Evaluating network interventions requires understanding how treatment affects people through their social relationships. Counting treated neighbors without distinguishing supportive and antagonistic ties can conceal opposing influences. We define effects through positive and negative ties, their interaction, and a sign-composition effect of reallocating treatment between the two types at a fixed total, and give their identification formulas. Under sign-blind assignment, we show how ignoring signs mixes the effects of the two tie types. We propose SiDE (Signed-exposure Doubly robust Estimator), which combines sign-specific outcome models with exposure probabilities induced by individual treatment assignment. We establish double robustness of its score and assess approximate intervals that account for overlapping neighborhoods. Semi-synthetic experiments on six real signed networks demonstrate accurate effect estimation and examine the limits of interval coverage. An exploratory reanalysis of published school-experiment data yields a positive estimate of the peer effect through spend-time ties on wristband wearing, but the intervals for all four effects include zero after adjustment for multiple comparisons. This framework can inform network intervention design by showing when influences through the two tie types reinforce or offset one another.
Figures & tables
Figure 1: Exposure patterns with four treated neighbors. The grid counts treated trusted neighbors horizontally and treated avoided neighbors vertically. Mean outcome changes along horizontal and vertical arrows define the positive- and negative-tie effects. Their interaction subtracts the solid horizontal increment from the dotted one. The sign-composition effect compares outcomes under exposures A= (3,1) and B= (4,0) .
Figure 2: Causal graph for one ego. Shaded nodes are the numbers of treated neighbors within Ni+ and within Ni− .
Slashdot
Epinions
Reddit
Wiki-Elec
Method
τ+
τ−
τint
κ
τ+
τ−
τint
κ
τ+
τ−
τint
κ
τ+
τ−
τint
κ
Sat. reg.
0.059 ± 0.033
0.123 ± 0.048
0.068 ± 0.060
0.180 ± 0.042
0.056 ± 0.021
0.123 ± 0.048
0.044 ± 0.049
0.178 ± 0.055
0.048 ± 0.034
0.139 ± 0.120
0.148 ± 0.078
0.156 ± 0.108
0.064 ± 0.041
0.157 ± 0.089
0.143 ± 0.075
0.208 ± 0.128
GPS
0.100 ± 0.065
1.532 ± 0.074
0.270 ± 0.066
1.579 ± 0.077
0.109 ± 0.032
2.191 ± 0.065
0.405 ± 0.090
2.331 ± 0.064
0.880 ± 0.193
2.490 ± 0.243
0.384 ± 0.153
2.540 ± 0.297
0.154 ± 0.079
0.717 ± 0.167
0.235 ± 0.146
0.760 ± 0.154
TARNet
0.042 ± 0.018
0.045 ± 0.029
0.048 ± 0.029
0.099 ± 0.030
0.019 ± 0.010
0.053 ± 0.017
0.170 ± 0.029
0.082 ± 0.031
0.032 ± 0.026
0.532 ± 0.083
0.298 ± 0.041
0.473 ± 0.081
0.046 ± 0.033
0.065 ± 0.038
0.161 ± 0.023
0.191 ± 0.063
CFR
0.032 ± 0.017
0.036 ± 0.024
0.039 ± 0.023
0.108 ± 0.038
0.017 ± 0.011
0.017 ± 0.010
0.106 ± 0.028
0.075 ± 0.024
0.036 ± 0.027
0.319 ± 0.072
0.240 ± 0.037
0.324 ± 0.066
0.043 ± 0.032
0.098 ± 0.064
0.176 ± 0.033
0.185 ± 0.088
SAGE-HSIC
0.134 ± 0.053
0.097 ± 0.042
0.162 ± 0.096
0.259 ± 0.057
0.073 ± 0.026
0.226 ± 0.047
0.096 ± 0.081
0.298 ± 0.069
0.440 ± 0.074
0.312 ± 0.279
0.245 ± 0.182
0.808 ± 0.312
0.131 ± 0.029
0.148 ± 0.090
0.107 ± 0.070
0.288 ± 0.099
Table 1: MAE on four networks for methods that use tie signs, with standard deviations of the absolute error below. Bold and underline mark the best and second-best values.
Figure 3: Effects with and without tie signs. Markers show mean estimates, bars show standard deviations, and crosses show true effects.
(a) MAE and standard deviation
Slashdot
Epinions
Reddit
Wiki-Elec
Variant
τ+
τ−
τint
κ
τ+
τ−
τint
κ
τ+
τ−
τint
κ
τ+
τ−
τint
κ
SiDE
0.013 ± 0.009
0.049 ± 0.028
0.085 ± 0.045
0.064 ± 0.034
0.012 ± 0.008
0.014 ± 0.009
0.032 ± 0.033
0.027 ± 0.018
0.042 ± 0.031
0.118 ± 0.075
0.134 ± 0.104
0.128 ± 0.083
0.042 ± 0.022
0.060 ± 0.052
0.195 ± 0.135
0.093 ± 0.097
w/o weighting
0.015 ± 0.004
0.035 ± 0.021
0.053 ± 0.046
0.050 ± 0.023
0.012 ± 0.009
0.017 ± 0.011
0.060 ± 0.038
0.035 ± 0.022
0.040 ± 0.027
0.073 ± 0.055
0.112 ± 0.089
0.143 ± 0.060
0.033 ± 0.025
0.086 ± 0.074
0.121 ± 0.065
0.167 ± 0.117
w/o outcome
0.114 ± 0.109
0.156 ± 0.145
0.218 ± 0.165
0.248 ± 0.293
0.060 ± 0.028
0.147 ± 0.102
0.233 ± 0.273
0.156 ± 0.116
0.238 ± 0.108
0.829 ± 0.674
0.507 ± 0.436
0.991 ± 0.557
0.115 ± 0.053
0.243 ± 0.204
0.369 ± 0.400
0.319 ± 0.194
w/o PB
0.025 ± 0.024
0.138 ± 0.103
0.204 ± 0.132
0.106 ± 0.075
0.014 ± 0.009
0.042 ± 0.023
0.186 ± 0.080
0.091 ± 0.066
0.189 ± 0.161
0.714 ± 0.858
0.842 ± 0.890
0.378 ± 0.397
0.131 ± 0.294
0.164 ± 0.120
0.436 ± 0.302
0.281 ± 0.112
Table 2: Ablation results on four networks: (a) MAE with standard deviations of the absolute error below and (b) bias. The variant w/o weighting drops the residual correction, w/o outcome drops the outcome model, and w/o PB predicts joint-state probabilities with a softmax model. Bold and underline mark the smallest and second-smallest MAE in (a) and absolute bias in (b).
