Measuring Who Connects with Whom: Homophily & Dyadic Analysis

vignette summary

This vignette shows how to use netify to explore relationships between international alliance patterns and country characteristics using data from the Correlates of War (COW) project and Alliance Treaty Obligations and Provisions (ATOP) data. The examples focus on four common international relations questions:

Democratic Peace: Do democracies cooperate more with each other than with non-democracies?
Economic Interdependence: Do countries with similar economic development levels cooperate more?
Geographic Proximity: Does geographic distance affect cooperation patterns?
Regional Clustering: Do countries primarily cooperate within their own regions?

We’ll focus on how to do some exploratory statistical analysis with netify:

homophily(): Do Birds of a Feather Flock Together?
- Tests whether similar countries tend to form alliances with each other. For example, do democracies primarily ally with other democracies? Do rich countries mainly partner with other rich countries?
- Calculates correlations between attribute similarity and network tie presence using multiple similarity metrics. The optional permutation test compares the observed dyad-level association to a permuted similarity vector.
- Function notes:
  - Flexible similarity metrics: Correlation, euclidean, categorical, cosine and other methods
  - Permutation summaries: P-values and confidence intervals for exploratory comparisons
  - Missing data handling: Uses dyads with the needed edge and attribute values
  - Multi-network ready: Works across time periods and layers
mixing_matrix(): Who Actually Partners With Whom?
- Creates detailed “who allies with whom” tables. Shows not just whether democracies ally with democracies, but how much they interact with autocracies, hybrid regimes, and other categories.
- Constructs mixing matrices showing tie distributions across attribute combinations. Calculates assortativity coefficients, modularity scores, and entropy measures to quantify mixing patterns with optional normalization schemes.
- Function notes:
  - Cross-dimensional analysis: How regime types mix across regions
  - Rich summary statistics: Assortativity, modularity, entropy, diagonal proportions
  - Flexible normalization: Raw counts, proportions, or row-normalized
  - Weighted network support: Incorporates alliance strength, not just presence
dyad_correlation(): Which Relationship Factors Are Associated with Alliances?
- Tests how characteristics of country pairs (like geographic distance, trade volume, or cultural similarity) are associated with alliance ties. Answers questions like “Do nearby countries ally more often?”
- Correlates dyadic (pairwise) variables with network ties using multiple correlation methods. Supports partial correlation analysis to control for confounding dyadic factors while handling missing data through pairwise deletion.
- Function notes:
  - Partial correlation support: Isolate specific effects while controlling for others
  - Multiple correlation methods: Pearson, Spearman, Kendall with significance testing
  - Binary network options: Analyze tie presence vs. strength separately
  - Attribute diagnostics: Descriptive stats for all variables
attribute_report(): Attribute Report Summary
- Runs the main attribute-oriented analyses from one interface and returns results on actor position, mixing patterns, and dyadic associations.
- Runs homophily analysis, mixing matrices, dyadic correlations, and centrality-attribute relationships from one call. Tries to automatically determine appropriate methods based on variable types.

library(netify)
library(ggplot2)
library(peacesciencer)
library(dplyr)
library(countrycode)

fmt_p <- function(p, digits = 3) {
  cutoff <- 10^-digits
  ifelse(
    is.na(p),
    NA_character_,
    ifelse(
      p < cutoff,
      paste0("< ", formatC(cutoff, format = "f", digits = digits)),
      formatC(p, format = "f", digits = digits)
    )
  )
}

data preparation

We’ll use the Correlates of War data via the peacesciencer package to build a network of international alliances. This data includes measures of democracy, economic development, military capabilities, geographic relationships between countries, and alliance commitments from the ATOP dataset.

cow data

# download peacesciencer external data if needed
peacesciencer::download_extdata()

# build dyadic dataset for a recent 5-year period
cow_dyads <- create_dyadyears(subset_years = c(2010:2014)) |>
  add_cow_mids() |>
  add_capital_distance() |>
  add_democracy() |>
  add_sim_gdp_pop(keep = "pwtrgdp") |>
  add_nmc() |>
  add_atop_alliance()

# build alliance cooperation measure from atop alliance types
cow_dyads <- cow_dyads |>
  mutate(
    alliance_score = atop_defense + atop_offense + atop_neutral + atop_nonagg + atop_consul,
    alliance_norm = alliance_score / 5,
    cooperation = alliance_norm,
    region1 = countrycode(ccode1, "cown", "region"),
    region2 = countrycode(ccode2, "cown", "region"),
    log_gdp1 = log(pwtrgdp1 + 1),
    log_gdp2 = log(pwtrgdp2 + 1),
    log_capdist = log(capdist + 1),
    alliance_intensity = alliance_norm,
    defense_alliance = atop_defense
  )

# filter to 2012 for cross-sectional analysis
cow_2012 <- cow_dyads |>
  filter(year == 2012)

# create alliance network
alliance_net <- netify(
  cow_2012,
  actor1 = 'ccode1', actor2 = 'ccode2',
  symmetric = TRUE,
  weight = 'cooperation'
)

alliance_net

Nodal data is one row per actor (here, per country) describing actor-level attributes — distinct from dyadic (pair-level) variables.

# prepare nodal data with country attributes
nodal_data <- cow_2012 |>
  select(
    ccode1, region1, v2x_polyarchy1,
    log_gdp1, cinc1
    ) |>
  distinct() |>
  rename(
    actor = ccode1,
    region = region1,
    democracy = v2x_polyarchy1,
    log_gdp = log_gdp1,
    mil_capability = cinc1
  ) |>
  mutate(
    regime_type = case_when(
      democracy >= 0.6 ~ "Democracy",
      democracy >= 0.4 ~ "Hybrid",
      democracy < 0.4 ~ "Autocracy",
      TRUE ~ "Unknown"
    ),
    development = case_when(
      log_gdp >= quantile(log_gdp, 0.75, na.rm = TRUE) ~ "High",
      log_gdp >= quantile(log_gdp, 0.25, na.rm = TRUE) ~ "Medium",
      TRUE ~ "Low"
    )
  )

nodal_data$country_name <- countrycode(nodal_data$actor, "cown", "country.name")

alliance_net <- add_node_vars(alliance_net, nodal_data, actor = "actor")

Add dyadic (relationship-level) variables. A dyad is a pair of actors and a dyadic variable describes the relationship itself (e.g., distance between two capitals) rather than either actor alone.

