Network Effect Decomposition for Causal Effect Estimation under Network Interference

Zhirong Huang; Wenzhen Zhang; Guangquan Lu; Boyan Chen; Debo Cheng; Shichao Zhang

stat.MLNov 5, 2026

Network Effect Decomposition for Causal Effect Estimation under Network Interference

Authors: Zhirong Huang, Wenzhen Zhang, Guangquan Lu, Boyan Chen, Debo Cheng, Shichao Zhang

Organizations: School of Computer Science and Engineering, Guangxi Normal University; the Guangxi Key Lab of MIMS; and the MOE Key Lab of EBIT, Guilin, Guangxi 541004, China · School of Computer Science and Technology, Hainan University, Haikou, Hainan 570228, China

Abstract

Estimating causal effects from observational network data requires accounting for both individual treatment and neighborhood exposure. Existing outcome models commonly incorporate these intervention variables into a joint predictor and estimate causal effects through prediction contrasts, leaving the corresponding response components implicit. We propose the $\underline{N}$ etwork $\underline{E}$ ffect $\underline{D}$ ecomposition $\underline{Net}$ work (NEDNet), which parameterizes the conditional outcome response as a query-invariant pre-treatment baseline plus separate treatment- and exposure-response pathways. Path-specific neighborhood contextualization supplies local pre-treatment information to the two pathways. We characterize the fitted response through a unique reference-normalized decomposition, which is equivalent to a zero mixed treatment--exposure contrast. Experiments on semi-synthetic benchmarks derived from BlogCatalog and Flickr show that NEDNet achieves the lowest PEHE among the compared methods for individual, peer, and total effects across homogeneous and covariate-heterogeneous response settings.

Figures & tables

Fig. 1: Illustrative causal graph for two connected units

i

and

j

under network interference. Here,

x

t

, and

y

denote pre-treatment covariates, treatment, and outcome, respectively. Black arrows denote within-unit causal relationships, whereas red arrows denote spillover effects from one unit’s treatment to the other unit’s outcome.

Fig. 2: Overview of the proposed NEDNet framework. Given the network structure

A

, pre-treatment covariates

X

, individual treatment

T

, and neighborhood exposure

Z

, NEDNet first learns a graph-dependent context representation

H

and a compact covariate representation

X^{w}

. Treatment and exposure are then processed by separate response encoders and contextualized through separate graph-aggregation operations to obtain

(R^{T},\bar{R}^{T})

and

(R^{Z},\bar{R}^{Z})

. Finally, the pre-treatment baseline and two intervention-specific response components are integrated to predict factual or queried potential outcomes, from which the IE, PE, and TE are estimated.

Fig. 3: Counterfactual outcome estimation performance (CF-MSE) under different treatment Flip Rates on BC and Flickr with the homogeneous outcome-generation setting. From left to right, the panels show within-sample and out-of-sample results on BC, followed by those on Flickr. Lower values indicate better performance.

Datasets	Effects	TARNet+Z	CFR+Z	NetDeconf+Z	NetEst	RRNet	TNet	gDIS	NEDNet
BC (Within-sample)	Ind.	$0.2000_{\pm 0.0674}$	$0.1506_{\pm 0.0678}$	$0.2690_{\pm 0.1379}$	$\underline{0.0782_{\pm 0.0249}}$	$0.0972_{\pm 0.0915}$	$0.0809_{\pm 0.0360}$	$0.1287_{\pm 0.0882}$	$\mathbf{0.0472_{\pm 0.0319}}$
	Peer	$0.4977_{\pm 0.0520}$	$0.3854_{\pm 0.0857}$	$0.4962_{\pm 0.1162}$	$0.0782_{\pm 0.0336}$	$\underline{0.0501_{\pm 0.0248}}$	$0.0823_{\pm 0.0337}$	$0.0922_{\pm 0.0474}$	$\mathbf{0.0251_{\pm 0.0098}}$
	Total	$0.7855_{\pm 0.1743}$	$0.5132_{\pm 0.1986}$	$0.6457_{\pm 0.2817}$	$0.1871_{\pm 0.0898}$	$0.1682_{\pm 0.0843}$	$\underline{0.0860_{\pm 0.0464}}$	$0.2428_{\pm 0.1805}$	$\mathbf{0.0539_{\pm 0.0292}}$
BC (Out-of-sample)	Ind.	$0.2058_{\pm 0.0686}$	$0.1521_{\pm 0.0683}$	$0.2635_{\pm 0.1365}$	$\underline{0.0706_{\pm 0.0278}}$	$0.0957_{\pm 0.0951}$	$0.0819_{\pm 0.0356}$	$0.1308_{\pm 0.0880}$	$\mathbf{0.0504_{\pm 0.0342}}$
	Peer	$0.4958_{\pm 0.0550}$	$0.3853_{\pm 0.0858}$	$0.4962_{\pm 0.1164}$	$0.0742_{\pm 0.0352}$	$\underline{0.0501_{\pm 0.0250}}$	$0.0821_{\pm 0.0336}$	$0.0941_{\pm 0.0455}$	$\mathbf{0.0256_{\pm 0.0097}}$
	Total	$0.7802_{\pm 0.1745}$	$0.5124_{\pm 0.2004}$	$0.6342_{\pm 0.2842}$	$0.1786_{\pm 0.0874}$	$0.1667_{\pm 0.0800}$	$\underline{0.0884_{\pm 0.0482}}$	$0.2522_{\pm 0.1732}$	$\mathbf{0.0568_{\pm 0.0327}}$

