2.1. Scale-free networks: Hubs and disease transmission

Scale-free networks are characterized by power-law degree distributions, where a small number of highly connected hubs disproportionately influence overall network dynamics [3,4,14]. These hubs are key to disease transmission, as they serve as central nodes that facilitate the rapid spread of infections [15].

Degree-based assortative mixing, namely, the tendency for nodes with similar degrees to connect, is a common feature of social networks [16]. This pattern entails a dense core of highly interconnected hubs that significantly influence disease dynamics. In networks characterized by assortative mixing, hubs tend to connect with other hubs, intensifying transmission within the network. Targeting such hubs for interventions has been shown to reduce infection rates and mitigate disease spread [17].

Scale-free network models emphasize global connectivity through hubs and often downplay the importance of local dynamics. Real-world disease transmission involves both global and localized processes. For example, rapid transmission within closely knit groups can occur due to clustering. Consequently, scale-free models may inadequately represent the dynamics between global hub-driven spread and localized outbreaks.

2.2. Small-world networks: Balancing local and global transmission

In small-world networks, real-world contact patterns are effectively represented through the integration of global connectivity with local clustering dynamics [5]. This balance is achieved through the implementation of densely connected local clusters linked by “shortcuts” that bridge otherwise distant parts of the network [18]. These shortcuts enable the rapid spread of disease across the entire network, whereas the high level of clustering within the network means that most infections still occur locally [18].

Although valuable for understanding disease spread, small-world networks may not completely account for the influence of community structure on epidemic dynamics. Research has demonstrated that community structure plays a key role in shaping infection patterns within networks [19]. Individuals who connect otherwise separate communities are shown to be important for disease transmission. Targeting these bridging individuals for immunization can substantially increase intervention effectiveness, particularly in networks with strong community structures. Combining modularity analysis with small-world network models could provide a more comprehensive approach to understanding and managing disease transmission.

2.3. Modularity: The role of communities and containment

Modularity, also referred to as community structure [6, 7], is a key concept for understanding disease transmission in networks. It quantifies the extent to which a network is partitioned into densely connected communities with relatively sparse connections between them [6,20]. These network boundaries, defined by the limits of social groups or communities, shape disease transmission dynamics by balancing localized containment with the potential for broader disease spread. Dense intracommunity ties allow infection to spread easily within a community, acting as an incubator [21], whereas sparse intercommunity links can hinder spread between communities [22].

In highly modular networks characterized by strong subgroup cohesion and fragmentation, the spread of infection across community boundaries can be hindered, potentially slowing or even containing outbreaks [23]. However, these protective effects can be offset by global shortcuts, where long-range connections bridge otherwise disconnected communities [23]. Intercommunity edges are essential for infection to reach other communities, acting as conduits for wider dissemination or enabling global spread, as seen in a spatial modular network [21,24,25].

In addition to basic dynamics, studies of the epidemic threshold indicate complexities in modular networks. Investigations into the epidemic threshold have revealed various impacts. Some studies suggest that, under otherwise identical conditions, the epidemic threshold (to first order) remains unchanged by community structure [26, 27]. However, the effects of community structure on epidemic dynamics can vary on the basis of specific network features, such as the interplay between clustering and the degree distribution, as well as the conditions of balanced communities [26,27]. Analyses of spatial models identify distinct local and global thresholds [25], and empirical studies have shown significant increases, primarily in extremely modular networks [23]. Regardless, the mesoscopic community structure is essential for predicting post-invasion dynamics [24].

An epidemiological study on HIV transmission among intravenous drug users in New York City illustrated how network structure influences disease dynamics [28]. This study identified smaller, interconnected subgroups of uninfected individuals within a larger network that included both infected and uninfected individuals. These subgroups, while not entirely isolated, had limited connections to the wider network. This limited connectivity potentially restricted HIV spread. If a new infection emerged, the outbreak was likely contained within the immediate contacts of the newly infected person in the subgroup. Longer-term HIV-infected individuals on the periphery of these islands acted as “firewalls” owing to their lower infectivity, potentially restricting the spread of HIV from newly infected individuals with high infectivity. This example illustrates how individual traits (e.g., infectivity) and network position interact to influence disease spread within a modular network.

Although this study focused on local dynamics within modular components [28], it did not explore how connections with a broader network might influence disease spread. Building on this work, small-world theory suggests that even a few long-range links can facilitate rapid transmission across communities [29], explaining how long-range connections can drive the broader spread of a disease [5,30]. More specifically, an infected individual bridging an isolated subgroup to the broader network could act as a small-world shortcut, triggering rapid transmission within previously protected communities. This implies how long-range connections can disrupt the containment provided by modularity, balancing local cohesion with global connectivity.

