LaPCoM: Latent Position Co-clustering Model

• LaPCoM is a Bayesian nonparametric hierarchical latent-space model designed for multiplex networks that simultaneously clusters network views and their nodes.
• It employs a two-level mixture approach where each network-level cluster has its own latent Euclidean space, enabling tailored grouping of nodes within similar networks.
• The model uses dynamic mixture-of-finite-mixtures priors and a Metropolis-within-Gibbs sampler to automatically select cluster numbers and ensure robust latent space recovery.

Latent Position Co-clustering Model (LaPCoM) is a Bayesian nonparametric hierarchical latent-space model for multiplex networks in which the observed data are a collection of networks on a common node set, and the inferential target is simultaneous clustering at two levels: clustering of network views and clustering of nodes within groups of views. In LaPCoM, each network-level cluster is associated with its own latent Euclidean representation, and node positions inside that representation are themselves generated by a lower-level Gaussian mixture. The resulting model performs co-clustering of networks and nodes, dimension reduction, and latent-space-based interpretation of topological similarity, while accommodating both binary and count-valued multiplex data (Clarke et al., 12 Jul 2025).

1. Concept and scope

LaPCoM is designed for multiplex data written as

$$\mathcal{Y}=\{\boldsymbol{Y}^{(1)},\ldots,\boldsymbol{Y}^{(M)}\},$$

where each $\boldsymbol{Y}^{(m)}$ is an $N\times N$ adjacency matrix on a common set of $N$ nodes. Its central premise is that multiplexes can exhibit structure at two distinct levels. Some network views may be globally similar and should therefore be grouped together, while within those groups the nodes may organize into communities or role-based clusters. Existing methods described in the literature summarized here typically address only one of these levels: either clustering whole networks or clustering nodes. LaPCoM addresses both simultaneously by assigning networks to top-level components and, within each such component, clustering the constituent nodes in a component-specific latent space (Clarke et al., 12 Jul 2025).

The model is explicitly formulated as a latent position co-clustering model rather than merely a multilayer latent space model. At the network level, it identifies groups of networks that share a common latent geometric structure. At the node level, it identifies communities or social roles inside each shared latent space. This two-level organization is the defining distinction between LaPCoM and earlier multiplex latent position models that cluster actors only, or co-clustering methods that use discrete latent blocks without continuous latent geometry.

The model builds directly on the latent position model (LPM) and the latent position cluster model (LPCM). In the LPM formulation used here, nodes occupy latent positions $\boldsymbol{z}_i\in\mathbb{R}^p$, and connectivity is driven by relative positions through

$$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$

where $\lambda_{ij}=\mathbb{E}[y_{ij}]$, $f$ is a link function, and $\alpha$ controls overall connectivity. LaPCoM extends this by associating each network-level cluster with its own latent space $\boldsymbol{Z}_g$, so that shared topology is interpreted as shared latent geometry.

2. Hierarchical generative specification

At the top level, LaPCoM uses a finite mixture over network-level components. Networks assigned to the same component $\boldsymbol{Y}^{(m)}$0 share the same latent positions

$\boldsymbol{Y}^{(m)}$1

The model therefore treats topological similarity among networks as equivalence of latent geometric representation rather than mere similarity of summary statistics. For binary multiplexes, the Bernoulli/logit specification is

$\boldsymbol{Y}^{(m)}$2

For count-valued multiplexes, the Poisson/log formulation is

$\boldsymbol{Y}^{(m)}$3

Thus the same latent-distance mechanism is used for both data types, with the link function determining whether the model targets binary or count edges (Clarke et al., 12 Jul 2025).

The lower level introduces node clustering inside each network-level component by assuming that node positions arise from a finite Gaussian mixture,

$\boldsymbol{Y}^{(m)}$4

with diagonal covariance

$\boldsymbol{Y}^{(m)}$5

The latent dimension is fixed to $\boldsymbol{Y}^{(m)}$6 for visualization and interpretability. These subcomponents define node communities or role-based groups within each shared latent space.

