a shared set of relational schema. In this sense, too, the NNTuck provides an explicit operationalization of Howard’s theory of social cognition, because it jointly models the connection between the cognitive and social structure in the network. In Section 4, we interpret possible relational schema underlying the shared cognitive spaces in three empirical datasets, and in Section 4.3, we directly tie our analysis using the NNTuck to Howard’s theories. For a visualization of the connection between the NNTuck and relational schema, see Figure 2.

Figure 2. A visualization and example showing the connection between the stochastic block model (SBM), the nonnegative Tucker decomposition (NNTuck), 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.

To formalize our interpretation of the NNTuck in terms of established social theories, we now turn to possible different assumptions of the structure and dimension of the “layer” factor matrix $\boldsymbol{{Y}} \in{\mathbb{R}}^{L \times C}$ in the NNTuck of a CSS. The following definitions apply specifications for multilayer network models from Aguiar, et al. (Reference Aguiar, Taylor and Ugander2022) to the CSS context. They form the basis of our primary exploration of empirical CSS data in Section 4.

Definition 1 (Cognitively independent NNTuck). A cognitively independent NNTuck is a nonnegative Tucker decomposition where $C = L$ and $\boldsymbol{{Y}}$ has the constraint $\boldsymbol{{Y}} = \boldsymbol{{I}}$ , meaning each individual in the network relies on a distinct relational schema.

Definition 2 (Cognitively dependent NNTuck). A cognitively dependent NNTuck is a nonnegative Tucker decomposition where $\boldsymbol{{Y}}$ has the constraint $C\lt L$ , meaning each individual’s social cognition can be described by a mixture of $C$ relational schema.

Definition 3 (Cognitively redundant NNTuck). A cognitively redundant NNTuck is a nonnegative Tucker decomposition where $C=1$ and we constrain $\boldsymbol{{Y}}$ to be the ones vector, $\boldsymbol{{Y}} = [1, \dots, 1]^{\top }$ , meaning each individual in the network relies on the exact same relational schema in perceiving their network.

3.3. Social-cognitive agreement

In addition to the above model specifications, we next propose a new specification that is particularly well suited to the CSS context. Considering that the NNTuck of a CSS can tell us both about the latent social and cognitive structures, a natural question to ask is how the two spaces are related to one another. Do our social surroundings directly influence how we conceptualize our social network, does our conceptualization influence how we socialize, and if so, can we empirically assess the ways in which they do?

The hypothesis that our cognitions are influenced by our peers was proposed in Carley (Reference Carley1986), where a main goal of the work was to “relate cognitive structure to social structure at an empirical level.” As Carley writes, “the social and cognitive processes cannot be decoupled.” Indeed, through a combination of quantitative and qualitative data collection, Carley identifies specific tightly knit groups in a community who share similar cognitive mappings of a concept. A similar idea was later echoed in Freeman (Reference Freeman1992), where the author writes, “the individuals involved in any particular local community would be expected ultimately to produce very similar mental images of group structure in that community.”

These related hypotheses have a natural analog to a specification of the NNTuck. Namely, the social structure is the same as the cognitive structure if ${\boldsymbol{{U}}}=\boldsymbol{{Y}}$ . We refer to this structural assumption as one of social-cognitive agreement.

We note that constraining $\boldsymbol{{Y}}$ and $\boldsymbol{{U}}$ to be the same does not imply that everybody in the same social/cognitive group will have the exact same perceptions. Rather, the model is constrained so that the structure we observe in the social space is also reflected in the structure of the cognitive space. The implication of this constraint is that the perceptions of people in the same social/cognitive space are modeled with the same generative model, which will result in similar (but not the same) perceived networks.

3.4. Model optimization details

Algorithmically, our primary method for maximizing the log-likelihood of observing a CSS dataset under a given NNTuck model specification is to employ the multiplicative updates algorithm of Kim and Choi (Reference Kim and Choi2007), an effective extension of the multiplicative updates algorithm for nonnegative matrix factorization (NMF) given by Lee and Seung (Reference Lee and Seung2001). For more details, see Algorithm 1 in Appendix B or Aguiar et al. (Reference Aguiar, Taylor and Ugander2022). The Kim and Choi (Reference Kim and Choi2007) algorithm, like the NMF algorithm it builds on, is guaranteed to find a local maxima of the nonconvex log-likelihood, but not necessarily the global maxima. That said, extensive empirical evaluations have shown it to perform very well when employed using best practices for nonconvex optimization (e.g., choosing the best of many random restarts of the optimization routine).

