Social Influence Regression (SIR) Model

Fits a Social Influence Regression model for network data with social influence effects. The SIR model captures how network connections influence outcomes through bilinear interaction terms, allowing for both sender and receiver effects in directed networks.

The model decomposes network influence into two components:

• Sender influence (A matrix): A[i,k] measures how much node k's behavior (via X) shapes node i's outgoing ties.

• Receiver influence (B matrix): B[j,l] measures how node l's position shapes node j's incoming ties.

Usage

sir(
  Y,
  W = NULL,
  X = NULL,
  Z = NULL,
  family,
  method = "ALS",
  calc_se = TRUE,
  fix_receiver = FALSE,
  symmetric = FALSE,
  bipartite = NULL,
  kron_mode = FALSE,
  ...
)

Arguments

Y

A three-dimensional array of dimensions (m x m x T) containing the network outcomes. Y[i,j,t] represents the directed outcome from node i to node j at time t. Can contain NA values for missing observations. The diagonal (self-loops) can be included or excluded depending on the application.

W

Optional influence covariate array, either:

• 3D array (m x m x p): Static influence covariates. W[i,j,r] represents the r-th covariate for the edge from i to j. The same W is used for all time periods.

• 4D array (m x m x p x T): Dynamic (time-varying) influence covariates. W[i,j,r,t] allows the influence structure to change over time. Parameters (alpha, beta) are still estimated jointly across all periods, but the influence matrices A_t and B_t vary with t. Only ALS method is supported for 4D W.

Common choices include graph Laplacians, geographic distance matrices, or node-level covariates expanded to edge-level. If NULL or p=0, the model uses only identity matrices (no network influence structure).

X

Optional three-dimensional array of dimensions (m x m x T) representing the network state that carries influence. Typically this is a lagged version of Y (e.g., X[,,t] = Y[,,t-1]). If NULL and W is provided, an error is thrown. X determines which network patterns influence future outcomes.

Z

Optional array of exogenous covariates. Can be either:

• 3D array (m x m x T): Single covariate varying across edges and time

• 4D array (m x m x q x T): Multiple (q) covariates

Examples include dyadic covariates (trade agreements, geographic distance) or node-level attributes (GDP, population) expanded to edge-level.

family

Character string specifying the distribution family and link function. Must be one of "poisson", "normal", or "binomial". The choice depends on the nature of your outcome variable.

method

Character string specifying the estimation method. Either "ALS" (Alternating Least Squares) or "optim" (direct optimization via BFGS). Default is "ALS" which is generally more stable.

calc_se

Logical indicating whether to calculate standard errors for the parameters. Standard errors are computed using the observed information matrix. Setting to FALSE speeds up computation when uncertainty quantification is not needed.

fix_receiver

Logical. If TRUE, fixes B = I (identity matrix) and estimates only (theta, alpha). This eliminates the bilinear identification problem (scaling ambiguity between A and B) by removing the receiver influence channel. The model becomes a standard GLM, yielding proper standard errors. Appropriate when receiver effects are negligible. Default is FALSE.

symmetric

Logical. If TRUE, treats the network as undirected (symmetric). The function symmetrizes Y by averaging upper and lower triangles, uses only upper-triangle observations for fitting, and sets fix_receiver = TRUE (since sender/receiver distinction is meaningless for undirected networks). Default is FALSE.

bipartite

Logical or NULL. Indicates whether the network is bipartite (senders and receivers are distinct node sets). If NULL (the default), bipartite status is inferred from Y: non-square arrays (n1 != n2) are treated as bipartite. Set to TRUE explicitly for square arrays where senders and receivers are nonetheless distinct populations. Setting FALSE on a non-square Y raises an error. Bipartite networks require fix_receiver = TRUE.

kron_mode

Logical. If TRUE, replaces separate (alpha, beta) with a single p x p coefficient matrix C, where C[r,s] is the weight on W_r X W_s'. This is a general fix for the bilinear identification problem. Not yet implemented. Default is FALSE.

...

Additional arguments passed to the fitting functions:

• trace: Logical or integer controlling output verbosity.

• tol: Convergence tolerance for ALS (default 1e-8).

• max_iter: Maximum ALS iterations (default 100).

Value

An object of class "sir" with the following components:

summ

Data frame of parameter estimates with columns coef, se (classical SE), rse (robust/sandwich SE), t_se (z-statistic using classical SE), t_rse (z-statistic using robust SE). Row names identify each parameter.

A

Sender influence matrix. For static W: n1 x n1 matrix. For dynamic (4D) W: n1 x n1 x T array. Off-diagonal entry A[i,k] measures how much node k's behavior (via X) shapes node i's outgoing ties. Diagonal is set to zero.

