pith. sign in

arxiv: 2606.31997 · v1 · pith:ZM42ZZ2Inew · submitted 2026-06-30 · ❄️ cond-mat.stat-mech · physics.bio-ph

Multifractal Scaling in Hi-C Maps

Seong-Gyu Yang , Lucas Hedstr\"om , Jan Smrek , Ludvig Lizana This is my paper

Pith reviewed 2026-07-01 02:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-ph
keywords Hi-C mapsmultifractal scalingcontact probabilitypower-law decaymass exponentchromatin organizationpolymer contacts
0
0 comments X

The pith

The multifractal structure of Hi-C maps arises directly from the power-law contact probability P(s) with a single exponent gamma.

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

The paper shows that multifractal scaling in Hi-C chromosome contact maps stems from the empirical power-law decay of contact probability with genomic distance. Starting from P(s) proportional to s to the power of negative gamma, the authors derive the mass exponent tau(q) analytically to describe how the q-th moment of contact density scales with box size. This behavior reflects the geometric competition between contacts inside and between genomic segments. The large-q slope of tau(q) equals 2 minus gamma when gamma is less than 1 and equals 1 otherwise. The result is robust to noise and appears consistent across organisms, linking the multifractality to basic polymer contact statistics.

Core claim

Starting from the power-law contact probability P(s) with a single exponent gamma measured from Hi-C maps, the mass exponent tau(q) is derived analytically. This exponent describes the scaling of the q-th moment of contact density with the box size l used to coarse-grain the genomic coordinate. The multifractal behavior originates from the geometric competition between intra- and inter-segment contacts, with the slope of tau(q) at large q given by 2 - gamma when gamma < 1 and by 1 when gamma >= 1.

What carries the argument

The analytical derivation of the mass exponent tau(q) from the single-exponent power-law contact probability P(s), capturing how moments of contact density scale with coarse-graining length.

If this is right

  • The large-q slope of tau(q) maps directly onto the known polymer contact exponent gamma when gamma is less than 1.
  • Multifractal scaling is a universal feature of chromatin organization that holds across diverse organisms.
  • The scaling remains robust even when noise is added to the contact data.
  • For double-exponent forms of P(s), the multifractal behavior acquires an explicit dependence on the coarse-graining length l.

Where Pith is reading between the lines

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

  • Multifractal spectra extracted from Hi-C maps could serve as an independent route to estimating the contact exponent gamma without separate fitting of P(s).
  • Deviations from the predicted tau(q) would signal the presence of additional structural features beyond pure power-law contacts.
  • The same mapping supplies a direct bridge between multifractal observables in contact maps and standard polymer-physics models of chromatin.

Load-bearing premise

A pure single-exponent power-law form for the contact probability P(s) is sufficient by itself to produce the observed multifractal scaling without additional contributions from other chromatin structural features, measurement biases, or non-power-law corrections.

What would settle it

Measuring tau(q) directly from Hi-C maps whose contact probability follows a clean single power law with exponent gamma, but finding that the large-q slope deviates from the predicted value of 2 - gamma or 1.

Figures

Figures reproduced from arXiv: 2606.31997 by Jan Smrek, Lucas Hedstr\"om, Ludvig Lizana, Seong-Gyu Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. Multifractal analysis results for human (GM12878) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic overview of the multifractal analysis of Hi-C maps. (A) Relation between box mass [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Region diagram and its validation. (A) Diagram [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Multifractal analysis of human Chr 1 with additive [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Multifractal behavior across six Australian species. For each species, we show the Hi-C contact map (top), the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Scale dependence of multifractal behavior and mass [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

The three-dimensional organization of the genome exhibits rich, scale-dependent structure, as revealed by both chromosome contact maps (e.g., Hi-C maps) and chromatin density measured by microscopy. Recent studies have reported multifractal scaling in these data. Yet, the origin of this scaling behavior remains unclear: existing efforts describe it through postulated models. Here, we show that the multifractal structure of Hi-C maps is a direct consequence of the power-law contact probability $P(s)$, which is itself an empirical observable measured from Hi-C maps. Starting from $P(s)$ with a single exponent $\gamma$, we analytically derive the mass exponent $\tau(q)$, which characterizes how the $q$-th moment of contact density scales with box size $l$ used to coarse-grain the genomic coordinate. This multifractal behavior reflects the geometric competition between intra- and inter-segment contacts. We find that the slope of $\tau(q)$ at large $q$ is given by $2 -\gamma$ when $\gamma <1$, and by $1$ when $\gamma \geq 1$. We further show that this behavior is robust to noise and consistent across diverse organisms, indicating that it is a universal feature of chromatin organization. We extend our analysis into double-exponent $P(s)$, and show the $l$ dependence in multifractal behavior. Taken together, these results provide a physical explanation for multifractal scaling and establish a direct link between the multifractality in Hi-C maps and polymer contact statistics, with the large-$q$ slope of $\tau(q)$ mapping onto a known polymer contact exponent.