For the cognitively independent, dependent, and redundant NNTuck model assumptions (Definitions 1 to 3), this algorithm transfers easily to the relevant specifications. Briefly, when constraining $\boldsymbol{{Y}} ={\boldsymbol{{I}}}$ or $\boldsymbol{{Y}} = \mathbf{1}$ as in the cognitively independent and redundant NNTucks, respectively, the algorithm maintains monotonic convergence to a local minimum by simply initializing these constraints and never updating $\boldsymbol{{Y}}$ after initialization. Similarly, when constraining ${\boldsymbol{{U}}} ={\boldsymbol{{V}}}$ , as in the case of an undirected network, by initializing both ${\boldsymbol{{U}}}={\boldsymbol{{V}}}$ and symmetry in the frontal slices of $\boldsymbol{\mathcal{G}}$ , monotonic convergence to a local minimum is maintained. The symmetric structure in the frontal slices of adjacency tensor $\boldsymbol{\mathcal{A}}$ (as is the case in an undirected network) and core tensor $\boldsymbol{\mathcal{G}}$ (which is an interpretable assumption for an undirected network) result in equivalence between the multiplicative update derived for $\boldsymbol{{U}}$ and the multiplicative update derived for $\boldsymbol{{V}}$ . Thus, by simply initializing a symmetric $\boldsymbol{\mathcal{G}}$ and ${\boldsymbol{{U}}}={\boldsymbol{{V}}}$ , Algorithm 1 maintains this symmetry and equivalence.

In the case of estimating a SCA NNTuck (Definition 4), however, constraining $\boldsymbol{{Y}} ={\boldsymbol{{U}}}$ is a nontrivial constraint on the multiplicative updates algorithm. The structure in both the data $\boldsymbol{\mathcal{A}}$ and core tensor $\boldsymbol{\mathcal{G}}$ that would be necessary in order to ensure equivalence in the multiplicative updates of $\boldsymbol{{Y}}$ and $\boldsymbol{{U}}$ (as we noted for the symmetric network specification above), are contextually unreasonable to expect or assume. Specifically, equivalence in the multiplicative updates for $\boldsymbol{{Y}}$ and $\boldsymbol{{U}}$ requires that the 1-unfolding and the 3-unfoldingFootnote ¹ of $\boldsymbol{\mathcal{A}}$ are equivalent ( $\boldsymbol{{A}}_{(1)}=\boldsymbol{{A}}_{(3)}$ ) and that the 1-unfolding and the 3-unfolding of $\boldsymbol{\mathcal{G}}$ are equivalent ( $\boldsymbol{{G}}_{(1)}=\boldsymbol{{G}}_{(3)}$ ). In the context of the CSS, this structural assumption on $\boldsymbol{\mathcal{A}}$ means that, for every pair of people $i$ and $j$ , person $i$ ’s perception of $j$ ’s outgoing relationships is the same as person $j$ ’s perception of person $i$ ’s outgoing relationships. In order for this assumption to hold, each person would need to report their own outgoing relationships as their perceptions for everyone else’s outgoing relationships (that they assume everybody goes to the exact same people for advice/friendship).

Thus, in order to estimate a SCA NNTuck, seemingly minor changes to the Kim and Choi (Reference Kim and Choi2007) algorithm nontrivially change its behavior. In considering other algorithmic approaches, Cambre et al. (Reference Cambre, De Lathauwer and De Moor1999) (and more recently Jin et al. (Reference Jin, Kileel, Kolda and Ward2022)) develop and discuss a method for estimating a symmetric Tucker decomposition (with no nonnegativity constraint) wherein all three factors ( ${\boldsymbol{{U}}},{\boldsymbol{{V}}}, \boldsymbol{{Y}}$ in our vocabulary), are constrained to be equivalent. However, in this and other related work, the focus is on developing algorithms for decomposing a data tensor which already has an appropriately symmetric structure. Such data symmetry is something that, again, is not reasonable to assume in the case of CSSs.

