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Related Concept Videos

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Complex numbers, represented in Cartesian coordinates, can also be visualized as vectors. These vectors can be expressed in polar form, emphasizing their magnitude and angle. When a complex number is input into a function, the output is another complex number, highlighting the function's zero point from which the vector representation can originate.
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p-adic numbers encode complex networks.

Hao Hua1,2, Ludger Hovestadt3

  • 1School of Architecture, Southeast University, 2 Sipailou, Nanjing, 210096, China. whitegreen@163.com.

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|January 9, 2021
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Summary

This study introduces p-adic random graphs to model complex networks with multiple large components, enhancing the Erdős-Rényi model for real-world data like genetic and social networks.

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Area of Science:

  • Network Science
  • Complex Systems
  • Mathematical Physics

Background:

  • The Erdős-Rényi (ER) random graph model is foundational for complex network analysis but often lacks the sophistication to capture real-world network structures.
  • Real-world networks frequently exhibit hierarchical organization and multiple large components, which are not adequately represented by standard ER models.
  • Existing models struggle to fit empirical data, particularly concerning the distribution of sizes of the largest components in networks.

Purpose of the Study:

  • To introduce and analyze the p-adic random graph model as an extension of the ER model.
  • To demonstrate the capability of the p-adic ultrametric to simulate multiple large components in network structures.
  • To explore the implications of community structures on component size distributions and their relevance to intervention strategies.

Main Methods:

  • Utilizing the p-adic number system to represent hierarchical structures in complex networks.
  • Interpreting the parameter 'n' in the ER model as the cardinality of a set of p-adic numbers.
  • Applying the p-adic ultrametric to the ER model to simulate networks with multiple large components.

Main Results:

  • The p-adic random graph model provides a natural framework for representing hierarchical organization in complex networks.
  • The model successfully simulates the presence of multiple large components, a feature often observed in empirical network data.
  • Community structures within networks are shown to result in multimodal distributions of large component sizes.

Conclusions:

  • The p-adic random graph model offers a more realistic approach to modeling complex networks compared to the standard ER model.
  • The ability to simulate multiple large components is crucial for accurately fitting observational data from genetic, social, and epidemiological networks.
  • Understanding multimodal distributions of component sizes is key for designing effective interventions in spreading processes within networks.