High-order Knowledge Based Network Controllability Robustness Prediction: A Hypergraph Neural Network Approach

Shibing Mo; Jiarui Zhang; Jiayu Xie; Xiangyi Teng; Jing Liu

cs.LGFeb 28, 2026

High-order Knowledge Based Network Controllability Robustness Prediction: A Hypergraph Neural Network Approach

Authors: Shibing Mo, Jiarui Zhang, Jiayu Xie, Xiangyi Teng, Jing Liu

Organizations: School of Artifcial Intelligence, Xidian University, Xian 710126, China. · School of Computer Science and Technology, Xidian University, Xian 710126, China. · Guangzhou Institute of Technology, Xidian University, Guangzhou 510555, China.

Abstract

In order to evaluate the invulnerability of networks against various types of attacks and provide guidance for potential performance enhancement as well as controllability maintenance, network controllability robustness (NCR) has attracted increasing attention in recent years. Traditionally, controllability robustness is determined by attack simulations, which are computationally time-consuming and only applicable to small-scale networks. Although some machine learning-based methods for predicting network controllability robustness have been proposed, they mainly focus on pairwise interactions in complex networks, and the underlying relationships between high-order structural information and controllability robustness have not been explored. In this paper, a dual hypergraph attention neural network model based on high-order knowledge (NCR-HoK) is proposed to accomplish robustness learning and controllability robustness curve prediction. Through a node feature encoder, hypergraph construction with high-order relations, and a dedicated dual hypergraph attention module, the proposed method can effectively learn three types of network information simultaneously: explicit structural information in the original graph, high-order connection information in local neighborhoods, and hidden features in the embedding space. Notably, we explore for the first time the impact of high-order knowledge on network controllability robustness. Compared with state-of-the-art methods for network robustness learning, the proposed method achieves superior performance on both synthetic and real-world networks with low computational overhead.

Figures & tables

Fig. 1: The Overview of the proposed NCR-HoK framework. NCR-HoK can effectively learn three types of network information simultaneously: the explicit structural information in the original graph, the high-order connection information in local neighborhoods and more hidden structures and features in the embedding space.

Fig. 2: The Principle of Hypergraph Generation by K-Hop and K-NN.

Fig. 3: The predictions of the controllable robustness curves for different network types under the RAs condition by NCR-HoK, PCR, and iPCR models. $P_{N}$ represents the number of nodes having been removed from the network; and $n_{D}$ is calculated by 1 . RA (x)- $\textless{}k\textgreater{}$ =(y) represents the true values of the controllable robustness curve under the RA condition for a network of type (x) with an average degree of (y), as well as the prediction results of each model.

		<k>=2	<k>=5	<k>=8	<k>=10	Average
	$\overline{er}$	0.020	0.016	0.014	0.014	0.016(#2)
PCR	$\overline{\sigma}$	0.015	0.019	0.020	0.022	0.019(#2)
	$\overline{er}$	0.043	0.015	0.017	0.013	0.022(#3)
iPCR	$\overline{\sigma}$	0.054	0.018	0.022	0.020	0.029(#4)
	$\overline{er}$	0.029	0.034	0.020	0.014	0.024(#4)
CRL-SGNN	$\overline{\sigma}$	0.016	0.043	0.021	0.022	0.026(#3)

TABLE I: The average errors

\overline{er}

and average standard deviations

\overline{\sigma}

of the controllable robustness curve predicting for various types of networks under the RA condition are compared among the NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] models and CRL-SGNN [ 29 ] , along with a comprehensive ranking.

Fig. 4: The predictions of the controllable robustness curves for different types of network under the RA condition by NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] and CRL-SGNN [ 29 ] models. $P_{N}$ represents the number of nodes having been removed from the network; and $n_{D}$ is calculated by 1 . RA (x)- $<k>$ =(y) represents the true values of the controllable robustness curve under the RA condition for a network of type (x) with an average degree of (y), as well as the prediction results of each model.