In addition to the influence of long-range links, assortative mixing, whether by grouping individuals with similar numbers of contacts or clustering those with shared traits or behaviors, plays a key role in influencing disease dynamics. It creates clusters within networks that experience varying levels of disease prevalence compared with the general population [18,31]. These clusters often serve as focal points for localized outbreaks and influence epidemic trajectories.

2.4. Multilayer approaches: Progress and challenges in transmission modeling

Building on foundational network models, this subsection explores multilayer frameworks developed to account for the multiscale complexity in real-world social interactions and their heterogeneous contact patterns [27]. Hierarchical metapopulation models represent populations as nested systems of distinct contexts, such as local settings (e.g., hospitals, schools, or workplaces, organized within broader communities such as neighborhoods, cities, and national regions). The movement of individuals between these levels is essential for widespread pathogen diffusion [30].

Extending this concept, multilayer networks represent diverse forms of interactions as separate layers, including connections both within (intralayer) and across (interlayer) these layers [20]. For example, empirical studies have shown that age-specific social preferences lead to high infection prevalence among schoolchildren, whereas connections with individuals of different ages facilitate the wider spread of disease through the community [31]. However, conventional network investigations, which are frequently limited to single-mode structures and direct links, often fail to provide a complete representation of disease propagation. A more comprehensive framework is therefore needed to analyze direct and indirect connections across multiple, overlapping social contexts. By acknowledging the influence of contact patterns and hierarchical social structures on disease transmission, multilevel intervention strategies integrate considerations of individual relationships, local groups, and broad system-level configurations to guide effective public health responses [32].

Despite recent advances, the potential of multilayer networks to represent modular social structures in epidemiology is underexplored [20,33,34]. Prior studies have advanced the modeling of disease dynamics across network layers [22,35] but have not yet fully accounted for the dynamic and modular nature of social structures. Epidemic spread in a two-layer system with an information layer and an infection layer has been analyzed; for example, a large number of communities in the infection layer increase the propagation threshold (thus inhibiting spread), whereas a higher average degree decreases it (thus facilitating transmission) [22]. However, this model treats community structure as static within each layer, with limited consideration of how modularity evolves across layers or social contexts. Concurrent contagions (disease and opinion diffusion) across a spatial lattice and a scale-free network have similarly been examined, with a focus on centrality-based interventions that target high-degree nodes in the opinion layer to mitigate disease spread [35]. However, their focus on influence strategies underrepresents the role of structural modularity in shaping transmission patterns across layers.

To address these gaps, this study introduces the multilayer modular fusion graph attention network (MMF-GAT), a framework that positions modularity as a core component of epidemic prediction. This study makes a dual contribution. Our first contribution is to multilayer network science by adapting established tensorial formalism and providing a novel graph neural network architecture designed to preserve and integrate context-specific information. The second contribution is to operationalize public health by introducing a high-performing predictive tool.

By integrating data from personal contact (P), household coresidency (H), and community colocation (C) layers, the MMF-GAT uses layerwise attention and a late fusion mechanism to model complex interactions. As demonstrated with COVID-19 surveillance data from Houston, Texas, U.S., this approach significantly improves prediction accuracy compared with baseline models. Although sampling biases inherent in contact tracing data preclude causal claims about disease transmission, the ability of the framework to identify high-risk individuals and contexts makes it a valuable tool for guiding public health interventions such as prioritizing contact tracing and testing resources.

3.1. Sociological concepts of group membership and structural overlap

This study employs a multilayer modular network framework grounded in Breiger’s (1974) theory of the “duality of persons and groups,” which conceptualizes social structures as emerging from the interplay between individual relationships and group affiliations [8]. Individuals are shaped by the groups to which they belong and, in turn, influence the structure and dynamics of those groups through repeated interactions. In our framework, personal networks (e.g., neighborhood-based contacts), residential settings (e.g., households), and community collectives (e.g., outbreak sites) are modeled as interrelated layers within a dynamic social system.

Sociological perspectives further clarify that social groups are defined not only by relational ties but also by shared attributes, distinct community traditions, and social contexts [36,37]. These categories establish social boundaries embedded in broader systems of resource distribution, which regulate access to opportunities and structure patterns of association [9,10,37–39]. Such boundaries can contribute to imbalances in community health outcomes by creating varied levels of exposure and sensitivity to health risks, including diseases [36,37,40–42]. In fact, social boundaries act as foundational drivers of inconsistent access to health-promoting resources [36,37,41–43]. This connection persists and influences health outcomes even as specific risk factors change over time [40].