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

2 major / 2 minor

Summary. The paper claims that the multifractal structure of Hi-C maps, characterized by the mass exponent τ(q), is a direct consequence of the empirical power-law contact probability P(s) ~ s^{-γ} with a single exponent γ. Starting from this P(s), the authors analytically derive τ(q) for the scaling of q-th moments of contact density under box coarse-graining of genomic coordinate l, attributing the behavior to geometric competition between intra- and inter-segment contacts. They report that the large-q slope of τ(q) is 2-γ when γ<1 and 1 when γ≥1, demonstrate robustness to noise and consistency across organisms, and extend the analysis to double-exponent P(s) and l-dependence.

Significance. If the analytical mapping holds without hidden assumptions, the result provides a parameter-free physical explanation for observed multifractality in chromatin contact maps, directly linking it to the well-measured polymer contact exponent γ. This is a strength: the derivation is from an empirical observable, yields falsifiable predictions for the slope, and avoids postulated models, establishing a universal feature of chromatin organization grounded in polymer statistics.

major comments (2)
  1. [Analytical derivation of τ(q)] The central derivation (main text, the analytical mapping from P(s) to τ(q) via q-moments under box coarse-graining) starts from the ensemble-averaged P(s) and treats contacts as statistically homogeneous at fixed s. This does not incorporate position-dependent heterogeneities such as TADs, compartments, or loops that modulate local contact densities in real Hi-C maps. The paper must demonstrate explicitly (e.g., via direct comparison of predicted τ(q) to measured multifractal spectra from actual maps) that the global P(s) alone reproduces the observed scaling, or the claim that multifractality is a direct consequence of P(s) alone is not supported.
  2. [Robustness and double-exponent P(s)] § on robustness and double-exponent extension: the reported large-q slope (2-γ or 1) and its mapping to the polymer exponent are load-bearing for the universality claim, but the derivation's sensitivity to non-power-law corrections or local variations is not quantified. A concrete test (e.g., adding measured TAD-level fluctuations to the moment calculation) is needed to confirm the slope remains unchanged.
minor comments (2)
  1. [Methods] Notation for the box size l and the coarse-graining procedure should be clarified with an explicit equation for the q-moment definition to avoid ambiguity in the scaling analysis.
  2. [Results] The abstract states the slope is 'given by 2-γ when γ<1, and by 1 when γ≥1'; the main text should include a short derivation sketch of this asymptotic to make the result self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments point by point below, agreeing that additional explicit demonstrations will strengthen the manuscript. We will incorporate the suggested comparisons and tests in a revised version.

read point-by-point responses

  1. Referee: [Analytical derivation of τ(q)] The central derivation (main text, the analytical mapping from P(s) to τ(q) via q-moments under box coarse-graining) starts from the ensemble-averaged P(s) and treats contacts as statistically homogeneous at fixed s. This does not incorporate position-dependent heterogeneities such as TADs, compartments, or loops that modulate local contact densities in real Hi-C maps. The paper must demonstrate explicitly (e.g., via direct comparison of predicted τ(q) to measured multifractal spectra from actual maps) that the global P(s) alone reproduces the observed scaling, or the claim that multifractality is a direct consequence of P(s) alone is not supported.

