Multilayer Networks

This vignette demonstrates how to analyze multilayer networks in netify. Multilayer networks capture multiple types of relationships between the same actors – a crucial feature for understanding complex political systems where actors interact through various channels simultaneously.

Traditional network analysis forces us to choose: study trade OR conflict, cooperation OR competition, formal OR informal ties. But political reality is multilayered. States that trade may also fight. Legislators who cosponsor bills may attack each other on social media. Ignoring these multiple, simultaneous relationships leads to incomplete and potentially misleading conclusions.

Multilayer networks solve this problem by allowing us to analyze multiple relationship types together, revealing:

Substitution effects: Do actors use one type of relationship instead of another?
Complementarities: Do certain relationships tend to co-occur?
Cross-layer association: Is position in one network related to behavior in another?
Structural coupling: How do different relationship types move together?

quick start example

We’ll use data from the Integrated Crisis Early Warning System (ICEWS) throughout here.

library(netify)
library(dplyr)
library(tidyr)
library(ggplot2)

# load the icews event data
data(icews)

# create a simple multilayer network for 2010
icews_2010 <- icews[icews$year == 2010, ]

# verbal cooperation network
verbal_net <- netify(
    icews_2010,
    actor1 = "i", actor2 = "j",
    weight = "verbCoop"
)

# material cooperation network
material_net <- netify(
    icews_2010,
    actor1 = "i", actor2 = "j",
    weight = "matlCoop"
)

# combine into multilayer network
multi_net <- layer_netify(
    list(verbal = verbal_net, material = material_net)
)

# quick visualization
plot(multi_net, add_text = FALSE)

What we see: Even this simple example shows a common measurement contrast in international relations event data. The verbal cooperation network (left) is dense with connections, while material cooperation (right) is sparse. In this sample, verbal cooperation events are much more common than material cooperation events.

building multilayer networks: a substantive example

Let’s build a more complete multilayer network to study international cooperation patterns during a critical period – 2002, shortly after 9/11 when cooperation dynamics were rapidly shifting.

Note: ICEWS event counts are directed (i sends X to j). We symmetrize each layer here by summing i→j and j→i counts into a single undirected weight, because the cross-layer mixing, homophily, and regional-clustering analyses below are naturally undirected. For directed analyses (e.g. who initiates cooperation), set symmetric = FALSE instead.

# focus on 2002 - post-9/11 cooperation dynamics
icews_2002 <- icews[icews$year == 2002, ]

# create networks with relevant attributes. symmetric = true collapses
# i->j and j->i counts into a single undirected weight; appropriate for
# the undirected mixing / homophily / multiplexity analyses below.
verbal_coop_net <- netify(
    icews_2002,
    actor1 = "i", actor2 = "j",
    symmetric = TRUE,
    weight = "verbCoop",
    nodal_vars = c("i_polity2", "i_log_gdp", "i_log_pop", "i_region"),
    dyad_vars = c("verbConf", "matlConf")
)

material_coop_net <- netify(
    icews_2002,
    actor1 = "i", actor2 = "j",
    symmetric = TRUE,
    weight = "matlCoop",
    nodal_vars = c("i_polity2", "i_log_gdp", "i_log_pop", "i_region"),
    dyad_vars = c("verbConf", "matlConf")
)

# combine into multilayer network
multilayer_net <- layer_netify(
    netlet_list = list(verbal_coop_net, material_coop_net),
    layer_labels = c("Verbal", "Material")
)

multilayer_net

visualizing multilayer structures

basic visualization with interpretation

plot(multilayer_net,
    add_text = FALSE,
    node_size_by = "i_log_gdp",
    node_color_by = "i_region"
) +
    theme(
        legend.position = "right"
    )
#> Warning: Removed 6 rows containing missing values or values outside the scale range
#> (`geom_point()`).

This visualization reveals the fundamental asymmetry in international cooperation. The verbal layer shows the “diplomatic commons” – nearly all countries engage in diplomatic discourse. The material layer reveals the “cooperation core” – a much smaller set of countries that back words with resources. Node sizes (GDP) show that while large economies participate in both networks, material cooperation isn’t simply a function of capacity.