This induces a hierarchical mixture-of-mixtures. The first mixture clusters networks into $\boldsymbol{Y}^{(m)}$7 groups. Each such group has a latent space $\boldsymbol{Y}^{(m)}$8. Inside each latent space, a second finite mixture clusters nodes into $\boldsymbol{Y}^{(m)}$9 subgroups. The allocation structure is explicit at both levels. Network memberships are represented by an $N\times N$0 binary allocation matrix $N\times N$1, with

$N\times N$2

For each $N\times N$3, node memberships are represented by an $N\times N$4 binary matrix $N\times N$5, with

$N\times N$6

A network-level cluster therefore corresponds to a set of views that share global topology through a common latent geometry, whereas a node-level cluster corresponds to a local grouping of nodes inside one such geometry. This layered interpretation is central to the model’s use in multiplex analysis.

3. Prior structure and automatic determination of cluster numbers

LaPCoM places priors on both the network-level and node-level mixing weights through dynamic mixtures of finite mixtures. The network-level and node-level weights satisfy

$N\times N$7

with hyperpriors

$N\times N$8

The component counts themselves are random: $N\times N$9 The paper emphasizes the distinction between the total numbers of components $N$0 and the active non-empty clusters $N$1. The dynamic MFM prior makes the Dirichlet concentration depend on the current number of components through $N$2 and $N$3, inducing shrinkage toward sparse mixing proportions and helping redundant components empty automatically (Clarke et al., 12 Jul 2025).

The prior structure on node-level Gaussian components is standard-conjugate: $N$4 for $N$5, and the intercept obeys

$N$6

The allocation priors are multinomial: $N$7

A practical consequence is automatic selection of both the number of network clusters and the number of node clusters within each network cluster. This is not achieved by reversible-jump moves. Instead, the model uses the telescoping mixture-of-finite-mixtures sampler with empty-component augmentation, so dimension changes occur by sampling $N$8 and $N$9 directly while sparse weights encourage superfluous components to be emptied.

The paper also gives explicit practical hyperparameter guidance. It recommends $\boldsymbol{z}_i\in\mathbb{R}^p$0 for both $\boldsymbol{z}_i\in\mathbb{R}^p$1 and $\boldsymbol{z}_i\in\mathbb{R}^p$2, preferring it over $\boldsymbol{z}_i\in\mathbb{R}^p$3 because the former places prior mass on a moderate number of components rather than shrinking too aggressively toward one or two. The intercept prior is $\boldsymbol{z}_i\in\mathbb{R}^p$4. Cluster-variance priors are $\boldsymbol{z}_i\in\mathbb{R}^p$5 for $\boldsymbol{z}_i\in\mathbb{R}^p$6 and $\boldsymbol{z}_i\in\mathbb{R}^p$7 for $\boldsymbol{z}_i\in\mathbb{R}^p$8, chosen to keep node clusters tight. Initial values are $\boldsymbol{z}_i\in\mathbb{R}^p$9 and $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$0, except in the primary-school application where model-based clustering via mclust gave a better $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$1.

4. Posterior inference, identifiability, and computation

Posterior inference is performed by a Metropolis-within-Gibbs MCMC algorithm combined with the telescoping sampler of Frühwirth-Schnatter et al. (2021). The update cycle is explicit: sample network allocations $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$2; identify active network components $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$3 and relabel non-empty ones first; for each active network cluster update latent positions $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$4 in a block Metropolis–Hastings step, then update node allocations $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$5, active node components $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$6, Gaussian parameters $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$7, then $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$8, then $$f(\lambda_{ij})=\alpha-\|\boldsymbol{z}_i-\boldsymbol{z}_j\|_2^2,$$9, then add empty node components if needed and sample $\lambda_{ij}=\mathbb{E}[y_{ij}]$0; after looping over $\lambda_{ij}=\mathbb{E}[y_{ij}]$1, update $\lambda_{ij}=\mathbb{E}[y_{ij}]$2 by Metropolis–Hastings, update $\lambda_{ij}=\mathbb{E}[y_{ij}]$3, then $\lambda_{ij}=\mathbb{E}[y_{ij}]$4, add empty top-level components if needed, and sample $\lambda_{ij}=\mathbb{E}[y_{ij}]$5 (Clarke et al., 12 Jul 2025).