As such, we move forward with altering Algorithm 1 to accommodate the SCA constraint, resigning ourselves to a estimation algorithm without guaranteed monotonic convergence to a local optima. That said, this was only ever a weak guarantee relative to guarantees of reaching a global optima, as enjoyed by many optimization procedures (e.g., for convex log-likelihoods).

We alter the multiplicative updates from Kim and Choi (Reference Kim and Choi2007) in the following ways. First, we initialize nonnegative factors $\boldsymbol{{Y}},{\boldsymbol{{V}}}$ , and nonnegative core tensor $\boldsymbol{\mathcal{G}}$ . Then, we set ${\boldsymbol{{U}}} = \boldsymbol{{Y}}$ . For each multiplicative iteration, we proceed by first updating $\boldsymbol{{V}}$ , then updating $\boldsymbol{{U}}$ , and then updating $\boldsymbol{{Y}}$ , all according to the same multiplicative updates as in Algorithm 1. As our key modification, after updating $\boldsymbol{{Y}}$ and before updating $\boldsymbol{\mathcal{G}}$ , we set both $\boldsymbol{{U}}$ and $\boldsymbol{{Y}}$ to be equal to their average. This step ensures that the model remains “feasible” within the constrained model class, essentially updating both factor matrices to reflect the gradients in both roles (as $\boldsymbol{{U}}$ and $\boldsymbol{{Y}}$ factors). We then update $\boldsymbol{\mathcal{G}}$ as usual. As a final minor modification, given the lack of monotonicity guarantee, Algorithm 2 also features a modified termination criteria to ensure the return of the minimum solution found during a given solution trajectory. We describe this approach in full in Algorithm 2 in Appendix B, and provide a python implementation in our code repository (see our Data Availability Statement). In Appendix B, we also provide further details on our decision to choose this modification over others.

This heuristic optimization procedure for returning a maximum likelihood estimate of an NNTuck with SCA is provided as an initial feasible approach to the question of modeling SCA in CSSs. It is an interesting open question if the multiplicative updates algorithm of Kim and Choi (Reference Kim and Choi2007), or some other algorithm, can be modified to produce a maximum likelihood estimate with local or global optimality guarantees. For example, Arora et al. (Reference Arora, Ge, Kannan and Moitra2012) showed that under “approximate separability” (Donoho and Stodden, Reference Donoho and Stodden2003) assumptions on a data matrix, the NMF has a guaranteed solution and an associated convex optimization problem. Their proposed algorithm overcame prior limitations of heuristics for approximating an NMF by minimizing the Frobenius norm error of the approximation. Estabilishing identifiability and convergence conditions for the nonnegative Tucker decomposition has been addressed in, e.g., Xu (Reference Xu2015) and Sun and Huang (Reference Sun and Huang2023). In Xu (Reference Xu2015), the author proposes an algorithm for estimating the nonnegative Tucker decomposition with global convergence guarantees, but does so for minimizing only the Frobenius loss. In Sun and Huang (Reference Sun and Huang2023), the authors propose a regularization that makes the nonnegative Tucker decomposition identifiable, but the associated loss based on KL-divergence is nonconvex. Extending this work to estimating the nonnegative Tucker decomposition (and the nonnegative Tucker decomposition with SCA) with global convergence guarantees under loss given by the KL-divergence is promising future work.

3.5. Statistical tests for cognitive structure

The above structural assumptions are all articulated in terms of constraints on the parameters of the given models, where the constrained models all lie within a subspace of the more general (cognitively independent) model. As such, these assumptions are highly testable from data using likelihood ratio tests (LRTs). We adapt three statistical tests from Aguiar et al. (Reference Aguiar, Taylor and Ugander2022) to have vocabulary specific to the interpretation of the CSS and introduce an additional statistical test for SCA, all for studying the cognitive structure of a social network through its CSS.

The use of the standard LRT, as in the analogous tests for layer interdependence in Aguiar et al. (Reference Aguiar, Taylor and Ugander2022), relies on Wilks’ Theorem (Wilks, Reference Wilks1938) and its assumptions. Wilks’ Theorem provides the standard theory for the asymptotic validity of LRTs. In the context of statistically comparing different NNTuck models, we do not satisfy the necessary assumptions or regularity conditions in order for Wilks’ Theorem to hold: the nonnegative Tucker decomposition is neither identifiableFootnote ² (Kolda and Bader, Reference Kolda and Bader2009; Chen et al., Reference Chen, Moustaki and Zhang2020), nor can we be certain that our estimation of the NNTuck has reached a global maximum of the log-likelihood. Because the NNTuck is not identifiable, the concept of which models are nested within one another is subtle, as one can rewrite the NNTuck by absorbing the parameters of the factor matrices into the core tensor, or vice versa. As such, the notion of nested models and the difference in parameters (and subsequently, determining the degrees of freedom of the $\chi ^2$ distribution for the standard LRT) is nuanced, especially in the context of a three-mode tensor decomposition.