Building on this foundation, our study posits that an individual’s risk of infection is influenced by their structural position both within and across multiple layers of the social environment. Our model represents each individual with a feature set that includes their demographic profile, behavioral risk, and group affiliation. These features are designed to reflect how individuals are embedded in an overlapping modular setting, defined by households, institutions, and social ties, and how their roles within those settings contribute to disease transmission.

3.2. Social mechanisms forming modularity

To understand that infections spread through the social system, it is essential to examine the mechanisms that generate modularity, i.e., dense clusters of social ties within groups and sparser connections between them. Several sociological concepts shed light on the development of modular structures and interactions within layers (intralayer interactions).

Feld’s concept of foci (1981, 1982) refers to shared settings or routine activities (e.g., schools, workplaces, or religious centers) that serve as focal points for repeated interaction [9,10]. These foci foster localized clustering of interactions and set up the structural conditions for the formation of modular communities.

Bourdieu’s (1990) concept of habitus adds depth by emphasizing how internalized dispositions, formed through past experiences, guide behaviors within specific settings [43]. For example, hygiene practices within households or cultural norms governing physical proximity in public settings reflect habitual routines that may increase or reduce transmission risk.

Homophily, or the tendency for individuals to associate with others who share similar characteristics, contributes to the formation of modular structures [39]. It promotes clustering within groups that are socially, demographically, and behaviorally alike, which may intensify localized outbreaks but also limit immediate spread to dissimilar groups.

These mechanisms not only support the internal cohesion of modules (e.g., households or social circles) but also affect the potential for disease contamination or acceleration, depending on the scope and type of connections that link one module to another.

3.3. Cross-layer connectivity and brokerage roles

Although modularity represents the internal structure of social groupings, transmission across groups and layers is shaped by how individuals link otherwise connected segments of the network. To understand how infections cross both modular and multilayer boundaries, this study focuses on brokerage roles or specific structural positions that facilitate transmission between otherwise disconnected groups.

Gould and Fernandez (1989) provide a typology of brokerage roles on the basis of the predefined, nonoverlapping subgroup affiliations of the broker and actors being connected [44]. These roles include coordinator (within-group), itinerants (outsiders brokering within-group), gatekeepers (insiders controlling access), representatives (insiders reaching out), and liaisons (connecting members of distinct groups). This typology is especially relevant for multilayer networks, where subgroups may exist both within and across layers (e.g., households in the residential layer and shared sites in the community layer). For example, an individual may act as a representative of a household entering a school environment (crossing residential and community layers) or serve as a gateway controlling what flows back into the household.

This brokerage framework was further extended to multilayer network contexts through the concept of “brokerage-centrality conjugated,” which emphasizes how individuals with high degree centrality (those with many connections to their immediate neighbors) are presumed to be effective as brokers across social contexts [45]. Prominent brokerage serves as a specific form of these conjugated roles, involving individuals who hold multiple brokerage positions at once through their broad, multilayer connectivity. Such positions give them disproportionate influence on epidemic trajectories by linking distinct contexts and allowing transmission to move across otherwise disconnected or weakly connected layers.

In the context of epidemic prediction, considering these structural positions allows a model such as MMF-GAT to incorporate relevant patterns of connectivity across layers. Through its late fusion strategy, MMF-GAT integrates information from each social layer in a way that reflects the underlying modular and cross-layer structures, representing not only who is connected but also how those connections span boundaries across modular and layered social systems.

3.4. Metrics for structural positions in personal and modular layers

In the personal contact network (layer P), an individual’s position is assessed via standard network analysis metrics, including degree centrality to measure the number of direct connections for each individual [46], clustering coefficients to measure the density of connections within an individual’s immediate neighborhood [47], and network exposure to measure the extent to which an individual is connected to others who are already infected [48] to determine their effects on transmission patterns. For example, a high degree centrality in layer P identifies hubs with wide-reaching ties that may accelerate pathogen spread. On the other hand, clustering coefficients quantify the local density of connections, indicating the potential for disease containment in cohesive neighbor groups.

These individual-level metrics are then contextualized within the broader multilayer framework. Affiliation in the household (layer H) and community (layer C) networks establishes modularity of the system, forming structural boundaries around family units or shared sites or their types. Although dense connections within these modules can facilitate local transmission, the sparse connections between them can limit widespread outbreaks. The social mechanism for transmission across these modular boundaries is brokerage.

