Learning graph topology from metapopulation epidemic encoder-decoder

Xin Li; Jonathan Cohen; Shai Pilosof; Rami Puzis

cs.LGMar 2, 2026

Learning graph topology from metapopulation epidemic encoder-decoder

Authors: Xin Li, Jonathan Cohen, Shai Pilosof, Rami Puzis

Organizations: Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Be’er Sheva, South District, Israel · Department of Life Sciences, Ben-Gurion University of the Negev, Be’er Sheva, South District, Israel · The Goldman Sonnenfeldt School of Sustainability and Climate Change, Ben-Gurion University of the Negev, Be’er Sheva, South District, Israel

Abstract

Metapopulation epidemic models are a valuable tool for studying large-scale outbreaks. With the limited availability of epidemic tracing data, it is challenging to infer the essential constituents of these models, namely, the epidemic parameters and the relevant mobility network between subpopulations. Either one of these constituents can be estimated while assuming the other; however, the problem of their joint inference has not yet been solved. Here, we propose two encoder-decoder deep learning architectures that infer metapopulation mobility graphs from time-series data, with and without the assumption of epidemic model parameters. Evaluation across diverse random and empirical mobility networks shows that the proposed approach outperforms the state-of-the-art topology inference. Further, we show that topology inference improves dramatically with data on additional pathogens. Our study establishes a robust framework for simultaneously inferring epidemic parameters and topology, addressing a persistent gap in modeling disease propagation.

Figures & tables

Fig 1: Representative graph topologies used in this study. Top row: synthetic random graphs - er ( er ), ba ( ba ), ws ( ws ), rgg ( rgg ).

Panel of Figure 2 — Fig 2: Left: A metapopulation epidemic process showing the infection both within and among the populations. The blue blocks indicate the state evolution of subpopulation 1. Right: An example of one pathogen metapopulation epidemic time-series data $\Delta\bar{S}$ in typical ba and rgg graphs with 100 nodes, as well as empirical contiguous US [ 35 ] , where the nodes are geometric states (two nodes establish a link if they share a border), and global air transportation graphs, which represent the commute between two airports. The x-axis represents the number of days since the start of the epidemic. The nodes are ordered along the y-axis according to the distance from the seed node. The color indicates the fraction of daily new cases $\Delta\bar{S}$ .

Fig 3: Architecture of the self-supervised DTEF model. The input is the daily new infection data $\Delta\bar{S}$ . The encoder extracts node–pathogen embeddings and infers the infection matrix $\hat{Z}$ , and the decoder uses $\hat{Z}$ to reconstruct the predicted daily new infections $\Delta\hat{{S}}$ . Green ovals and rectangles denote modules with learnable parameters, blue ovals denote parameter-free operations, the black box in $\sigma$ is the scalar output of $F$ transform operation, $\sigma$ is the sigmoid activation enforcing $\hat{Z}_{ij}\in[0,1]$ (red cell), arrows indicate input sequences, and the highlighted red cell shows a specific reconstructed entry of $\Delta\hat{S}$ .

Name	Possible values
$Graph$	er , ba , ws , rgg , or real graphs
$RandomSeed$	Integer
$n$	Positive integer
$Number\ of\ Pathogen$	Positive integer
$DLModel$	dtef , ftef , Infer2018
$Dense$	Positive integer

Table 1: Configurable parameter settings for training.

Fig 4: The loss (Eq. ( 18 )) as a function of the number of optimization epochs (left); four evaluation metrics as a function of the number of optimization epochs (right).

	Spectral similarity				Pearson correlation
	DTEF	FTEF	Infer2018	RND	DTEF	FTEF	Infer2018	RND
ER	0.17	0.26	0.11	0.03	0.62	0.02	0.03	0
BA	0.22	0.31	0.14	0.05	0.72	0.03	0.03	0
WS	0.74	0.55	0.1	0.1	0.88	0.14	0.12	0
RGG	0.82	0.62	0.44	0.09	0.95	0.18	0.22	0
Contiguous US	0.8	0.55	0.4	0.01	0.98	0.14	0.17	0.01

Table 2: Topology reconstruction benchmarks on representative random and real graphs. This experiment is performed using four pathogens. We show the performance of the five fitting models and a random Baseline (RND), based on four evaluation metrics as described in the text. The absence of PR-AUC values for Mobility US is due to them being weighted graphs. The bottom-line win rate represents the percentage of tasks where the model achieves a higher accuracy score than the other models.