    Authors: The derivation begins from the definition of the ensemble-averaged P(s), which by construction integrates over all position-dependent variations present in the map. The multifractal moments are likewise computed globally, so the analytical τ(q) already corresponds to the scaling that would be measured from an ensemble whose average contact probability follows the observed power law. The geometric competition between intra- and inter-segment contacts at different scales is encoded directly in this average. Nevertheless, we agree that an explicit side-by-side comparison would make the link unambiguous. We will add a new panel (or supplementary figure) that overlays the analytically predicted τ(q) obtained by inserting the measured P(s) into the derived formula against the τ(q) extracted directly from the same Hi-C maps for each organism examined. This will be accompanied by a brief discussion of residual discrepancies attributable to finite-size effects or higher-order corrections. revision: yes

  2. Referee: [Robustness and double-exponent P(s)] § on robustness and double-exponent extension: the reported large-q slope (2-γ or 1) and its mapping to the polymer exponent are load-bearing for the universality claim, but the derivation's sensitivity to non-power-law corrections or local variations is not quantified. A concrete test (e.g., adding measured TAD-level fluctuations to the moment calculation) is needed to confirm the slope remains unchanged.

    Authors: We already report robustness under additive noise and consistency of the large-q slope across organisms whose P(s) differ in detail. However, we concur that a targeted quantification of sensitivity to localized, non-power-law perturbations (such as TAD-scale fluctuations) is desirable. In the revision we will implement the suggested test: we will modulate the contact matrix by measured TAD-level density variations drawn from the same Hi-C data, recompute the q-moments under box coarse-graining, and verify that the asymptotic slope of τ(q) remains 2−γ (or 1) within statistical error. The results will be presented in an expanded robustness section together with the double-exponent extension already present in the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic derivation from measured P(s) to tau(q)

full rationale

The paper's central step is an analytic mapping that takes the empirical single-exponent form P(s) ~ s^{-gamma} (an external observable) as input and produces the mass exponent tau(q) that governs the scaling of contact-density moments under box coarse-graining. This is a forward derivation whose output is not identical to the input by the paper's own equations; tau(q) encodes the geometric competition between intra- and inter-segment contacts and yields a concrete prediction for the large-q slope (2-gamma or 1). No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation appears in the derivation chain. The result is therefore independent of the input quantity and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating the empirical power-law P(s) as the sole input for an analytical derivation of tau(q); no additional free parameters are introduced in the abstract description.

axioms (1)
  • domain assumption Contact probability follows a power-law P(s) ~ s^{-gamma} with single exponent gamma.
    This is stated as the empirical starting point measured from Hi-C maps.

pith-pipeline@v0.9.1-grok · 5839 in / 1332 out tokens · 46088 ms · 2026-07-01T02:16:05.299737+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

66 extracted references · 1 canonical work pages

  1. [1]

    We then approximate the hypotenuse of the resulting triangle by a staircase boundary shown as yellow steps and arrows

    Since P(s) of a single-exponent model depends only on the genomic distances=|x−y|, the two halves of the triangle contribute equally toZ(q, l), and we restrict to one half. We then approximate the hypotenuse of the resulting triangle by a staircase boundary shown as yellow steps and arrows. The step indexfdenotes the distance from the diagonal of the orig...

  2. [2]

    5 as µl(f >0)≈Cl(l/s 0)1−γ f −γ,(9) 5 0 1 2 3 0 1 2 3 q γ A I II III IV γ=1 q=1/γ 0 3 6τ(q) B γ=0.5 0 3 0 1 2 3 4 τ(q) q C γ=1.5 FIG

    from Eq. 5 as µl(f >0)≈Cl(l/s 0)1−γ f −γ,(9) 5 0 1 2 3 0 1 2 3 q γ A I II III IV γ=1 q=1/γ 0 3 6τ(q) B γ=0.5 0 3 0 1 2 3 4 τ(q) q C γ=1.5 FIG. 3. Region diagram and its validation. (A) Diagram of the dominantτ(q) behavior in (γ, q)-space. The dominant behavior ofτ(q) is divided into four regions: (I) yellow:τ(q) = 2(q−1); (II) red:τ(q) =q(2−γ)−1; (III) bl...

