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arxiv: 2606.19860 · v1 · pith:DXQHYAPFnew · submitted 2026-06-18 · ⚛️ physics.comp-ph · cond-mat.stat-mech· physics.soc-ph

The Heat Kernel Expansion: Curvature for Shock Detection in Higher-Order Financial Networks

Mohammad Elsayed , Sara Najem This is my paper

Pith reviewed 2026-06-26 15:17 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.stat-mechphysics.soc-ph
keywords heat kernel expansioncurvaturesimplicial complexesfinancial networksshock detectionboard interlockshigher-order networksNorway legislation
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The pith

Curvature extracted from the heat kernel expansion on simplicial complexes detects legislative shocks in Norwegian financial board networks where Euler characteristic and torsion do not.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tracks nine years of monthly Norwegian financial data by representing directors as nodes and companies as faces in a simplicial complex. Standard topological invariants such as the Euler characteristic, computed via Betti numbers, and torsion, derived from higher-order Laplacians, fail to register the impact of new legislation on board interlocks. In contrast, the local curvature obtained from the short-time coefficients of the heat kernel expansion registers clear changes tied to the legislation. Inflection points in this curvature align with external forcing, while its minima mark the arrival of shocks. This demonstrates that integrated topological summaries lose the local geometric signal needed to identify network disruptions.

Core claim

The coefficients of the heat kernel expansion yield a curvature that registers variation in board interlock caused by legislation and functions as a sensitive detector of shocks; inflection points correspond to external forcing and minima to shock arrival times, whereas the Euler characteristic integrates away local information and torsion and clustering coefficients do not isolate the legislative effect.

What carries the argument

Curvature obtained from the coefficients in the short-time series expansion of the heat kernel on the simplicial complex, which supplies local geometric information absent from global topological invariants.

If this is right

  • Curvature tracks the effect of imposed law on director-company representation while the Euler characteristic does not.
  • Inflection points in curvature mark periods of external forcing on the network.
  • Minima in curvature coincide with the arrival times of shocks in board interlock structure.
  • Torsion components display sharp transitions at shock times while higher-order clustering coefficients vary smoothly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same heat-kernel curvature construction could be tested on other director-company or ownership networks to check whether legislative or regulatory shocks produce comparable signatures.
  • If curvature minima reliably precede or coincide with observable market or governance disruptions, the measure might support earlier monitoring of policy impacts.
  • The distinction between local curvature and integrated topology suggests that geometric probes may outperform purely combinatorial ones when networks are driven by external rules.

Load-bearing premise

The simplicial-complex model with directors as nodes and companies as faces accurately encodes the higher-order interactions that legislation actually alters.

What would settle it

Curvature coefficients extracted from the same Norwegian board data show no detectable shift at the dates of known legislative changes, or exhibit shifts of comparable size at times with no legislative event.

Figures

Figures reproduced from arXiv: 2606.19860 by Mohammad Elsayed, Sara Najem.

Figure 1
Figure 1. Figure 1: Work Flow Adopted in the Topological and Geometric Characterization of our Simplicial Complex [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sample Construction Depicting a Small Realization of Interlocking Boards [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The average node curvature R¯(t) is shown, fitted to a fifth-order polynomial ¯˙R. R¯ = 9.968 − 5.290 × 10−2 t + 1.538 × 10−3 t 2 − 7.047 × 10−5 t 3 + 9.894 × 10−7 t 4 − 4.154 × 10−9 t 5 , with R-squared is 0.98. The dashed lines correspond to the introduction of the legislation in January 2006, and its full implementation in January 2008, which coincide with the curve’s inflection point and minimum. The i… view at source ↗
Figure 4
Figure 4. Figure 4: The figures follow the evolution of the total number of directors as well as evolution per gender. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The figure shows the temporal evolution of the number of boards. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The dynamics of the higher order clustering [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The evolution of the Euler characteristic [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The evolution of the logarithm of spanning trees for the 0-order and first order Laplacian are shown in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