Interlayer links, which represent an individual’s participation in multiple social contexts, allow them to act as brokers connecting otherwise isolated groups. Prominent brokers are identified by operationalizing the “brokerage-centrality conjugated” pattern, which targets individuals who have high degree centrality and spans multiple layers to bridge distinct social contexts [45]. These positions can facilitate disease transmission, as they create channels that cross boundaries that separate different clusters. Therefore, the interaction between personal network metrics and multilayer affiliations is important. For example, high degree centrality in the personal contact network (P) can increase an individual’s brokerage potential when they are embedded in the household (H) and community (C) layers. This provides a more comprehensive picture for understanding an individual’s risk and potential influence on epidemic dynamics.

3.5. Visual representation of multilayer modular dynamics

A multilayer modular network is operationalized via conventions from multilayer network science [33] to represent the complex organization of social interactions. This framework breaks down how individuals navigate overlapping social contexts and that their structural positions create channels for disease transmission. It lays the groundwork for the mathematical formalism presented later.

Fig 1 illustrates the abstract structure of this multilayer network, featuring three primary layers (P, H, C) and seven site-type sublayers ). Fig 1A shows how individuals traverse these layers through overlapping affiliations, a concept grounded in the sociological theory of the duality of persons and groups [8]. Shared community sites, such as schools and workplaces within the C layer, are described as foci for repeated interactions and local clustering [9,10]. This setup can accelerate disease spread, especially when connected to a high-exposure personal network (P).

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Fig 1. A. Multilayer modular network illustrating primary layers and brokerage roles in disease transmission. B. Site-type decomposition of the community layer.   

https://doi.org/10.1371/journal.pcsy.0000070.g001

As illustrated in Fig 1A, “Person X” serves as an example that overlapping brokerage roles can bridge social contexts and facilitate pathogen transmission. By spanning the personal, household, and community layers, Person X’s position combines centrality and brokerage through high connectivity within and across layers. This structural position allows Person X to fulfill several key functions in disease spread, as outlined below.

3.6. Mathematical formalism and MMF-GAT model architecture

This section introduces the computational framework developed for this study. It begins by establishing the multilayer modular network with a mathematical formalism and then specifies the architecture of the MMF-GAT model.

3.6.1. Tensorial formulation of the multilayer social graph.

To formally represent the complex social system relevant to epidemic modeling, this study adopts the tensorial framework of multilayer network science described by Artime et al. [33]. In this approach, a multilayer network consists of physical nodes (individuals), each interacting across distinct layers, where each layer represents a unique social context such as personal contact, household coresidency, or various categories of community sites.

The network is mathematically encoded as a fourth-order adjacency tensor . Each element specifies the presence and weight of an edge from node in layer to node in layer . Here, the Latin indices () and denote physical nodes, and the Greek indices ( and denote layers [33]. Each node is represented by a set of state nodes (replicas), (), corresponding to its presence in each layer . This tensorial formulation distinguishes intralayer edges (), which connect nodes within the same layer, from interlayer edges (), which connect nodes across different layers, preserving the modularity of the network’s structure.

Following Artime et al., the tensor is conceptually decomposed into four components: self-interactions (), endogenous interactions (), exogenous interactions (), and intertwining interactions (). Self-interactions () refer to ties from a node to itself within a layer (), endogenous interactions () represent ties within a layer ( ), exogenous interactions () describe ties between distinct nodes in different layers (), and intertwining interactions () connect replicas of the same node across layers (). Thus, the tensor can be decomposed as . In this study, endogenous and intertwining () interactions are the focus, whereas self-interaction and exogenous interaction tensors are set to zero. This modeling assumption simplifies the interaction structure and is not a general rule from Artime et al. (2022), who include all components in their comprehensive framework.

Intralayer connections are denoted by adjacency matrices for each layer , where = . Each is a symmetric binary matrix of size N × N, with entries =1 if there is an edge between node and node within layer and 0 otherwise. These matrices represent endogenous interactions within each layer. Interlayer edges, representing intertwining interactions , are encoded in the tensor element for ≠ , representing connections between node in layer and node in layer . Although this study emphasizes intralayer processing before fusion in the MMF-GAT architecture, the tensorial framework provides the mathematical basis for both intralayer and interlayer information flow.

The network constructed for this study comprises L = 10 distinct layers, each corresponding to a specific social context and defined as a graph. Each layer graph is associated with an intralayer adjacency matrix , where belongs to the set {P, H, C, , , …, }. For each layer , is a symmetric binary matrix of size N × N (N = 2,264 individuals), where = 1 if there is an edge from node to node within layer and 0 otherwise. Each thus represents the intralayer component of the fourth-order adjacency tensor for =.

