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The second derivative of a function provides essential information about a graph's curvature and how it changes over an interval. It helps determine whether a function is concave upward or concave downward and identifies points where the curvature changes. These properties are fundamental in analyzing real-world scenarios, such as changes in road elevation, population growth, and economic trends.A function f(x) is considered concave upward on an interval if its graph lies above all its tangent...
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Curvature on Graphs with Negative Edge Weights.

Daniel Grange1, Yifan Sun1, Corey Weistuch2

  • 1Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA.

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Summary
This summary is machine-generated.

This study introduces graph frustration and new Ollivier-Ricci-inspired fragility indices for analyzing signed networks. These indices effectively identify critical pathways in gene regulatory and social networks, outperforming existing measures.

Keywords:
Discrete CurvatureGeometry of GraphsGraph RobustnessOptimal Mass TransportSigned Graphs

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Area of Science:

  • Network Science
  • Graph Theory
  • Mathematical Modeling

Background:

  • Discrete curvature measures offer insights into network fragility.
  • Ollivier-Ricci (OR) curvature extends geometric concepts to graphs.
  • Existing methods do not capture antagonistic relationships in networks.

Purpose of the Study:

  • Introduce "balanced" graphs and "graph frustration" to model signed network interactions.
  • Develop modified Ollivier-Ricci-inspired fragility indices.
  • Evaluate the utility of these new indices in identifying critical pathways.

Main Methods:

  • Defined balanced graphs and graph frustration based on signed edge weights (promotion/inhibition).
  • Developed modified Ollivier-Ricci-inspired fragility indices.
  • Applied indices to gene regulatory and social networks.

Main Results:

  • The new fragility indices effectively identify critical edges in unbalanced graphs.
  • These indices outperform commonly used measures in quantifying network criticality.
  • Demonstrated utility in gene regulatory and social network analysis.

Conclusions:

  • Graph frustration and modified OR-inspired indices provide novel tools for analyzing signed networks.
  • These methods enhance the identification of critical nodes and pathways.
  • The approach offers improved understanding of network functionality and disruption.