# prepare dyadic data
dyad_data <- cow_2012 |>
  select(ccode1, ccode2, log_capdist, alliance_norm, atop_defense) |>
  rename(
    actor1 = ccode1,
    actor2 = ccode2,
    geographic_distance = log_capdist,
    alliance_intensity = alliance_norm,
    defense_alliance = atop_defense
  )

alliance_net <- add_dyad_vars(
  alliance_net,
  dyad_data = dyad_data,
  actor1 = "actor1",
  actor2 = "actor2",
  dyad_vars = c("geographic_distance", "alliance_intensity", "defense_alliance"),
  dyad_vars_symmetric = c(TRUE, TRUE, TRUE)
)

1. testing the democratic peace with `homophily()`

Democratic peace arguments often lead researchers to ask whether democracies are more likely to cooperate with one another. Here we use alliance data to examine a narrower descriptive question: are allied country pairs more similar in democracy scores than non-allied pairs?

🔍 using `homophily()` for continuous variables

Homophily is the tendency for connected actors to be similar to each other on some attribute – “birds of a feather flock together.” The homophily() function is a tool in netify that tests whether similar actors tend to connect more in a network. It can handle both continuous and categorical attributes.

The function’s optional permutation test permutes the dyad-level similarity values relative to tie indicators. The p-value is useful as an exploratory comparison, not as evidence that the network came from independent dyads or that similarity caused ties.

# test whether countries with similar democracy levels form more alliances
democracy_homophily <- homophily(
  alliance_net,
  attribute = "democracy",
  method = "correlation",
  significance_test = TRUE
)

knitr::kable(democracy_homophily, digits=3, align='c')

net	layer	attribute	method	threshold_value	homophily_correlation	mean_similarity_connected	mean_similarity_unconnected	similarity_difference	p_value	ci_lower	ci_upper	n_connected_pairs	n_unconnected_pairs	n_missing	n_pairs
1	cooperation	democracy	correlation	0	0.142	-0.245	-0.316	0.07	0.001	0.127	0.157	3571	11480	21	15051

democracy_summary <- paste0(
  "**Democracy Homophily Results:**\n\n",
  "- Homophily correlation: ", round(democracy_homophily$homophily_correlation, 3), "\n",
  "- Avg similarity among allies: ", round(democracy_homophily$mean_similarity_connected, 3), "\n",
  "- Avg similarity among non-allies: ", round(democracy_homophily$mean_similarity_unconnected, 3), "\n",
  "- P-value: ", fmt_p(democracy_homophily$p_value), "\n",
  if(democracy_homophily$p_value < 0.05) {
    "→ Alliance ties are more common among similarly democratic country pairs\n"
  } else {
    "→ No detectable democracy-similarity association in these alliance ties\n"
  }
)

Democracy Homophily Results:

Homophily correlation: 0.142
Avg similarity among allies: -0.245
Avg similarity among non-allies: -0.316
P-value: < 0.001 → Alliance ties are more common among similarly democratic country pairs

understanding the output:

homophily_correlation: Measures tendency for similar values to connect (0 to 1)
mean_similarity_connected: Average similarity among connected pairs
mean_similarity_unconnected: Average similarity among unconnected pairs
p_value: Statistical significance of the homophily pattern

The democracy homophily analysis returns a small but statistically detectable association: countries with more similar democracy scores are slightly more likely to be alliance partners. The homophily correlation of about 0.14 (p < 0.05) is a modest positive association, not a causal estimate of why alliances form. The negative similarity values (roughly -0.25 for allies vs -0.32 for non-allies) reflect the correlation method’s transformed distance metric, where higher (less negative) values indicate greater similarity. The ~0.07 difference between allied and non-allied pairs means that allied countries are, on average, modestly closer in regime score than non-allied ones. This descriptive comparison does not adjust for geography, power, timing, or other network dependence.

visualizing homophily patterns

We can visualize the homophily pattern to better understand how democracy similarity relates to alliance formation:

plot_homophily(democracy_homophily, alliance_net,
               type = "distribution",
               attribute = "democracy",
               method = "correlation",
               sample_size = 5000) +
  labs(subtitle = "Allied countries show greater similarity in democracy scores") +
  xlim(c(-1, 1))

understanding the distribution shape

The distribution plot reveals the empirical density of pairwise similarity scores computed using the correlation method from calculate_similarity_matrix(). The distinctly non-normal shape arises from the specific calculation procedure:

Details of the Similarity Calculation:

For continuous attributes like democracy scores, when method = "correlation" is specified, the homophily function computes pairwise similarities as a negative absolute difference:

# for each dyad (i,j), similarity is calculated as:
similarity[i,j] = -abs(attr[i] - attr[j])

This produces similarity scores that:

Equal 0 when two actors have the same attribute value
Become more negative as the attribute difference grows
Generate the observed multimodal distribution due to the discrete clustering of democracy scores

interpretation of the result

The observed homophily correlation of about 0.14 indicates that, despite these distributional complexities, allied countries have higher democracy similarity scores than non-allied pairs in this snapshot. The mean difference (roughly -0.25 vs -0.32) is detectable in the dyad-level permutation summary even though both distributions have similar non-normal shapes.

