Multivariate Hawkes processes on inhomogeneous random graphs. (English) Zbl 1502.05226
This paper constructs and describes the limiting behavior of a system of randomly interacting neurons, with a combination of a random graph embedded in the space and random interactions corresponding to a Hawkes process. More precisely, first, we consider \(N\) vertices in \(\mathbb R^n\), and add edges randomly, with probabilities depending on the positions of the endpoints in the \(n\)-dimensional space. Once we have this random graph, a multivariate Hawkes process is constructed, where only connected vertices can interact, and the firing rate of the neurons depends on the past of their neighbors. In this sense, the original Hawkes process corresponds to the mean-field case, where the underlying graph is a complete graph without any spatial structure, while here they are connected randomly with probabilities depending on their position, and interactions are restricted to neighboring vertices. The significance of this model is that it incorporates the most important properties of interactions of neurons (spatial structure, inhomogeneous graph structure, random interactions with varying intensity), by building an extension of the Hawkes process.
The first goal of the paper is to find sufficient conditions on the intensities and the kernel function defining the random graph such that the model becomes well-defined even if the number of vertices is infinite. If the number of vertices is finite, the Lipschitz continuity of the function which gives the intensity of the interaction of a vertex \(i\) given the past of the process at the neighboring vertices of \(i\) turns out to be sufficient. Then stronger conditions are formulated on the macroscopic degree of the vertices, which are sufficient for the definition of the model if the number of vertices is infinite.
Once the model is well-defined, we can ask whether it is true that the finite-vertex version converges to the version where the number of vertices is infinite. By using concepts of the theory of limits of dense graph sequences (e.g. cut distance), the paper finds sufficient conditions for this convergence as well. The paper also examines the limit of the intensity of the interactions when time goes to infinity.
The proofs build together various tools from the theory of differential equations, probability theory (e.g. concentration inequalities), and graph limit theory.
The first goal of the paper is to find sufficient conditions on the intensities and the kernel function defining the random graph such that the model becomes well-defined even if the number of vertices is infinite. If the number of vertices is finite, the Lipschitz continuity of the function which gives the intensity of the interaction of a vertex \(i\) given the past of the process at the neighboring vertices of \(i\) turns out to be sufficient. Then stronger conditions are formulated on the macroscopic degree of the vertices, which are sufficient for the definition of the model if the number of vertices is infinite.
Once the model is well-defined, we can ask whether it is true that the finite-vertex version converges to the version where the number of vertices is infinite. By using concepts of the theory of limits of dense graph sequences (e.g. cut distance), the paper finds sufficient conditions for this convergence as well. The paper also examines the limit of the intensity of the interactions when time goes to infinity.
The proofs build together various tools from the theory of differential equations, probability theory (e.g. concentration inequalities), and graph limit theory.
Reviewer: Ágnes Backhausz (Budapest)
MSC:
| 05C80 | Random graphs (graph-theoretic aspects) |
| 44A35 | Convolution as an integral transform |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 68T07 | Artificial neural networks and deep learning |
Keywords:
multivariate nonlinear Hawkes processes; mean-field systems; neural networks; spatially extended system; random graph; graph limitReferences:
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