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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.13423 |
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| _version_ | 1866909040241541120 |
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| author | Ledezma, Ángel Alfredo Morán |
| author_facet | Ledezma, Ángel Alfredo Morán |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13423 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ultrametric Graphons and Hierarchical Community Networks: Spectral Theory and Applications Ledezma, Ángel Alfredo Morán Spectral Theory Probability 05C80, 47A75, 60B20 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. |
| title | Ultrametric Graphons and Hierarchical Community Networks: Spectral Theory and Applications |
| topic | Spectral Theory Probability 05C80, 47A75, 60B20 |
| url | https://arxiv.org/abs/2605.13423 |