{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T09:49:43Z","timestamp":1791884983609,"version":"4.3.4"},"posted":{"date-parts":[[2026]]},"group-title":"SSRN","reference-count":16,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>We study the local geometry of time-varying correlation networks in financial markets and introduce the Ultrametric Defect, a closed-form statistic that quantifies a correlation network's departure from hierarchical tree-representability by summing the intransitive gaps of all triadic motifs-configurations where two pairwise correlations are strong but the third is weak. The aggregate defect factorizes at leading order into a metric factor that depends only on the mean correlation and a structural factor that depends only on the centered deviations. For the Mantegna metric the exact aggregate defect admits a closed form in terms of harmonic means, which reduces to the product form at leading order. The spectral gap of the correlation matrix decomposes into a synchronization channel and a frustration channel, and in a minimal three-unit model the crossover between them is state-dependent. Applying the measure to daily returns of the 49 Fama-French industry portfolios (1976-2025), we verify the factorization on the data, find that it holds as a leading-order approximation, and show that the Defect Level and the first PCA share are weakly correlated. A joint classification into four regimes produces clustered states. In this sample, the regime in which a strong market factor coincides with high structural intransitivity has the highest contemporaneous volatility of the four, although its first PCA share is close to that of the synchronized benchmark. The analysis is descriptive and does not test predictive content.<\/jats:p>","DOI":"10.2139\/ssrn.7591178","type":"posted-content","created":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T04:47:28Z","timestamp":1791607648000},"source":"Crossref","is-referenced-by-count":0,"title":["Triadic Motifs and Hidden Fragility in Financial Markets"],"prefix":"10.2139","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-4867-1079","authenticated-orcid":true,"given":"Ruslan","family":"Yagufarov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"3","key":"ref1","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.jfineco.2011.12.010","article-title":"Econometric measures of connectedness and systemic risk in the finance and insurance sectors","volume":"104","author":"M Billio","year":"2012","journal-title":"Journal of Financial Economics"},{"key":"ref2","doi-asserted-by":"crossref","DOI":"10.4159\/9780674029095","author":"R S Burt","year":"1992","journal-title":"Structural Holes: The Social Structure of Competition"},{"issue":"5","key":"ref3","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1037\/h0046049","article-title":"Structural balance: A generalization of Heider's theory","volume":"63","author":"D Cartwright","year":"1956","journal-title":"Psychological Review"},{"issue":"5","key":"ref4","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/j.physa.2008.10.047","article-title":"Cross-correlation dynamics in financial time series","volume":"388","author":"T Conlon","year":"2009","journal-title":"Physica A"},{"issue":"6","key":"ref5","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1086\/225469","article-title":"The strength of weak ties","volume":"78","author":"M S Granovetter","year":"1973","journal-title":"American Journal of Sociology"},{"key":"ref6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1080\/00223980.1946.9917275","article-title":"Attitudes and cognitive organization","volume":"21","author":"F Heider","year":"1946","journal-title":"Journal of Psychology"},{"issue":"4","key":"ref7","first-page":"112","article-title":"Principal components as a measure of systemic risk","volume":"37","author":"M Kritzman","year":"2011","journal-title":"Journal of Portfolio Management"},{"issue":"7","key":"ref8","doi-asserted-by":"crossref","first-page":"1467","DOI":"10.1103\/PhysRevLett.83.1467","article-title":"Noise dressing of financial correlation matrices","volume":"83","author":"L Laloux","year":"1999","journal-title":"Physical Review Letters"},{"issue":"1","key":"ref9","doi-asserted-by":"crossref","DOI":"10.1007\/s100510050929","article-title":"Hierarchical structure in financial markets","volume":"11","author":"R N Mantegna","year":"1999","journal-title":"European Physical Journal B"},{"key":"ref10","author":"M M\u00e9zard","year":"1987","journal-title":"Spin Glass Theory and Beyond"},{"issue":"5594","key":"ref11","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1126\/science.298.5594.824","article-title":"Network motifs: Simple building blocks of complex networks","volume":"298","author":"R Milo","year":"2002","journal-title":"Science"},{"issue":"6","key":"ref12","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.65.066126","article-title":"Random matrix approach to cross correlations in financial data","volume":"65","author":"V Plerou","year":"2002","journal-title":"Physical Review E"},{"issue":"20","key":"ref13","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1051\/jphyslet:019850046020094500","article-title":"On the degree of ultrametricity","volume":"46","author":"R Rammal","year":"1985","journal-title":"Journal de Physique Lettres"},{"key":"ref14","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1103\/RevModPhys.58.765","article-title":"Ultrametricity for physicists","volume":"58","author":"R Rammal","year":"1986","journal-title":"Reviews of Modern Physics"},{"issue":"2","key":"ref15","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2307\/1217208","article-title":"The comparison of dendrograms by objective methods","volume":"11","author":"R R Sokal","year":"1962","journal-title":"Taxon"},{"issue":"1","key":"ref16","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.jebo.2010.01.004","article-title":"Correlation, hierarchies, and networks in financial markets","volume":"75","author":"M Tumminello","year":"2010","journal-title":"Journal of Economic Behavior & Organization"}],"container-title":[],"original-title":[],"deposited":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T09:39:58Z","timestamp":1791884398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ssrn.com\/abstract=7591178"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":16,"URL":"https:\/\/doi.org\/10.2139\/ssrn.7591178","relation":{},"subject":[],"published":{"date-parts":[[2026]]},"subtype":"preprint"}}