TABLE I: Causal effect estimation results on the homogeneous (homo) datasets measured by

\sqrt{\epsilon_{\mathrm{PEHE}}}

. Results are reported as mean

\pm

standard deviation over five independent runs. Lower values are better; the best and second-best results are highlighted in bold and underlined, respectively. “Ind.”, “Peer”, and “Total” denote the Individual, Peer, and Total Effects.

Datasets	Effects	NEDNet	w/o T-Rep.	w/o Z-Rep.	Shared encoder	Joint decoder
BC (Within-sample)	Ind.	$\mathbf{0.0472_{\pm 0.0319}}$	$1.2008_{\pm 0.0254}$	$2.0336_{\pm 0.0667}$	$0.0678_{\pm 0.0406}$	$\underline{0.0555_{\pm 0.0213}}$
	Peer	$\mathbf{0.0251_{\pm 0.0098}}$	$0.6734_{\pm 0.1077}$	$0.7929_{\pm 0.0286}$	$\underline{0.0283_{\pm 0.0076}}$	$0.0367_{\pm 0.0104}$
	Total	$\mathbf{0.0539_{\pm 0.0292}}$	$0.7512_{\pm 0.0448}$	$0.8665_{\pm 0.0341}$	$\underline{0.0761_{\pm 0.0355}}$	$0.0870_{\pm 0.0400}$
BC (Out-of-sample)	Ind.	$\mathbf{0.0504_{\pm 0.0342}}$	$1.2019_{\pm 0.0279}$	$1.8051_{\pm 0.0629}$	$0.0689_{\pm 0.0417}$	$\underline{0.0583_{\pm 0.0218}}$
	Peer	$\mathbf{0.0256_{\pm 0.0097}}$	$0.6203_{\pm 0.0805}$	$0.7932_{\pm 0.0276}$	$\underline{0.0294_{\pm 0.0083}}$	$0.0372_{\pm 0.0102}$
	Total	$\mathbf{0.0568_{\pm 0.0327}}$	$0.7359_{\pm 0.0884}$	$0.5933_{\pm 0.0173}$	$\underline{0.0773_{\pm 0.0351}}$	$0.0862_{\pm 0.0356}$