B

Receiver influence matrix. Same dimensions as A. Off-diagonal entry B[j,l] measures how node l's position shapes node j's incoming ties. Identity when fix_receiver = TRUE. Diagonal is zeroed.

tab

Numeric vector of all estimated parameters in order: [theta_1, ..., theta_q, alpha_2, ..., alpha_p, beta_1, ..., beta_p]. When fix_receiver = TRUE: [theta_1, ..., theta_q, alpha_1, ..., alpha_p].

theta

Coefficients for exogenous covariates Z (length q).

alpha

Full alpha vector including the fixed alpha_1 = 1 (length p). When fix_receiver = TRUE, all alpha are free.

beta

Coefficients for receiver influence covariates (length p). Empty when fix_receiver = TRUE.

ll

Log-likelihood at convergence.

family

The distribution family used ("poisson", "normal", or "binomial").

method

The estimation method used ("ALS" or "optim").

p

Number of influence covariates in W.

q

Number of exogenous covariates in Z.

m

Number of sender nodes (same as n1).

n1

Number of sender (row) nodes.

n2

Number of receiver (column) nodes.

bipartite

Logical, TRUE if the network is bipartite (n1 != n2).

n_periods

Number of time periods.

nobs

Number of non-missing observations used in estimation.

fitted.values

Array (n1 x n2 x T) of fitted values on the response scale (counts for Poisson, probabilities for binomial, means for normal).

residuals

List with three components: response (Y - fitted), pearson (standardized by variance function), and deviance (signed square root of deviance contributions).

vcov

Variance-covariance matrix of parameters from the Hessian (classical SEs). NULL if calc_se = FALSE.

vcov_robust

Sandwich (robust) variance-covariance matrix. NULL if calc_se = FALSE or computation failed.

Y

The outcome array as used in fitting (with NAs from symmetric masking or Z missingness applied).

W

The influence covariate array.

X

The network state array (NAs replaced with 0).

Z

The exogenous covariate array (converted to 4D if 3D input).

fix_receiver

Logical, whether receiver effects were fixed.

symmetric

Logical, whether the network was treated as undirected.

kron_mode

Logical, whether Kronecker mode was used.

iterations

Number of iterations until convergence.

history

List with matrices ALPHA, BETA, THETA, DEV tracking parameter trajectories across iterations (useful for convergence diagnostics).

convergence

Logical, TRUE if the algorithm converged.

call

The matched function call.

sigma2

Estimated error variance (only for family = "normal").

Details

The SIR model specifies the expected outcome for the directed edge from node i to node j at time t as:

$$\mu_{i,j,t} = \theta^T z_{i,j,t} + \sum_{k,l} X_{k,l,t} A_{i,k} B_{j,l}$$

Where:

• \(\mu_{i,j,t}\) is the expected value of the outcome Y_ijt

• \(\theta\) is a q-dimensional vector of coefficients for exogenous covariates

• \(z_{i,j,t}\) is a q-dimensional vector of exogenous covariates

• \(X_{k,l,t}\) represents the network state (often lagged Y) that carries influence

• \(A_{i,k}\) represents how node i is influenced by the behavior of node k

• \(B_{j,l}\) represents how node j's reception is affected by node l's position

The bilinear term \(\sum_{k,l} X_{k,l,t} A_{i,k} B_{j,l}\) captures network influence and can be parameterized using influence covariates W through:

• \(A = \sum_{r=1}^{p} \alpha_r W_r\) (sender effects, \(\alpha_1 = 1\) fixed)

• \(B = \sum_{r=1}^{p} \beta_r W_r\) (receiver effects)

This parameterization reduces the number of parameters from \(O(m^2)\) to \(O(p)\), where \(p \ll m\).

Estimation Methods

Alternating Least Squares (ALS):

• Iteratively optimizes A given B, then B given A

• Generally more stable for high-dimensional problems

• Better for sparse networks or when p is large

• May converge to local optima

Direct Optimization (optim):

• Uses BFGS to optimize all parameters simultaneously

• Can be faster for small problems

• May provide better solutions when good starting values are available

• More prone to numerical issues in high dimensions

Distribution Families

Poisson: For count data (e.g., number of interactions)

• Link function: log

• Variance function: \(V(\mu) = \mu\)

• Use when: Y_ijt represents counts

Normal: For continuous data (e.g., trade volumes, distances)

• Link function: identity

• Variance function: \(V(\mu) = \sigma^2\)

• Use when: Y_ijt is continuous and approximately normal

Binomial: For binary data (e.g., presence/absence of ties)

• Link function: logit

• Variance function: \(V(\mu) = \mu(1 - \mu)\)

• Use when: Y_ijt is binary (0/1)

References

Minhas, S. & Hoff, P. D. (2025). Social Influence Regression. Political Analysis.

Examples

# \donttest{
set.seed(123)
m <- 8; T_len <- 5; p <- 2
Y <- array(rpois(m * m * T_len, lambda = 2), dim = c(m, m, T_len))
X <- array(0, dim = c(m, m, T_len))
X[,,2:T_len] <- Y[,,1:(T_len - 1)]
W <- array(rnorm(m * m * p), dim = c(m, m, p))
model <- sir(Y = Y, W = W, X = X, family = "poisson",
             method = "ALS", calc_se = FALSE, max_iter = 10)
print(model)
#> 
#> Social Influence Regression Model
#> 8 nodes, 5 time periods (directed)
#> Config: poisson | ALS
#> Status: converged | N = 280 | Log-Lik: -576.96 | AIC: 1159.9
#> Coefficients:
#>             Estimate
#> (alphaW) W2   3.9283
#> (betaW) W1    0.0010
#> (betaW) W2    0.0020
#> (SEs not computed)
#> Use `summary()` for detailed results
coef(model)
#> [1] 3.928314350 0.001010846 0.001972906
# }