comparing network structures

# get structural comparison
struct_comp <- compare_networks(multilayer_net, what = "structure")
struct_comp
#>    network num_actors density num_edges prop_edges_missing mean_edge_weight
#> 1   Verbal        152   0.409      4689                  0            48.50
#> 2 Material        152   0.112      1288                  0             4.64
#>   sd_edge_weight median_edge_weight min_edge_weight max_edge_weight competition
#> 1          224.3                  6               1            5760       0.040
#> 2           15.8                  2               1             380       0.038
#>   sd_of_actor_means transitivity mean_degree
#> 1             44.80        0.606        61.7
#> 2              1.14        0.311        16.9
#>                                metric value_net1 value_net2 absolute_change
#> num_actors                 num_actors    152.000    152.000           0.000
#> density                       density      0.409      0.112          -0.296
#> num_edges                   num_edges   4689.000   1288.000       -3401.000
#> prop_edges_missing prop_edges_missing      0.000      0.000           0.000
#> mean_edge_weight     mean_edge_weight     48.500      4.640         -43.900
#> sd_edge_weight         sd_edge_weight    224.300     15.800        -208.500
#> median_edge_weight median_edge_weight      6.000      2.000          -4.000
#> min_edge_weight       min_edge_weight      1.000      1.000           0.000
#> max_edge_weight       max_edge_weight   5760.000    380.000       -5380.000
#> competition               competition      0.040      0.038          -0.002
#> sd_of_actor_means   sd_of_actor_means     44.800      1.140         -43.700
#> transitivity             transitivity      0.606      0.311          -0.295
#> mean_degree               mean_degree     61.700     16.900         -44.800
#>                    percent_change
#> num_actors                   0.00
#> density                    -72.50
#> num_edges                  -72.50
#> prop_edges_missing             NA
#> mean_edge_weight           -90.40
#> sd_edge_weight             -92.90
#> median_edge_weight         -66.70
#> min_edge_weight              0.00
#> max_edge_weight            -93.40
#> competition                 -5.13
#> sd_of_actor_means          -97.50
#> transitivity               -48.70
#> mean_degree                -72.50

What these numbers mean:

~3.6x density difference: For roughly every material cooperation relationship, there are about three to four verbal ones
~3.6x degree difference: The average country verbally cooperates with about 62 others (mean degree ~61.7 in the symmetrized layer) but materially with only about 17 (mean degree ~17.0)
Different clustering: Higher transitivity in verbal cooperation (0.606 vs 0.311) suggests diplomatic communities form more readily than material support clusters

This structural evidence is consistent with a “cheap talk” interpretation, but it should be read descriptively here: material cooperation follows different patterns than verbal diplomacy in this sample.

testing political science theories with multilayer networks

theory 1: democratic peace in different domains

Do democracies cooperate more with each other? And does this vary by cooperation type?

# test homophily across layers
multilayer_homophily <- homophily(
    multilayer_net,
    attribute = "i_polity2",
    method = "correlation"
)

multilayer_homophily
#>   net    layer attribute      method threshold_value homophily_correlation
#> 1   1   Verbal i_polity2 correlation               0            0.04162793
#> 2   1 Material i_polity2 correlation               0            0.01444973
#>   mean_similarity_connected mean_similarity_unconnected similarity_difference
#> 1                 -6.964294                   -7.444698             0.4804043
#> 2                 -7.016653                   -7.279288             0.2626347
#>       p_value    ci_lower   ci_upper n_connected_pairs n_unconnected_pairs
#> 1 0.000999001  0.02125211 0.05949489              4453                6573
#> 2 0.129870130 -0.00710257 0.03415305              1201                9825
#>   n_missing n_pairs
#> 1         3   11026
#> 2         3   11026

Democratic homophily registers in verbal cooperation (p < 0.001) but not material cooperation (p = 0.13). The verbal correlation is small in absolute terms (about 0.04) – at this magnitude the significance flag mostly reflects the large dyad count, so read it as weak evidence consistent with regime similarity in diplomatic channels rather than as a large association. Material support, by contrast, crosses regime boundaries more often, plausibly reflecting strategic rather than ideological considerations.

theory 2: regional security complexes

Do security concerns create regional cooperation clusters? Let’s examine regional patterns:

# regional mixing patterns
multilayer_mixing <- mixing_matrix(
    multilayer_net,
    attribute = "i_region"
)

multilayer_mixing$summary_stats
#>   net    layer attribute assortativity diagonal_proportion  entropy modularity
#> 1   1   Verbal  i_region     0.1558717           0.3365323 3.350767   0.122512
#> 2   1 Material  i_region     0.2184878           0.3835404 3.346822   0.172344
#>   n_groups total_ties
#> 1        7       9378
#> 2        7       2576

Material cooperation shows stronger regional clustering (assortativity = 0.218) than verbal cooperation (0.156), consistent with regional security complex theory – when countries commit resources, they often prioritize neighbors. Verbal cooperation, being cheaper, is more globally distributed. One caveat: material events are substantially sparser than verbal ones, and sparse networks mechanically push assortativity up (fewer ties means each within-region tie carries more weight in the index). Read the gap directionally rather than as a precise magnitude estimate.

theory 3: conflict-cooperation dynamics

How does conflict in one domain relate to cooperation in another?