However, the extensive overlap between distributions shows that democracy similarity alone does not explain alliance formation. Many democratic countries ally with non-democracies (left side of the blue distribution), while many similar democracies remain unallied (right side of the gold distribution). Geographic, security, economic, and temporal factors would need to be modeled separately before making stronger claims.

using `homophily()` for categorical variables

Now let’s move onto the categorical regime type variable we made:

head(attr(alliance_net, 'nodal_data'))

##   actor                    region democracy   log_gdp mil_capability
## 1   100 Latin America & Caribbean     0.664 13.376060   6.695519e-03
## 2   101 Latin America & Caribbean     0.401 13.129611   4.859170e-03
## 3   110 Latin America & Caribbean     0.579  8.887165   3.355918e-05
## 4   115 Latin America & Caribbean     0.753  8.912825   4.996089e-05
## 5   130 Latin America & Caribbean     0.589 12.028537   1.530854e-03
## 6   135 Latin America & Caribbean     0.823 12.706243   3.378733e-03
##   regime_type development country_name
## 1   Democracy        High     Colombia
## 2      Hybrid        High    Venezuela
## 3      Hybrid         Low       Guyana
## 4   Democracy         Low     Suriname
## 5      Hybrid      Medium      Ecuador
## 6   Democracy        High         Peru

regime_homophily <- homophily(
  alliance_net,
  attribute = "regime_type",
  method = "categorical",
  significance_test = TRUE)

knitr::kable(regime_homophily, digits=3, align='c')

net	layer	attribute	method	threshold_value	homophily_correlation	mean_similarity_connected	mean_similarity_unconnected	similarity_difference	p_value	ci_lower	ci_upper	n_connected_pairs	n_unconnected_pairs	n_missing	n_pairs
1	cooperation	regime_type	categorical	0	0.122	0.394	0.259	0.135	0.001	0.107	0.137	4046	14869	0	18915

regime_summary <- paste0(
  "**Regime Type Homophily Results:**\n\n",
  "- Homophily score: ", round(regime_homophily$homophily_correlation, 3), "\n",
  "- Same-regime alliances: ", round(regime_homophily$mean_similarity_connected * 100, 1), "%\n",
  "- Different-regime alliances: ", round((1 - regime_homophily$mean_similarity_connected) * 100, 1), "%\n",
  "- Expected if random: ", round(regime_homophily$mean_similarity_unconnected * 100, 1), "%\n",
  "- P-value: ", fmt_p(regime_homophily$p_value), "\n",
  if(regime_homophily$p_value < 0.05 && regime_homophily$homophily_correlation > 0.15) {
    "→ Same-regime alliance ties are clearly more common in this network\n"
  } else if(regime_homophily$p_value < 0.05 && regime_homophily$homophily_correlation > 0) {
    "→ Same-regime alliance ties are modestly more common in this network\n"
  } else {
    "→ No detectable regime-type association in these alliance ties\n"
  }
)

Regime Type Homophily Results:

Homophily score: 0.122
Same-regime alliances: 39.4%
Different-regime alliances: 60.6%
Expected if random: 25.9%
P-value: < 0.001 → Same-regime alliance ties are modestly more common in this network

The regime type analysis shows a small but detectable pattern of political homophily in alliance formation. With a homophily score of about 0.12 (p < 0.05), alliance ties are modestly more common within the same regime type. The similarity scores show that roughly 39% of allied pairs share the same regime type, compared to only 26% of non-allied pairs – a roughly 13 percentage point difference. Treat this as a descriptive association unless a design explicitly accounts for alternative explanations.

The categorical nature of this analysis provides a clearer interpretation than continuous measures: when countries form alliances, there is about a 39% chance their partner shares the same regime type, compared to 26% for non-allied pairs. The modest association leaves plenty of room for cross-regime alliances tied to strategic, geographic, and economic considerations. Note that we are not testing the democratic peace idea specifically here, since we are amalgamating autocracy-autocracy and democracy-democracy pairs into the same-regime bucket.

visualizing categorical homophily

For categorical variables like regime type, the visualization looks different. Instead of continuous similarity distributions, we see discrete categories:

plot_homophily(regime_homophily, alliance_net,
               type = "distribution",
               attribute = "regime_type",
               method = "categorical",
               sample_size = 5000) +
  labs(title = "Regime Type Homophily in Alliance Networks",
       subtitle = "Do similar political systems form more alliances?")

Unlike the continuous democracy score, regime type similarity is binary: country pairs either share the same regime type (similarity = 1) or they don’t (similarity = 0). The visualization shows two bars comparing the proportion of alliances within each category. A higher blue bar at similarity = 1 indicates that countries with the same regime type are more likely to form alliances than those with different regime types. This categorical approach provides a clearer test of the “democracies ally with democracies” hypothesis, though it loses the nuance of how similar countries are on the democracy spectrum.

2. economic interdependence and development

International relations theory often asks whether countries at similar levels of economic development are more likely to cooperate. We can summarize that relationship with homophily():

gdp_homophily <- homophily(
  alliance_net,
  attribute = "log_gdp",
  method = "correlation",
  significance_test = TRUE)

knitr::kable(gdp_homophily, digits=3, align='c')

net	layer	attribute	method	threshold_value	homophily_correlation	mean_similarity_connected	mean_similarity_unconnected	similarity_difference	p_value	ci_lower	ci_upper	n_connected_pairs	n_unconnected_pairs	n_missing	n_pairs
1	cooperation	log_gdp	correlation	0	0.118	-2.209	-2.796	0.587	0.001	0.106	0.131	4046	14869	0	18915

