Graph Representation via Elements of Discrete Morse and Cobordism Theories

Jennifer Rozenblit; Chenguang Yang; Yuxin Liu; Yuzhou Chen; Yulia Gel

cs.LGOct 1, 2026

Graph Representation via Elements of Discrete Morse and Cobordism Theories

Authors: Jennifer Rozenblit, Chenguang Yang, Yuxin Liu, Yuzhou Chen, Yulia Gel

Organizations: Department of Mathematics; University of Texas, Austin · Department of Statistics; University of California, Riverside · Department of Statistics; Virginia Tech

Abstract

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Figures & tables

Figure 2 : MG-Diff (left) vs. higher-order guided diffusion (right). MG-Diff identifies critical cells (red), computes descending manifolds (dotted), and perturbs outside these regions (blue). Higher-order diffusion preserves only 2-cell boundaries (teal triangles), missing critical 1-cells that govern topological connectivity.

Method	Data	PEMS-03			PEMS-BAY			AIR-BJ			AIR-GZ
Method	Metric	MAE	RMSE	CRPS	MAE	RMSE	CRPS	MAE	RMSE	CRPS	MAE	RMSE	CRPS
TimeGrad		21.55	36.57	0.101	2.62	5.30	0.034	33.40	54.93	0.363	15.45	21.93	0.376
MC Dropout		18.87	29.81	0.093	3.50	5.43	0.040	37.92	55.49	0.391	13.10	19.26	0.290
CSDI		23.46	39.60	0.098	2.67	4.10	0.031	38.94	57.81	0.417	14.78	22.24	0.361
DiffSTG		17.58	28.75	0.095	2.03	4.22	0.025	38.03	56.87	0.373	13.06	18.25	0.319
PriSTI		22.30	37.58	0.092	2.51	3.99	0.026	36.81	54.34	0.388	14.04	21.03	0.352

Table 1 : Prediction comparison over spatio-temporal graphs based on PEMS-03 [ 13 ] , PEMS-BAY [ 42 ] , AIR-BJ [ 75 ] , and AIR-GZ [ 75 ] .

Method	Degree $\downarrow$	Clus. $\downarrow$
GDSS	0.0373	0.0723
DiGress	0.0899	0.1920
GraphVAE-MM	0.0587	0.3560
HOG-Diff	0.4700	0.3650
CoPHo	0.0523	0.0911
SwinGNN	0.00366	0.0862

Table 2 : Graph generation performance on community-small data in MMD along with the average Wasserstein distance

W_{2}

and mean absolute error in Betti

\beta_{0}

and

\beta_{1}

numbers between the observed and generated graphs.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Graphs and complexes
Symbol	Meaning
$G=(V,E,X)$	Simple graph: vertices $V$ , edges $E$ , node features $X$
$N=\|V\|,\ M=\|E\|$	Number of vertices and edges
$\deg(v)$	Degree of vertex $v$ ; $\deg_{\max}=\max_{u\in V}\deg(u)$
$\Lambda$	Finite simplicial / regular CW complex
$\mathrm{Cl}(G)$	Clique complex of $G$ (simplices $=$ cliques of $G$ )

Appendix

Table 3 : Table of Notations

Method	Topological signal	DMT	Persistence	Domain	Task
TAGG [ 53 ]	PH (diagram-matching loss, $\beta_{i}$ attention)	No	full	abstract graph	graph generation (diffusion)
TopoDiffusionNet [ 26 ]	PH (Betti number)	No	full	image/cubical	image generation (diffusion)
Prob. Topo. Rep. [ 34 ]	DMT+PH	Yes	prunes Morse complex	image likelihood field	segmentation (reconstruction)
Dey–Wang–Wang [ 19 ]	DMT+PH	Yes	prunes Morse complex	density field $\rho$	graph/curve reconstruction
TOGL [ 31 ]	PH (learned filtration)	No	full	abstract graph	classification
PDGNN [ 73 ]	PH (EPD surrogate)	No	full	abstract graph	classification

Appendix

Table 4 : Positioning of MG-Diff among topology-aware diffusion and discrete-Morse (DMT) methods in ML.