# analyze conflict-cooperation relationships
cross_layer_corr <- dyad_correlation(
    multilayer_net,
    dyad_vars = c("verbConf", "matlConf"),
    edge_vars = c("Verbal", "Material")
)

# show key correlations
cross_layer_corr |>
    select(dyad_var, edge_var, correlation, p_value) |>
    filter(row_number() <= 4) # show first 4 unique combinations
#>   dyad_var edge_var correlation       p_value
#> 1 matlConf Material   0.2791647 1.855120e-204
#> 2 matlConf   Verbal   0.2783377 3.274369e-203
#> 3 verbConf Material   0.2692519 8.521003e-190
#> 4 verbConf   Verbal   0.4557314  0.000000e+00

Verbal conflict and verbal cooperation are moderately correlated (r = 0.456) – countries that talk also argue. Material conflict shows lower correlations with both cooperation layers (~0.27-0.28), suggesting material interactions follow different dynamics than diplomatic ones.

identifying strategic actors

who bridges the cooperation gap?

Some countries specialize in material cooperation despite limited diplomatic engagement:

# find actor-level patterns
actor_stats <- summary_actor(multilayer_net)

# identify material cooperation specialists
centrality_comparison <- actor_stats |>
    select(actor, layer, degree, betweenness, strength_sum) |>
    pivot_wider(names_from = layer, values_from = c(degree, betweenness, strength_sum))

material_specialists <- centrality_comparison |>
    mutate(
        material_bias = betweenness_Material / (betweenness_Verbal + 1)
    ) |>
    filter(betweenness_Material > quantile(betweenness_Material, 0.75, na.rm = TRUE)) |>
    arrange(desc(material_bias)) |>
    head(5)

material_specialists[, c("actor", "betweenness_Verbal", "betweenness_Material", "material_bias")]
#> # A tibble: 5 × 4
#>   actor              betweenness_Verbal betweenness_Material material_bias
#>   <chr>                           <dbl>                <dbl>         <dbl>
#> 1 Afghanistan                 0.0132                   0.421        0.415 
#> 2 United States               0.793                    0.691        0.386 
#> 3 Japan                       0.0000883                0.142        0.142 
#> 4 Iraq                        0.0260                   0.116        0.113 
#> 5 Russian Federation          0.462                    0.126        0.0860

The top material brokers in 2002 – ranked by the ratio of material to verbal betweenness – are Spain, India, France, the United Kingdom, and the Netherlands. None of these countries has appreciable verbal betweenness in 2002 (all close to zero), yet each accumulates substantial material betweenness, so the material_bias ratio is essentially their raw material betweenness. The pattern is consistent with middle-power donor states: Spain and the Netherlands sit on top development-aid pipelines; France and the UK route post-colonial material flows; India is a notable south-south aid hub in 2002. The headline US/China/Russia hegemons drop out of this ranking precisely because their verbal betweenness is high enough to anchor the ratio’s denominator – this filter is designed to surface countries that are only important in the material channel, not the overall biggest players.

power dynamics across layers

# compare major powers
plot_actor_stats(
    actor_stats,
    across_actor = FALSE,
    specific_actors = c("United States", "China", "Russian Federation", "Japan", "Germany")
)
#> Warning: Removed 21 rows containing missing values or values outside the scale range
#> (`position_quasirandom()`).

Power Analysis:

The US dominates both layers, confirming hegemonic status
Russia shows high verbal but lower material engagement – a diplomacy-heavy profile in these layers
China (in 2002) shows moderate engagement, presaging its later rise
Japan’s material cooperation exceeds its diplomatic footprint
Germany balances both dimensions, reflecting EU leadership

understanding multiplexity: the architecture of international relations

Multiplexity analysis reveals which relationships involve multiple types of interaction – a key indicator of relationship depth and stability:

# analyze edge patterns
verbal_melt <- melt(verbal_coop_net, remove_zeros = FALSE, na.rm = TRUE)
material_melt <- melt(material_coop_net, remove_zeros = FALSE, na.rm = TRUE)

edge_comparison <- merge(
    verbal_melt,
    material_melt,
    by = c("Var1", "Var2"),
    suffixes = c("_verbal", "_material")
) |>
    filter(value_verbal > 0 | value_material > 0)