	Spectral similarity				Pearson correlation
	DTEF	FTEF	Infer2018	RND	DTEF	FTEF	Infer2018	RND
ER	0.16	0.28	0.09	0.03	0.61	0.65	0.57	0
BA	0.14	0.06	0.41	0.05	0.75	0.77	0.66	0
WS	0.8	0.71	0.75	0.1	0.92	0.94	0.91	0
RGG	0.8	0.83	0.76	0.09	0.97	0.93	0.85	0
Contiguous US	0.97	0.9	0.83	0.01	0.99	1.0	0.98	0.01

Table 3: Topology reconstruction benchmarks on representative random and real graphs with epidemic parameters set to their ground truth values. Other settings are the same as table 2 .

	RMSE( $\beta$ )					RMSE(1/ $\gamma$ )
—	DTEF	FTEF	Infer2018	Bayes	NLS	DTEF	FTEF	Infer2018	Bayes	NLS
ER	0	1.05	0.69	0.02	0.01	0.01	1.79	3.01	1.49	0
BA	0	1.04	0.55	0.02	0.01	0.02	1.84	2.76	1.30	0
WS	0	1.01	0.62	0.02	0.03	0.01	0.4	2.57	1.79	0
RGG	0	0.97	0.57	0.02	0.01	0.02	0.5	6.84	1.30	0
Contiguous US	0	0.16	0.21	0	0.01	0	0.12	6.93	0.47	0

Table 4: Epidemic parameter estimations of

\hat{\beta}

(upper Table) and

\hat{\gamma}

(lower Table) on both synthetic random and real-world graphs. The performance of our different model configurations is evaluated based on the rmse metric. In addition, the results of Infer2018, Bayes and NLS are shown for comparison (see text).

Fig 5: DTEF model performance PR curves for various random graph models, with 1-4 pathogens and a random baseline.

Fig 6: Evolution of the DTEF model’s topology inference with different numbers of pathogens. Each row contains four pairs of graphs: the spring layout of the graph titled with the Jaccard similarity where the blue color represents the ground-truth graph, while the orange color is the inferred graph (left); the degree distribution histogram comparison with the x-label degree and y-label probability (right).

Fig 7: Effect of different mobility rates on the topology inference accuracy of the DTEF model.

	Nodes	Links	Weighted
Contiguous US	49	107	no
Contiguous China	33	69	no
Contiguous EU	24	38	no
Contiguous Africa	47	103	no
Mobility German	52	-	yes
Mobility US	401	-	yes

Table S1: Real graphs’ attributions, including the node number, link number, and whether the graph is weighted or not.

Fig S1: Adjacency matrix comparison between ground-truth and inferred rgg graphs containing 100 nodes with four pathogens: the real binary graph (left); the inferred adjacency topology according to the DTEF model with four pathogens of data (right).

Fig S2: Sparsity index of the mobility matrix as a function of the number of optimization epochs.

	Spectral similarity					Pearson correlation					Jaccard similarity
Avg degree	4	5	6	7	8	4	5	6	7	8	4	5	6	7	8
er	0.2	0.22	0.32	0.19	0.33	0.61	0.56	0.52	0.51	0.49	0.25	0.23	0.21	0.22	0.22
ba	0.21	0.38	0.43	0.36	0.48	0.72	0.63	0.64	0.6	0.59	0.34	0.28	0.29	0.28	0.27
ws	0.78	0.75	0.83	0.83	0.83	0.88	0.87	0.84	0.82	0.79	0.59	0.57	0.54	0.52	0.5
rgg	0.83	0.83	0.84	0.88	0.9	0.95	0.96	0.95	0.95	0.94	0.75	0.76	0.76	0.75	0.73
	PR-AUC

Table S2: Effect of different average degrees on the topology inference accuracy of the DTEF model.