Building on this tensorial framework, both intralayer and interlayer connections inform the fusion strategies and architectural design choices in the MMF-GAT model, guiding how information is aggregated and integrated across social contexts.

Fig 2 illustrates the overall structure and flow of the MMF-GAT model, describing the input from multilayer social graphs, GAT modular processing, late fusion integration, and output predictions with key evaluation metrics.

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Fig 2. Workflow of the MMF-GAT architecture for COVID-19 prediction.

https://doi.org/10.1371/journal.pcsy.0000070.g002

3.6.2. Graph attention network integration for multilayer fusion.

The MMF-GAT is a computational framework designed to predict infection status by learning from the relational patterns in the multilayer social graph. Its architecture translates the rank-4 tensor formalism into the graph attention network (GAT) formulation [11]. The MMF-GAT first processes each social layer individually before integrating their outputs through a late fusion mechanism.

A GAT layer learns to assign importance to different nodes within a neighborhood on a specific layer . The initial features for all N nodes in the graph are compiled into a node feature matrix , where F is the number of features per node, which distinguishes the input features from the hidden representations () generated by subsequent neural network layers. The attention mechanism computes unnormalized scores, , which are then passed through a Softmax function to obtain the final attention weights, , to be described momentarily. The updated feature vector for node , specific to layer , is then computed as:

(1)

where is the feature vector for a neighboring node , for a layer , is a shared learnable weight matrix, is a nonlinear activation function (ELU), and is the set of neighbors of node within layer . This layer-specific notation illustrates how the GAT mechanism operates independently on each layer. The architecture proceeds in steps. First, it processes intralayer connections by computing the attention coefficient between node and its neighbor for layer :

(2)

These scores are normalized across the neighborhood of node within layer to obtain attention weights:

(3)

where is the set of neighbors of node as defined by the connections in layer .

Second, multiple heads (with = 8 heads) are used to discern diverse relational patterns within each layer [49]. Thus, each head generates a distinct embedding, , and the corresponding outputs are concatenated and transformed via a learnable output weight matrix, , to form the final layer-specific embedding:

(4)

Here, , where is the dimensionality of each head’s output and where is the desired dimensionality of the final embedding.

Third, a deep late fusion architecture with a sparsity-optimal design based on the principles in [50] (see Model 3 in the ablation experiment) processes interlayer connections. It employs two L1-regularized fusion layers and multihop processing to integrate information, accounting for both 1-hop (direct) and 2-hop (indirect) neighborhood information. The final embeddings from each layer are then concatenated into a comprehensive vector:

(5)

where is the total number of layers indexed by . This unified vector, , represents an individual’s composite identity, which is then fed to a final classification layer.

3.7. Model validation through ablation experiments

This subsection describes the assessment of the impact of different fusion strategies and network depths, designing three architectural variants of the MMF-GAT. All the models operate on ten network layers representing distinct social contexts (P, H, C, and for k ranging from 1–7). Each model applies to GAT blocks [11] to extract neighborhood-aware embeddings, and all use L1-regularized fusion layers for feature integration and sparsity. Fig 3 illustrates the three models that differ in depth and fusion strategy, as described below.

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Fig 3. Ablation experiment architecture for the MMF-GAT.

https://doi.org/10.1371/journal.pcsy.0000070.g003

3.7.3. Model 3: Deep GAT with double fusion (Intermediate + late).

Model 3 advances the previous designs by introducing an additional intermediate fusion stage between the two GAT layers, resulting in a double-fusion architecture that integrates cross-layer information both earlier and later while preserving modularity. After the first GAT layer computes 1-hop embeddings for each node in each layer (using attention weights , weight matrices , and ELU activation), these layer-specific embeddings are concatenated across all L = 10 layers to form . This unified vector is then passed through the first L1-regularized dense fusion layer to produce an intermediate fused embedding that is then input into the second GAT layer for each layer , which computes 2-hop embeddings via updated attention weights and weight matrices . These outputs are again concatenated across all layers to form a refined , which is subsequently passed through a second L1-regularized fusion layer to yield the final fused embedding for classification.

This double-fusion architecture enables both early and late integration of cross-layer information, supporting modular and interpretable learning. By leveraging multihead attention mechanisms (with K = 8 heads), the model is able to discern diverse relational patterns within and across layers and to represent dynamic interactions such as brokerage roles that span the tensor’s intralayer () and interlayer () components, as defined in Section 3.6.