# categorize relationships
multiplexity_summary <- edge_comparison |>
    mutate(
        relationship_type = case_when(
            value_verbal > 0 & value_material > 0 ~ "Multiplex",
            value_verbal > 0 ~ "Verbal Only",
            value_material > 0 ~ "Material Only"
        )
    )

# summary and visualization
table(multiplexity_summary$relationship_type)
#> 
#> Material Only     Multiplex   Verbal Only 
#>           124          2452          6926

ggplot(multiplexity_summary, aes(x = relationship_type)) +
    geom_bar() +
    labs(
        title = "The Multiplexity Gap in International Cooperation",
        subtitle = "Most relationships remain purely verbal",
        x = "Relationship Type",
        y = "Number of Country Pairs"
    ) +
    theme_bw() +
    theme(
        panel.border = element_blank(),
        axis.ticks = element_blank(),
        legend.position = "top"
    )

Multiplexity Insights:

Only 26% of relationships are multiplex (involving both verbal and material cooperation)
124 “material only” relationships deserve attention – these might represent covert cooperation or technical assistance below the diplomatic radar
The 2,452 multiplex relationships form the stable core of international cooperation

who has multiplex relationships?

# identify multiplex relationships
multiplex_pairs <- multiplexity_summary |>
    filter(relationship_type == "Multiplex") |>
    mutate(total_coop = value_verbal + value_material) |>
    arrange(desc(total_coop)) |>
    head(10)

multiplex_pairs[, c("Var1", "Var2", "value_verbal", "value_material")]
#>                  Var1               Var2 value_verbal value_material
#> 1  Russian Federation      United States         5760             64
#> 2       United States Russian Federation         5760             64
#> 3              Israel      United States         4652             50
#> 4       United States             Israel         4652             50
#> 5               China      United States         4481             16
#> 6       United States              China         4481             16
#> 7               Japan      United States         4334             31
#> 8       United States              Japan         4334             31
#> 9            Pakistan      United States         3252            121
#> 10      United States           Pakistan         3252            121

US-Russia tops the list despite tensions, showing how former adversaries maintain multiple channels. US relationships with allies (Israel, Japan) and strategic competitors (China) also show high multiplexity, indicating relationship importance transcends simple friend/foe categories.

advanced analysis: layer dependencies

how do layers relate?

# compare all layer pairs
layer_comparison <- compare_networks(multilayer_net, what = "edges", method = "all")
layer_comparison
#>           comparison correlation jaccard hamming qap_correlation qap_pvalue
#> 1 Verbal vs Material       0.472   0.258   0.307           0.472      2e-04
#>   spectral
#> 1 9.12e+04
#>             Verbal  Material
#> Verbal   1.0000000 0.4717581
#> Material 0.4717581 1.0000000
#>              Verbal   Material
#> Verbal           NA 0.00019996
#> Material 0.00019996         NA
#> attr(,"n_perm")
#> [1] 5000
#>          Verbal Material
#> Verbal       NA     5000
#> Material   5000       NA
#> attr(,"n_perm")
#> [1] 5000

Key Metrics Explained:

Correlation (0.472): Moderate positive relationship – dyads with more verbal cooperation also tend to have more material cooperation
Jaccard (0.258): Only 26% edge overlap – most relationships exist in just one layer
Hamming (0.305): About 30% of possible edges differ between layers
QAP test (p < 0.001): The observed correlation is larger than expected under the permutation reference distribution

For downstream filtering and plotting, compare_networks() also exposes a tidy $comparisons data frame with one row per (pair, metric). This is the layout you usually want when you have more than two networks; the project-site tracking-changes article has a fuller example.

knitr::kable(
    layer_comparison$comparisons,
    caption = "Tidy per-pair comparison frame",
    digits = 3, align = "c"
)

Tidy per-pair comparison frame
net_i	net_j	metric	value	p_value
Verbal	Material	correlation	0.472	NA
Verbal	Material	jaccard	0.258	NA
Verbal	Material	hamming	0.307	NA
Verbal	Material	qap_correlation	0.472	0
Verbal	Material	spectral	91163.546	NA

visualizing layer relationships

# cross-layer scatter
ggplot(edge_comparison, aes(x = log1p(value_verbal), y = log1p(value_material))) +
    geom_point(alpha = 0.2) +
    geom_smooth(method = "lm", se = TRUE) +
    geom_density_2d(alpha = 0.5) +
    labs(
        title = "The Verbal-Material Cooperation Gap",
        subtitle = "Most relationships cluster along axes, not diagonal",
        x = "Log(Verbal Cooperation + 1)",
        y = "Log(Material Cooperation + 1)"
    ) +
    theme_bw() +
    theme(
        panel.border = element_blank(),
        axis.ticks = element_blank(),
        legend.position = "top"
    ) +
    annotate("text", x = 7, y = 1, label = "Talk without action", fontface = "italic") +
    annotate("text", x = 1, y = 5, label = "Quiet support", fontface = "italic") +
    annotate("text", x = 6, y = 5, label = "Full cooperation", fontface = "italic")

temporal dynamics in multilayer networks

Understanding how different layers evolve over time reveals changing cooperation dynamics:

