There is no reason to think that Markwayne Mullin will change course in any significant way at DHS.
Like Kristy Noem, he will be answering primarily to White House deputy chief of staff Stephen Miller, a man who spends all day on X implying that non-white immigrants are the primary source of crime and economic deprivation.
Miller’s demand that Immigration and Customs Enforcement, which is overseen by DHS,
arrest 3,000 people a day is the force behind Noem’s worst choices a…
Amazing image from Artemis II
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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Finally an election graphic that is more confusing than the #LibDem dodgy bar charts. I think this is implying that if you don't vote Green then Reform will pick up the last of the proportionally distributed seats for the #Senedd. I don't think PR works that way.
Oh, so all of these MAGA shops that made 86 46 shirts, hats, stickers, etc. wanted to kill Joe Biden?
@cdarwin@c.im
