A Review of Dynamic Network Models with Latent Variables

• Latent variables capture dependence between edges while yielding conditional independence in error terms, a central challenge in network modeling.Source: §1 · p. 2
• Latent-space models place nodes in a space where closer positions tend to form edges, supporting visualization and interpretation of networks.Source: §1 · p. 2
• The review outlines prominent dynamic latent-space models and recent dynamic stochastic blockmodels, an area the authors say had not yet been reviewed to their knowledge.Source: §1 · p. 3
• The review presents static models before their dynamic extensions, discusses real-world applications, and provides model-relationship diagrams, open problems, and an Appendix list of data sources.Source: §1 · p. 3

• Y denotes the N×N adjacency matrix, whose edge values may be binary, discrete, or continuous.Source: §2 · p. 3
• X denotes observed pair-specific covariates, while Z denotes unobserved node-level latent variables: positions in latent-space models and classes in stochastic blockmodels.Source: §2 · p. 3
• A t subscript marks a variable at time t for discrete time points t = 1, . . . , T; uppercase denotes random variables and lowercase fixed quantities or realizations.Source: §2 · p. 3

3. Latent Space Models

• The review presents static latent-space models and then their dynamic extensions, tracing the two frameworks in chronological order.Source: §3 · p. 4

3.1.1. Latent Space Approaches to Social Network Analysis

• In the social-space approach, each node is a point representing unobserved characteristics, and edges are conditionally independent given the two nodes’ latent positions.Source: §3.1.1 · p. 4
• Distance and projection formulations favor edges between nodes with shorter distances or narrower angles, respectively, and can model binary, discrete, or continuous edges through a generalized linear model’s linked mean.Source: §3.1.1 · p. 4

Latent Distance Model

• Latent distance models link binary edges to dyadic covariates and Euclidean separation between node positions.Projection alternatives favor same-direction positions and can yield asymmetric edge probabilities even with symmetric dyadic covariates.Source: Latent Distance Model · pp. 4–5

3.1.2. Latent Position Cluster Model

• The latent position cluster model combines latent-space modeling with model-based grouping, drawing positions from a finite Gaussian mixture with group-specific means and covariance matrices.Source: §3.1.2 · p. 5

3.1.3. Bilinear Mixed-effects Model

• Bilinear mixed-effects models include sender, receiver, and within-dyad effects plus a bilinear term to represent second- and third-order dependence in dyadic data.The bilinear term is associated with patterns such as transitivity, balance, and clusterability.Source: §3.1.3 · p. 6
• The review calls GBME descendants latent factor models and notes that this framework generalizes latent distance and latent class models [29].Source: §3.1.3 · p. 6

3.2.1. Dynamic Social Network in Latent Space Model

• NIPS coauthorship embeddings showed noticeable researcher-network changes across three periods, and the model was extended to author–word bipartite data.The DSNL model makes each time’s adjacency matrix depend on current latent positions, which evolve under Markovian assumptions; estimation uses generalized MDS followed by conjugate-gradient optimization and Procrustes alignment.Source: §3.2.1 · pp. 7–8

3.2.2. Dynamic Latent Distance Model with Popularity and Activity Effects

• Dutch classroom and U.S. House applications revealed gender and ethnic separation and political ideology in latent positions; weighted extensions cover count and continuous edges.The model adds popularity and activity effects to temporal latent trajectories and supports directed as well as undirected networks.Source: §3.2.2 · pp. 8–9

3.2.3. Dynamic Latent Distance Model for Bipartite Network

• Irish director–company networks showed increasing board interlocks before and during the financial crisis, then stabilization, across 761 directors and 59 companies.The model represents edge evolution using separate Markov processes for parameters, latent positions, and edges; the application fixed the directors’ latent positions.Source: §3.2.3 · pp. 9–10

3.3.1. Dynamic Gravity Model

• Bilateral trade analyses found strong reciprocity and third-order dependencies; GBME outperformed conventional trade models in prediction and understanding unobserved dyadic relationships.A later dynamic approach carries previous-year fitted positions into current edges, unlike earlier separate year-by-year fits.Source: §3.3.1 · p. 11

3.3.2. Hierarchical Multilinear Model

• Analysis of 66 countries’ conflict and cooperation records found most Cold War conflicts involved economically large countries, with conflict and cooperation patterns varying over time.The reduced-rank model uses time-invariant eigenvectors and time-varying eigenvalues to represent symmetric dynamic networks.Source: §3.3.2 · pp. 11–12

3.3.3. Dynamic Bilinear Effects Model

• In 23 stock-market indices from 2004–2013, time-varying predictors reflected the global and Greek debt crises and revealed international contagion and opposite verbal and material cooperation effects [15].The Gaussian-process formulation relaxes the Markovian assumption and allows unequally spaced observation times.Source: §3.3.3 · p. 12

4.1. Static Stochastic Blockmodels

• Static stochastic blockmodels [74, 59] assign each node one latent class and make edge probabilities depend on endpoint classes, representing unobserved heterogeneity in network relations.The review describes Bayesian and frequentist inference approaches for estimating model parameters and classes.Source: §4.1 · pp. 13–14

