Social Relations Models for Network Analysis

Package Overview

This vignette introduces the srm package, which estimates social relations models for network data. It breaks down and analyzes networks into nodal and dyadic dependencies and allows users to understand actor, partner, and dyadic effects in network data (Dorff and Ward, 2013; Dorff and Minhas, 2017).

What users can do:

Identify the most active actors in the network - both in terms of sender and receiver patterns. Users can identify which actors are more active than expected after accounting for overall network structure and partner characteristics.
Identify overall network patterns where users can separate systematic actor behaviors from relationship-specific effects.
Generate specific summary statistics for the network data that can reveal whether active actors are also active receivers.
Create visualizations to visualize actor behaviors and their relationship-specific patterns.

In comparison to netify, which provides general network statistics and visualizations, the srm package helps users break down network relationships allowing for deeper analysis. With srm, users are able to identify which actors are more or less active than typical and helps users explore why, relationally, some connections are stronger or weaker than expected. While netify focuses on overall network patterns, srm centers on explaining the individual behaviors and relationships that drive those patterns.

Note: This vignette uses symmetric (undirected) network data, which constrains the SRM decomposition. If you have directed data where sender and receiver roles are distinct, the pipeline vignette provides a more complete walkthrough with all five SRM components. For bipartite (two-mode) networks, see the bipartite vignette.

Load Packages

library(netify)
library(srm)

Prepare Data

We will make use of data from ATOP (Alliance Treaty Obligations and Provisions) to demonstrate how to use the srm package. The data can be downloaded at http://www.atopdata.org. For this tutorial, we have selected the atop5_1dy dataset and focus on military security consultation agreements from 2010 to 2018 involving countries in the broader East and Central Asian region. The dataset includes 18 countries: the East Asian core (China, India, Japan, South Korea, North Korea, Philippines, Thailand, Vietnam) along with their consultation treaty partners (Armenia, Kazakhstan, Kyrgyzstan, Pakistan, Russia, Tajikistan, Turkmenistan, Ukraine, USA, Uzbekistan).

data("atop_EA")
head(atop_EA)
  year country1 country2
1 2010      USA      KOR
2 2011      USA      KOR
3 2012      USA      KOR
4 2013      USA      KOR
5 2014      USA      KOR
6 2015      USA      KOR

We will make use of the netify package to make a network object from the ATOP data. We will create two network objects, one for cross-sectional data and the other for longitudinal data as examples for this vignette. Note that these consultation agreements are symmetric (undirected) – if country A has a consultation agreement with country B, the reverse is also true.

# cross-sectional
atop_EA_short = netify::netify(
  input = atop_EA,
  actor1 = 'country1',
  actor2 = 'country2',
  symmetric = TRUE,
  sum_dyads = TRUE,
  diag_to_NA = TRUE,
  missing_to_zero = TRUE
)

# longitudinal
atop_EA_long = netify::netify(
  input = atop_EA,
  actor1 = 'country1',
  actor2 = 'country2',
  time = 'year',
  symmetric = TRUE,
  sum_dyads = TRUE,
  diag_to_NA = TRUE,
  missing_to_zero = TRUE
)

See ?netify::netify for details on the parameters used above. The netify function creates a network object that can be used with the srm package. The resulting object is a matrix for cross-sectional data and a list of matrices for longitudinal data, where each matrix represents the network at a specific time point.

A note on symmetric networks: Because these consultation agreements are undirected, the resulting matrices are symmetric. In symmetric networks, actor effects and partner effects are identical by construction – a country’s tendency to “send” ties is the same as its tendency to “receive” them. The SRM is most informative for directed (asymmetric) networks where sender and receiver roles are distinct. We use this symmetric example here for illustration, but see the classroom dataset for an example with asymmetric data where the full SRM decomposition is more revealing.

SRM Summary Statistics

Now that we have our network objects, we can analyze relationships in the network via the srm package. Let’s begin by generating summary statistics for the network data. We will look at the row means and actor-partner covariance for the cross-sectional data.

sort(srm_stats(atop_EA_short, type = "rowmeans"), decreasing = TRUE)
 [36mℹ [39m Converting  [34m [34m<netify> [34m [39m object to standard matrices.
      CHN       IND       RUS       USA       KAZ       KGZ       TJK       UZB 
3.7647059 2.4705882 2.1176471 2.1176471 0.7647059 0.7647059 0.7647059 0.7647059 
      ARM       JPN       KOR       PHL       PRK       THA       VNM       PAK 
0.5294118 0.5294118 0.5294118 0.5294118 0.5294118 0.5294118 0.5294118 0.4705882 
      UKR       TKM 
0.3529412 0.2941176

The row means immediately reveal the most active countries: China (3.76) and India (2.47) have far more consultation agreements on average than countries like Turkmenistan (0.29) or Ukraine (0.35). Note the sharp drop-off after the top four (China, India, Russia, USA), with most countries clustered between 0.3 and 0.8.

srm_stats(atop_EA_short, type = "actor_partner_cov")
 [36mℹ [39m Converting  [34m [34m<netify> [34m [39m object to standard matrices.
[1] 0.6474946

The actor-partner covariance of 0.65 is positive, but this is a direct consequence of the network being symmetric: because every consultation tie is mutual, actor and partner effects are identical and the actor-partner covariance equals the actor variance by construction. In a directed network, this covariance would reveal whether active senders also tend to be active receivers; here, that question is not separable from general activity.

