NetworkNet: A Deep Neural Network Approach for Random Networks with Sparse Nodal Attributes and Complex Nodal Heterogeneity

Zhaoyu Xing; Xiufan Yu

stat.MEApr 13, 2026

NetworkNet: A Deep Neural Network Approach for Random Networks with Sparse Nodal Attributes and Complex Nodal Heterogeneity

Authors: Zhaoyu Xing, Xiufan Yu

Organizations: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame

Abstract

Heterogeneous network data with rich nodal information become increasingly prevalent across multidisciplinary research, yet accurately modeling complex nodal heterogeneity and simultaneously selecting influential nodal attributes remains an open challenge. This problem is central to many applications in economics and sociology, when both nodal heterogeneity and high-dimensional individual characteristics highly affect network formation. We propose a statistically grounded, unified deep neural network approach for modeling nodal heterogeneity in random networks with high-dimensional nodal attributes, namely ``NetworkNet''. A key innovation of NetworkNet lies in a tailored neural architecture that explicitly parameterizes attribute-driven heterogeneity, and at the same time, embeds a scalable attribute selection mechanism. NetworkNet consistently estimates two types of latent heterogeneity functions, i.e., nodal expansiveness and popularity, while simultaneously performing data-driven attribute selection to extract influential nodal attributes. By unifying classical statistical network modeling with deep learning, NetworkNet delivers the expressive power of DNNs with methodological interpretability, algorithmic scalability, and statistical rigor with a non-asymptotic approximation error bound. Empirically, simulations demonstrate strong performance in both heterogeneity estimation and high-dimensional attribute selection. We further apply NetworkNet to a large-scale author-citation network among statisticians, revealing new insights into the dynamic evolution of research fields and scholarly impact.

Figures & tables

\mathcal{L}(\bm{W},\bm{\theta})=\sum_{i\neq j}\left[e^{L_{\alpha}}-A_{ij}L_{\alpha}\right]+\lambda\|\bm{\theta}\|_{1}+\gamma\left[\sum_{k}f_{\bm{W}}(\bm{x}_{k})-\sum_{k}\lambda_{\text{fixed},k}\right]^{2}

Algorithm 2 The Update Procedure (Steps 3 & 4 in Algorithm 1 )

\bm{w}_{k,\cdot,\text{new}}\leftarrow\bm{w}_{k,\cdot}\min\left(1,\frac{M\cdot|\theta_{\text{new},k}|}{\|\bm{w}_{k,\cdot}\|_{2}}\right)

Algorithm 3 Hierarchical Prox Procedure (Step 9 in Algorithm 2 )

Measures	Notation	Formula
Estimation
Average root mean squared error for $\alpha(\cdot)$	$\alpha$ -RMSE	$\{100^{-1}\sum_{r}[\frac{1}{n}\sum_{i=1}^{n}(\hat{\alpha}_{i}^{(r)}-\alpha_{i,0})^{2}]\}^{1/2}$
Average root mean squared error for $\beta(\cdot)$	$\beta$ -RMSE	$\{100^{-1}\sum_{r}[\frac{1}{n}\sum_{i=1}^{n}(\hat{\beta}_{i}^{(r)}-\beta_{i,0})^{2}]\}^{1/2}$
Selection
Average precision for $\alpha$ -related attributes	$\alpha$ -Precision	$100^{-1}\sum_{r}\|\hat{\mathcal{A}}_{\alpha}^{(r)}\cap\mathcal{A}_{\alpha}\|/\|\hat{\mathcal{A}}_{\alpha}^{(r)}\|$
Average precision for $\beta$ -related attributes	$\beta$ -Precision	$100^{-1}\sum_{r}\|\hat{\mathcal{A}}_{\beta}^{(r)}\cap\mathcal{A}_{\beta}\|/\|\hat{\mathcal{A}}^{(r)}_{\beta}\|$

Table 1: The definitions of the simulation performance measures to assess the performance of estimation of

\alpha(\cdot)

and

\beta(\cdot)

, and selection of nodal attributes.