Fig S3: Effect of different average degrees on the topology inference accuracy of the DTEF model.

	Spectral similarity				Pearson correlation				Jaccard similarity				PR-AUC
Nodes	50	100	200	400	50	100	200	400	50	100	200	400	50	100	200	400
er	0.18	0.2	0.12	0.21	0.85	0.61	0.46	0.35	0.52	0.25	0.14	0.07	0.82	0.56	0.37	0.22
ba	0.45	0.21	0.12	0.22	0.87	0.72	0.55	0.49	0.56	0.34	0.19	0.13	0.82	0.69	0.48	0.38
ws	0.76	0.78	0.76	0.68	0.97	0.88	0.78	0.73	0.81	0.59	0.41	0.34	0.87	0.86	0.78	0.71
rgg	0.9	0.83	0.77	0.78	0.98	0.95	0.94	0.92	0.85	0.75	0.7	0.65	0.88	0.89	0.9	0.89

Table S3: Effect of different numbers of nodes on the topology inference accuracy of the DTEF model.

Fig S4: Effect of different numbers of nodes on the topology inference accuracy of the DTEF model.

	Spectral similarity									Pearson correlation
$M_{R}$	2e-4	5e-4	1e-3	2e-3	5e-3	1e-2	2e-2	5e-2	1e-1	2e-4	5e-4	1e-3	2e-3	5e-3	1e-2	2e-2	5e-2	1e-1
er	0.22	0.25	0.17	0.2	0.19	0.2	0.24	0.19	0.24	0.53	0.57	0.62	0.63	0.63	0.61	0.6	0.57	0.23
ba	0.34	0.3	0.3	0.2	0.3	0.21	0.3	0.21	0.25	0.58	0.67	0.68	0.71	0.71	0.72	0.71	0.52	0.24
ws	0.72	0.76	0.76	0.79	0.75	0.78	0.76	0.76	0.77	0.8	0.86	0.9	0.9	0.89	0.88	0.87	0.86	0.73
rgg	0.78	0.82	0.8	0.82	0.84	0.83	0.84	0.85	0.59	0.87	0.93	0.93	0.95	0.94	0.95	0.95	0.94	0.36
	Jaccard similarity									PR-AUC

Table S4: Effect of different mobility rates

M_{R}

on the topology inference accuracy of the DTEF model.

	Spectral similarity				Pearson correlation				Jaccard similarity				PR-AUC
$Pathogen$	1	2	3	4	1	2	3	4	1	2	3	4	1	2	3	4
contiguous_US	0.46	0.78	0.82	0.84	0.58	0.93	0.98	0.98	0.26	0.69	0.86	0.83	0.53	0.82	0.83	0.83
contiguous_China	0.55	0.81	0.71	0.98	0.64	0.93	0.99	1	0.3	0.69	0.9	0.96	0.57	0.82	0.83	0.83
contiguous_EU	0.17	0.35	0.63	0.58	0.7	0.96	0.99	1	0.38	0.78	0.92	0.96	0.66	0.8	0.8	0.8
contiguous_Africa	0.36	0.73	0.81	0.85	0.53	0.9	0.96	0.99	0.22	0.63	0.77	0.89	0.44	0.81	0.83	0.83
mobility_german	0.21	0.13	0.08	0.04	0.05	0.32	0.4	0.54	0	0.12	0.17	0.22	0.31	0.44	0.51	0.52

Table S5: Real-world graph evaluation results for pathogens one to four, as inferred by the DTEF model.

Fig S5: The inference accuracy of the DTEF model on real contiguous region graphs: spring layout topology (left) and degree distribution (right).

Fig S6: The inference accuracy of the DTEF model on real global plane graphs: spring layout topology (left) and degree distribution (right).

Fig S7: The inference accuracy of the DTEF model on real Spanish graphs: spring layout topology (upper Figures) and degree distribution (lower Figures).

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