Figure 3 : Side-by-side comparison of the Morse-guided graph evolution (left) and the corresponding human motion phases (right).

Figure 4 : The torus viewed as a cobordism between two empty sets. The green marks indicate the gradient flow lines that constitute the connections between critical points along the gradient vector field.

Panel of Figure 5 — Figure 5 : Example graph with two 4-cycles connected by a bridge. Top: the input graph. Middle: critical cells in red. Bottom: descending paths in green, flowing from critical edges to the critical vertex.

Edge $e$	$f(e)$	Pairing $(v,e)\in\mathcal{V}$
$(4,5)$	0.078	$(4,(4,5))$
$(2,4)$	0.222	$(2,(2,4))$
$(1,2)$	0.421	$(1,(1,2))$
$(5,6)$	0.554	$(6,(5,6))$
$(4,7)$	0.756	$(7,(4,7))$
$(0,1)$	0.831	$(0,(0,1))$

Appendix

Table 5 : Vertex–edge pairings in the discrete gradient vector field. Each paired edge

e

is matched with its higher-valued endpoint. Unpaired cells are critical.

Panel of Figure 6 — Figure 6 : Same persistence, different gradient routing. Both panels show the same filtered hexagon-fan complex with identical $H_{1}$ persistence $[1,2)$ . The critical 2-cell $\Delta_{0}=(c,0,1)$ (darkly shaded) is identical in both matchings. However, V-path reachability differs: in $V_{1}$ , gradient paths reach $\Delta_{0}$ counterclockwise (support excludes $s_{5}$ ); in $V_{2}$ , clockwise (support excludes $s_{2}$ ). This directed incidence information-invisible to persistence-determines which edges are downstream of a cycle-filling event in topology-guided generation.

Figure 7 : A filtration on

\Lambda

: the path graph enters in stages, with the final edge

e_{12}

merging two components at time

t=3

Figure 8 : Two distinct gradient vector fields

V_{1}

(left) and

V_{2}

(right) on the path graph.

Figure 9 : A filtration on the star graph

T

: three leaves appear at

t=0

, the center and one edge at

t=1

, and two more edges simultaneously at

t=2

Dataset	# Node	# Time Step	Granularity	Attribute
PEMS-03	358	26,208	5 min	Flow
PEMS-BAY	325	52,116	5 min	Speed
AIR-BJ	36	8,760	1 hour	PM2.5
AIR-GZ	42	8,760	1 hour	PM2.5

Appendix

Table 6 : Summary statistics of the datasets.

Architecture	MAE		CRPS
Architecture	PEMS-03	AIR-BJ	PEMS-03	AIR-BJ
MG-Diff	15.67	29.91	0.077	0.290
MG-Diff W/o MoMoE	16.20	30.22	0.082	0.348
MG-Diff W/o MDDN	16.56	30.83	0.234	0.534

Appendix

Table 7 : Ablation studies on MG-Diff variants.

Method	PEMS-03	AIR-BJ
MG-Diff	15.67	29.91
MG-Diff w. deg.	16.20	30.22
MG-Diff w. bet.	16.56	30.83

Appendix

Table 8 : Ablation studies with different Morse function values on PEMS-03 and AIR-BJ.

Method	Clean AIR-GZ		Noisy AIR-GZ
Method	MAE	RMSE	MAE	RMSE
USTD	9.99	15.41	10.72	15.71
MG-Diff (Ours)	9.80	15.07	10.25	15.26

Appendix

Table 9 : Robustness analysis (noise ratio

=5\%

) on AIR-GZ.

	PEMS-03	PEMS-BAY	AIR-BJ	AIR-GZ
MG-Diff	136	67.5	1	1

Appendix

Table 10 : Running time in seconds (s) (average per epoch).