As a means of addressing the first shortcoming, we utilize the split-LRT from Wasserman et al. (Reference Wasserman, Ramdas and Balakrishnan2020), which does not require identifiability or nested models. Although the split-LRT does not require the same strict regularity conditions as does the standard LRT, it does lead to lower powered tests. Notably, although the power deficit of the split-LRT is somewhat inherent in the test, it may be exacerbated in our particular setting due to our sample size and the presence of many nuisance parameters in our modelFootnote ³ (Tse and Davison, Reference Tse and Davison2022). In Tse and Davison (Reference Tse and Davison2022), Strieder and Drton (Reference Strieder and Drton2022), and Spector et al. (Reference Spector, Candes and Lei2023), the authors propose ways to improve the power when testing statistical models with nuisance parameters. For instance, both Tse and Davison (Reference Tse and Davison2022) and Strieder and Drton (Reference Strieder and Drton2022) discuss the importance of an optimal splitting of the data when conducting the split-LRT. In the context of the CSS, wherein we are splitting the data of a 3-mode tensor, we have many choices for how to do so, and our naïve approach of splitting the data at random may not be optimal. Determining the optimal split of a data tensor and implementing other suggestions for increasing the power of the split-LRT in the context of the NNTuck is a topic for future work.

To address the second shortcoming, of reaching a global maximum, we use the NNTuck corresponding to the highest log-likelihood over multiple initializations of Algorithm 1. However, when estimating a SCA NNTuck we use Algorithm 2, which, as discussed in the previous section, is not monotonic in KL-divergence nor does it come with any convergence guarantees, even to a local minima. As such, we caution against strong interpretation of the results of the split-LRT when testing for SCA and again highlight addressing this algorithmic limitation as subject of future work. For more discussion on the standard and split-likelihood ratio tests, see Appendix C.

3.6. Modeling limitations

We conclude this section by discussing some limitations of using the NNTuck for studying CSS datasets. First, the NNTuck models the CSS using a multilayer stochastic block model. As such, the latent cognitive space identifies sets of perceptions of the network can be well modeled by the same (degree-corrected, mixed-membership) SBM. It is in this sense that we say the NNTuck identifies people who share relational schema, because their perceptions can be modeled by same generative process. A significant focus of the active literature on SBMs works to enrich their modeling structure to more realistically model empirical networks, noting that the dc-mm-SBM is still quite coarse (e.g, (Peixoto, Reference Peixoto2014)). However, as we discuss above, both theoretical social theory as well as experimental and empirical studies on how people store and recall large social networks suggests that a coarse model of relationship well represents this cognitive process (see also, e.g., Mullainathan et al., Reference Mullainathan, Schwartzstein and Shleifer2008). In the context of CSS datasets, wherein each person is asked to report their perception of the social network, we believe that the coarse reasoning expressed in the SBM models this situation well.

Although the NNTuck does not tell us what the relational schema are, we believe that the identified spaces can be contextually interpreted (as we do in the empirical applications below). This modeling approach allows for flexibility in identifying people who share relational schema and interpreting that schema a posteriori, as opposed to deciding a priori which schema to study in a given dataset, which other work on relational schema necessitates (e.g., Kilduff et al., Reference Kilduff, Crossland, Tsai and Krackhardt2008). However, there may be relational schema that may not be well described using a stochastic block model, and latent space models which support these schema, or other generative models of multilayer network formation, would be better suited for modeling such cases. For example, a relational schema based on the assumption of triadic closure would likely be better modeled by a multilayer extension of the recent SBM developments presented in Peixoto (Reference Peixoto2022).

With these limitations in mind, we turn to four empirical applications of the NNTuck.

