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Ultrametric Graphons and Hierarchical Community Networks: Spectral Theory and Applications
\'Angel Alfredo Mor\'an Ledezma
https://arxiv.org/abs/2605.13423 https://arxiv.org/pdf/2605.13423 https://arxiv.org/html/2605.13423
arXiv:2605.13423v1 Announce Type: new
Abstract: We develop a theory of ultrametric graphons as limiting objects for random networks with nested hierarchical community structure. A graphon $W:[0,1]^2\to[0,1]$ is called ultrametric if $W(x,y)=w(d(x,y))$, where $d$ is an ultrametric on $[0,1]$ induced by a family of nested partitions and $w$ is a positive kernel. The resulting random graphs exhibit a nested hierarchical community structure in which the density of connections is governed by the ultrametric distance between vertices. The Laplacian $L_d^k$ of the deterministic graph sampled from an ultrametric graphon is itself an ultrametric Laplacian, whose eigenvalues and spectral projectors admit completely explicit closed-form expressions in terms of the community sizes and inter-community connection densities. We show that the normalized eigenvalues and spectral projectors of the random Laplacian $L_r^k$ are arbitrarily close to those of $L_d^k$ with high probability as $k\to\infty$, so that the explicit formulas for $L_d^k$ provide closed-form approximations for the spectrum and spectral projectors of $L_r^k$. As applications: a sign structure theorem generalizes the Fiedler vector criterion to hierarchical networks with arbitrarily many communities; a detectability threshold $p^*=\min_i\rho_i$ governs spectral community detection for one-level hierarchical graphons; the pseudo-inverse Laplacian $L_W^ $ is constructed and shown to be the almost sure limit of the pseudo-inverse of $L_r^k$, implying that hitting and commute times collapse to quantities depending only on the expected degrees of the endpoints; and explicit closed-form stability conditions for the SIS disease-free equilibrium reveal a fundamental tension between homogeneous and heterogeneous community structures, confirmed by numerical experiments.
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A structural reduction for the symmetric hit problem in four variables
Dang Vo Phuc
https://arxiv.org/abs/2606.02626 https://arxiv.org/pdf/2606.02626 https://arxiv.org/html/2606.02626
arXiv:2606.02626v1 Announce Type: new
Abstract: Let $\mathcal{A}$ be the mod $2$ Steenrod algebra, and $P(n) = \mathbb{F}_2[x_1, \dots, x_n]$ be the polynomial algebra viewed as an unstable module over $\mathcal{A}$. The symmetric hit conjecture asks whether the symmetrization of a hit monomial in $P(n)$ is always hit in the symmetric invariant subalgebra $B(n) = P(n)^{\Sigma_n}$. While resolved for $n \leq 3$, the case $n=4$ presents significant obstructions due to combinatorial complexity, orbit cancellations intrinsically tied to $\Sigma_4$-stabilizers, and the emergence of strongly spike-free survivor modules. This paper introduces a conditional structural reduction to overcome these obstructions in the domain where the numerical weight satisfies $\mu(d) \leq 4$. By integrating Walker-Wood duality with a new $\Sigma_4$-stabilizer parity analysis, we reduce the global conjecture to localized algebraic conditions: a symmetric lower-spike reduction and a strengthened four-row digital-engineering hypothesis. Assuming these inputs, the conjecture follows by lexicographic induction on the column-sum and row-sum sequences of the binary exponent matrices.
Our approach isolates the four-variable repeated-row anomaly into exact local identities, utilizing global Steenrod-kernel functionals lifted from local spike-free quotients to detect potential survivor elements. Finally, we provide explicit monomial-level computations in degrees $8$, $12$, and $14$, explicitly illustrating the stabilizer mechanism in practice and framing the precise algebraic identities required for a future unconditional proof.
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