Entropy-Driven Sensor Deployment and Source Detection in Hypergraphs

TL;DR

This work tackles the challenge of locating diffusion sources under partial observations in networks with higher-order interactions by introducing Sensor-based Source Detection in Hypergraphs (SSDH). SSDH couples an entropy-driven sensor deployment strategy with a two-stage, path-entropy–based source localization algorithm, enabling robust source identification in hypergraphs. Across synthetic and empirical datasets, SSDH outperforms strong baselines by substantial margins (5–20% accuracy gains, lower AED) and demonstrates resilience to varying spreading parameters and sensor budgets. The framework highlights the importance of leveraging higher-order diffusion dynamics and information-theoretic sensor placement for effective outbreak tracing and related applications.

Abstract

Identifying the diffusion source in complex networks is critical for understanding and controlling epidemic spread. In realistic settings, full observation of node states is rarely available, making sensor-based source detection a practical alternative. However, existing sensor-based methods are often confined to simple networks, failing to capture the higher-order group dynamics of real-world spreading process. By deploying a limited number of sensors to monitor the diffusion process, one can infer the origin from partial observations. Yet, determining optimal sensor placement is challenging, i.e., poor deployment leads to redundant or noisy data, while optimal placement must balance coverage diversity and information value under limited resources. To address these challenges, we propose a dedicated framework termed Sensor-based Source Detection in Hypergraphs (SSDH). Specifically, we introduce a novel entropy-driven sensor deployment strategy that effectively captures critical early-stage diffusion signals by maximizing information gain under limited resources. Furthermore, we develop a source localization algorithm that quantifies propagation uncertainty through a newly defined path uncertainty-based score. By integrating this score with topological distance, SSDH enables accurate and robust source identification. Extensive experiments on both synthetic and empirical hypergraphs demonstrate that SSDH consistently outperforms competing algorithms by 5%--30% across different sensor ratios, final spreading ratios, and infection probabilities. These results validate the effectiveness of SSDH and highlight its superior capability to tackle source localization in complex systems characterized by higher-order interactions.

Figures (19)

• Figure 1: Example of the hypergraph-based SI spreading process.
• Figure 2: Overall workflow of the SSDH algorithm. (a) A proportion of nodes is selected as sensor nodes using the entropy-based deployment strategy, and their infection information is extracted from the resulting infected hypergraph. (b) Candidate screening based on the earliest sensor infection time $t^1$, followed by source refinement through the integration of topological distance and entropy-based transmission. (c) The effectiveness of the proposed algorithm is evaluated through comparative experiments.
• Figure 3: Visualization and quantitative assessment of sensor monitoring coverage across the empirical hypergraphs. (a--c) Node observation intensity ($\tilde{M_i}$) in Algebra, Restaurants-Rev, and Geometry. Sensor nodes are shown in blue, while non-sensor nodes are colored based on their normalized observation intensity: $\tilde{M_i} \leq 0.4$, $0.4 < \tilde{M_i} \leq 0.5$, $0.5 < \tilde{M_i} \leq 0.6$, and $\tilde{M_i} > 0.6$, represented by a gradient from red to pink. Node size corresponds to the $k$-core value, with a larger size indicating a higher $k$-core value. The sensor ratio is set to 10%. (d--f) Average first-sensor infection time (AFIT, $\bar{t_i^{1}}$) for the same hypergraphs. The AFIT values are visualized using a color gradient from dark to light green, corresponding to $\bar{t_i^{1}} \ge 2.5$, $2 < \bar{t_i^{1}} \le 2.5$, and $\bar{t_i^{1}} \le 2$, respectively. The same sensor set and node sizes as in (a--c) are used, with the sensor ratio fixed at 10%. (g--l) Distribution of $\tilde{M_i}$ across six empirical hypergraphs under a 10% sensor ratio. (m) Variation of the average observation intensity $\langle \tilde{M_i}\rangle$ with different sensor ratios in the empirical hypergraphs. (n--s) Distribution of AFIT ($\bar{t_i^{1}}$) across six empirical hypergraphs with a 10% sensor ratio. (t) Variation of the average AFIT $\langle \bar{t_i^{1}} \rangle$ with sensor ratios in the empirical hypergraphs. All results are obtained using the hypergraph SI spreading model with $\lambda = 0.5$ and averaged over 100 independent Monte Carlo simulations.
• Figure 4: Illustration of the candidate screening process under different earliest sensor infection times ($t^1$). (a) $t^1 = 1$; (b) $t^1 = 2$; and (c) $t^1 = 3$. In each case, node $v_8$ represents the sensor node, while the yellow nodes indicate the corresponding candidate source nodes identified based on the temporal constraint.
• Figure 5: Comparison of source detection performance across different empirical hypergraphs under varying sensor deployment ratios: (a) Algebra; (b) Restaurants-Rev; (c) Geometry; (d) Email-Eu; (e) Music-Rev; (f) Bars-Rev. The performance of the synthetic hypergraphs is shown in Fig. \ref{['fig:sensor1']} in the Appendix \ref{['addexp']}.