4.1. Krackhardt advice CSS

For this CSS with $N=21$ managers, the test-AUC under different model specifications is shown in Figure 3 (left). We choose to examine the factor matrices corresponding to the cognitively dependent NNTuck (see Definition 2) with $K=3$ and $C=3$ . We choose these parameters over others with a higher test-AUC due to the observation that higher values of either $K$ or $C$ result in a minimal increase in test-AUC.Footnote ⁵ We perform a LRT to compare the goodness-of-fit of the nested, cognitively dependent NNTuck with $K=C=3$ , to the full cognitively independent NNTuck. In doing so, we fail to reject the null hypothesis that the data was generated from the smaller cognitively dependent model (see Table 1).

Table 1. The determinations from the split-LRT for the Krackhardt and Hunter CSS datasets. See Appendix C for a discussion on the likelihood ratio test (LRT), for motivation for why we use the split-LRT as opposed to the regular LRT, and to compare these determinations to those of the standard LRT . Of note is that the split-LRT suggests that both the Krackhardt Friendship CSS and the last week of the Hunter Friendship CSS are well modeled with social-cognitive agreement (SCA). The other two CSS datasets are better explained when we allow for the social space to differ from the cognitive space

Figure 3. The test-AUC averaged across a tubular fivefold cross-validation task for the Krackhardt advice (left) and friendship (right) CSS datasets. The pink and black lines correspond to the NNTuck 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 cross-validation task, we choose to examine the social and cognitive factor matrices of the advice and friendship CSS datasets corresponding to the cognitively dependent NNTuck with $K=C=3$ , and $K = 3,C= 5$ , respectively.

We inspect the latent spaces identified by this NNTuck in Figure 4. We see that the groupings of the social space of the advice network can be mostly attributed according to departmental affiliation within the firm: the three social groups mostly correspond to the left, bottom, and right, departmental groupings (visualized here in separate clusters), whereas the president (person 6) doesn’t strongly belong to any of the three groups. The exception to this is the departmental grouping that we view in the upper center of the network visualization, containing nodes 9, 10, and 17, which doesn’t clearly belong to any of the three social groups. The cognitive space, however, seems to group employees according to different attributes.

Figure 4. The latent social and cognitive spaces in the high tech firm from Krackhardt (Reference Krackhardt1987), identified by estimating a cognitively dependent NNTuck 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.

As is, these cognitive groupings aren’t entirely interpretable. To further interpret the identified cognitive space, we can rewrite the $\boldsymbol{{Y}}$ factor matrix and the core tensor in the basis of $C=3$ individuals in the network. We choose the three individuals according to the heuristic proposed in Aguiar et al. (Reference Aguiar, Taylor and Ugander2022), where the aim is to choose $C$ layers such that the corresponding rows of $\boldsymbol{{Y}}$ are (nearly) linearly independent. Doing so, we identify person 6, person 14, and person 10 as the reference perspectives to consider. We transform $\boldsymbol{{Y}}$ such that the $C=3$ frontal slices of $\boldsymbol{\mathcal{G}}$ correspond exactly to the affinity matrix from which persons 6, 14, and 10 generate their perceptions, respectively. Thus, the $6th$ , $14th$ , and $10th$ rows of the transformed $\boldsymbol{{Y}}^*$ matrix will be $[1,0,0]$ , $[0,1, 0]$ , and $[0,0,1]$ , respectively, and all other rows will represent each individual’s relational schema relative to these three people. We inspect this transformed $\boldsymbol{{Y}}^*$ matrix, which identifies the relative cognitive space, by plotting each individual’s proportional cognitive membership in Figure 5.

Figure 5. The latent cognitive space of the Krackhardt (Reference 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.

Whereas we know that person 6 is the president of the firm, all we nominally know about person 14 is that he is a supervisor of a department, and that person 10 is a long-standing employee. Digging further into the original analysis of this dataset, however, we find that person 14 is identified as having a particularly interesting perception of his advice network (the friendship network is not discussed). Quoting from Krackhardt (Reference Krackhardt1987) (emphasis our own),

Thus, the relational schema that the 21 different individuals in this firm rely upon when recalling the advice network can be concisely described by the relational schema of just three distinct people: the president of the firm, this supervisor with an overly optimistic view of his surrounding advice network, and someone who seems to represent the remaining employees. We continue to analyze this network and these interpretations in the next section, as we explore and contrast the social and cognitive spaces of the friendship network.

