Causal Effect Estimation under Networked Interference without Networked Unconfoundedness Assumption

Weilin Chen; Ruichu Cai; Jie Qiao; Yuguang Yan; José Miguel Hernández-Lobato

cs.LGFeb 27, 2025

Causal Effect Estimation under Networked Interference without Networked Unconfoundedness Assumption

Authors: Weilin Chen, Ruichu Cai, Jie Qiao, Yuguang Yan, José Miguel Hernández-Lobato

Organizations: School of Computer Science, Guangdong University of Technology, Guangzhou, 510006, China · School of Computer Science, Guangdong University of Technology, Guangzhou, 510006, China and Peng Cheng Laboratory, Shenzhen, China · Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom

Abstract

Estimating causal effects under networked interference from observational data is a crucial yet challenging problem. Most existing methods mainly rely on the networked unconfoundedness assumption, which guarantees the identification of networked effects. However, this assumption is often violated due to the latent confounders inherent in observational data, thereby hindering the identification of networked effects. To address this issue, we leverage the rich interaction patterns between units in networks, which provide valuable information for recovering these latent confounders. Building on this insight, we develop a confounder recovery framework that explicitly characterizes three categories of latent confounders in networked settings: those affecting only the unit, those affecting only the unit's neighbors, and those influencing both. Based on this framework, we design a networked effect estimator using identifiable representation learning techniques. From a theoretical standpoint, we prove the identifiability of all three types of latent confounders and, by leveraging the recovered confounders, establish a formal identification result for networked effects. Extensive experiments validate our theoretical findings and demonstrate the effectiveness of the proposed method.

Figures & tables

Fig. 1: A simple example illustrating networked interference among three units. The networked interference introduces interactions between units, represented by the solid red arrows. These interactions violate the traditional SUTVA assumption, rendering the networked effects non-identifiable.

Fig. 2: Assumed causal graph in this paper.

x

denotes observed proxies,

u

denotes latent confounders,

t

denotes the treatment, and

y

denotes the outcome of interest. We assume that latent confounders

u

contain three types of variables, i.e.,

u^{i}

affecting the unit itself,

u^{n}

affecting the unit’s neighbors, and

u^{c}

affecting both.

Fig. 3: Model architecture of the proposed CaLaNet. The representation learning module learns the latent confounders, the feature module aggregates the confounders of unit

i

and its neighbors, and the outcome estimator module estimates the potential outcomes of unit

i

. The same colors denote the same latent roles

u^{i}

u^{c}

, and

u^{n}

across modules, while darker colors indicate the corresponding neighbor-aggregated features.

