Directed graphs are crucial in modeling complex real-world systems, from gene regulatory networks and flow networks to stochastic processes and graph metanetworks. Representing these directed graphs presents significant challenges, particularly in causal reasoning applications where understanding cause-and-effect relationships is paramount. Current methodologies face a fundamental limitation in balancing directional and distance information within the representation space. They often sacrifice the ability to effectively encode distance information, leading to incomplete or inaccurate representations of the underlying graph structures. This trade-off limits the effectiveness of directed graph embeddings in applications requiring causal understanding and spatial relationships.Various approaches have been developed to address the challenge of embedding graphs in continuous spaces, focusing on adapting to different graph structures through non-Euclidean geometries. Hyperbolic embeddings have been utilized for tree-like graphs, while spherical and toroidal embeddings serve graphs with cycles. Product Riemannian geometries and combinations of constant curvature Riemannian manifolds have been employed to handle graphs with multiple characteristics. Despite these advances, the fundamental challenge of simultaneously representing causal relationships and spatial structures remains. Current solutions either prioritize one aspect over the other or use complex geometric combinations.In this paper, Neural SpaceTimes (NSTs) have been proposed by Anonymous authors, an innovative approach to represent weighted Directed Acyclic Graphs (DAGs) in spacetime manifolds. This novel methodology addresses the dual challenge of encoding spatial and temporal dimensions through a unique product manifold architecture. The framework combines a quasi-metric structure for spatial relationships with a partial order system for temporal dimensions, enabling a comprehensive representation of edge weights and directionality. It offers a significant advancement by providing a universal embedding theorem that guarantees any k-point DAG can be embedded with minimal distortion while maintaining its causal structure intact.The NST architecture is implemented through three specialized neural networks working in concert. The first network serves as an embedding network that optimizes node positions within the spacetime manifold. The second network implements a neural quasi-metric for spatial relationships, while the third network handles temporal aspects through a neural partial order system. A key architectural feature is using multiple time dimensions to model anti-chains in the graph structure effectively. The framework operates by optimization of one-hop neighborhoods for each node, while inherently maintaining transitive causal connectivity across multiple hops through the partial order definition. This implementation bridges theoretical guarantees with practical computation through gradient descent optimization.Experimental evaluations demonstrate NSTs superior performance across both synthetic and real-world datasets. In synthetic weighted DAG embedding tests, NSTs consistently achieve perfect edge directionality preservation while maintaining lower metric distortion when compared with traditional approaches like Minkowski and De Sitter spaces. The framework shows strong performance in low-dimensional embedding spaces, with distortion decreasing as embedding dimensions increase. In real-world network tests using the WebKB datasets (Cornell, Texas, and Wisconsin), NSTs effectively encode both hyperlink directionality and connectivity strength between webpages, achieving low distortions despite the complexity of the network structures.In conclusion, this paper introduces Neural SpaceTimes (NSTs) which represents a significant advancement in DAG representation learning through its innovative use of multiple time dimensions and neural network-based geometry construction. The framework successfully decouples spatial and temporal aspects using a product manifold approach and combining quasi-metrics for space and partial orders for time relationships. However, current implementation is restricted to DAGs rather than general digraphs, and optimization becomes challenging with larger graphs due to computational constraints in calculating shortest-path distances and global causal structures. Despite these limitations, NSTs offer promising directions for future research in graph embedding and causal representation learning.Check outthePaper.All credit for this research goes to the researchers of this project. Also,dont forget to follow us onTwitterand join ourTelegram ChannelandLinkedIn Group. Dont Forget to join our75k+ ML SubReddit. Sajjad AnsariSajjad Ansari is a final year undergraduate from IIT Kharagpur. As a Tech enthusiast, he delves into the practical applications of AI with a focus on understanding the impact of AI technologies and their real-world implications. 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