In all three models, the L1 penalty promotes sparse representations that improve interpretability by prioritizing key features from the multilayer graph. Model 3 was selected as the final MMF-GAT architecture on the basis of its better performance in ablation experiments (reported in Table 1), particularly in representing complicated cross-layer patterns essential for epidemic forecasting tasks.

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Table 1. Ablation experiment results for the MMF-GAT architectures.

https://doi.org/10.1371/journal.pcsy.0000070.t001

4.1. Study population and data collection

Epidemiological surveillance data for social network analysis were collected by the Houston Health Department (HHD) between March 1, 2020, and December 31, 2021. Following the receipt of a positive PCR result, index cases were interviewed by HHD surveillance investigators to document health status, personal contacts, and potential exposures at identified outbreak sites. When the manual outreach capacity was exceeded, data collection was automated via third-party contractors and supplemented by Qualtrics and telephone surveys. All data, including PCR test results, addresses, and reported contacts, were consolidated within HHD’s Houston Electronic Disease Surveillance System (HEDSS). Among individuals present in all three network layers (personal contact, household coresidency, and outbreak site colocation), a final analytical cohort of 2,264 was identified on the basis of having at least one definitive PCR test recorded during Houston’s delta-variant surge (April 4–September 4, 2021). Any inconclusive test results were excluded from this classification.

4.2. Outcome measures

The primary outcome was COVID-19 positivity, defined as one or more positive PCR results recorded between April 4 and September 4, 2021.

4.3. Network layer construction and graph definitions

The construction of the multilayer network was guided by operational definitions and data collected by HHD for epidemiological surveillance. In the early stages of the pandemic, HHD used traditional contact tracing methods (i.e., direct interviews) to gather information. When the manual outreach capacity was exceeded, this process was supplemented by automated systems, including Qualtrics and telephone surveys. The following sections detail the construction of each network layer and provide its formal graph definition with a multilayer social graph framework.

4.4. Structural characteristics of the network layers

To assess the structural characteristics of the multilayer network, degree distributions for personal contact (P), household coresidency (H), and overlap (P ∩ H) were computed for the full surveillance period (March 2020–December 2021) via the entire analytic sample (N = 2,264). Although individuals were selected on the basis of their presence in all three layers, the analysis demonstrated significant network sparsity. This sparsity is explained by the temporal constraints of the outcome variable, i.e., individuals could have a degree of zero if their contacts were not tested during the delta surge period (April–September 2021), even if connections existed at other times. The analysis revealed that 47% of individuals had a degree of zero in the personal contact network, and 62% had a degree of zero in the overlaps (P ∩ H).

In the denser household network, 63% of individuals had a degree between one and three. More reports with a detailed breakdown of these distributions, including histograms and stratification by infection status, can be found in the online supplementary material (S1 Section, S1 Fig, S2 Fig, S3 Fig, S4 Fig, S5 Fig, S1 Table, S2 Table, S3 Table, S4 Table, S5 Table, S1 Text, S2 Text, S3 Text, S4 Text, S5 Text). Statistical significance was evaluated using the Wilcoxon signed-rank test for differences between the P and H networks, and the Wilcoxon rank-sum tests for comparisons of overlapping degrees by COVID-19 status.

4.5. Construction of node-level features for predictive modeling

The initial feature vector, , for each individual was constructed from a combination of risk factors for COVID-19, such as age, sex, and race, converting variables to appropriate measurement scales. Age was categorized into four levels (0–17, 18–44, 45–64, 65+), sex was encoded as a binary variable for males and females, and race (excluding Hispanic ethnicity due to data limitations) was one-hot encoded into multicategory variables with levels White, Black, Asian, and Other (including multiple races). From the personal contact network (P), three network features were extracted on the basis of network theory as predictors: the number of contacts (degree centrality), the interconnectedness among neighboring nodes (clustering coefficient), and the number of COVID-19 positive contacts to measure the network exposure level. From the household network (H), two coresidency features, the number of coresidents (household degree) and the number of COVID-19 positive coresidents (household exposure), were extracted. To represent location-based modularity at the node level within the community layer (C), seven binary dummy variables were created. These variables correspond to each outbreak site type and indicate whether an individual was affiliated with at least one site of a given type, also representing membership in aggregated outbreak-type modules.

These node features served as covariates in subsequent statistical machine learning and graph-based deep learning models to predict COVID-19 positivity. On the basis of the logistic regression results, two significant interaction terms were included: [1] the interaction between degree in the P network and affiliation with education centers and [2] the interaction between the number of infected coresidents in the H network and affiliation with education center outbreak sites. The detailed process of feature engineering via logistic regression is reported in the online supplementary material (S3 Section).