This work follows the evolution of financial networks in Norway over a period of nine years at a monthly rate. The data consist of board directors and their affiliations to companies, which we model as simplicial complexes. In this framework, directors are represented as nodes and companies as faces of the complex. To characterize the latter, we focus on three topological measures: the Euler characteristic, computed through the Betti numbers, torsion computed through the reduced determinant of the higher-order Laplacians, and higher-order clustering coefficients. The first two fail to capture the effect of imposed law on representation, unlike our notion of curvature which is a geometrical measure computed from the coefficients of the series expansion of the heat kernel in powers of time, which is our major contribution in this work. In particular, the Euler characteristic integrates curvature, and thus local information is lost. Subsequently, not every topological measure can reliably capture shocks in networks. Further, the number of spanning trees may undergo significant changes at the lowest order, yet these changes need not be reflected in the torsion. Conversely, the change in the curvature revealed variation in the board interlock due to legislation, and serves as a sensitive measure for detecting shocks in networks. Inflection points in curvature are associated with external forcing, and minima with shock arrival times. Sharp transitions are also observed in the components of torsion, while smooth changes are observed in higher-order clustering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript models Norwegian board-interlock networks over nine years as simplicial complexes (directors as 0-simplices, companies as faces). It computes the Euler characteristic via Betti numbers, torsion via reduced determinants of higher-order Laplacians, higher-order clustering coefficients, and a curvature derived from the coefficients of the heat-kernel expansion on the complex. The central claim is that only the curvature measure detects variations in board interlock induced by legislation, with inflection points linked to external forcing and minima to shock arrival times, while the other invariants fail to capture these effects.

Significance. If the explicit heat-kernel coefficients can be shown to isolate local geometric responses to legislation without circular dependence on global density or degree sequence, the approach would supply a geometrically grounded detector for shocks in higher-order networks that retains information lost when curvature is integrated into the Euler characteristic. The longitudinal financial application is a concrete test case, but the current absence of formulas and validation prevents assessment of whether this advantage is realized.

major comments (3)
  1. [Abstract] Abstract: the claim that curvature is 'a geometrical measure computed from the coefficients of the series expansion of the heat kernel' supplies neither the explicit form of those coefficients, the definition of the higher-order Laplacian, nor the truncation order employed. Without these, it is impossible to determine whether the reported sensitivity to legislation is independent of modeling choices or reduces to a fitted global quantity.
  2. The comparison asserting that Euler characteristic and torsion 'fail to capture the effect of imposed law' while curvature succeeds rests entirely on narrative description; no quantitative error estimates, statistical tests, or cross-validation against known legislation dates are provided to establish that the curvature changes are significantly associated with the legislation rather than other network variations.
  3. The simplicial-complex construction is stated only at the level of the abstract (directors as nodes, companies as faces). No explicit description of the 1-skeleton, the precise definition of the faces, or the choice of higher-order structure is given, so it cannot be verified whether the complex faithfully encodes the board-interlock dynamics affected by gender-quota legislation.
minor comments (1)
  1. The phrase 'reduced determinant of the higher-order Laplacians' for torsion is used without clarifying the reduction procedure or its relation to standard definitions of Reidemeister torsion in algebraic topology.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below and will revise the manuscript accordingly to improve clarity, provide missing technical details, and add quantitative validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that curvature is 'a geometrical measure computed from the coefficients of the series expansion of the heat kernel' supplies neither the explicit form of those coefficients, the definition of the higher-order Laplacian, nor the truncation order employed. Without these, it is impossible to determine whether the reported sensitivity to legislation is independent of modeling choices or reduces to a fitted global quantity.

    Authors: We agree that the abstract is too concise and omits these elements. In the revised manuscript we will expand the abstract (or add a short methods paragraph) to state the explicit heat-kernel coefficients through order t², the definition of the higher-order combinatorial Laplacian on the simplicial complex, and the truncation at order t³. These additions will make clear that the curvature is obtained from local geometric terms in the expansion and is not a global fit. revision: yes

  2. Referee: The comparison asserting that Euler characteristic and torsion 'fail to capture the effect of imposed law' while curvature succeeds rests entirely on narrative description; no quantitative error estimates, statistical tests, or cross-validation against known legislation dates are provided to establish that the curvature changes are significantly associated with the legislation rather than other network variations.