		NetworkNet	MLE+Lasso	MLE+LassoNet	MLE
$\alpha$	RMSE	0.5870 (0.2623)	1.3196 (0.0835)	1.4785 (0.0941)	1.3758 (0.0833)
	Precision	0.9983 (0.0167)	0.0050 (0.0286)	0.0298 (0.0602)	/
	TPR	0.9680 (0.1497)	0.0060 (0.0343)	0.0480 (0.0990)	/
	F1	0.9737 (0.1218)	0.0055 (0.0312)	0.0365 (0.0739)	/
$\beta$	RMSE	0.6163 (0.1142)	1.3081 (0.0876)	1.2672 (0.0896)	1.3654 (0.0885)
	Precision	0.7179 (0.1783)	0.0033 (0.0235)	0.0121 (0.0563)	/

Table 2: Performance of estimation and nodal attribute selection for linear nodal heterogeneity

		NetworkNet	MLE+Lasso	MLE+LassoNet	MLE
$\alpha$	RMSE	1.8608 (0.3217)	4.0095 (0.3761)	3.6234 (0.4250)	5.2026 (0.4876)
	Precision	0.9105 (0.1791)	0.0131 (0.0468)	0.0503 (0.0646)	/
	TPR	0.9880 (0.0844)	0.0180 (0.0642)	0.1620 (0.1943)	/
	F1	0.9327 (0.1414)	0.0151 (0.0539)	0.0737 (0.0870)	/
$\beta$	RMSE	2.0015 (0.4917)	4.2462 (0.4712)	3.7348 (0.3889)	5.5397 (0.7502)
	Precision	0.8291 (0.2872)	0.0150 (0.0480)	0.0375 (0.0760)	/

Table 3: Performance of estimation and nodal attribute selection for non-linear nodal heterogeneity

Figure 2: The construction of the academic author-citation networks

Figure 3: Example of count-valued author-citation networks with nodal attributes

	80-90	90-00	00-10	10-20
Publication counts	11279	18212	25495	38432
Author counts	2967	4998	8283	14347
Keyword counts	393	23761	36400	57927
Edges counts	78640	211668	447434	632485
Network density	0.009	0.008	0.007	0.003
Average path length	3.290	3.113	3.106	3.336

Table 4: Network information of the four author-citation networks

Attributes	80-90	90-00	00-10	10-20
Expansiveness-related Attributes
bootstrap	2	1	2	6
consistency	7	7	7	10
Popularity-related Attributes
bootstrap	2	1	2	4
consistency	6	6	8	15

Table 5: The ranks of the consistently selected influential research fields in all four decades

Nodal expansiveness( $\alpha$ )-related fields	80-90	90-00	00-10	10-20
simulation	1	20
bootstrap	2	1	2	6
reliability	3
logistic regression	4	8	15
mean squared error	5
empirical bayes	6

Table 6: The ranks of the selected influential research fields for nodal expansiveness in the four decades

Nodal popularity( $\beta$ )-related fields	80-90	90-00	00-10	10-20
simulation	1
bootstrap	2	1	2	4
reliability	3
poisson regression	4
empirical bayes	5
consistency	6	6	8	15

Table 7: The ranks of the selected influential research fields for nodal popularity in the four decades

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RSNet is an open-source R package that provides a resampling-based framework for robust and interpretable network inference, designed to address the limited-sample-size challenges common in high-dimensional data. It supports both the estimation of partial correlation networks modeled as Gaussian networks and conditional Gaussian Bayesian networks for mixed data types that combine continuous and discrete variables. The framework incorporates multiple resampling strategies, including bootstrap, subsampling, and cluster-based approaches, to accommodate both independent and correlated observations. To enhance interpretability, RSNet integrates graphlet-based topology analysis that captures higher-order connectivity and edge sign information, enabling single-node and subnetwork-level insights. Notably, RSNet is the first R package to efficiently construct signed graphlet degree vector matrices (GDVMs) in near-constant time for sparse networks, providing scalable analysis of higher-order network structure. Collectively, RSNet offers a versatile tool for statistically reliable and interpretable network inference in high-dimensional data.

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