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Detecting Mutual Excitations in Non-Stationary Hawkes Processes

Elchanan Mossel, Anirudh Sridhar

TL;DR

The paper tackles learning the sparse mutual excitation structure in non-stationary, multivariate Hawkes processes with time-varying baselines and non-shift-invariant kernels. It introduces pairwise and triple-event statistics that, when aggregated into $D^{(1)}$ and $D^{(2)}$, reveal the presence of directed excitations between node pairs. By combining these statistics and leveraging concentration results, the authors prove that the ground-truth dependency graph $G^*$ can be exactly recovered from $T = \mathrm{polylog}(n)$ time with high probability, even if only a subset of nodes is observed and event times are quantized. The approach avoids full parameter estimation, offers robustness to non-stationarity, and yields practical guarantees for structure learning in large-scale networks with self- and mutual-excitation dynamics.

Abstract

We consider the problem of learning the network of mutual excitations (i.e., the dependency graph) in a non-stationary, multivariate Hawkes process. We consider a general setting where baseline rates at each node are time-varying and delay kernels are not shift-invariant. Our main results show that if the dependency graph of an $n$-variate Hawkes process is sparse (i.e., it has a maximum degree that is bounded with respect to $n$), our algorithm accurately reconstructs it from data after observing the Hawkes process for $T = \mathrm{polylog}(n)$ time, with high probability. Our algorithm is computationally efficient, and provably succeeds in learning dependencies even if only a subset of time series are observed and event times are not precisely known.

Detecting Mutual Excitations in Non-Stationary Hawkes Processes

TL;DR

The paper tackles learning the sparse mutual excitation structure in non-stationary, multivariate Hawkes processes with time-varying baselines and non-shift-invariant kernels. It introduces pairwise and triple-event statistics that, when aggregated into and , reveal the presence of directed excitations between node pairs. By combining these statistics and leveraging concentration results, the authors prove that the ground-truth dependency graph can be exactly recovered from time with high probability, even if only a subset of nodes is observed and event times are quantized. The approach avoids full parameter estimation, offers robustness to non-stationarity, and yields practical guarantees for structure learning in large-scale networks with self- and mutual-excitation dynamics.

Abstract

We consider the problem of learning the network of mutual excitations (i.e., the dependency graph) in a non-stationary, multivariate Hawkes process. We consider a general setting where baseline rates at each node are time-varying and delay kernels are not shift-invariant. Our main results show that if the dependency graph of an -variate Hawkes process is sparse (i.e., it has a maximum degree that is bounded with respect to ), our algorithm accurately reconstructs it from data after observing the Hawkes process for time, with high probability. Our algorithm is computationally efficient, and provably succeeds in learning dependencies even if only a subset of time series are observed and event times are not precisely known.
Paper Structure (15 sections, 14 theorems, 76 equations)

This paper contains 15 sections, 14 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Models and results
  3. The multivariate Hawkes process
  4. Detecting mutual excitations
  5. Estimation algorithm
  6. Analysis of pairwise events
  7. Analysis of event triples
  8. Algorithm description
  9. Analysis of Algorithm \ref{['alg:structure_learning']}
  10. Conclusion
  11. Additional notation
  12. Preliminary results
  13. Bounding the maximum rate: Proof of Lemma \ref{['lemma:maximum_rate']}
  14. Analysis of short waiting times: Proof of Lemma \ref{['lemma:EX']}
  15. Analysis of key statistics: Proof of Lemma \ref{['lemma:D_analysis']}

Key Result

Theorem 2.7

Suppose that the point processes $N_1, \ldots, N_n$ are observed in $[0,T]$, with $T = \log^{100}(n)$. Then there is an estimator $\widehat{G}$ such that $\widehat{G} = G^*$ with probability $1 - o(1)$.

Theorems & Definitions (25)

  • Theorem 2.7
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof : Proof of Theorem \ref{['thm:alg']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 15 more