{"title":"The rich are loopy: a spectral framework for loop structure in the rich club","authors":"Leo Torres","abstract":"The rich club coefficient $\\phi(k)$ quantifies whether high-degree nodes interconnect more densely than expected, but it is blind to the structure of those interconnections and offers no principled way to select the degree threshold $k$.  We introduce the *cycle core* $H_k$, the maximal subgraph of vertices with degree at least $k$ that forms cycles among themselves.  The nested sequence $H_1 \\supseteq H_2 \\supseteq \\cdots$ reveals how the cycle core evolves as the degree threshold increases, capturing structure that $\\phi(k)$ cannot see.  We develop two spectral tools for this sequence.  The first is structural: the *$k$-blocking operator* $\\mathbf{M}_k$ generalizes the non-backtracking matrix by blocking $k - 1$ directions at each node. We prove that its spectrum is equivalent to that of a modified non-backtracking matrix weighted with a specific function of the nodes' degrees. Thus, $\\mathbf{M}_k$ couples cycle structure with the degree sequence of the graph. We show the spectral radius of $\\mathbf{M}_k$ peaks at an intrinsic threshold $k^*$, yielding the *optimal rich club* $H_{k^*}$ without null models or parameter tuning.  The second tool is dynamical: the $k$-blocking walk defines a sub-stochastic Markov chain whose leading eigenvalue $\\lambda_1(k)$ measures how long a process can persist on the cycle core before being expelled to the periphery.  We evaluate the framework on fifteen networks spanning different domains, wherein $H_{k^*}$ closely recovers the published rich clubs.","keywords":[],"year":2026,"slug":"rich-are-loopy","version":2,"url_hash":"d777d2bda4b2","license":"cc-by-4.0","doi":"10.5281/zenodo.20491050","published_at":"2026-06-01T13:51:42.381780+00:00","subject":"Computer Science","storage_type":"inline","canonical_url":"https://scroll.press/2026/rich-are-loopy","version_url":"https://scroll.press/2026/rich-are-loopy/v2","paper_url":"https://scroll.press/2026/rich-are-loopy/paper","paper_version_url":"https://scroll.press/2026/rich-are-loopy/v2/paper","html_url":"https://scroll.press/2026/rich-are-loopy/paper","html_sha256":"d777d2bda4b20491acb353092f08367ec5418cfb61b4685f9a771fb0bf0c0ce2","html_bytes":888797,"cite_as":{"bibtex":"@misc{rich-are-loopy2026,\n  author = {Torres, Leo},\n  title = {The rich are loopy: a spectral framework for loop structure in the rich club},\n  year = {2026},\n  publisher = {Scroll Press},\n  url = {https://scroll.press/2026/rich-are-loopy},\n  version = {2},\n  doi = {10.5281/zenodo.20491050}\n}","csl_json":{"type":"article","id":"d777d2bda4b2","title":"The rich are loopy: a spectral framework for loop structure in the rich club","author":[{"family":"Torres","given":"Leo"}],"publisher":"Scroll Press","URL":"https://scroll.press/2026/rich-are-loopy","version":"2","issued":{"date-parts":[[2026]]},"DOI":"10.5281/zenodo.20491050","license":"cc-by-4.0"}}}