4.6. Experimental design and evaluation

The network data were preprocessed to prepare adjacency matrices for analysis. Normalization was applied to the P, H, and C matrices by converting them to a sparse format and row-normalizing them to address differences in network density. The adjacency matrices for the seven site-type specific sublayers are not normalized, as their binary structure (of all entries confined to the unit interval ) does not require weight scaling. To evaluate the performance of the MMF-GAT model for COVID-19 prediction, the dataset (N = 2,264) was randomly divided into training (70%, N = 1,585) and testing (30%, N = 679) sets.

The performance of the MMF-GAT was compared against that of five baseline models: logistic regression (LR), random forest (RF), graph convolutional network (GCN) [54], GAT without fusion, and supra-graph GAT [51]. Models were evaluated in terms of accuracy, area under the receiver operating characteristic curve (AUC), F1 score, and precision-recall AUC (PRAUC). The statistical significance of the performance differences was assessed via the DeLong test for AUC and PRAUC and paired t tests for accuracy and F1 scores across cross-validation folds. The negative log-likelihood function was used as the loss function, which is mathematically equivalent to optimizing the cross-entropy loss for a multiclass classification problem where the final layer uses a log softmax activation function.

The hyperparameters for the GCN and GAT baseline models were optimized via grid search. For the GCN, the following settings were used: learning rate = 0.001, weight decay = 0.005, dropout = 0.5, hidden dimension = 64, and early stopping patience = 50. For the GAT, the configuration included a learning rate = 0.001, weight decay = 5 × 10 ⁻ ⁵, dropout = 0.3, hidden dimension = 64, number of attention heads = 12, and attention slope (α) = 0.2. For the proposed GAT fusion model, the optimal hyperparameters were determined as follows: training epochs = 500; learning rate = 0.0005; weight decay = 5 × 10 ⁻ ⁵; dropout = 0.3; hidden dimensions (layers 1 and 2) = 16; fusion layer dimension = 4; number of attention heads = 8; attention slope (α) = 0.2; L1 regularization coefficient (λ) = 0.0001; and early stopping patience = 50.

5.1. Descriptive statistics

Among the 2,264 individuals, 797 (35%) tested positive for COVID-19, and 1,467 (65%) tested negative. There were significant differences in infection rates across age groups: the youngest group had a 40% infection rate, whereas the 18–44 age group had a significantly lower infection rate of 30% (p = 0.0005), and the 65 + group had an even lower infection rate of 22% (p = 0.016). The “Other” racial/ethnic category also had a significantly lower infection rate of 30% (p = 0.038). No statistically significant differences were observed by sex.

In the personal contact network (P), t tests revealed that infected individuals had a significantly greater average degree (1.47 vs. 0.71; p < 0.0001) and more infected contacts (0.88 vs. 0.32; p < 0.001). Similarly, in the household network (H), infected individuals had a higher household degree (0.99 vs. 0.69; p < 0.0001) and greater exposure to infected coresidents (0.69 vs. 0.16; p < 0.001). With respect to outbreak site affiliations, settings such as detention centers (16%; p = 0.024) and assisted living facilities (20%; p = 0.001) had significantly lower infection rates. In contrast, commercial offices (53% infected; p = 0.036) and government facilities (56% infected; p = 0.020) presented higher infection rates. However, the small sample sizes for these latter two settings (N = 38 and N = 32) warrant caution in interpretation. Descriptive statistics for the study sample across all features can be found in the online supplementary material (S2 Section, S6 Table, S6 Text).

5.2. Visualization of the multilayered network structure

The multilayer networks were visualized via the igraph package in R [55]. Fig 4 visually demonstrates how integrating multiple social contexts refines the network’s structure. Starting with the personal contact network (P) in Panel A, distinct social clusters are visible. Adding household coresident data (H) in Panel B increases network density within these clusters by introducing additional ties. Incorporating outbreak site colocation information (C) in Panel C creates new connections that bridge previously separate clusters, enhancing connectivity across the network. Panel D further categorizes these colocation ties by outbreak site type (), showing that education centers (purple nodes) and assisted living facilities (red nodes) are the predominant settings forming these bridging links. These layers demonstrate how household affiliations strengthen local cohesion, whereas community affiliations promote broader integration through institutional settings.