# gate substantive claims on magnitude (|r| >= 0.15) so a near-zero
# correlation does not get described as substantively large just because n is large
economic_summary <- paste0(
  "**Economic Development Homophily Results:**\n\n",
  "- Homophily correlation: ", round(gdp_homophily$homophily_correlation, 3), "\n",
  "- Similarity among allies: ", round(gdp_homophily$mean_similarity_connected, 3), "\n",
  "- Similarity among non-allies: ", round(gdp_homophily$mean_similarity_unconnected, 3), "\n",
  "- P-value: ", fmt_p(gdp_homophily$p_value), "\n",
    dplyr::case_when(
      gdp_homophily$p_value >= 0.05 ~
        "→ No detectable association between economic development similarity and alliance patterns\n",
      gdp_homophily$homophily_correlation >= 0.15 ~
        "→ Alliance ties are more common among countries at similar development levels\n",
      gdp_homophily$homophily_correlation > 0 ~
        "→ Detectable but very small (|r| < 0.15); the dyad count is doing most of the work behind the p-value\n",
      TRUE ~
        "→ Slight association with alliances between countries at different development levels (small heterophily)\n"
    )
)

Economic Development Homophily Results:

Homophily correlation: 0.118
Similarity among allies: -2.209
Similarity among non-allies: -2.796
P-value: < 0.001 → Detectable but very small (|r| < 0.15); the dyad count is doing most of the work behind the p-value

3. regional clustering in international cooperation

Do countries primarily form alliances within their own regions, or are alliances more globally distributed? Regional patterns provide another example of categorical homophily:

region_homophily <- homophily(
  alliance_net,
  attribute = "region",
  method = "categorical",
  significance_test = TRUE)

knitr::kable(region_homophily, digits=3, align='c')

net	layer	attribute	method	threshold_value	homophily_correlation	mean_similarity_connected	mean_similarity_unconnected	similarity_difference	p_value	ci_lower	ci_upper	n_connected_pairs	n_unconnected_pairs	n_missing	n_pairs
1	cooperation	region	categorical	0	0.783	0.793	0.034	0.759	0.001	0.772	0.794	4046	14869	0	18915

regional_summary <- paste0(
  "**Regional Clustering Results:**\n\n",
  "- Homophily score: ", round(region_homophily$homophily_correlation, 3), "\n",
  "- Within-region alliances: ", round(region_homophily$mean_similarity_connected, 3), "\n",
  "- Cross-region alliances: ", round(region_homophily$mean_similarity_unconnected, 3), "\n"
)

Regional Clustering Results:

Homophily score: 0.783
Within-region alliances: 0.793
Cross-region alliances: 0.034

4. who forms alliances with whom? using `mixing_matrix()`

The mixing_matrix() function reveals detailed interaction patterns between different types of actors in your network. This is crucial for understanding not just if certain types connect, but how much and with whom.

A few summary statistics will show up below: assortativity is a -1 to 1 coefficient that captures how strongly ties prefer same-attribute partners (positive = same-type, 0 = random, negative = opposites-attract). Modularity rewards within-group edges over what you’d expect by chance, so higher values mean ties cluster more strongly within attribute groups. Entropy quantifies how spread out (versus concentrated) the mixing pattern is across the cells. Diagonal proportion is the share of ties that fall on the diagonal of the mixing matrix (i.e., within the same category).

📊 democracy mixing matrix

regime_mixing <- mixing_matrix(
  alliance_net,
  attribute = "regime_type",
  normalized = TRUE
)

knitr::kable(round(regime_mixing$mixing_matrices[[1]], 3),
             caption = "Regime Type Alliance Matrix (normalized)",
             align = "c")

Regime Type Alliance Matrix (normalized)
	Autocracy	Democracy	Hybrid	Unknown
Autocracy	0.123	0.091	0.073	0.006
Democracy	0.091	0.217	0.085	0.037
Hybrid	0.073	0.085	0.045	0.011
Unknown	0.006	0.037	0.011	0.008

regime_mixing_summary <- paste0(
  "**Key Insights from mixing_matrix():**\n\n",
  "- Assortativity: ", round(regime_mixing$summary_stats$assortativity, 3), "\n",
  "  (Positive = similar types connect more; Negative = different types connect more)\n",
  "- Proportion of within-type alliances: ", round(regime_mixing$summary_stats$diagonal_proportion, 3), "\n",
  "  (Higher values indicate more homophily)\n"
)

Key Insights from mixing_matrix():

Assortativity: 0.108 (Positive = similar types connect more; Negative = different types connect more)
Proportion of within-type alliances: 0.394 (Higher values indicate more homophily)

how to read the mixing matrix:

Rows: Source regime type
Columns: Target regime type
Values: Proportion of ties from row type to column type
Diagonal: Within-type alliances (homophily)

🌍 regional alliance patterns with row normalization

region_mixing <- mixing_matrix(
  alliance_net,
  attribute = "region",
  normalized = TRUE,
  by_row = TRUE)

regional_mixing_header <- "**Regional Alliance Matrix (row-normalized):**\n\n"

Regional Alliance Matrix (row-normalized):

knitr::kable(round(region_mixing$mixing_matrices[[1]], 3),
             caption = "Regional Alliance Matrix (row-normalized)",
             align = "c")

Regional Alliance Matrix (row-normalized)
	East Asia & Pacific	Europe & Central Asia	Latin America & Caribbean	Middle East & North Africa	North America	South Asia	Sub-Saharan Africa
East Asia & Pacific	0.549	0.221	0.032	0.005	0.063	0.123	0.008
Europe & Central Asia	0.047	0.882	0.004	0.022	0.035	0.006	0.004
Latin America & Caribbean	0.019	0.012	0.931	0.000	0.031	0.004	0.003
Middle East & North Africa	0.005	0.111	0.000	0.411	0.005	0.002	0.467
North America	0.208	0.542	0.172	0.016	0.010	0.047	0.005
South Asia	0.639	0.148	0.033	0.008	0.074	0.098	0.000
Sub-Saharan Africa	0.002	0.005	0.001	0.114	0.000	0.000	0.877

visualizing regional alliance patterns

plot_mixing_matrix(
    region_mixing,
    show_values = TRUE,
    value_digits = 2,
    text_size = 3,
    text_color_threshold = .7,
    diagonal_emphasis = TRUE,
    reorder_categories = FALSE
) +
  labs(title = "Regional Alliance Patterns",
       subtitle = "Within-region vs cross-region alliance formation",
       x = "Allied with region",
       y = "From region") +
  theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 10),
        axis.text.y = element_text(size = 10))

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## ℹ The deprecated feature was likely used in the netify package.
##   Please report the issue at <https://github.com/netify-dev/netify/issues>.
## This warning is displayed once per session.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

The regional mixing matrix shows strong regional clustering in alliance formation. The emphasized diagonal shows that most regions primarily form alliances within their own geographic area, with some notable cross-regional partnerships.