Several full conditionals are standard. Network allocations satisfy

$\lambda_{ij}=\mathbb{E}[y_{ij}]$6

and node allocations obey

$\lambda_{ij}=\mathbb{E}[y_{ij}]$7

The node-cluster means and variances have Gaussian- and inverse-gamma-conjugate updates, and the concentration hyperparameters $\lambda_{ij}=\mathbb{E}[y_{ij}]$8 and $\lambda_{ij}=\mathbb{E}[y_{ij}]$9 are updated by Metropolis–Hastings on the log scale.

The model has two major non-identifiability classes. First, standard mixture label switching occurs at both network and node levels. The paper addresses this by combining posterior similarity matrix and variation-of-information summarization with relabelling methods from Frühwirth-Schnatter (2011). It first obtains an optimal network partition $f$0 from the posterior similarity matrix, then clusters vectorized latent spaces $f$1 using $f$2-means into $f$3 groups to define valid relabellings, retains iterations satisfying the permutation condition, and relabels the corresponding component-specific parameters. The same procedure is applied at the node level using cluster means $f$4.

Second, Euclidean latent spaces are invariant under rotation, reflection, and translation. As in standard latent position models, the sampled latent spaces are aligned post hoc by offline Procrustes transformation, aligning each sampled latent space to the first retained sample for that cluster. The same transformation is applied to the means, with scaling only for diagonal variances.

The paper does not give a formal asymptotic computational complexity expression, but states that computational cost is dominated by likelihood evaluations over all dyads in all networks assigned to each cluster, so the burden scales with the number of MCMC iterations, the number of active latent spaces, and roughly quadratically in the number of nodes due to all $f$5 pairs. Empirically, LaPCoM took about 3 hours in the smallest comparison scenario and 15 hours in the largest, versus 6 and 25 hours for PopNet. The authors note that they used long chains for convergence, though fewer iterations may suffice in practice.

5. Empirical behavior and applications

The empirical studies support both levels of the model. In Simulation Study 1, which examined eight count-valued scenarios under varying $f$6, $f$7, $f$8, and node-cluster structures, both LaPCoM and its simplified network-only variant mono-LaPCM recovered the correct number of network-level clusters in nearly all scenarios, but LaPCoM improved clustering in harder settings. Mean Procrustes correlations for latent-space recovery were $f$9 across scenarios. Network-level ARI was perfect except in one harder setting, where mono-LaPCM had mean ARI $\alpha$0 and LaPCoM $\alpha$1. Node-level ARIs for LaPCoM had medians above $\alpha$2 and IQRs below $\alpha$3, remaining strong even in more complex scenarios (Clarke et al., 12 Jul 2025).

Simulation Study 2 considered five binary scenarios and compared LaPCoM to PopNet, graphclust, a mixture of measurement-error models, and a mixture of generalized linear (mixed) models. PopNet and LaPCoM were the only consistently competitive methods for network clustering. PopNet recovered $\alpha$4 exactly with narrower intervals, while LaPCoM also recovered the truth but sometimes with slightly wider credible intervals, which the authors attribute to greater flexibility. Mean network-level ARIs for LaPCoM were all above $\alpha$5, with PopNet near $\alpha$6 throughout. For node clustering, LaPCoM dramatically outperformed methods that also returned node partitions, with median ARIs above $\alpha$7 versus near-zero medians for Mantziou et al.’s method.

The real-data analyses illustrate the interpretation of the two clustering levels. In the Krackhardt advice multiplex, with $\alpha$8, $\alpha$9, and directed binary edges, LaPCoM found $\boldsymbol{Z}_g$0 network-level clusters corresponding to different perceived advice densities. One cluster had diffuse latent structure and low density $\boldsymbol{Z}_g$1, another tightly packed structure and high density $\boldsymbol{Z}_g$2, and a third intermediate density $\boldsymbol{Z}_g$3. The model did not find evidence of node-level clustering in this case, which the authors judged reasonable given the lack of visually distinct groups in the latent spaces.

