arXiv subjects

Ángel Alfredo Morán Ledezma

Publications and source records attributed to Ángel Alfredo Morán Ledezma.

4 recordsLinked to original sources

arXiv2026

Ultrametric Graphons and Hierarchical Community Networks: Spectral Theory and Applications

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 $p^*=\min_iρ_i$ 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.

math.SP↗

arXiv2026

Dynamic centrality of headwater sources in river networks: a stochastic approach via ultrametric Laplacians

River networks are hierarchical transport systems in which the timing and position of headwater confluences govern hydrologic response, solute transport, and ecological connectivity. Despite the recognized importance of headwater sources in structuring downstream processes, no mathematically grounded centrality index exists that captures their dynamic role in the transport hierarchy. We apply the dynamic centrality index $C_{\mathrm{CTMC}}$ [Morán Ledezma, arXiv:2603.20922], originally introduced in the context of phylogenetic trees, to the problem of headwater centrality in river networks via the dynamic tree representation of [Zaliapin et al., https://doi.org/10.1029/2009JF001281]. Through a topological analysis of the ultrametric structure induced by the dynamic tree, we show that high-centrality headwaters are the tributaries that most efficiently transmit water into the rest of the network, in the sense that their flows merge earliest and most broadly with surrounding sources as transport proceeds downstream. The index admits a fully explicit closed-form expression computable in $O(n)$ time from the tree structure alone, without simulation. Comparing $C_{\mathrm{CTMC}}$ rankings against the number of downstream junctions reached during transport, a direct measure of hydrological influence, on a dataset of 49 natural river basins across the United States, we find that top-ranked headwaters consistently reach a disproportionately large number of junctions across all transport times. This indicates that high-centrality headwaters are not merely early contributors but consistently influential throughout the entire transport process. These results suggest that ultrametric spectral analysis provides an interpretable and scalable framework for identifying hydrologically influential headwaters, with potential applications in ecological monitoring and watershed management.

physics.geo-ph↗

arXiv2026

Spectral Geometry and Heat Kernels on Phylogenetic Trees

We develop a unified spectral framework for finite ultrametric phylogenetic trees, grounding the analysis of phylogenetic structure in operator theory and stochastic dynamics in the finite setting. For a given finite ultrametric measure space $(X,d,m)$, we introduce the ultrametric Laplacian $L_X$ as the generator of a continuous time Markov chain with transition rate $q(x,y)=k(d(x,y))m(y)$. We establish its complete spectral theory, obtaining explicit closed-form eigenvalues and an eigenbasis supported on the clades of the tree. For phylogenetic applications, we associate to any ultrametric phylogenetic tree $\mathcal{T}$ a canonical operator $L_{\mathcal{T}}$, the ultrametric phylogenetic Laplacian, whose jump rates encode the temporal structure of evolutionary divergence. We show that the geometry and topology of the tree are explicitly encoded in the spectrum and eigenvectors of $L_{\mathcal{T}}$: eigenvalues aggregate branch lengths weighted by clade mass along ancestral paths, while eigenvectors are supported on the clades, with one eigenspace attached to each internal node. From this we derive three main contributions: a spectral reconstruction theorem with linear complexity $O(|X|)$; a rigorous geometric interpretation of the spectral gaps of $L_{\mathcal{T}}$ as detectors of distinct evolutionary modes, validated on an empirical primate phylogeny; an eigenmode decomposition of biological traits that resolves trait variance into contributions from individual splits of the phylogeny; and a closed-form centrality index for continuous-time Markov chains on ultrametric spaces, which we propose as a mathematically grounded measure of evolutionary distinctiveness. All results are exact and biologically interpretable, and are supported by numerical experiments on empirical primate data.

q-bio.PE↗

arXiv2023

A probabilistic interpretation of Weil's explicit sums and arithmetic spectral measures

In this paper we study the connections of three paradigms in number theory: the adelic formulation of the Riemann zeta function, the Weil explicit formula and the concepts of the so called probabilistic number theory initiated by Harald Bohr. We give a different reformulation, rooted in the adelic framework, of the theory of distribution values of the Riemann zeta function. By introducing the Bohr compactification of the real numbers as a natural probability space for this theory, we show that the Weil explicit sum can be expressed in terms of covariances and expected values attached to random variables defined on this space. Moreover, we express the explicit formula as a limit of spectral integrals attached to operators defined on the Hilbert space of square-integrable functions on the Bohr compactification. This gives a probabilistic and a geometrical interpretation of the Weil explicit formula.

math.NT↗