Nonlocal dynamics on networks via fractional graph Laplacians: theory and numerical methods

Abstract: Nonlocal diffusive dynamics on large, sparse networks can be modeled by means of systems of differential equations involving fractional graph Laplacians. The solution of such systems leads to non-analytic matrix functions, due to the singularity of the graph Laplacian. Off-diagonal decay estimates for these and related matrix functions will be presented, together with explicit (closed form) expressions for some simple but important examples. The case of directed networks (leading to nonsymmetric Laplacians) will also be discussed.

The numerical approximation of the dynamics can be implemented by means of Krylov subspace methods. The lack of smoothness of the underlying function suggests the use of rational approximation techniques. Some results using a shift-and-invert approach will be presented.

Applications include the efficient exploration of large spatial networks and consensus dynamics in multi-agent systems.

This is joint work with Daniele Bertaccini (U. of Rome `Tor Vergata’), Fabio Durastante (IAC-CNR), and Igor Simunec (Scuola Normale Superiore).

numerical analysis

Audience: researchers in the topic

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