    2025

  3. [3]

    Lieberman-Aiden, N

    E. Lieberman-Aiden, N. L. Van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B. R. La- joie, P. J. Sabo, M. O. Dorschner,et al., Science326, 289 (2009)

    2009

  4. [4]

    S. S. Rao, M. H. Huntley, N. C. Durand, E. K. Sta- menova, I. D. Bochkov, J. T. Robinson, A. L. Sanborn, I. Machol, A. D. Omer, E. S. Lander,et al., Cell159, 1665 (2014)

    2014

  5. [5]

    Wang, J.-H

    S. Wang, J.-H. Su, B. J. Beliveau, B. Bintu, J. R. Moffitt, C.-t. Wu, and X. Zhuang, Science353, 598 (2016)

    2016

  6. [6]

    Bintu, L

    B. Bintu, L. J. Mateo, J.-H. Su, N. A. Sinnott- Armstrong, M. Parker, S. Kinrot, K. Yamaya, A. N. Boettiger, and X. Zhuang, Science362, eaau1783 (2018)

    2018

  7. [7]

    Takei, J

    Y. Takei, J. Yun, S. Zheng, N. Ollikainen, N. Pierson, J. White, S. Shah, J. Thomassie, S. Suo, C.-H. L. Eng, et al., Nature590, 344 (2021)

    2021

  8. [8]

    Takaki, Y

    R. Takaki, Y. Savich, J. Brugu´ es, and F. J¨ ulicher, Phys- ical review letters134, 128401 (2025)

    2025

  9. [9]

    A. N. Boettiger, B. Bintu, J. R. Moffitt, S. Wang, B. J. Beliveau, G. Fudenberg, M. Imakaev, L. A. Mirny, C.-t. Wu, and X. Zhuang, Nature529, 418 (2016)

    2016

  10. [10]

    S. S. Rao, S.-C. Huang, B. G. St Hilaire, J. M. Engreitz, E. M. Perez, K.-R. Kieffer-Kwon, A. L. Sanborn, S. E. Johnstone, G. D. Bascom, I. D. Bochkov,et al., Cell171, 305 (2017)

    2017

  11. [11]

    Schwarzer, N

    W. Schwarzer, N. Abdennur, A. Goloborodko, A. Pekowska, G. Fudenberg, Y. Loe-Mie, N. A. Fonseca, W. Huber, C. H. Haering, L. Mirny,et al., Nature551, 51 (2017)

    2017

  12. [12]

    S. Kim, I. Liachko, D. G. Brickner, K. Cook, W. S. Noble, J. H. Brickner, J. Shendure, and M. J. Dunham, Elife6, e23623 (2017)

    2017

  13. [13]

    T.-H. S. Hsieh, C. Cattoglio, E. Slobodyanyuk, A. S. Hansen, X. Darzacq, and R. Tjian, Nature genetics54, 1919 (2022)

    1919

  14. [14]

    De Gennes,Scaling concepts in polymer physics (Cornell university press, 1979)

    P.-G. De Gennes,Scaling concepts in polymer physics (Cornell university press, 1979)

    1979

  15. [15]

    Rubinstein and R

    M. Rubinstein and R. H. Colby,Polymer physics(Oxford university press, 2003)

    2003

  16. [16]

    L. A. Mirny, Chromosome Res.19, 37 (2011)

    2011

  17. [17]

    Grosberg, Y

    A. Grosberg, Y. Rabin, S. Havlin, and A. Neer, Europhys. Lett.23, 373 (1993). 12

    1993

  18. [18]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Phys. Rev. A33, 1141 (1986)

    1986

  19. [19]

    Ott,Chaos in dynamical systems(Cambridge univer- sity press, 2002)

    E. Ott,Chaos in dynamical systems(Cambridge univer- sity press, 2002)

    2002

  20. [20]

    Salat, R

    H. Salat, R. Murcio, and E. Arcaute, Physica A: Statis- tical Mechanics and its Applications473, 467 (2017)

    2017

  21. [21]

    A. L. Sanborn, S. S. Rao, S.-C. Huang, N. C. Durand, M. H. Huntley, A. I. Jewett, I. D. Bochkov, D. Chinnap- pan, A. Cutkosky, J. Li,et al., Proc. Natl. Acad. Sci. U.S.A.112, E6456 (2015)

    2015

  22. [22]

    Chan and M

    B. Chan and M. Rubinstein, Proc. Natl. Acad. Sci. U.S.A.121, e2401494121 (2024)

    2024

  23. [23]

    Frisch and G

    U. Frisch and G. Parisi, New York Academy of Sciences, Annals357, 359 (1980)

    1980

  24. [24]

    Benzi, G

    R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, Journal of Physics A: Mathematical and General17, 3521 (1984)

    1984

  25. [25]

    She and E

    Z.-S. She and E. Leveque, Physical review letters72, 336 (1994)

    1994

  26. [26]

