Lee DeVille

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  The Spectrum of Signed Laplacian, or Feuds, Coalitions and Love Triangles
  Lee DeVille

  The stability of any fixed point of a dynamical system defined on a network is determined by the spectrum of the Jacobian at that point.

  
  

  For a wide variety of networked dynamical systems, this Jacobian takes the form of a "graph Laplacian". In contrast to the classical Laplacian, for many fixed points, the network configuration will be such that we need to consider negatively-weighted edges, i.e. configurations with repelling pairs. We present the spectral theory of such operators, give a natural description using the language of social networks, and show that many of the dynamical properties of such networks can be intuited using this conceptual framework.