		ER	SF	QSN	SW	Average
	$\overline{er}$	0.019	0.030	0.018	0.021	0.022(#2)
PCR	$\overline{\sigma}$	0.017	0.018	0.016	0.018	0.017(#2)
	$\overline{er}$	0.030	0.109	0.029	0.030	0.050(#4)
iPCR	$\overline{\sigma}$	0.021	0.140	0.018	0.024	0.051(#4)
	$\overline{er}$	0.052	0.044	0.035	0.035	0.042(#3)
CRL-SGNN	$\overline{\sigma}$	0.057	0.045	0.040	0.041	0.028(#3)

TABLE II: The average errors and standard deviations of the controllable robustness curve predicting for various types of networks with different average degrees under the RA condition are compared among the NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] and CRL-SGNN [ 29 ] models, along with a comprehensive ranking.

Fig. 5: The predictions of the controllable robustness curves for various types of network with different node sizes under the RAs condition by the NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] and CRL-SGNN [ 29 ] models.

		ER	SF	QSN	SW	Average
	$\overline{er}$	0.012	0.026	0.013	0.011	0.016(#1)
PCR	$\overline{\sigma}$	0.022	0.023	0.020	0.021	0.022(#2)
	$\overline{er}$	0.012	0.043	0.013	0.011	0.020(#2)
iPCR	$\overline{\sigma}$	0.022	0.044	0.019	0.021	0.027(#4)
	$\overline{er}$	0.012	0.026	0.016	0.011	0.016(#1)
CRL-SGNN	$\overline{\sigma}$	0.024	0.029	0.022	0.023	0.025(#3)

TABLE III: The average errors and standard deviations of the controllable robustness curve predicting for various types of networks with different node sizes under the RA condition are compared among NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] and CRL-SGNN [ 29 ] models, along with a comprehensive ranking.

Real-World Networks	Brief description	Node Number	Edge Number
DDG(DD-g97)	Protein	1021	5778
DEL(delaunay-n10)	DIMACS10 problem	1024	3056
DW5(dwt-1005)	Symmetric connection from Washington	1005	4813
DW7(dwt-1007)	Symmetric connection from Washington	1007	4791
LSH(lshp1009)	Alan George’s L-shape problem	1009	3937
ORS(orsirr-1)	oil reservoir simulation	1030	6858

TABLE IV: The details of real-world networks used in the experiment.

Real-World Networks		DDG	DEL	DW5	DW7	LSH	ORS	Average
PCR	$\overline{er}$	0.053	0.210	0.157	0.146	0.195	0.147	0.151(#3)
PCR	$\overline{\sigma}$	0.037	0.076	0.222	0.211	0.223	0.210	0.163(#3)
iPCR	$\overline{er}$	0.039	0.285	0.154	0.145	0.228	0.118	0.162(#4)
iPCR	$\overline{\sigma}$	0.023	0.116	0.221	0.214	0.223	0.193	0.165(#4)
CRL-SGNN	$\overline{er}$	0.033	0.043	0.119	0.091	0.123	0.205	0.102(#2)
CRL-SGNN	$\overline{\sigma}$	0.028	0.039	0.186	0.164	0.184	0.063	0.111(#2)

TABLE V: The evaluation performance of NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] and CRL-SGNN [ 29 ] on real-world datasets.

Fig. 6: Comparison of controllability robustness curve predicting results of NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] and CRL-SGNN [ 29 ] on real-world networks.

Fig. 7: NCR-HoK’s controlled robustness curve prediction results for various networks under 20%, 40%, 60%, and 80% reduction of the training dataset, respectively.

Fig. 8: The prediction results of NCR-HoK’s controllable robustness curves for various network topologies for K-Hop’s K-values of 2, 3, and 4 cases, respectively.

Fig. 9: The prediction results of NCR-HoK’s controllable robustness curves for various network topologies for K-NN’s K-values of 5, 10, 20 and 30 cases, respectively.