Let’s make use of the time argument and look at the column means for select years in the longitudinal data. New treaties entered the ATOP data between 2012 and 2015, so we select years that span these changes.

yrs = c("2010", "2013", "2015")

col_means = srm_stats(atop_EA_long, type = "colmeans", time = yrs)
 [36mℹ [39m Converting  [34m [34m<netify> [34m [39m object to standard matrices.

With 18 countries across 3 years, the full output is large. We can compare a few key countries:

focus = c("CHN", "IND", "RUS", "USA")
sapply(col_means, function(x) round(x[focus], 2))
    2010 2013 2015
CHN 0.29 0.35 0.53
IND 0.12 0.12 0.47
RUS 0.24 0.24 0.24
USA 0.24 0.24 0.24

China’s consultation activity increases from 0.29 in 2010 to 0.35 in 2013 and 0.53 by 2015, reflecting new agreements entering the dataset. India holds steady at 0.12 through 2013 and then jumps to 0.47 in 2015. Russia and the USA remain stable at 0.24 throughout.

SRM Effects

We can analyze the actor, partner, and unique effects in the network data using the srm_effects() function. The actor effect for observation i captures how much that actor deviates from the network average as a sender of ties. It is calculated as:

$\hat{a}_i = \frac{(n-1)^2}{n(n-2)} X_{i.} + \frac{(n-1)}{n(n-2)} X_{.i} - \frac{n-1}{n-2} \bar{X}$

where $X_{i.}$ is the row mean for actor i, $X_{.i}$ is the column mean, and $\bar{X}$ is the overall network mean.

Similarly, the partner effect shows how much actor i tends to receive ties above or below average:

$\hat{b}_i = \frac{(n-1)^2}{n(n-2)} X_{.i} + \frac{(n-1)}{n(n-2)} X_{i.} - \frac{n-1}{n-2} \bar{X}$

We look at the actor effects for the cross-sectional data:

EA_actor = srm_effects(atop_EA_short, type = "actor")
 [36mℹ [39m Converting 'netify' matrix to standard matrix.
sort(EA_actor[, 1], decreasing = TRUE)
       CHN        IND        RUS        USA        KAZ        KGZ        TJK 
 2.9166667  1.5416667  1.1666667  1.1666667 -0.2708333 -0.2708333 -0.2708333 
       UZB        ARM        JPN        KOR        PHL        PRK        THA 
-0.2708333 -0.5208333 -0.5208333 -0.5208333 -0.5208333 -0.5208333 -0.5208333 
       VNM        PAK        UKR        TKM 
-0.5208333 -0.5833333 -0.7083333 -0.7708333

The actor effects show how much an actor deviates from the network average in security consultation pacts. China has the highest positive actor effect (+2.92), meaning it engages in security consultation pacts far more than the network average. Other countries such as Thailand, Japan, and Korea have negative effects (around -0.52), suggesting they participate below the network average.

We can also look at the unique effects. The unique dyadic effect is the residual after removing both actor and partner tendencies:

$\hat{g}_{ij} = X_{ij} - \hat{a}_i - \hat{b}_j - \bar{X}$

We run this from the srm_effects function with type = "unique".

EA_unique = srm_effects(atop_EA_short, type = "unique")
 [36mℹ [39m Converting 'netify' matrix to standard matrix.

The unique effects are an 18x18 matrix. Rather than printing the full matrix, we can identify the strongest bilateral relationships by sorting the off-diagonal values:

g_vec = EA_unique[row(EA_unique) != col(EA_unique)]
names(g_vec) = outer(
  rownames(EA_unique), colnames(EA_unique), paste, sep = "-"
)[row(EA_unique) != col(EA_unique)]
head(sort(g_vec, decreasing = TRUE), 6)
 USA-JPN  USA-KOR  USA-PHL  RUS-PRK  PRK-RUS  VNM-RUS 
7.334559 7.334559 7.334559 7.334559 7.334559 7.334559

The top unique effects all have the same value (+7.3) because the ATOP consultation data is binary and symmetric: once actor and partner tendencies are removed, the remaining dyadic effects are strongly constrained. The pairs highlighted — USA-Japan, USA-Korea, USA-Philippines, and Russia-North Korea — are the alliances that exist despite both partners having relatively low overall activity. In a directed, continuous-valued network, unique effects would show more differentiation across dyads; see the pipeline vignette for such an example.

For the longitudinal data, we extract actor effects for selected years. Because the network is symmetric, actor and partner effects are identical, so we focus on actor effects only.

EA_actor_long = srm_effects(atop_EA_long, type = "actor", time = yrs)
 [36mℹ [39m Converting one or more 'netify' matrices in the list to standard matrices.