\varepsilon_{average}

\varepsilon_{individual}

Within Sample

Out-of Sample

Within Sample

Out-of Sample

Methods

AME

ASE

ATE

AME

ASE

ATE

IME

ISE

ITE

IME

ISE

ITE

TARNET+z

0.1573_{\pm 0.0405}

0.0824_{\pm 0.0149}

0.2046_{\pm 0.0193}

0.1492_{\pm 0.0370}

0.0855_{\pm 0.0147}

0.1982_{\pm 0.0380}

0.2096_{\pm 0.0250}

0.1161_{\pm 0.0159}

0.2444_{\pm 0.0239}

0.2809_{\pm 0.0507}

0.1209_{\pm 0.0152}

0.3062_{\pm 0.0627}

CFR+z

\underline{0.0788}_{\pm 0.0096}

0.1157_{\pm 0.0076}

0.2323_{\pm 0.0106}

\underline{0.0770}_{\pm 0.0099}

0.1157_{\pm 0.0075}

0.2306_{\pm 0.0106}

\underline{0.0796}_{\pm 0.0091}

0.1158_{\pm 0.0076}

0.2325_{\pm 0.0106}

0.1000_{\pm 0.0388}

0.1157_{\pm 0.0075}

0.2405_{\pm 0.0166}

GEst

0.1872_{\pm 0.0672}

0.2369_{\pm 0.0607}

0.1422_{\pm 0.0562}

0.1955_{\pm 0.0611}

0.2391_{\pm 0.0617}

0.1302_{\pm 0.0524}

0.2307_{\pm 0.0493}

0.2603_{\pm 0.0586}

0.1877_{\pm 0.0495}

0.2388_{\pm 0.0431}

0.2623_{\pm 0.0592}

0.1790_{\pm 0.0440}

ND+z

0.2375_{\pm 0.0450}

\underline{0.0316}_{\pm 0.0104}

0.0790_{\pm 0.0226}

0.2380_{\pm 0.0458}

\underline{0.0323}_{\pm 0.0122}

0.0768_{\pm 0.0254}

0.2377_{\pm 0.0448}

\underline{0.0321}_{\pm 0.0101}

0.0792_{\pm 0.0226}

0.2477_{\pm 0.0460}

\underline{0.0379}_{\pm 0.0099}

0.1068_{\pm 0.0172}

TABLE I: Experimental results on BC(homo) Dataset. The top result is highlighted in bold, and the runner-up is underlined.

\varepsilon_{average}

\varepsilon_{individual}

Within Sample

Out-of Sample

Within Sample

Out-of Sample

Methods

AME

ASE

ATE

AME

ASE

ATE

IME

ISE

ITE

IME

ISE

ITE

TARNET+z

0.2538_{\pm 0.1127}

0.1657_{\pm 0.0563}

0.3866_{\pm 0.0711}

0.2619_{\pm 0.1054}

0.1701_{\pm 0.0594}

0.4044_{\pm 0.1334}

0.3455_{\pm 0.0654}

0.2122_{\pm 0.0558}

0.4605_{\pm 0.0720}

0.5590_{\pm 0.2643}

0.2207_{\pm 0.0510}

0.6349_{\pm 0.3280}

CFR+z

0.1580_{\pm 0.0189}

0.2071_{\pm 0.0237}

0.4067_{\pm 0.0407}

0.1559_{\pm 0.0203}

0.2076_{\pm 0.0245}

0.4061_{\pm 0.0449}

0.1825_{\pm 0.0129}

0.2092_{\pm 0.0233}

0.4316_{\pm 0.0351}

0.2058_{\pm 0.0432}

0.2098_{\pm 0.0242}

0.4422_{\pm 0.0456}

GEst

0.2734_{\pm 0.1240}

0.4257_{\pm 0.0973}

0.2916_{\pm 0.1119}

0.2722_{\pm 0.1308}

0.4277_{\pm 0.1012}

0.2873_{\pm 0.1220}

0.3334_{\pm 0.1082}

0.4592_{\pm 0.0934}

0.3546_{\pm 0.0947}

0.3832_{\pm 0.1474}

0.4612_{\pm 0.0972}

0.3958_{\pm 0.1486}

ND+z

0.4124_{\pm 0.0702}

\underline{0.0451}_{\pm 0.0201}

0.1330_{\pm 0.0205}

0.4111_{\pm 0.0737}

0.0486_{\pm 0.0206}

0.1326_{\pm 0.0261}

0.4226_{\pm 0.0673}

\underline{0.0562}_{\pm 0.0146}

0.1941_{\pm 0.0246}

0.4330_{\pm 0.0662}

0.0666_{\pm 0.0137}

0.2211_{\pm 0.0321}

TABLE II: Experimental results on BC(hete) Dataset. The top result is highlighted in bold, and the runner-up is underlined.