Conditioning skeleton	MAE	RMSE	MAPE	CRPS
Random skeleton (matched size)	10.19767 $\pm$ 0.00002	15.46738 $\pm$ 0.00003	0.41979 $\pm$ 0.00000	0.25223 $\pm$ 0.00105
Morse, $g(v)=\deg(v)+\varepsilon_{v}$	9.72684 $\pm$ 0.00000	15.20191 $\pm$ 0.00003	0.38107 $\pm$ 0.00000	0.23853 $\pm$ 0.00084
Morse, $g(v)=\deg_{\mathrm{med}}+\deg(v)+\varepsilon_{v}$	10.44039 $\pm$ 0.00004	15.30176 $\pm$ 0.00003	0.46580 $\pm$ 0.00000	0.24502 $\pm$ 0.00112
Morse, $g(v)=\deg_{\max}-\deg(v)+\varepsilon_{v}$ (Ours)	9.58643 $\pm$ 0.00001	14.92188 $\pm$ 0.00002	0.38942 $\pm$ 0.00000	0.23099 $\pm$ 0.00108

Appendix

Table 11 : Isolating the Morse skeleton on AIR-GZ (mean

\pm

std over 5 seeds): a random skeleton of matched size vs. Morse skeletons under different vertex scoring rules

g

, with identical architecture, training configuration, and checkpoint.

Method $\downarrow$	Degree $\downarrow$	Wavelet $\downarrow$	Spec. $\downarrow$	Orbit $\downarrow$
DiGress	0.000374	0.00567	0.0175	0.000611
DeFoG	0.000349	0.00517	0.0149	0.000335
BWFlow	0.003180	0.01920	0.0300	0.000840
MG-Diff (Ours)	0.000204	0.00500	0.0112	0.000175

Appendix

Table 12 : Graph generation performance on tree data

#Nodes	#Edges	Construction time (s)	Start memory (MB)	End memory (MB)
1,000	1,333	0.0049	666.00	666.00
5,000	13,530	0.0333	666.64	666.64
10,000	44,175	0.0912	666.89	666.89
25,000	353,582	0.6185	666.90	666.90
50,000	669,889	1.1644	670.53	680.37
169,343	1,157,799	2.2019	681.35	706.65

Appendix

Table 13 : Runtime and memory consumption of the one-time Morse construction on BFS-induced subgraphs of ogbn-arxiv. Memory is the process RSS before and after construction.

Corruption	Rate	MG-Diff (Ours)			USTD
Corruption	Rate	MAE	RMSE	CRPS	MAE	RMSE	CRPS
Clean	0%	10.259	15.201	0.2452	10.725	15.713	0.2750
Missing events	5%	10.449	15.458	0.2508	11.271	16.196	0.2938
Missing events	10%	10.815	15.837	0.2612	11.936	16.725	0.3174
Missing events	20%	12.132	16.945	0.2953	13.628	18.073	0.3718
Degradation at 5%		$-1.85\%$	$-1.69\%$	$-2.28\%$	$-5.09\%$	$-3.07\%$	$-6.84\%$

Appendix

Table 14 : Robustness to missing events on AIR-GZ. Top: absolute performance under increasing missing-event rates. Bottom: relative degradation with respect to the clean setting.

Method $\downarrow$	Degree $\downarrow$	Wavelet $\downarrow$	Spec. $\downarrow$	Orbit $\downarrow$
DiGress	0.000374	0.00567	0.0175	0.000611
DeFoG	0.000349	0.00517	0.0149	0.000335
BWFlow	0.003180	0.01920	0.0300	0.000840
MG-Diff (Ours)	0.000204	0.00500	0.0112	0.000175

Appendix

Table 15 : Graph generation performance on tree data

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Open paper AskPDF arXiv

After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of geometric deep learning, and paradigms that were once considered to be firmly in the realm of the abstract-like sheaves-have been "tamed" to serve as novel inductive biases for model architectures in topological deep learning. The veritable diversity of models, however, is in stark contrast to the scarcity of suitable benchmark datasets. As a result, researchers often resort to lifting existing graph datasets to include higher-order information. In this opinion paper, I want to encourage the community to also source new datasets, which may be used to prop up the foundations of our research field.

Abstract

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