		<k>=2	<k>=5	<k>=8	<k>=10
	$\overline{er}$	0.037	0.027	0.017	0.017
w.o. BC	$\overline{\sigma}$	0.020	0.014	0.012	0.011
	$\overline{er}$	0.214	0.098	0.061	0.047
w.o. S-HGNN	$\overline{\sigma}$	0.018	0.018	0.012	0.011
	$\overline{er}$	0.340	0.105	0.056	0.044
w.o. Dual HGNN	$\overline{\sigma}$	0.036	0.021	0.012	0.010

TABLE VI: Results of ablation experiments with

BC_{GAT}

module, w.o. BC indicates no

BC_{GAT}

module, w.o. S-HGNN indicates only using a single HGNN module, w.o. Dual HGNN indicates no Dual HGNN module, Raw indicates original setting.

Fig. 10: The training process of NCR-HoK for 3200 networks with 1000 nodes.

Controllability Robustness	Unit: Second
RAS	23.581
PCR	0.306
iPCR	0.325
CRL-SGNN	0.029
NCR-HoK	0.085

TABLE VII: The average time taken to process each graph for RAS, PCR [ 27 ] , iPCR [ 28 ] , CRL-SGNN [ 29 ] and NCR-HoK.

Panel of Figure 18 — Fig. S1: Visualization results of the corresponding matrix to compare large entries of $\psi$ and $B^{\prime}$ (with top pairs overlapping).

Layer Name	Layer Type	Input Channels	Output Channels	Number of Parameters
gat1	GATConv (4 heads)	1	64	$1\times 64\times 4+2\times 64\times 4+64\times 4=\textbf{1024}$
gat2	GATConv (1 head)	$64\times 4=256$	64	$256\times 64+2\times 64+64=\textbf{16576}$
fc	Fully Connected	64	1	$64\times 1+1=\textbf{65}$

TABLE S1:

BC_{GAT}

Model Network Structure

		<k>=2	<k>=5	<k>=8	<k>=10
	$\overline{er}$	0.019	0.015	0.016	0.012
ER	$\overline{\sigma}$	0.014	0.011	0.010	0.008
	$\overline{er}$	0.020	0.022	0.023	0.023
SF	$\overline{\sigma}$	0.016	0.016	0.017	0.017
	$\overline{er}$	0.016	0.018	0.016	0.016
QSN	$\overline{\sigma}$	0.012	0.012	0.011	0.011

TABLE S2: The mean error

\overline{er}

and mean standard deviation

\overline{\sigma}

of the controlled robustness curves of NCR-HoK’s prediction for different network topologies are reduced by 20%, 40%, 60%, and 80% for the training dataset, respectively.

		<k>=2	<k>=5	<k>=8	<k>=10
	$\overline{er}$	0.016	0.014	0.012	0.011
ER	$\overline{\sigma}$	0.013	0.011	0.010	0.008
	$\overline{er}$	0.016	0.019	0.021	0.021
SF	$\overline{\sigma}$	0.013	0.015	0.016	0.016
	$\overline{er}$	0.015	0.015	0.013	0.013
QSN	$\overline{\sigma}$	0.011	0.011	0.010	0.009

TABLE S3: The mean error er and mean standard deviation a of the controlled robustness curves of NCR-HoK’s prediction for different network topologies for K-Hop’s K-values of 2, 3, and 4 cases, respectively.

		<k>=2	<k>=5	<k>=8	<k>=10
	$\overline{er}$	0.016	0.014	0.012	0.010
ER	$\overline{\sigma}$	0.012	0.011	0.009	0.008
	$\overline{er}$	0.016	0.019	0.021	0.021
SF	$\overline{\sigma}$	0.012	0.015	0.016	0.015
	$\overline{er}$	0.015	0.014	0.013	0.012
QSN	$\overline{\sigma}$	0.011	0.011	0.010	0.009

TABLE S4: The mean error er and mean standard deviation a of the controlled robustness curves of NCR-HoK’s prediction for different network topologies for K-NN’s K-values of 5, 10, 20 and 30 cases, respectively.

