Signature-native graph recovery for sparse multivariate Hawkes kernels

For a sparse multivariate Hawkes process, let $A_T^{ij}$ denote the cross-area between coordinates $i$ and $j$ over a time horizon $T$, and let the associated excitation graph encode the directed excitation relationships between coordinates. Signature-native graph recovery. For sparse multivariate Hawkes kernels, the sign pattern of the expected cross-areas

$$\{\mathbb{E}[A_T^{ij}]\}_{i<j}$$

over one or several horizons recovers the dominant directed excitation graph after controlling for common baselines and self-excitation. This would provide a signature-based method for recovering directional interaction structure from Hawkes-process data; the statement is presented as an open conjecture, and its precise identifiability conditions remain to be established.