4.2. Krackhardt friendship CSS

We turn now to analyze the same firm as above, with the CSS now representing perceptions of the friendship relationships. In modeling this CSS, the test-AUC for different model specifications suggests using $K=3$ , $C=5$ as preferable to $K=3$ , $C=3$ , what we used to analyze the advice network. The former has a much lower test-AUC (see Figure 3, right). We see that for these values of $K=3$ and $C=5$ , the standard LRT fails to reject the null hypothesis that the CSS is explained by the simpler cognitively dependent NNTuck.

We inspect the social and cognitive spaces with this model specification in Figure 6. We observe that this decomposition identifies a similar three-dimensional social space to that identified in the advice CSS above. Again, the three social dimensions mostly identify three different department affiliations within the firm. Differently from the advice network, however, we see that the president (person 6) almost entirely belongs to the third social space, whereas in the decomposition of the advice CSS, his social membership was mostly spread across all three groups. Additionally, we see that person 7’s social membership does not correspond to his departmental affiliation, but to that of another department. Whereas we lack ethnographic data that might support these friendship alliances, it is curious to wonder how and why these two people belong to different social groups when considering relationships defined by friendship as opposed to advice.

Figure 6. The latent social and cognitive spaces in the friendship CSS of the high tech firm from Krackhardt (Reference Krackhardt1987), identified by estimating a cognitively dependent NNTuck.

In comparing the identified cognitive spaces in the advice and friendship CSSs for this firm, we see that, again, person 6 and person 5 belong to the same cognitive space. We also observe again that person 14 is identified as belonging to a different cognitive space than most of the others. Although the friendship CSS is not discussed in Krackhardt (Reference Krackhardt1987), it is reasonable to assume that person 14 might have been just as overly optimistic in reporting his perception of the friendship network as he was in the advice network. Why is his relational schema, which sets him apart from many others in the firm, so notably different from that of his colleagues? While there are any number of reasons that could describe this (internal firm politics, his socioeconomic status, his upbringing), it’s also possible that he is singled out by the NNTuck because of the noise that he contributed to the CSS by exaggerating connections in his perceived network. In this sense, the NNTuck has the potential to identify a cleaner model of the data, if we were to remove the noisy perceptions.

We see that using the NNTuck to understand the advice and friendship CSSs gives us a much cleaner and richer framework for analyzing these social networks. The original data analysis of this firm was able to identify person 14 as someone who had a noticeably abnormal perception of his social network by comparing his perception of the advice network to an aggregation of his coworkers’ perceptions. The NNTuck, however, and the sociological theories that it builds upon and operationalizes, gives us the vocabulary to say that he has a different relational schema than most of his coworkers. Likewise, for the advice network, we are able to identify two other groups with shared relational schema, as well as visualize the employees who share relational schema across the groups (for instance, person 8 partially shares the relational schema of both person 14 and person 10). Furthermore, we are able to visualize how both the social and cognitive spaces change when considering the advice network in contrast to the friendship network.

A clear limitation of this present analysis is the lack of data about the individuals in this organization: each person is a manager in a high tech firm in the pacific northwest; we don’t know much about the internal dynamics of the company; each person is a male aged between 27 and 59. According to Howard (Reference Howard1994), “Gender, race, and age are social systems of differentiation that are especially prone to cognitive categorization.” In this present dataset, where each individual has the same gender and race, we are unable to confirm or refute whether the latent cognitive space identifies these relational schema.

4.3. Hunter friendship CSS

We focus the analysis of this rich dataset by contrasting the identified social and cognitive spaces of week one and week six of the longitudinal study.

4.3.1. Week one

When considering the predictive power of various NNTuck models of the week one CSS, we see that the cognitively dependent NNTuck with $K=2$ and $C=3$ performs just as well as the cognitively independent NNTuck (see Figure 7, left). However, the LRT comparing these two models fails to reject the smaller, cognitively dependent, model (see Table 1). Considering this alongside the observation that the two models have nearly identical predictive power and similar log-likelihoods, we choose to examine the cognitively dependent model further.

Figure 7. The test-AUC averaged across a tubular fivefold cross-validation task for the Hunter friendship week one (left) and week six (right) CSS datasets. We choose to examine the social and cognitive factor matrices of both CSS datasets corresponding to the cognitively dependent NNTuck with $K = 2$ and $C=3$ .