We can compare China and India’s actor effects across these three years to see how their consultation activity evolved:

sapply(EA_actor_long, function(x) round(x[c("CHN", "IND"), 1], 2))
    2010 2013 2015
CHN 0.22 0.28 0.41
IND 0.03 0.03 0.35

China’s actor effect grows from +0.22 in 2010 to +0.41 by 2015 as new agreements enter the data. India starts near the network average in 2010 (+0.03) and jumps to +0.35 in 2015, reflecting the same wave of new consultation agreements visible in the column means above.

Visualization

We can visualize the actor, partner, or dyadic effects using the srm_plot function. The actor/partner effect plots show actors sorted by the absolute magnitude of their effects, with positive effects in dark bars and negative effects in light gray bars. The default number of actors shown is the top 10, adjustable via the n argument.

Things to note:

Users need to put in results from the srm_effects function as input to the srm_plot function.
For undirected networks, use type = "actor" when plotting since partner effects are identical.

We look at the actor effects for the cross-sectional data:

srm_plot(EA_actor, type = "actor", n = 9)

Positive effects are shown in dark bars (right side) and negative effects in light gray bars (left side). China has the highest positive effect, engaging in military consultation obligations significantly more than the network average. Countries such as Turkmenistan have the strongest negative effects.

We can also look at the dyadic effects. When users supply the unique effects, the function returns a heatmap with a diverging color scale. Dark cells indicate stronger-than-expected relationships (positive unique effects), blue cells indicate weaker-than-expected relationships (negative unique effects), and white cells are close to what we would predict from actor tendencies alone.

srm_plot(EA_unique, type = "dyadic", n = 12)

The heatmap reveals which bilateral relationships are unusually strong or weak after controlling for each actor’s general tendencies. For example, the USA has strong military consultation relationships with Japan, Korea, and Thailand beyond what their general activity levels would predict. Most cells are white, showing that many bilateral relationships are close to what we would expect.

For the longitudinal data, we can set facet = TRUE to compare actor effects across time periods:

srm_plot(EA_actor_long, type = "actor", facet = TRUE, time = yrs, n = 8)

The faceted plot shows how actor effects evolve over time. China consistently has the highest positive actor effect. India’s effect grows across the periods as new consultation agreements enter the data. Other countries with light gray bars have negative effects, suggesting below-average consultation activity.

Using the Unified Interface

The examples above use the original component functions (srm_effects, srm_stats, srm_plot). The package also provides a unified srm() function that computes everything at once, returning a single object with print, summary, and plot methods. The srm() approach is more concise for a typical analysis:

fit = srm(atop_EA_short)
 [36mℹ [39m Converting  [34m [34m<netify> [34m [39m object to standard matrices.
 [36mℹ [39m NAs found in input; replacing with zero.
summary(fit)
Social Relations Model - Variance Decomposition
================================================== 

Component                Variance  % Total
------------------------------------------ 
Actor                      0.6475     9.0%
Partner                    0.6475     9.0%
Unique                     5.8607    81.9%
Relationship (cov)         5.8607       --
Actor-Partner (cov)        0.6475       --

Because this is a symmetric (undirected) network, the variance decomposition is heavily constrained: actor and partner variance are equal (both 9%), while unique variance dominates at 82%. The relationship covariance equals the unique variance and the actor-partner covariance equals the actor variance, all by construction. The SRM cannot distinguish sender behavior from receiver behavior here.

plot(fit, type = "variance")

The lopsided 9% / 9% / 82% split is typical for symmetric networks and reflects the structural constraints rather than a substantive finding. For directed networks, the decomposition is far more informative. The classroom dataset (a directed friendship network) produces a 42% / 26% / 32% split across distinct actor, partner, and unique components, with sender tendencies dominating. See the pipeline vignette for a complete directed-data walkthrough.

Conclusion

This vignette demonstrated the core srm workflow using ATOP consultation data: computing row and column means, decomposing actor and dyadic effects, and visualizing the results. China and India stand out as the most active consultation partners, and the strongest relationship-specific effects belong to alliances like USA-Japan and USA-Korea. Because this dataset is symmetric (undirected), the SRM decomposition is structurally constrained — actor and partner effects are identical and the variance partition is dominated by unique effects. For directed-data examples where all five SRM components are distinct, see the pipeline and methodology vignettes.

References:

Dorff, Cassy, and Michael D. Ward. (2013) Networks, Dyads, and the Social Relations Model. Political Science Research Methods 1(2):159-178.

Dorff, Cassy, and Shahryar Minhas. (2017). When Do States Say Uncle? Network Dependence and Sanction Compliance. International Interactions 43(4): 563-588.

Leeds, Brett Ashley, Jeffrey M. Ritter, Sara McLaughlin Mitchell, and Andrew G. Long. 2002. Alliance Treaty Obligations and Provisions, 1815-1944. International Interactions 28: 237-260.

Cassy Dorff, Shahryar Minhas, and Tosin Salau

2026-03-29