    Authors: The referee correctly notes the absence of formal statistical support. We will add, in a new subsection, Pearson correlations between the curvature time series and binary legislative-event indicators, together with a bootstrap permutation test against degree-preserved randomizations, to quantify the association of curvature minima and inflection points with the known legislation dates. revision: yes

  3. Referee: The simplicial-complex construction is stated only at the level of the abstract (directors as nodes, companies as faces). No explicit description of the 1-skeleton, the precise definition of the faces, or the choice of higher-order structure is given, so it cannot be verified whether the complex faithfully encodes the board-interlock dynamics affected by gender-quota legislation.

    Authors: We acknowledge that the construction details are insufficiently prominent. The revised version will include an expanded Methods section that explicitly defines the 1-skeleton (edges between directors sharing at least one board), the 2-simplices (complete cliques for each company board), and the inclusion of selected 3-simplices for multi-company interlocks, together with a schematic figure illustrating the encoding of board-interlock relations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained at modeling level

full rationale

The abstract introduces curvature via heat-kernel expansion coefficients on a simplicial-complex model of board interlocks as the central contribution, contrasting it with the Euler characteristic (which integrates curvature) and other topological measures. No equations, parameter fits, self-citations, or uniqueness theorems are supplied in the provided text, so no load-bearing step can be exhibited that reduces by construction to its own inputs. The modeling choice that the simplicial representation and heat-kernel truncation register legislation-induced shocks is presented as an empirical observation rather than a self-referential derivation. This is the normal case of an honest non-finding when explicit algebraic steps are absent.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, background axioms, or newly postulated entities; assessment is therefore empty.

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Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages

  1. [1]

    Cambridge University Press, 2016

    Albert-L ´aszl´o Barab´asi.Network Science, chapter 1. Cambridge University Press, 2016. Section 1.5

    2016

  2. [2]

    Cambridge University Press, Cambridge, 2016

    Albert-L ´aszl´o Barab´asi and M´arton P´osfai.Network science. Cambridge University Press, Cambridge, 2016. URL:http://barabasi.com/networksciencebook/

    2016

  3. [3]

    The physics of financial networks.Nature Reviews Physics, 3(7):490–507, 2021

    Marco Bardoscia, Paolo Barucca, Stefano Battiston, Fabio Caccioli, Giulio Cimini, Diego Garlaschelli, Fabio Saracco, Tiziano Squartini, and Guido Caldarelli. The physics of financial networks.Nature Reviews Physics, 3(7):490–507, 2021

    2021

  4. [4]

    Using discrete ricci curvatures to infer covid-19 epidemic network fragility and systemic risk.Journal of Statistical Mechanics: Theory and Experiment, 2021(5):053501, 2021

    Danillo Barros de Souza, Jonatas TS Da Cunha, Everlon Figueirˆoa dos Santos, Jailson B Correia, Hernande P da Silva, Jos ´e Luiz de Lima Filho, Jones Albuquerque, and Fernando AN Santos. Using discrete ricci curvatures to infer covid-19 epidemic network fragility and systemic risk.Journal of Statistical Mechanics: Theory and Experiment, 2021(5):053501, 2021

    2021

  5. [5]

    The physics of higher-order interactions in complex systems.Nature physics, 17(10):1093–1098, 2021

    Federico Battiston, Enrico Amico, Alain Barrat, Ginestra Bianconi, Guilherme Ferraz de Arruda, Benedetta Franceschiello, Iacopo Iacopini, Sonia K ´efi, Vito Latora, Yamir Moreno, et al. The physics of higher-order interactions in complex systems.Nature physics, 17(10):1093–1098, 2021

    2021

  6. [6]

    Elements in the Structure and Dynamics of Complex Networks

    Ginestra Bianconi.Higher-Order Networks. Elements in the Structure and Dynamics of Complex Networks. Cambridge University Press, 2021

    2021

  7. [7]

    Kirchhoff’s theorems in higher dimensions and reidemeister torsion.Homology, Homotopy and Applications, 17(1):165–189, 2015

    Michael J Catanzaro, Vladimir Y Chernyak, and John R Klein. Kirchhoff’s theorems in higher dimensions and reidemeister torsion.Homology, Homotopy and Applications, 17(1):165–189, 2015

    2015

  8. [8]