We inspect the identified social and cognitive space, as well as the relative cognitive space, in Figure 8. Notably, we see that during week one of the course, the students’ social space can largely be described by their self-identified gender. Similarly, the identified cognitive spaces are well aligned with gender identity, where we see the first group is mostly one gender and the second group is mostly another gender. Interestingly, the third identified cognitive group, which mostly includes students 3 and 6, has students of both gender identities. While we do not have enough data at hand to understand why these students have different relational schema from the others with their same gender identity, our analysis provides an entry point for further analysis.

Figure 8. The latent social and cognitive spaces in week one of the Hunter Friendship CSS, identified by estimating a cognitively dependent NNTuck with $K=2$ and $C=3$ . In the first row, we plot the gender and race identifiers for each of the 20 students and in the last row we plot the cognitive space relative to the relational schema of persons 0, 6, and 18. Note that both the social and cognitive spaces in week one align well with the self-identified gender of each student. The plotted network is of the network’s locally aggregated structure, with an edge shown from node $i$ to $j$ if node $i$ perceived its existence.

4.3.2. Week six

The test-AUC for different model specifications describing the last week of the friendship CSS lead us to inspect the NNTuck with $K=2$ and $C=3$ (see Figure 7, right). With this model choice we fail to reject the null hypothesis that the advice CSS was generated from a cognitively dependent NNTuck, and we explore this dependent NNTuck in further detail. Figure 9 shows the identified social and cognitive spaces.

Figure 9. The latent social and cognitive spaces in the last week of the college leadership course friendship network from Hunter (Reference Hunter2019), identified by estimating a cognitively dependent NNTuck with $K=2$ and $C=3$ . In the last row, we also plot the cognitive space relative to students 9, 2, and 10. To visualize the differences in the cognitive spaces of these students, see Appendix D.

The social spaces that are identified in the NNTuck estimation are not easily explained given the metadata we have (see Figure  A1 in A for visualizations of available student attributes, including study groups, undergraduate institutions, and declared majors). Even without any external context about the students in each of the identified social and cognitive spaces in week six, the longitudinal nature of this dataset provides unique opportunities for other rich insights. Notably, we can learn a lot about both the students and the course by comparing these spaces to those identified in week one. We first observe that the identified social and cognitive spaces no longer correspond with the self-reported gender identity of the students. Secondly, we note that students 3 and 6, who shared relational schema in the first week, now have different relational schema, with person 3 mostly belonging to the first cognitive space and person 6 mostly belonging to the last cognitive space. While there are many possible interpretations for the social and cognitive spaces identified in the last week (see Appendix A for one interpretation), the most valuable observations would come from a contextual analysis of these findings done directly by the researchers (see, for example, Carley, Reference Carley1986, wherein the author compares the findings of a community detection algorithm with her own ethnographic observations of the study participants.). To develop an understanding of the differences between the cognitive spaces identified in the last week of this CSS, we consider two different visualizations in Appendix D which display different ways to understand the differences in the generative processes identified by each cognitive space.

Overall, the above observations have the potential to say a lot about this classroom setting, the instructional materials, and the students enrolled in this course. The course began with a relatively clear divide along gender lines, both in the social environment and the students’ relational schema. Again, we quote Howard (Reference Howard1994) in her theory of social cognition wherein she notes that gender is one of the “social systems of differentiation that are especially prone to cognitive categorization.” Furthermore, Howard offers additional theory about why these initial relational schema occurred along gender lines, writing that “systems of classification must be relatively simple in order to provide a beginning place for interaction. This interactional requirement generalizes the principle of cognitive efficiency: interaction is easier if there are fewer cognitive distinctions to consider … the constant use, in interaction, of dichotomies such as gender may help to keep them simple.”

Considering the last week of the course, however, after 6 weeks of instructional material, both the social and cognitive spaces became more complicated across identifiers. Whereas the students were prone to this easy heuristic of cognitive categorization—gender—in the first week of the course, we see richer and more complicated relational schema emerge by the end of the study. Again, in Howard’s (Reference Howard1994) introduction of the theory of social cognition she writes, “if schemas are to be sustained and reproduced over time …they must be validated by the accumulation of resources that their enactment engenders. Schemas and resources constitute structure only when they imply and sustain each other over time.” It could be argued that this departure from a gender-informed schema indicates the success of this particular course: that the students’ initial schema was not validated and engendered.