		<k>=2	<k>=5	<k>=8	<k>=10	Average
	$\overline{er}$	0.039	0.040	0.018	0.013	0.028(#3)
PCR	$\overline{\sigma}$	0.017	0.016	0.010	0.008	0.013(#2)
	$\overline{er}$	0.053	0.004	0.008	0.006	0.018(#2)
iPCR	$\overline{\sigma}$	0.070	0.004	0.007	0.006	0.022(#3)
	$\overline{er}$	0.013	0.070	0.019	0.049	0.038(#4)
CRL-SGNN	$\overline{\sigma}$	0.014	0.047	0.022	0.051	0.034(#4)

TABLE S5: The average errors

\overline{er}

and average standard deviations

\overline{\sigma}

of the controllable robustness curve predicting for various types of networks under the TDA condition are compared among the NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] models and CRL-SGNN [ 29 ] , along with a comprehensive ranking.

		<k>=2	<k>=5	<k>=8	<k>=10	Average
	$\overline{er}$	0.047	0.033	0.018	0.016	0.029(#3)
PCR	$\overline{\sigma}$	0.017	0.015	0.012	0.010	0.014(#2)
	$\overline{er}$	0.080	0.013	0.011	0.013	0.029(#3)
iPCR	$\overline{\sigma}$	0.086	0.009	0.011	0.013	0.030(#3)
	$\overline{er}$	0.014	0.041	0.015	0.038	0.027(#2)
CRL-SGNN	$\overline{\sigma}$	0.015	0.052	0.014	0.038	0.030(#3)

TABLE S6: The average errors

\overline{er}

and average standard deviations

\overline{\sigma}

of the controllable robustness curve predicting for various types of networks under the TBA condition are compared among the NCR-HoK, PCR [ 27 ] , iPCR [ 28 ] models and CRL-SGNN [ 29 ] , along with a comprehensive ranking.

Graph Robustness

See all 12 papers

Structural Controllability of Large-Scale Hypergraphs
Mar 20, 2026Joshua Pickard, Xin Mao, Can ChenDynamical Systems
Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard, Cambridge, MA 02142, USA · School of Data Science and Society, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA · Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA +1
Do Higher-Order Models Win for Higher-Order Reasons? Rethinking Performance Gains in Hypergraph Learning
Oct 6, 2026Fanchen Bu, Fan Li, Geon Lee +4Benchmark Design Hypergraph Neural Networks
KAIST · UNSW Sydney · University of Oxford
HGRP: Heterogeneous Graph Neural Network for Routability Prediction
Oct 14, 2026Yule Sheng, Yue Yuan, Shan Shen +1Electronic Design Automation Heterogeneous GNNs
Nanjing University of Science and Technology, Nanjing, China

Mar 20, 2026J/K move · Enter open · S save

Structural Controllability of Large-Scale Hypergraphs

Joshua Pickard, Xin Mao, Can Chen

Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard, Cambridge, MA 02142, USA · School of Data Science and Society, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA · Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA+1

Dynamical Systems

Open paper AskPDF arXiv

Controlling real-world networked systems, including ecological, biomedical, and engineered networks that exhibit higher-order interactions, remains challenging due to inherent nonlinearities and large system scales. Despite extensive studies on graph controllability, the controllability properties of hypergraphs remain largely underdeveloped. Existing results focus primarily on exact controllability, which is often impractical for large-scale hypergraphs. In this article, we develop a structural controllability framework for hypergraphs by modeling hypergraph dynamics as polynomial dynamical systems. In particular, we extend classical notions of accessibility and dilation from linear graph-based systems to polynomial hypergraph dynamics and establish a hypergraph-based criterion under which the topology guarantees satisfaction of classical Lie-algebraic and Kalman-type rank conditions for almost all parameter choices. We further derive a topology-based lower bound on the minimum number of driver nodes required for structural controllability and leverage this bound to design a scalable driver node selection algorithm combining dilation-aware initialization via maximum matching with greedy accessibility expansion. We demonstrate the effectiveness and scalability of the proposed framework through numerical experiments on hypergraphs with tens to thousands of nodes and higher-order interactions.

Abstract

Graph Robustness

Explore similar work

Structural Controllability of Large-Scale Hypergraphs

Do Higher-Order Models Win for Higher-Order Reasons? Rethinking Performance Gains in Hypergraph Learning

HGRP: Heterogeneous Graph Neural Network for Routability Prediction