    Systemic risk, contagion, and financial networks: A survey.Conta- gion, and Financial Networks: A Survey (June 3, 2015), 2015

    Matteo Chinazzi and Giorgio Fagiolo. Systemic risk, contagion, and financial networks: A survey.Conta- gion, and Financial Networks: A Survey (June 3, 2015), 2015

    2015

  9. [9]

    Systemic risk analysis on re- constructed economic and financial networks.Scientific reports, 5(1):15758, 2015

    Giulio Cimini, Tiziano Squartini, Diego Garlaschelli, and Andrea Gabrielli. Systemic risk analysis on re- constructed economic and financial networks.Scientific reports, 5(1):15758, 2015. 9

    2015

  10. [10]

    Simplicial matrix-tree theorems.Transactions of the American Mathematical Society, 361(11):6073–6114, 2009

    Art Duval, Caroline Klivans, and Jeremy Martin. Simplicial matrix-tree theorems.Transactions of the American Mathematical Society, 361(11):6073–6114, 2009

    2009

  11. [11]

    Forman–ricci communicability curvature of graphs and networks.European Journal of Applied Mathematics, 36(6):1096–1120, 2025

    Ernesto Estrada. Forman–ricci communicability curvature of graphs and networks.European Journal of Applied Mathematics, 36(6):1096–1120, 2025

    2025

  12. [12]

    Measuring road network topology vulnerability by ricci curvature.Physica A: Statistical Mechanics and Its Applications, 527:121071, 2019

    Lei Gao, Xingquan Liu, Yu Liu, Pu Wang, Min Deng, Qing Zhu, and Haifeng Li. Measuring road network topology vulnerability by ricci curvature.Physica A: Statistical Mechanics and Its Applications, 527:121071, 2019

    2019

  13. [13]

    Dey, Soham Mukherjee, Shreyas N

    Mustafa Hajij, Ghada Zamzmi, Theodore Papamarkou, Nina Miolane, Aldo Guzm ´an-S´aenz, Karthikeyan Natesan Ramamurthy, Tolga Birdal, Tamal K. Dey, Soham Mukherjee, Shreyas N. Samaga, Neal Livesay, Robin Walters, Paul Rosen, and Michael T. Schaub. Topological deep learning: Going beyond graph data, 2023.arXiv:2206.00606

    arXiv 2023

  14. [14]

    The hyperbolic geometry of financial networks.Scientific reports, 11(1):4732, 2021

    Martin Keller-Ressel and Stephanie Nargang. The hyperbolic geometry of financial networks.Scientific reports, 11(1):4732, 2021

    2021

  15. [15]

    Modeling shock propagation and resilience in financial temporal net- works.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(1), 2025

    Fabrizio Lillo and Giorgio Rizzini. Modeling shock propagation and resilience in financial temporal net- works.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(1), 2025

    2025

  16. [16]

    Spec- tral signatures of structural change in financial networks.Chaos, Solitons & Fractals, 193:116065, 2025

    Valentina Macchiati, Emiliano Marchese, Piero Mazzarisi, Diego Garlaschelli, and Tiziano Squartini. Spec- tral signatures of structural change in financial networks.Chaos, Solitons & Fractals, 193:116065, 2025

    2025

  17. [17]

    Higher-order network representation of j

    Dima Mrad and Sara Najem. Higher-order network representation of j. s. bach’s solo violin sonatas and partitas: Topological and geometrical explorations, 2025. URL:https://arxiv.org/abs/2506. 08540,arXiv:2506.08540

    arXiv 2025

  18. [18]

    Torsion in higher-order networks: Application to music variations.Chaos, Solitons & Fractals, 208:118170, 2026

    Sara Najem and Dima Mrad. Torsion in higher-order networks: Application to music variations.Chaos, Solitons & Fractals, 208:118170, 2026. URL:https://www.sciencedirect.com/science/ article/pii/S0960077926003115,doi:10.1016/j.chaos.2026.118170

    work page doi:10.1016/j.chaos.2026.118170 2026

  19. [19]

    Geometric features of higher-order networks via the spectral triplet.Communications Physics, 2026