\varepsilon_{average}

\varepsilon_{individual}

Within Sample

Out-of Sample

Within Sample

Out-of Sample

Methods

AME

ASE

ATE

AME

ASE

ATE

IME

ISE

ITE

IME

ISE

ITE

TARNET+z

0.0783_{\pm 0.0418}

0.0874_{\pm 0.0213}

0.2025_{\pm 0.0396}

0.0976_{\pm 0.0506}

0.0724_{\pm 0.0184}

0.1356_{\pm 0.0587}

0.1362_{\pm 0.0254}

0.1103_{\pm 0.0194}

0.2358_{\pm 0.0383}

1.0869_{\pm 1.2258}

0.1011_{\pm 0.0185}

1.0889_{\pm 1.2270}

CFR+z

0.0579_{\pm 0.0247}

0.0785_{\pm 0.0070}

0.1651_{\pm 0.0121}

0.0507_{\pm 0.0192}

0.0783_{\pm 0.0070}

0.1581_{\pm 0.0097}

0.0599_{\pm 0.0240}

0.0786_{\pm 0.0070}

0.1654_{\pm 0.0120}

0.3465_{\pm 0.4615}

0.0786_{\pm 0.0069}

0.4102_{\pm 0.4278}

GEst

0.1551_{\pm 0.0130}

0.2475_{\pm 0.0476}

0.0805_{\pm 0.0325}

0.1511_{\pm 0.0137}

0.2494_{\pm 0.0470}

0.0805_{\pm 0.0278}

0.1779_{\pm 0.0122}

0.2656_{\pm 0.0378}

0.1268_{\pm 0.0160}

0.2867_{\pm 0.2172}

0.2677_{\pm 0.0372}

0.2471_{\pm 0.2352}

ND+z

0.1416_{\pm 0.0240}

\underline{0.0204}_{\pm 0.0093}

0.0478_{\pm 0.0216}

0.1435_{\pm 0.0364}

\underline{0.0226}_{\pm 0.0101}

0.0485_{\pm 0.0236}

0.1427_{\pm 0.0246}

\underline{0.0221}_{\pm 0.0090}

0.0501_{\pm 0.0178}

0.3849_{\pm 0.2395}

0.0348_{\pm 0.0078}

0.3453_{\pm 0.2772}

TABLE III: Experimental results on Flickr(homo) Dataset. The top result is highlighted in bold, and the runner-up is underlined.

\varepsilon_{average}

\varepsilon_{individual}

Within Sample

Out-of Sample

Within Sample

Out-of Sample

Methods

AME

ASE

ATE

AME

ASE

ATE

IME

ISE

ITE

IME

ISE

ITE

TARNET+z

0.1315_{\pm 0.0740}

0.1673_{\pm 0.0423}

0.3590_{\pm 0.0785}

0.1554_{\pm 0.1110}

0.1319_{\pm 0.0307}

0.2728_{\pm 0.1321}

0.2320_{\pm 0.0432}

0.2042_{\pm 0.0479}

0.4254_{\pm 0.0802}

1.5274_{\pm 1.6256}

0.1760_{\pm 0.0351}

1.5957_{\pm 1.5779}

CFR+z

0.1131_{\pm 0.0476}

0.1437_{\pm 0.0081}

0.2960_{\pm 0.0182}

0.0998_{\pm 0.0458}

0.1412_{\pm 0.0085}

0.2789_{\pm 0.0309}

0.1445_{\pm 0.0407}

0.1463_{\pm 0.0083}

0.3242_{\pm 0.0224}

0.5946_{\pm 0.7379}

0.1448_{\pm 0.0087}

0.7104_{\pm 0.6761}

GEst

0.3283_{\pm 0.0426}

0.4717_{\pm 0.1336}

0.1074_{\pm 0.0255}

0.3356_{\pm 0.0270}

0.4723_{\pm 0.1312}

0.0969_{\pm 0.0099}

0.3697_{\pm 0.0386}

0.5123_{\pm 0.1231}

0.2178_{\pm 0.0214}

0.7124_{\pm 0.6463}

0.5144_{\pm 0.1202}

0.5914_{\pm 0.7073}

ND+z

0.2420_{\pm 0.0330}

0.0293_{\pm 0.0113}

0.0852_{\pm 0.0365}

0.2433_{\pm 0.0539}

0.0318_{\pm 0.0134}

0.0785_{\pm 0.0422}

0.2571_{\pm 0.0348}

0.0430_{\pm 0.0040}

0.1607_{\pm 0.0188}

0.5156_{\pm 0.2180}

0.0669_{\pm 0.0059}

0.4720_{\pm 0.2580}

TABLE IV: Experimental results on Flickr(hete) Dataset. The top result is highlighted in bold, and the runner-up is underlined.

Fig. 4: The figure presents scatter plots visualizing the relationship between the recovered latent confounders and ground-truth latent confounders

U^{i},U^{c}

, and

U^{n}

	$\hat{U}^{i}$	$\hat{U}^{c}$	$\hat{U}^{n}$
${U}^{i}$	0.9729	0.1742	0.2032
${U}^{c}$	0.3796	0.9090	0.3616
${U}^{n}$	0.1879	0.2420	0.9901

TABLE V: MCC results between the recovered latent confounders and ground-truth latent confounders

U^{i},U^{c}

, and

U^{n}

Fig. 5: Hyperparameter sensitivity result regarding individual effect estimation error on BC(homo) dataset.

Fig. 6: Hyperparameter sensitivity result regarding individual effect estimation error on Flickr(homo) dataset.

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