TABLE II: Ablation results on BC and Flickr measured by

\sqrt{\epsilon_{\mathrm{PEHE}}}

Datasets	Effects	TARNet+Z	CFR+Z	NetDeconf+Z	NetEst	RRNet	TNet †	gDIS	NEDNet
BC (Within-sample)	Ind.	$0.3438_{\pm 0.0876}$	$0.3240_{\pm 0.0703}$	$0.4856_{\pm 0.1378}$	$0.1571_{\pm 0.0532}$	$0.1108_{\pm 0.0446}$	$\underline{0.0567_{\pm 0.0217}}$	$0.2691_{\pm 0.0635}$	$\mathbf{0.0353_{\pm 0.0064}}$
	Peer	$0.5588_{\pm 0.0540}$	$0.5052_{\pm 0.0596}$	$0.6018_{\pm 0.0666}$	$0.0778_{\pm 0.0269}$	$\underline{0.0501_{\pm 0.0190}}$	$0.1037_{\pm 0.0324}$	$0.1398_{\pm 0.0672}$	$\mathbf{0.0229_{\pm 0.0099}}$
	Total	$0.8413_{\pm 0.1467}$	$0.6859_{\pm 0.1688}$	$0.7586_{\pm 0.1875}$	$0.2189_{\pm 0.0918}$	$0.2035_{\pm 0.0584}$	$\underline{0.1265_{\pm 0.0827}}$	$0.5089_{\pm 0.1609}$	$\mathbf{0.0636_{\pm 0.0316}}$
BC (Out-of-sample)	Ind.	$0.3444_{\pm 0.0942}$	$0.3278_{\pm 0.0660}$	$0.4781_{\pm 0.1398}$	$0.1519_{\pm 0.0582}$	$0.1091_{\pm 0.0433}$	$\underline{0.0562_{\pm 0.0217}}$	$0.2724_{\pm 0.0657}$	$\mathbf{0.0361_{\pm 0.0069}}$
	Peer	$0.5590_{\pm 0.0553}$	$0.5052_{\pm 0.0596}$	$0.6018_{\pm 0.0665}$	$0.0734_{\pm 0.0305}$	$\underline{0.0501_{\pm 0.0201}}$	$0.1027_{\pm 0.0301}$	$0.1409_{\pm 0.0662}$	$\mathbf{0.0235_{\pm 0.0099}}$
	Total	$0.8236_{\pm 0.1599}$	$0.6885_{\pm 0.1689}$	$0.7466_{\pm 0.1898}$	$0.2110_{\pm 0.0912}$	$0.2026_{\pm 0.0551}$	$\underline{0.1281_{\pm 0.0845}}$	$0.5120_{\pm 0.1592}$	$\mathbf{0.0663_{\pm 0.0330}}$

TABLE III: Causal effect estimation on the hete datasets measured by

\sqrt{\epsilon_{\mathrm{PEHE}}}

. TNet † denotes TNet with hyperparameters selected on the validation set for the corresponding heterogeneous setting.

Datasets	Effects	TARNet+Z	CFR+Z	NetDeconf+Z	NetEst	RRNet	TNet †	gDIS	NEDNet
BC (Within-sample)	Ind.	$0.3758_{\pm 0.0896}$	$0.2483_{\pm 0.0770}$	$0.4571_{\pm 0.0672}$	$0.1583_{\pm 0.0446}$	$0.1846_{\pm 0.0506}$	$\underline{0.0826_{\pm 0.0353}}$	$0.2327_{\pm 0.0925}$	$\mathbf{0.0483_{\pm 0.0308}}$
	Peer	$0.8738_{\pm 0.0907}$	$0.7210_{\pm 0.1095}$	$0.8763_{\pm 0.1040}$	$0.1180_{\pm 0.0276}$	$\underline{0.0665_{\pm 0.0106}}$	$0.1408_{\pm 0.0367}$	$0.1881_{\pm 0.0523}$	$\mathbf{0.0316_{\pm 0.0072}}$
	Total	$1.3591_{\pm 0.2050}$	$0.9243_{\pm 0.1675}$	$1.1592_{\pm 0.2410}$	$0.3703_{\pm 0.0922}$	$0.3486_{\pm 0.0995}$	$\underline{0.1793_{\pm 0.0669}}$	$0.4843_{\pm 0.1379}$	$\mathbf{0.1155_{\pm 0.0242}}$
BC (Out-of-sample)	Ind.	$0.3837_{\pm 0.0943}$	$0.2448_{\pm 0.0753}$	$0.4510_{\pm 0.0618}$	$0.1602_{\pm 0.0480}$	$0.1841_{\pm 0.0504}$	$\underline{0.0849_{\pm 0.0360}}$	$0.2317_{\pm 0.0936}$	$\mathbf{0.0516_{\pm 0.0329}}$
	Peer	$0.8722_{\pm 0.0913}$	$0.7201_{\pm 0.1103}$	$0.8753_{\pm 0.1044}$	$0.1169_{\pm 0.0323}$	$\underline{0.0655_{\pm 0.0100}}$	$0.1437_{\pm 0.0369}$	$0.1885_{\pm 0.0513}$	$\mathbf{0.0321_{\pm 0.0065}}$
	Total	$1.3453_{\pm 0.2103}$	$0.9179_{\pm 0.1658}$	$1.1471_{\pm 0.2358}$	$0.3682_{\pm 0.0949}$	$0.3493_{\pm 0.1035}$	$\underline{0.1794_{\pm 0.0739}}$	$0.4887_{\pm 0.1423}$	$\mathbf{0.1196_{\pm 0.0242}}$

TABLE IV: Causal effect estimation on the hete_z datasets measured by

\sqrt{\epsilon_{\mathrm{PEHE}}}

Causal Effect Estimation

Causal effect estimation measures how outcomes would change under interventions, using experiments or observational data and explicit assumptions.

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