In the Aarhus social multiplex, with $\boldsymbol{Z}_g$4, $\boldsymbol{Z}_g$5, and undirected binary edges, LaPCoM found $\boldsymbol{Z}_g$6: the Facebook network formed its own network-level cluster, while co-authorship, leisure, lunch, and work grouped together. In the first latent space the model found six stable node-level clusters interpretable in terms of departmental roles and academic group structure. In the Facebook latent space, posterior mass suggested two clusters corresponding roughly to connected versus unconnected nodes, though label-switching post-processing merged them; the authors regarded the two-cluster interpretation as more substantively meaningful.

In the primary school interaction multiplex, with $\boldsymbol{Z}_g$7, $\boldsymbol{Z}_g$8, and count-valued one-hour snapshots, LaPCoM found $\boldsymbol{Z}_g$9, grouping lunchtime hours together and the remaining time periods together. In the lunchtime latent space the model found two node-level clusters corresponding to the connected component versus disconnected nodes. In the other latent space it found 13 node-level clusters strongly aligned with class and teacher labels, with ARI $\boldsymbol{Y}^{(m)}$00 against known class/status labels. Posterior predictive checks indicated good recovery of overall network structure via mean absolute difference, network distance, and true negative rate, but the ECDF of positive counts revealed underestimation of large counts, suggesting that the Poisson model is too restrictive for overdispersed count data.

Across the Krackhardt and Aarhus analyses, posterior predictive checks against separate-network LPCMs fitted by latentnet showed that LaPCoM had better or comparable fit while using fewer latent spaces. This suggests that latent-space sharing across groups of networks can yield a parsimonious representation without forcing all views into one global geometry.

6. Relation to adjacent models, limitations, and interpretation

LaPCoM occupies a specific position in the methodological landscape. It is neither equivalent to actor-only latent position clustering for multiplexes nor reducible to classical co-clustering without latent geometry. The most immediate nearby model is the Infinite Latent Position Cluster Model for multidimensional social networks, which uses a single latent space shared across all layers and a Dirichlet process mixture on actor positions, but does not cluster the layers themselves and does not assign layers latent positions. It is therefore best understood as actor clustering in a common multiplex latent space, not as true actor-layer co-clustering (D'Angelo et al., 2020).

At the opposite extreme, the non-parametric co-latent model of contingency tables and weighted networks provides explicit row-column co-clustering through

$\boldsymbol{Y}^{(m)}$01

but it is not a latent position model: it has no continuous coordinates, no Euclidean geometry, and no distance-based link function. It is therefore relevant to LaPCoM on the co-clustering side, but not on the latent-position side (Bavaud, 2016).

Other adjacent models reinforce the same distinction. The Deep LPBM combines a continuous latent representation with block-structured one-mode network clustering and partial memberships, but it is not a multiplex co-clustering model over networks and nodes (Boutin et al., 2024). Fused spatial latent block models provide spatially regularized row-column co-clustering with strong recovery theory, yet remain discrete latent block models rather than latent position models (Cai et al., 3 Jun 2026). Scalable robust Bayesian co-clustering with compositional ELBOs learns row and column latent representations with Gaussian mixture priors through coupled VAEs, but it does not use a direct latent-position interaction law and is better characterized as a deep variational co-clustering framework (Vinod et al., 5 Apr 2025).

Within its own formulation, LaPCoM has clear limitations. The latent dimension is fixed to $\boldsymbol{Y}^{(m)}$02, which aids interpretation but may be restrictive when the true structure is higher-dimensional. Scalability remains a concern because MCMC over latent positions and two-level mixtures is computationally intensive, especially as $\boldsymbol{Y}^{(m)}$03 grows. All networks must share a common node set. The model does not explicitly handle temporal dependence, although it can still cluster temporally indexed layers by similarity. It does not model directionality through sender/receiver effects. For count data, the Poisson observation model may be inadequate under overdispersion or zero inflation, as seen in the primary-school application. Missing data and covariates are not modeled.

These limitations also clarify what LaPCoM contributes. Its distinctive feature is not simply latent positions, and not simply co-clustering, but a single probabilistic framework in which groups of networks are represented by distinct latent spaces and nodes are clustered inside those spaces. A plausible implication is that LaPCoM is most valuable when the analyst expects heterogeneity across network views but also expects recurring local community structure within subsets of those views. In that regime, the model offers an interpretable decomposition of multiplex structure into global structural regimes and local node organization (Clarke et al., 12 Jul 2025).