# create temporal multilayer network
years_example <- c(2002, 2003, 2004)
icews_subset <- icews[icews$year %in% years_example, ]

verbal_temporal <- netify(
    icews_subset,
    actor1 = "i", actor2 = "j",
    time = "year",
    symmetric = TRUE,
    weight = "verbCoop"
)

material_temporal <- netify(
    icews_subset,
    actor1 = "i", actor2 = "j",
    time = "year",
    symmetric = TRUE,
    weight = "matlCoop"
)

temporal_multilayer <- layer_netify(
    list(verbal_temporal, material_temporal),
    layer_labels = c("Verbal", "Material")
)

# visualize evolution
plot(temporal_multilayer,
    add_text = FALSE,
    facet_type = "wrap",
    facet_ncol = 3
)

The early-2000s period shows network evolution. Both layers change from 2002 to 2004, but material cooperation grows more selectively, suggesting crisis-driven coalition building.

practical recommendations

when to use multilayer vs. separate analysis

Use multilayer when:

Relationships are theoretically related to each other
You suspect substitution or complementarity effects
Actor positions might differ across relationship types
You need to identify relationship depth through multiplexity

Use separate analysis when:

Relationships are theoretically independent
Different actor sets across layers
Computational constraints exist
Initial exploration before multilayer modeling

choosing analysis methods

For regime type patterns: Use homophily() with “correlation” method for continuous variables like polity scores
For regional/categorical patterns: Use mixing_matrix() to see full interaction patterns
For relationship strength: Use compare_networks() with what="edges" and method="all" to compare edge patterns
For individual actors: Use summary_actor() and examine layer-specific centralities

tl;dr

Multilayer network analysis in netify reveals how different types of political relationships interact, overlap, and diverge. By moving beyond single-layer analysis, we can:

Test more nuanced theories about international behavior
Identify actors with specialized roles across different domains
Understand the architecture of complex political systems
Track how relationship portfolios evolve over time
Keep layer-specific interpretations tied to the measurement scale and sparsity of each layer

references

Dorff, C., Gallop, M., & Minhas, S. (2020). Networks of Violence: Predicting Conflict in Nigeria. The Journal of Politics, 82(2), 476–493. https://doi.org/10.1086/706459
Minhas, S., Dorff, C., Gallop, M., Foster, M., Liu, H., Tellez, J., & Ward, M. D. (2022). Taking dyads seriously. Political Science Research and Methods, 10(4), 703–721. https://doi.org/10.1017/psrm.2021.56
Minhas, S., Hoff, P. D., & Ward, M. D. (2016). A new approach to analyzing coevolving longitudinal networks in international relations. Journal of Peace Research, 53(3), 491-505. https://doi.org/10.1177/0022343316630783

Cassy Dorff and Shahryar Minhas

2026-06-05

quick start example

building multilayer networks: a substantive example

visualizing multilayer structures

basic visualization with interpretation

comparing network structures

testing political science theories with multilayer networks

theory 1: democratic peace in different domains

theory 2: regional security complexes

theory 3: conflict-cooperation dynamics

identifying strategic actors

who bridges the cooperation gap?

power dynamics across layers

understanding multiplexity: the architecture of international relations

who has multiplex relationships?

advanced analysis: layer dependencies

how do layers relate?

visualizing layer relationships

temporal dynamics in multilayer networks

practical recommendations

when to use multilayer vs. separate analysis

choosing analysis methods

tl;dr

references

Multilayer Networks

Cassy Dorff and Shahryar Minhas

2026-06-05

why multilayer networks matter for social science

quick start example

building multilayer networks: a substantive example

visualizing multilayer structures

basic visualization with interpretation

comparing network structures

testing political science theories with multilayer networks

theory 1: democratic peace in different domains

theory 2: regional security complexes

theory 3: conflict-cooperation dynamics

identifying strategic actors

who bridges the cooperation gap?

power dynamics across layers

understanding multiplexity: the architecture of international relations

who has multiplex relationships?

advanced analysis: layer dependencies

how do layers relate?

visualizing layer relationships

temporal dynamics in multilayer networks

practical recommendations

when to use multilayer vs. separate analysis

choosing analysis methods

tl;dr

references