    Ba lyga and J

    J. Ba lyga and J. Bourne, Chemical engineering science 50, 381 (1995)

    1995

  27. [27]

    Paladin and A

    G. Paladin and A. Vulpiani, Physical Review A35, 1971 (1987)

    1971

  28. [28]

    Paladin and A

    G. Paladin and A. Vulpiani, Physics Reports156, 147 (1987)

    1987

  29. [29]

    A. B. Chhabra, C. Meneveau, R. V. Jensen, and K. Sreenivasan, Physical Review A40, 5284 (1989)

    1989

  30. [30]

    Chhabra and R

    A. Chhabra and R. V. Jensen, Physical Review Letters 62, 1327 (1989)

    1989

  31. [31]

    A.-M. Wink, E. Bullmore, A. Barnes, F. Bernard, and J. Suckling, Human brain mapping29, 791 (2008)

    2008

  32. [32]

    Calvet and A

    L. Calvet and A. Fisher, Review of Economics and Statis- tics84, 381 (2002)

    2002

  33. [33]

    Jiang, W.-J

    Z.-Q. Jiang, W.-J. Xie, W.-X. Zhou, and D. Sornette, Reports on Progress in Physics82, 125901 (2019)

    2019

  34. [34]

    N. Jung, Q. A. Le, B. J. Mafwele, H. M. Lee, S. Y. Chae, and J. W. Lee, Journal of the Korean Physical Society 77, 186 (2020)

    2020

  35. [35]

    Murcio, A

    R. Murcio, A. P. Masucci, E. Arcaute, and M. Batty, Physical Review E92, 062130 (2015)

    2015

  36. [36]

    J.-L. Liu, J. Wang, Z.-G. Yu, and X.-H. Xie, Scientific reports7, 45588 (2017)

    2017

  37. [37]

    Pigolotti, M

    S. Pigolotti, M. H. Jensen, Y. Zhan, and G. Tiana, Phys. Rev. Res.2, 043078 (2020)

    2020

  38. [38]

    S. Lee, X. Liu, I. Ziabkin, and A. Zidovska, Biophys- ical Journal https://doi.org/10.1016/j.bpj.2025.02.014 (2025), in press

    work page doi:10.1016/j.bpj.2025.02.014 2025

  39. [39]

    Georges, Q

    A. Georges, Q. Li, J. Lian, D. O’Meally, J. Deakin, Z. Wang, P. Zhang, M. Fujita, H. R. Patel, C. E. Holleley, et al., Gigascience4, s13742 (2015)

    2015

  40. [40]

    J. A. Weber, S. G. Park, V. Luria, S. Jeon, H.-M. Kim, Y. Jeon, Y. Bhak, J. H. Jun, S. W. Kim, W. H. Hong, et al., Proceedings of the National Academy of Sciences 117, 20662 (2020)

    2020

  41. [41]

    Hirakawa, P

    H. Hirakawa, P. Kaur, K. Shirasawa, P. Nichols, S. Nagano, R. Appels, W. Erskine, and S. N. Isobe, Sci- entific Reports6, 30358 (2016)

    2016

  42. [42]

    org/waga(2025), accessed: 2025-11-14

    Western Australia Genome Atlas,https://www.dnazoo. org/waga(2025), accessed: 2025-11-14

    2025

  43. [43]

    Dudchenko, S

    O. Dudchenko, S. S. Batra, A. D. Omer, S. K. Nyquist, M. Hoeger, N. C. Durand, M. S. Shamim, I. Machol, E. S. Lander, A. P. Aiden,et al., Science356, 92 (2017)

    2017

  44. [44]

    Dudchenko, M

    O. Dudchenko, M. S. Shamim, S. S. Batra, N. C. Durand, N. T. Musial, R. Mostofa, M. Pham, B. Glenn St Hi- laire, W. Yao, E. Stamenova, M. Hoeger, S. K. Nyquist, V. Korchina, K. Pletch, J. P. Flanagan, A. Tomaszewicz, D. McAloose, C. P´ erez Estrada, B. J. Novak, A. D. Omer, and E. L. Aiden, bioRxiv , 254797 (2018)

    2018

  45. [45]

    Hoencamp, O

    C. Hoencamp, O. Dudchenko, A. M. Elbatsh, S. Brahmachari, J. A. Raaijmakers, T. van Schaik, ´A. Sede˜ no Cacciatore, V. G. Contessoto, R. G. van Hees- been, B. van den Broek,et al., Science372, 984 (2021)