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Fig 4. Progressive multilayer network and transmission channels.

https://doi.org/10.1371/journal.pcsy.0000070.g004

5.3. Results of important network features from logistic regression analysis

A logistic regression analysis was conducted to identify key predictors and inform the feature set for the MMF-GAT model. The main model (Model 1) revealed that a higher degree in the personal contact network (P) was associated with increased odds of infection (OR = 1.72, p < 0.001), as was a greater number of infected coresidents (OR = 5.57, p < 0.001). Conversely, the 18–44 (OR = 0.64, p < 0.01) and 65+ (OR = 0.39, p < 0.05) age groups had lower odds of infection than did the 0–17 reference group. Affiliation with certain outbreak sites, such as education centers (OR = 0.51, p < 0.01) and detention centers (OR = 0.31, p < 0.05), was also associated with lower odds of infection. The second model (Model 2) incorporated significant interaction terms, demonstrating the combined effects of personal contact degree with education center affiliation (P × ) (OR = 1.42, p < 0.05) and the number of infected coresidents with education center affiliation (H × ) (OR = 2.74, p < 0.05). The results can be found in the online supplementary material (S7 Table, S7 Text).

5.4. Performance outcomes of the ablation experiment

As presented in Table 1, Model 3 consistently outperformed the other configurations across multiple metrics, achieving an accuracy of 0.78, an AUC of 0.90, an F1 score of 0.72, and a PRAUC of 0.89. These results support the superiority of its multiphase processing approach, which preserves the modular organization of social contexts such as households, outbreak sites, and specific site types by processing each layer independently before integration. This method also enables the model to detect patterns associated with bridging roles across networks through its multihop strategy.

Model 3 was selected as the optimal architecture because of its ability to balance comprehensive integration with computational focus on significant features. By employing two L1-regularized fusion layers, it effectively consolidates embeddings, emphasizing the most relevant social interactions for prediction. Although this approach increases computational demands, it addresses limitations in multilayer network frameworks for epidemic modeling by preserving the unique structures of community contexts [20,22]. This design improves predictive accuracy and offers a computational framework to better comprehend the complex social dynamics driving disease transmission.

5.5. Performance evaluation of the MMF-GAT model

Table 2 compares the performance of the MMF-GAT model against baseline models, including logistic regression (LR), random forest (RF), graph convolutional network (GCN), GAT without fusion, and supra-graph GAT. The MMF-GAT model (P + H + C + ) achieved superior results, with an accuracy of 0.78, an AUC of 0.90, an F1 score of 0.72, and a PRAUC of 0.89. The performance slightly decreased in 100-fold cross-validation (accuracy = 0.77, AUC = 0.83, PRAUC = 0.76), perhaps due to dataset imbalances and variability in test set sizes. Statistical tests (DeLong for AUC and PRAUC; paired t tests for accuracy and F1 score) confirmed the model’s significant improvement over baselines (p < 0.05 to p < 0.001). Adding household (H) and outbreak site (C) layers incrementally enhanced performance, with notable gains in the F1 score and AUC, indicating the value of multilayer data integration.

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Table 2. Model performance and node classification.

https://doi.org/10.1371/journal.pcsy.0000070.t002

5.6. XAI feature analysis

Table 3 presents the distribution of COVID-19 positive individuals located at various outbreak site types, illustrating the prevalence of disease transmission at specific locations. Education centers accounted for the highest frequency of colocation events, representing 87.0% of all individuals, 87.4% of empirical positive cases, and 92.1% of predicted positive cases. Smaller outbreak sites, such as assisted living facilities (4.6%), commercial offices (1.7%), and detention centers (1.7%), had lower representation but still contributed to transmission dynamics.

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Table 3. Empirical and MMF-GAT predicted COVID-19 positive cases by outbreak site type.

https://doi.org/10.1371/journal.pcsy.0000070.t003

Fig 5. illustrates the relative importance of features identified by the GNNExplainer applied to the MMF-GAT model.

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Fig 5. Ranked feature importance scores estimated by the MMF-GAT model.

https://doi.org/10.1371/journal.pcsy.0000070.g005

Among the 21 features considered, the top 10 relative feature importance scores ranged from 10–350. The distribution of these scores had a mean of 178, a median of 180, and an interquartile range from 40 (25^th percentile) to 310 (75^th percentile). Household degree (H) emerged as the most influential predictor, followed by personal contact degree (P), education center affiliation (), and the clustering coefficient (P). The interaction effect between the degree of personal contact and education center affiliation (P × ) was also ranked highly. Additional important features included exposure to infected coresidents (H), exposure to infected personal contacts (P), sex, interaction between infected coresidents and education center affiliation (H × ), and 18–44 years of age. These findings illustrate how infection prediction depends on a combination of network structures, institutional contexts, and demographic factors, reflecting the multilayer modular nature of social interactions.