    Sara Najem, Dima Mrad, and Mohammad Elsayed. Geometric features of higher-order networks via the spectral triplet.Communications Physics, 2026

    2026

  20. [20]

    Oxford university press, 2018

    Mark Newman.Networks. Oxford university press, 2018

    2018

  21. [21]

    Higher-order laplacian renormalization, 2024

    Marco Nurisso, Marta Morandini, Maxime Lucas, Francesco Vaccarino, Tommaso Gili, and Giovanni Petri. Higher-order laplacian renormalization, 2024. URL:https://arxiv.org/abs/2401.11298, arXiv:2401.11298

    arXiv 2024

  22. [22]

    ´Angeles Serrano, Mari ´an Bogu ˜n´a, and Dmitri Krioukov

    Fragkiskos Papadopoulos, Maksim Kitsak, M. ´Angeles Serrano, Mari ´an Bogu ˜n´a, and Dmitri Krioukov. Popularity versus similarity in growing networks.Nature, 489(7417):537–540, September 2012. URL: http://dx.doi.org/10.1038/nature11459,doi:10.1038/nature11459

    work page doi:10.1038/nature11459 2012

  23. [23]

    Network-centric indicators for fragility in global financial indices.Frontiers in Physics, 8:624373, 2021

    Areejit Samal, Sunil Kumar, Yasharth Yadav, and Anirban Chakraborti. Network-centric indicators for fragility in global financial indices.Frontiers in Physics, 8:624373, 2021

    2021

  24. [24]

    Ricci curvature: An economic indicator for market fragility and systemic risk.Science advances, 2(5):e1501495, 2016

    Romeil S Sandhu, Tryphon T Georgiou, and Allen R Tannenbaum. Ricci curvature: An economic indicator for market fragility and systemic risk.Science advances, 2(5):e1501495, 2016

    2016

  25. [25]

    Discrete curvatures and network analysis

    Emil Saucan, Areejit Samal, Melanie Weber, and J ¨urgen Jost. Discrete curvatures and network analysis. MATCH Commun. Math. Comput. Chem, 80(3):605–622, 2018

    2018

  26. [26]

    Cathrine Seierstad and Tore Opsahl. For the few not the many? the effects of affirmative action on pres- ence, prominence, and social capital of women directors in norway.Scandinavian journal of management, 27(1):44–54, 2011

    2011

  27. [27]

    Stationarity, non-stationarity and early warning signals in eco- nomic networks.Journal of Complex Networks, 3(1):1–21, 2015

    Tiziano Squartini and Diego Garlaschelli. Stationarity, non-stationarity and early warning signals in eco- nomic networks.Journal of Complex Networks, 3(1):1–21, 2015

    2015

  28. [28]

    Graph curvature and the robustness of cancer networks.arXiv preprint arXiv:1502.04512, 2015

    Allen Tannenbaum, Chris Sander, Liangjia Zhu, Romeil Sandhu, Ivan Kolesov, Eduard Reznik, Yasin Sen- babaoglu, and Tryphon Georgiou. Graph curvature and the robustness of cancer networks.arXiv preprint arXiv:1502.04512, 2015. 10

    Pith/arXiv arXiv 2015

  29. [29]

    Simplicial complexes: higher-order spectral dimension and dynam- ics.Journal of Physics: Complexity, 1(1):015002, 2020

    Joaqu ´ın J Torres and Ginestra Bianconi. Simplicial complexes: higher-order spectral dimension and dynam- ics.Journal of Physics: Complexity, 1(1):015002, 2020

    2020

  30. [30]

    Forman-ricci flow for change detection in large dynamic data sets.Axioms, 5(4):26, 2016

    Melanie Weber, J ¨urgen Jost, and Emil Saucan. Forman-ricci flow for change detection in large dynamic data sets.Axioms, 5(4):26, 2016

    2016

  31. [31]

    Curvature-based analysis of brain networks.arXiv preprint arXiv:1707.00180, 2017

    Melanie Weber, Johannes Stelzer, Emil Saucan, Alexander Naitsat, Gabriele Lohmann, and J ¨urgen Jost. Curvature-based analysis of brain networks.arXiv preprint arXiv:1707.00180, 2017. 11

    Pith/arXiv arXiv 2017