    2021

  46. [46]

    G¨ ursoy, Y

    G. G¨ ursoy, Y. Xu, A. L. Kenter, and J. Liang, Nucleic acids research42, 8223 (2014)

    2014

  47. [47]

    Chan and M

    B. Chan and M. Rubinstein, Proc. Natl. Acad. Sci. U.S.A.120, e2222078120 (2023)

    2023

  48. [48]

    Nagano, Y

    T. Nagano, Y. Lubling, T. J. Stevens, S. Schoenfelder, E. Yaffe, W. Dean, E. D. Laue, A. Tanay, and P. Fraser, Nature502, 59 (2013)

    2013

  49. [49]

    J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, The Journal of chemical physics134 (2011)

    2011

  50. [50]

    A. Y. Grosberg, Soft Matter10, 560 (2014)

    2014

  51. [52]

    Smrek and A

    J. Smrek and A. Y. Grosberg, Physica A: Statistical Me- chanics and its Applications392, 6375 (2013)

    2013

  52. [53]

    J. D. Halverson, J. Smrek, K. Kremer, and A. Y. Gros- berg, Rep. Prog. Phys.77, 022601 (2014)

    2014

  53. [54]

    Rosa and R

    A. Rosa and R. Everaers, The European Physical Journal E42, 7 (2019)

    2019

  54. [55]

    Fudenberg, M

    G. Fudenberg, M. Imakaev, C. Lu, A. Goloborodko, N. Abdennur, and L. A. Mirny, Cell reports15, 2038 (2016)

    2038

  55. [56]

    J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Physica A: Sta- tistical Mechanics and its Applications316, 87 (2002)

    2002

  56. [57]

    J. H. Gibcus and J. Dekker, Molecular cell49, 773 (2013)

    2013

  57. [58]

    Bernenko, S

    D. Bernenko, S. H. Lee, P. Stenberg, and L. Lizana, PLOS Computational Biology19, e1011185 (2023)

    2023

  58. [59]

    P. A. Knight and D. Ruiz, IMA Journal of Numerical Analysis33, 1029 (2013)

    2013

  59. [60]

    N. C. Durand, J. T. Robinson, M. S. Shamim, I. Machol, J. P. Mesirov, E. S. Lander, and E. L. Aiden, Cell Systems 3, 99 (2016)

    2016

  60. [61]

    K. J. Falconer, Journal of theoretical Probability7, 681 (1994)

    1994

  61. [62]

    Fornberg, Math

    B. Fornberg, Math. Comput.51, 699 (1988)

    1988

  62. [63]

    All code for numerical calculations in this Git repository: [https://github.com/lizanalab/yang2026MF]. (2026). Supporting Information: Multifractal Scaling in Hi-C Maps Seong-Gyu Yang (양성ᄀ ᅲ),1, 2,∗ Lucas Hedstr¨ om,3 Jan Smrek, 4 and Ludvig Lizana 1,† 1Integrated Science Lab, Department of Physics, Ume ˚ a University, SE-90187 Ume ˚ a, Sweden 2Depart...

    2026

  63. [64]

    Rosa and R

    A. Rosa and R. Everaers, Physical Review E95, 012117 (2017)

    2017

  64. [65]

    J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, The Journal of chemical physics134(2011)

    2011

  65. [66]

    Smrek and A

    J. Smrek and A. Y. Grosberg, Physica A: Statistical Mechanics and its Applications392, 6375 (2013)

    2013

  66. [67]

    J. D. Halverson, J. Smrek, K. Kremer, and A. Y. Grosberg, Rep. Prog. Phys.77, 022601 (2014). 5 0 3 0 1 2 3 4 τ(q) q A 200 kb 0 3 0 1 2 3 4 τ(q) q B 4 Mb 1 2 1.5 2 f(α) α C 1 2 1 1.5 2 f(α) α D 0 3 0 1 2 3 4 τ(q) q E 200 kb 0 3 0 1 2 3 4 τ(q) q F 4 Mb 1 2 1 1.5 2 f(α) α G 1 2 1 1.5 2 f(α) α H FIG. S1. (A, B, E, and F) Mass exponentτ(q) and (C, D, G, and H)...

    2014