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In geometry, measuring the direct distance between two points on a plane is essential in various practical and theoretical applications. Whether in navigation, engineering, or computer graphics, determining the shortest path between two locations involves using the distance formula. This formula is derived from the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. On a coordinate plane, the horizontal and vertical distances between two points serve as the legs of...
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Graph diffusion distance: Properties and efficient computation.

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  • 1Department of Computer Science, University of California, Irvine, CA, United States of America.

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Summary
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We introduce novel graph distance measures based on optimizing Laplacian matrices. Our efficient method accurately distinguishes graph lineages and offers potential for constructing graph limits.

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Area of Science:

  • Graph theory
  • Computational mathematics
  • Data science

Background:

  • Conventional graph distance metrics have limitations in capturing complex structural relationships.
  • Measuring similarity and distance between graphs is crucial for various data analysis tasks.

Purpose of the Study:

  • To define and explore a new family of graph similarity and distance measures.
  • To develop an efficient computational procedure for these novel measures.
  • To investigate the theoretical properties and applications of the proposed graph distance metrics.

Main Methods:

  • Defining new graph distance measures via optimization of graph Laplacian exponential matrices.
  • Developing a novel procedure for efficiently calculating optima under norm-preserving and sparsity constraints.
  • Deriving upper bounds for graph distances and graph products.

Main Results:

  • The proposed distance metrics can distinguish between related graph lineages.
  • The novel computational procedure is significantly faster (up to 103x) than existing methods.
  • Upper bounds of the distance metrics satisfy the triangle inequality under certain conditions.

Conclusions:

  • The new graph distance measures offer a powerful tool for analyzing graph structures.
  • The efficient calculation method enhances the practicality of these measures.
  • Potential applications include the construction of infinite graph limits and understanding graph evolution.