Origin of the atom in the dense KBRG spectral distribution

Determine whether the atom observed in the eigenvalue distribution of the dense kernel-based random graph realization is caused by the high connectivity of the kernel-based graph, in contrast to the Gaussian matrix model at \(\alpha=0\), where this trivialization does not occur.

Background

The paper compares numerical eigenvalue distributions for kernel-based random graphs with those of Gaussian matrices. In the displayed simulation discussion, the authors observe an atom in one kernel-based graph spectrum but not in the corresponding Gaussian setup. They explicitly conjecture that the atom results from the graph’s high connectivity, while noting that at α=0\alpha=0 the connection probabilities are identically one and the model becomes trivialized. The proposed causal explanation is not established analytically in the paper.

References

We conjecture that the atom appearing in the latter is due to high connectivity of the kernel-based realization (if \alpha=0, for all i, \,j we have that p_{ij} is identically one in~connection_proba), whilst in the Gaussian setup this trivialization does not arise.

connection_proba:

PW(i↔j)≔P(i↔j∣Wi,Wj)=κσ(Wi,Wj)∥i−j∥α∧1.P^W(i\leftrightarrow j) \coloneqq \mathbb{P}(i \leftrightarrow j \mid W_i, W_j) = \frac{\kappa_\sigma(W_i, W_j)}{\|i - j\|^\alpha} \wedge 1.

— The spectrum of dense kernel-based random graphs  (2502.09415 - Cipriani et al., 13 Feb 2025) in Section 2.3, “Examples, simulations and discussion”