Network
τ+
τ−
τint
κ
Coverage
95% CP interval
Coverage
95% CP interval
Coverage
95% CP interval
Coverage
95% CP interval
Slashdot
0.930
[0.885, 0.961]
0.955
[0.916, 0.979]
0.965
[0.929, 0.986]
0.945
[0.904, 0.972]
Epinions
0.905
[0.856, 0.942]
0.930
[0.885, 0.961]
0.965
[0.929, 0.986]
0.965
[0.929, 0.986]
Reddit
0.850
[0.793, 0.896]
0.935
[0.891, 0.965]
0.945
[0.904, 0.972]
0.985
[0.957, 0.997]
Wiki-Elec
0.960
[0.923, 0.983]
0.920
[0.873, 0.954]
0.970
[0.936, 0.989]
0.940
[0.898, 0.969]
Table 3: Coverage over 200 replications with 95% Clopper–Pearson (CP) confidence intervals. Underlining marks coverage below 0.919, two binomial standard errors below 0.95.
Peer effects are difficult to estimate when interaction graphs evolve because pre-assignment network history, dynamic peer exposure, and post-assignment network change have distinct causal roles. We introduce a controlled contrast framework that indexes potential outcomes by own treatment, temporally aggregated peer exposure, and a post-assignment evolution summary. Differences between the resulting means define own-treatment, peer-exposure, controlled network-evolution, and joint controlled contrasts rather than a mediation decomposition. We develop the Dynamic Network Doubly Robust estimator, DynaNet-DR, which combines a temporally factorized propensity with normalized augmentation. Under consistency, summary sufficiency, sequential exchangeability, positivity, nuisance convergence, and weak dependence, its canonical estimator is consistent when either the outcome regression or the propensity estimator is consistent. The reported implementation adds representative-score prediction, fixed clipping, and finite-sample stabilization. Semi-synthetic benchmarks on fixed real temporal graph sequences show favorable estimation accuracy among methods targeting the full profile. These benchmarks assess summary-indexed contrasts rather than counterfactual edge generation, and the MathOverflow study is an observational illustration under the stated assumptions.
Estimating causal effects under network interference typically assumes that the network used for training and the network used for deployment coincide. In practice, an intervention is run on one population while the question of interest concerns a different population, and the two generally differ in topology, node-covariate composition, and spillover pathways. Transporting a causal effect across networks is therefore a data-fusion problem that no existing algorithm solves. We employ a selection diagram, extended to the network setting so that covariate shift and structural network shift enter as separate selectors, and derive from it a transport formula for the direct, spillover, and total effects in the deployment population. Each formula makes explicit which interventional mechanism is assumed invariant and which observational distribution must be reweighted. We then turn the formulas into TranCE (Transported Causal Effects), a doubly-robust algorithm combining an interventional outcome model, a domain density-ratio correction, and cross-fitted inference. Extensive experiments on two semi-synthetic benchmarks derived from real-world social networks and on a fully real weather-insurance field experiment, where the transported effects are checked against held-out randomized estimates, confirm the effectiveness of our approach. Our findings have the potential to improve intervention strategies in networked systems, particularly in social networks and public health.
Xiaojing Du, Jiuyong Li, Lin Liu +3
Adelaide University, Adelaide, Australia · Hainan University, Haikou, China
Estimating individual treatment effect (ITE) from observational graph data is crucial for decision-making in the fields such as commerce and medicine. This task is challenging due to interference, where individual outcomes can be influenced by the treatments and covariates of their neighbors. Existing methods attempt to model such interference for accurate ITE estimation. However, a critical issue is often overlooked: differentiated networked effect (DNE), an effect caused by local networks consisting of neighbors with varying importance and scales. Capturing DNE is vital; otherwise, we will end up with imprecise ITE estimation due to an erroneous characterization of interference, which can result in misguided decisions. To address this challenge, we propose a novel interference modeling mechanism that incorporates two partial attention mechanisms and a message amplifier. The partial attention mechanisms automatically estimate the importance of different neighbors in contributing to interference, while the message amplifier adjusts the results of the interference modeling mechanism based on the scale of neighbors, all of which enables the model to capture DNE. Experiments on three real-world graphs demonstrate that our methods outperform existing approaches for ITE estimation from graph data, which corroborates the importance of explicitly capturing DNE.
Xiaofeng Lin, Han Bao, Hisashi Kashima
Kyoto University Kyoto, Japan · The Institute of Statistical Mathematics Tokyo, Japan · Tohoku University Sendai, Japan +1