4.1.1. Mixed Membership Stochastic Blockmodel

• The static mixed membership model relaxes single-group membership by assigning each node a distribution over communities and using interaction-specific roles to generate directed edges.A node’s membership vector gives its affiliation probabilities, while a community-pair matrix specifies edge probabilities.Source: §4.1.1 · p. 14

4.1.2. Degree-Corrected Stochastic Blockmodel

• The degree-corrected blockmodel [41] addresses within-community degree heterogeneity by adding node-specific degree parameters to Poisson edge counts.The expected adjacency value for nodes in groups g and h is θ_iθ_jω_gh.Source: §4.1.2 · pp. 15–16

4.2.1. Dynamic Stochastic Blockmodel with Varying Community Memberships

• Yang et al.’s dynamic blockmodel models changing node memberships with a transition matrix and detects membership shifts and changing community trends across three datasets.In the DBLP coauthorship data, the database community shrinks, data mining grows, and artificial intelligence remains relatively stable; some highly productive researchers change communities frequently.Source: §4.2.1 · pp. 16–18

4.2.2. Dynamic Stochastic Blockmodel with Varying Community Memberships and Connectivity Parameters

• Xu and Hero’s model [85] lets group connectivity probabilities evolve over time and finds rising CEO-to-president email probabilities as Enron’s financial situation worsened.The probabilities between other employees stayed near baseline until Enron faced federal investigation.Source: §4.2.2 · pp. 18–19
• Matias and Miele combine time-varying memberships with constrained connectivity, using independent Markov chains for node memberships and revealing distinct social structures across three applications.Their analyses find four interaction groups and gender homophily in a French high school, confirm stable sparrow communities, and reveal hierarchical social integration among onagers.Source: §4.2.2 · pp. 19–20

4.2.3. Dynamic Mixed Membership Stochastic Blockmodel

• Dynamic mixed membership models use state-space models for evolving node-role mixtures and connectivity, allowing nodes to take different roles over time and across interactions.The model combines dynamics for mixed-membership priors and connectivity parameters with a logistic-normal blockmodel for each time point.Source: §4.2.3 · pp. 20–21
• Applications to monk, Enron, and fruit-fly networks yielded results consistent with prior monk analyses except for Mark, five Enron roles, and changing gene roles across development.The model tracked an individual's mixed-membership trajectory in Enron; many fruit-fly genes showed sharp role transitions near the end of embryonic development, and role function composition varied over time.Source: §4.2.3 · p. 22
• Ho et al. extend Xing et al.'s evolving logistic-normal prior with a multimodal mixture of role trajectories, allowing nodes to switch trajectories over time.The model uses variational EM for parameter learning and latent-variable inference; the earlier model uses Laplace variational approximation based on generalized means field.Source: §4.2.3 · pp. 22–23
• Across eight quarterly networks of 100 senators from January 2005 to December 2006, the model recovered party clusters and identified a senator shifting from the Democrat cluster to the Republican cluster.Source: §4.2.3 · p. 23

5. Conclusions and Discussions

• The review distinguishes latent space models from stochastic blockmodels and outlines each family’s evolution from static to dynamic models.It also reviews their explicit connections.Source: §5 · p. 23
• The model maps use directed links for inspiration or influence, or to mark a dynamic extension of a sender model.Static models appear in italics; latent space models and stochastic blockmodels are mapped separately.Source: Figure 1 · p. 23, §5 · p. 24, Figure 2 · p. 24
• Latent-variable network modeling faces general challenges such as computation and visualization, alongside model-specific issues including identifiability and a lack of continuous-time models.Source: §5 · p. 24
• Distance-matrix calculations cost O(N^2) for static and O(T × N^2) for dynamic latent space models, making estimation impractical beyond a few thousand nodes.Stochastic blockmodel fitting is also challenging at large scale because optimizing over label assignments is NP-hard; variational methods can fit networks of thousands of nodes, but millions or tens of millions remain challenging.Source: §5 · pp. 24–25
• Latent space plots are usually limited to 2D or 3D, and long or continuous-time sequences make latent-position movement difficult to display.The review notes difficulty when a dynamic network spans T ≥50 time points.Source: §5 · p. 25
• Most reviewed dynamic latent-variable models use discrete-time networks or discrete snapshots of continuous processes, motivating future work on continuous-time models.The review notes that network data recorded on continuous-time scales are growing.Source: §5 · p. 25

Appendix: Dynamic Network Data Sources

• Listed examples include international commerce, militarized interstate disputes, stock-market indices, Italian soccer, international relations, and DBLP co-authorship data.Source: Appendix: Dynamic Network Data Sources · p. 31
• The Irish Stock Exchange and Irish Companies Registration Office are identified as websites for data sources.Source: Appendix: Dynamic Network Data Sources · p. 31
• Some entries direct readers to sources detailed in Appendix A, Section 3.4, or cited studies.Source: Appendix: Dynamic Network Data Sources · p. 31
• Other listed data sets include a Drosophila gene network, migratory sparrows, Indian onagers, and data from Rubenstein et al..Source: Appendix: Dynamic Network Data Sources · p. 31