🔀 cross-dimensional analysis: region × regime type

mixing_matrix() can analyze interactions across different attributes:

cross_mixing <- mixing_matrix(
  alliance_net,
  attribute = "regime_type",
  row_attribute = "region",
  normalized = TRUE)

cross_mixing_header <- "**How different regime types form alliances across regions:**\n\n"

How different regime types form alliances across regions:

knitr::kable(round(cross_mixing$mixing_matrices[[1]], 3),
             caption = "Cross-dimensional Analysis: Regime Types Across Regions",
             align = "c")

Cross-dimensional Analysis: Regime Types Across Regions
	Autocracy	Democracy	Hybrid	Unknown
East Asia & Pacific	0.023	0.035	0.018	0.003
Europe & Central Asia	0.058	0.227	0.061	0.026
Latin America & Caribbean	0.006	0.071	0.024	0.031
Middle East & North Africa	0.043	0.014	0.017	0.000
North America	0.004	0.012	0.005	0.002
South Asia	0.005	0.006	0.003	0.000
Sub-Saharan Africa	0.154	0.065	0.086	0.000

5. analyzing relationship-level factors with `dyad_correlation()`

A dyad is just a pair of actors (here, a pair of countries), and a dyadic variable describes the relationship itself (e.g., distance between two capitals, shared language) rather than either actor alone. The dyad_correlation() function examines how relationship-level (dyadic) variables correlate with network ties. This is useful for identifying pair-level factors associated with connections.

🌍 geographic distance and alliance formation

geo_correlation <- dyad_correlation(
  alliance_net,
  dyad_vars = "geographic_distance",
  method = "pearson",
  significance_test = TRUE
)

geo_summary <- paste0(
  "**Geographic Distance and Alliance Formation (dyad_correlation results):**\n\n",
  "- Correlation coefficient: ", round(geo_correlation$correlation, 3), "\n",
  "- P-value: ", fmt_p(geo_correlation$p_value), "\n",
  "- Number of dyads analyzed: ", geo_correlation$n_pairs[1], "\n\n",
  if(geo_correlation$correlation < -0.1 && geo_correlation$p_value < 0.05) {
    "→ Alliance ties are more common among geographically closer country pairs.\n  (Negative correlation = shorter distance, more alliances)\n"
  } else if(geo_correlation$correlation > 0.1 && geo_correlation$p_value < 0.05) {
    "→ Greater distance is associated with more alliance ties in this dyad-level summary.\n  (Positive correlation = greater distance, more alliances)\n"
  } else {
    "→ No clear geographic-distance association appears in these alliance ties.\n  (No detectable correlation)\n"
  }
)

The reported p-value comes from the ordinary correlation test on dyad vectors. Use it as a descriptive screen; network dependence and omitted dyadic structure require a more explicit modeling strategy for stronger inference.

Geographic Distance and Alliance Formation (dyad_correlation results):

Correlation coefficient: -0.586
P-value: < 0.001
Number of dyads analyzed: 18915

→ Alliance ties are more common among geographically closer country pairs. (Negative correlation = shorter distance, more alliances)

🤝 analyzing multiple dyadic variables

multi_dyad_correlation <- dyad_correlation(
  alliance_net,
  dyad_vars = c("geographic_distance", "alliance_intensity", "defense_alliance"),
  method = "pearson",
  significance_test = TRUE
)

multi_dyad_summary <- paste0(
  "**Multiple Dyadic Variables Analysis:**\n\n",
  paste(sapply(1:nrow(multi_dyad_correlation), function(i) {
    paste0(
      "**", multi_dyad_correlation$dyad_var[i], ":**\n",
      "  - Correlation: ", round(multi_dyad_correlation$correlation[i], 3), "\n",
      "  - P-value: ", fmt_p(multi_dyad_correlation$p_value[i]), "\n"
    )
  }), collapse = "\n")
)

Multiple Dyadic Variables Analysis:

geographic_distance: - Correlation: -0.586 - P-value: < 0.001

alliance_intensity: - Correlation: 1 - P-value: < 0.001

defense_alliance: - Correlation: 0.892 - P-value: < 0.001

6. attribute reports with `attribute_report()`

Centrality measures how important / well-positioned each actor is in the network – e.g., degree counts a node’s ties, betweenness counts how often a node lies on the shortest path between two others, and closeness measures how short the average distance is from a node to everyone else. Heterophily is the opposite of homophily: ties are more common between unlike partners.

The attribute_report() function combines the previous analyses into one report.

running the report

attribute_results <- attribute_report(
  alliance_net,
  node_vars = c("region", "regime_type", "democracy", "log_gdp", "mil_capability"),
  dyad_vars = c("geographic_distance", "alliance_intensity", "defense_alliance"),
  include_centrality = TRUE,
  include_homophily = TRUE,
  include_mixing = TRUE,
  include_dyadic_correlations = TRUE,
  centrality_measures = c("degree", "betweenness", "closeness"),
  significance_test = TRUE
)

attribute_report returns a list with multiple components:

homophily_analysis: Tests for each node attribute
mixing_matrices: Interaction patterns for categorical variables
centrality_correlations: How attributes relate to network position
dyadic_correlations: How dyad attributes are associated with ties

extracting results

homophily_header <- paste0(
  "**=== HOMOPHILY ANALYSIS ===**\n\n",
  "Do similar countries form more alliances?\n\n"
)

=== HOMOPHILY ANALYSIS ===

Do similar countries form more alliances?

Homophily Analysis Results
	Attribute	Method	Homophily Correlation	P-value	Significance	Interpretation
region	region	categorical	0.783	0.001	***	Strong homophily
regime_type	regime_type	categorical	0.122	0.001	***	Very weak (sig. but small)
democracy	democracy	correlation	0.142	0.001	***	Very weak (sig. but small)
log_gdp	log_gdp	correlation	0.118	0.001	***	Very weak (sig. but small)
mil_capability	mil_capability	correlation	-0.045	0.001	***	Heterophily

power_header <- paste0(
  "**=== POWER AND INFLUENCE ===**\n\n",
  "What makes countries central in the alliance network?\n\n"
)

=== POWER AND INFLUENCE ===

What makes countries central in the alliance network?

Top 10 Centrality-Attribute Correlations
	Node Variable	Centrality Measure	Correlation	P-value	Interpretation
cor7	mil_capability	betweenness	0.680	0.000	Strongly associated with centrality
cor5	log_gdp	closeness	0.425	0.000	Strongly associated with centrality
cor8	mil_capability	closeness	0.352	0.000	Strongly associated with centrality
cor4	log_gdp	betweenness	0.338	0.000	Strongly associated with centrality
cor3	log_gdp	degree	0.313	0.000	Strongly associated with centrality
cor2	democracy	closeness	0.318	0.000	Strongly associated with centrality
cor	democracy	degree	0.229	0.002	Moderately associated with centrality
cor6	mil_capability	degree	0.165	0.021	Moderately associated with centrality
cor1	democracy	betweenness	0.070	0.357	Not significantly related to centrality
NA	NA	NA	NA	NA	NA

relationship_header <- paste0(
  "**=== RELATIONSHIP FACTORS ===**\n\n",
  "What dyadic factors are associated with alliance formation?\n\n"
)

=== RELATIONSHIP FACTORS ===

What dyadic factors are associated with alliance formation?

Dyadic Variables Analysis
Dyadic Variable	Correlation	P-value
geographic_distance	-0.586	0
alliance_intensity	1.000	0
defense_alliance	0.892	0

7. testing specific ir hypotheses

hypothesis 1: democratic peace

# binary democracy indicator
nodal_data_binary <- nodal_data |>
  mutate(is_democracy = ifelse(regime_type == "Democracy", 1, 0))

alliance_net_binary <- add_node_vars(
  alliance_net,
  nodal_data_binary[, c("actor", "is_democracy")],
  actor = "actor")

dem_peace_test <- homophily(
  alliance_net_binary,
  attribute = "is_democracy",
  method = "categorical",
  significance_test = TRUE)

# flag both significance and magnitude so a tiny r does not get described
# as a clean "democracies prefer democracies" pattern just because p<0.05
dem_peace_summary <- paste0(
  "**Democratic Peace Hypothesis Test:**\n\n",
  "- Effect size: ", round(dem_peace_test$homophily_correlation, 3), "\n",
  "- P-value: ", fmt_p(dem_peace_test$p_value), "\n",
    "- Conclusion: ", dplyr::case_when(
      dem_peace_test$p_value >= 0.05 ~
        "No detectable democracy-based alliance pattern",
      dem_peace_test$homophily_correlation >= 0.15 ~
        "Democracy-democracy alliance ties are clearly more common",
      dem_peace_test$homophily_correlation > 0 ~
        "Detectable but very small same-regime pattern; the dyad count explains the significance flag more than the association size",
      TRUE ~
        "Detectable heterophily: democracy/non-democracy ties are more common"
    ), "\n"
)

Democratic Peace Hypothesis Test:

Effect size: 0.05
P-value: < 0.001
Conclusion: Detectable but very small same-regime pattern; the dyad count explains the significance flag more than the association size

hypothesis 2: power politics

Do powerful countries (high military capability) primarily form alliances with other powerful countries?

power_homophily <- homophily(
  alliance_net,
  attribute = "mil_capability",
  method = "correlation",
  significance_test = TRUE)

# gate substantive claims on |r| so a near-zero correlation does not get
# labeled "heterophily" simply because the dyad count makes every p tiny
power_politics_summary <- paste0(
  "**Power Politics Hypothesis:**\n\n",
  "- Correlation: ", round(power_homophily$homophily_correlation, 3), "\n",
  "- P-value: ", fmt_p(power_homophily$p_value), "\n",
    "- Interpretation: ", dplyr::case_when(
      power_homophily$p_value >= 0.05 ~
        "No detectable power-based alliance pattern",
      power_homophily$homophily_correlation >= 0.15 ~
        "Alliance ties are more common among similarly powerful countries",
      power_homophily$homophily_correlation <= -0.15 ~
        "Alliance ties are more common between high- and low-power countries (heterophily)",
      TRUE ~
        "Detectable but very small (|r| < 0.15); read as mild power-mixing rather than a clear homophily/heterophily pattern"
    ), "\n"
)

Power Politics Hypothesis:

Correlation: -0.045
P-value: < 0.001
Interpretation: Detectable but very small (|r| < 0.15); read as mild power-mixing rather than a clear homophily/heterophily pattern

8. visualizing network patterns

And as seen in other vignettes we can use the plot() function to visualize the network with node attributes and edge weights:

network visualization by attributes

# attach degree and other actor stats so we can map them to node aesthetics
alliance_net <- add_node_vars(
  alliance_net,
  summary_actor(alliance_net),
  actor = "actor"
)

plot(alliance_net,
     node_color_by = "region",
     node_color_label = "",
     node_shape_by = "regime_type",
     node_shape_label = "",
     node_size_by = "degree",
     node_size_label = "Degree",
     node_fill = "white",
     edge_color = "grey50",
     edge_linewidth = 0.5,
     edge_alpha_label = 'Alliance Strength (scaled)',
     layout = "nicely",
     seed = 6886) +
  ggtitle("ATOP Network") +
  theme(legend.position = 'right')

visualizing homophily results

plot_homophily(attribute_results$homophily_analysis,
               type = "comparison") +
  labs(title = "Alliance Formation Patterns: Which Attributes Matter?",
       subtitle = "Homophily analysis summarizes similarity patterns in international cooperation")

visualizing centrality patterns

centrality_viz <- attribute_results$centrality_correlations |>
  filter(p_value < 0.1) |>
  mutate(
    significant = p_value < 0.05,
    node_var = factor(node_var),
    centrality_measure = factor(
      centrality_measure,
      levels = c("degree", "betweenness", "closeness"))
  )

if(nrow(centrality_viz) > 0) {
  ggplot(centrality_viz, aes(x = correlation, y = node_var, color = significant)) +
    geom_segment(
      aes(
        x = 0, xend = correlation, y = node_var, yend = node_var),
      size = 1) +
    geom_point(size = 3) +
    facet_wrap(~centrality_measure, ncol = 1) +
    geom_vline(xintercept = 0, linetype = "dashed", color = "gray50") +
    scale_color_manual(
        values = c("FALSE" = "gray60", "TRUE" = "#2E86AB"),
        labels = c("FALSE" = "Not significant", "TRUE" = "(p < 0.05)")
    ) +
    labs(title = "What Makes Countries Central in the Alliance Network?",
         subtitle = "Correlation between node attributes and centrality measures",
         x = "Correlation with Centrality",
         y = "Node Attribute",
         color = "") +
    theme_bw() +
    theme(
        panel.border = element_blank(),
        axis.ticks = element_blank(),
        legend.position = "top",
        panel.grid.major.y = element_blank()
    )
} else {
  no_centrality_msg <- "**No significant centrality correlations to visualize.**\n"
  cat(no_centrality_msg)
}

mixing matrix heatmap

We can use the plot_mixing_matrix() function to visualize the mixing patterns as a heatmap:

plot_mixing_matrix(
    regime_mixing,
    show_values = TRUE,
    diagonal_emphasis = TRUE,
    text_color_threshold=.9
  ) +
  labs(title = "Regime Type Alliance Patterns",
       subtitle = "How different regime types interact in the network",
       x = "Allied with...",
       y = "Regime type")

The heatmap shows the alliance patterns between different regime types. The diagonal cells (emphasized with black borders) represent within-type alliances, while off-diagonal cells show cross-type alliances. Darker blue indicates higher proportions of alliances, consistent with the small but detectable regime-type homophily in alliance formation.

9. working with longitudinal networks

A longitudinal network is a network observed over multiple time periods (one snapshot per period), in contrast to a cross-sectional network that captures a single point in time. The attribute analysis functions in netify work with longitudinal networks. The example below creates a longitudinal alliance network and runs the same analyses across multiple time periods.

creating a longitudinal network

# longitudinal alliance network for 2010-2014
alliance_net_longit <- netify(
  cow_dyads,
  actor1 = 'ccode1',
  actor2 = 'ccode2',
  time = 'year',
  symmetric = TRUE,
  weight = 'cooperation'
)

alliance_net_longit

adding attributes to longitudinal networks

# nodal data for all time periods
nodal_data_longit <- cow_dyads |>
  select(year, ccode1, region1, v2x_polyarchy1, log_gdp1, cinc1) |>
  distinct() |>
  rename(
    time = year,
    actor = ccode1,
    region = region1,
    democracy = v2x_polyarchy1,
    log_gdp = log_gdp1,
    mil_capability = cinc1
  ) |>
  mutate(
    regime_type = case_when(
      democracy >= 0.6 ~ "Democracy",
      democracy >= 0.4 ~ "Hybrid",
      democracy < 0.4 ~ "Autocracy",
      TRUE ~ "Unknown"
    )
  )

alliance_net_longit <- add_node_vars(
  alliance_net_longit,
  nodal_data_longit,
  actor = "actor",
  time = "time"
)

dyad_data_longit <- cow_dyads |>
  select(year, ccode1, ccode2, log_capdist, alliance_intensity, defense_alliance) |>
  rename(
    time = year,
    actor1 = ccode1,
    actor2 = ccode2,
    geographic_distance = log_capdist
  )

alliance_net_longit <- add_dyad_vars(
  alliance_net_longit,
  dyad_data = dyad_data_longit,
  actor1 = "actor1",
  actor2 = "actor2",
  time = "time",
  dyad_vars = c("geographic_distance", "alliance_intensity", "defense_alliance"),
  dyad_vars_symmetric = c(TRUE, TRUE, TRUE)
)

homophily analysis across time

democracy_homophily_longit <- homophily(
  alliance_net_longit,
  attribute = "democracy",
  method = "correlation",
  significance_test = TRUE
)

democracy_homophily_longit

##    net       layer attribute      method threshold_value homophily_correlation
## 1 2010 cooperation democracy correlation               0             0.1544304
## 2 2011 cooperation democracy correlation               0             0.1537066
## 3 2012 cooperation democracy correlation               0             0.1416613
## 4 2013 cooperation democracy correlation               0             0.1292206
## 5 2014 cooperation democracy correlation               0             0.1243436
##   mean_similarity_connected mean_similarity_unconnected similarity_difference
## 1                -0.2443592                  -0.3226836            0.07832443
## 2                -0.2409122                  -0.3181384            0.07722618
## 3                -0.2452764                  -0.3156274            0.07035096
## 4                -0.2504444                  -0.3140721            0.06362762
## 5                -0.2520272                  -0.3131798            0.06115268
##       p_value  ci_lower  ci_upper n_connected_pairs n_unconnected_pairs
## 1 0.000999001 0.1395075 0.1700197              3497               11381
## 2 0.000999001 0.1383078 0.1682909              3497               11554
## 3 0.000999001 0.1270532 0.1562951              3571               11480
## 4 0.000999001 0.1137038 0.1437508              3645               11406
## 5 0.000999001 0.1089608 0.1399076              3646               11405
##   n_missing n_pairs
## 1        22   14878
## 2        21   15051
## 3        21   15051
## 4        21   15051
## 5        21   15051

# summary of trends across time periods
homophily_trends <- democracy_homophily_longit |>
  group_by(net) |>
  summarise(
    avg_homophily = mean(homophily_correlation, na.rm = TRUE),
    significant = any(p_value < 0.05, na.rm = TRUE)
  )

knitr::kable(homophily_trends, 
             caption = "Democracy Homophily Trends Over Time",
             digits = 3)

Democracy Homophily Trends Over Time
net	avg_homophily	significant
2010	0.154	TRUE
2011	0.154	TRUE
2012	0.142	TRUE
2013	0.129	TRUE
2014	0.124	TRUE

visualizing longitudinal homophily

plot_homophily(democracy_homophily_longit, type = "temporal") +
  labs(title = "Democracy Homophily in Alliance Networks Over Time",
       subtitle = "Tendency for democracies to ally with other democracies")

If you want to see the distribution for a specific time period, you can extract that period first:

# extract 2012 slice for a single-period distribution plot
alliance_2012 <- subset(alliance_net_longit, time = '2012')
democracy_homo_2012 <- homophily(
  alliance_2012,
  attribute = "democracy",
  method = "correlation"
)

plot_homophily(democracy_homo_2012, alliance_2012,
               type = "distribution",
               attribute = "democracy",
               method = "correlation") +
  labs(subtitle = "Distribution of democracy similarity scores in 2012 alliance network")

mixing matrices over time

regime_mixing_longit <- mixing_matrix(
  alliance_net_longit,
  attribute = "regime_type",
  normalized = TRUE
)

# summary statistics across each time period
mixing_summary <- regime_mixing_longit$summary_stats |>
  select(net, assortativity, diagonal_proportion) |>
  mutate(across(where(is.numeric), \(x) round(x, 3)))

knitr::kable(mixing_summary,
             caption = "Regime Type Mixing Patterns Over Time",
             col.names = c("Year", "Assortativity", "Within-Type %"))

Regime Type Mixing Patterns Over Time
Year	Assortativity	Within-Type %
2010	0.121	0.404
2011	0.120	0.400
2012	0.108	0.394
2013	0.096	0.392
2014	0.094	0.389

dyadic correlations across time

geo_correlation_longit <- dyad_correlation(
  alliance_net_longit,
  dyad_vars = "geographic_distance",
  method = "pearson",
  significance_test = TRUE
)

geo_summary_longit <- geo_correlation_longit |>
  select(net, correlation, p_value, n_pairs) |>
    mutate(
      significant = ifelse(p_value < 0.05, "*", ""),
      correlation = round(correlation, 3),
      p_value = fmt_p(p_value)
    )

knitr::kable(geo_summary_longit,
             caption = "Geographic Distance and Alliance Formation Over Time",
             col.names = c("Year", "Correlation", "P-value", "N Dyads", "Sig."))

Geographic Distance and Alliance Formation Over Time
Year	Correlation	P-value	N Dyads	Sig.
2010	-0.363	< 0.001	18915	*
2011	-0.592	< 0.001	18915	*
2012	-0.586	< 0.001	18915	*
2013	-0.590	< 0.001	18915	*
2014	-0.590	< 0.001	18915	*

longitudinal attribute report

longit_attribute_results <- attribute_report(
  alliance_net_longit,
  node_vars = c("region", "regime_type", "democracy", "log_gdp"),
  dyad_vars = c("geographic_distance", "alliance_intensity"),
  include_centrality = TRUE,
  include_homophily = TRUE,
  include_mixing = TRUE,
  include_dyadic_correlations = TRUE,
  centrality_measures = c("degree", "betweenness"),
  significance_test = TRUE
)

if (!is.null(longit_attribute_results$homophily_analysis)) {
  longit_patterns <- longit_attribute_results$homophily_analysis |>
    filter(attribute == "democracy") |>
    select(net, homophily_correlation, p_value) |>
    mutate(
      trend = case_when(
        net == min(net) ~ "Start",
        net == max(net) ~ "End",
        TRUE ~ "Middle"
      )
    )

  longit_patterns
} else {
  knitr::asis_output(
    "Note: homophily for longitudinal networks is currently limited. For full coverage, analyze each time period separately.\n"
  )
}

##              net homophily_correlation     p_value  trend
## democracy.1 2010             0.1544304 0.000999001  Start
## democracy.2 2011             0.1537066 0.000999001 Middle
## democracy.3 2012             0.1416613 0.000999001 Middle
## democracy.4 2013             0.1292206 0.000999001 Middle
## democracy.5 2014             0.1243436 0.000999001    End

tl;dr

From homophily():
- Whether democracies truly form more alliances with each other
- Whether economic similarity is associated with alliance patterns
- The strength of regional clustering in alliance formation
From mixing_matrix():
- Detailed patterns of who forms alliances with whom
- Whether alliances cross regime type boundaries
- How different regions form alliances globally
From dyad_correlation():
- The role of geographic distance in shaping alliance formation
- How alliance types (defense, offense, etc.) cluster
- Which relationship factors matter most for alliances
From attribute_report():
- What attributes make countries central/influential
- Complete homophily patterns across all variables
- One view of the main network-attribute relationships

references

Leeds, B. A., Ritter, J. M., Mitchell, S. M., & Long, A. G. (2002). Alliance Treaty Obligations and Provisions, 1815–1944. International Interactions, 28(3), 237–260. DOI 10.1080/03050620213653.
Miller, S. V. (2022). peacesciencer: An R Package for Quantitative Peace Science Research. Conflict Management and Peace Science, 39(6), 755–779. DOI 10.1177/07388942221077926.

Cassy Dorff and Shahryar Minhas

2026-06-05