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REVIEW 4 major objections 5 minor 53 references

Partitioning networks into clusters of synchronized nodes via the message-passing algorithm: an unbiased scalable approach

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A message-passing algorithm on a binary Ising surrogate can partition large networks into clusters of nodes that would synchronize, without knowing the microscopic dynamics.

desk verdict A clean noiseless symmetry-based partitioner with an internally inconsistent noisy variant and an overextended synchronization interpretation. read the letter →

arxiv 2601.15944 v1 pith:3HZBKKD6 submitted 2026-01-22 physics.soc-ph cond-mat.dis-nnphysics.data-an

Massimo Ostilli This is my paper · ORCID
classification physics.soc-phcond-mat.dis-nnphysics.data-an PACS 05.45.Xt89.75.Hc
keywords message-passingalgorithmbeliefpropagationclustersynchronizationnetworkpartitioningtopologicalequivalenceIsingmodelrandom-fieldcomplexnetworks
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the synchronization structure of a network can be read off from a surrogate model rather than from any specific oscillator dynamics. The surrogate is a system of binary Ising spins on the same graph with uniform couplings, evolved by the message-passing algorithm; nodes whose stationary local magnetizations coincide are declared a synchronized cluster. The author argues that topologically equivalent nodes (same degree and same external neighborhood) act as nucleation centers, and that the critical couplings where the zero-magnetization state destabilizes organize the cluster hierarchy. Because message passing scales near-linearly, the method is claimed to be usable on networks with hundreds of thousands of nodes, demonstrated on the US power grid and WordNet. A central reported finding is that abrupt desynchronization appears in simple graphs without higher-order interactions, while added noise smooths the transitions into plateaus at larger coupling.

What carries the argument

The machine is the message-passing (belief-propagation) update for binary variables: each directed edge carries a message ui→j, updated via tanh of the sum of incoming messages (Eq. 9), and the local magnetization of node i is tanh of the total incoming field (Eq. 10). With all couplings equal to J, the trivial zero-message solution loses stability at a critical Jc, and the sequence of critical couplings {J1, J2, ...} gives the cluster-splitting events. Topologically equivalent nodes — groups with identical degree and identical external neighborhoods — are the nucleation sites, and in the noisy variant frozen random fields hi destroy the symmetric structure and replace sharp transitions with

What would settle it

On a graph with known TE groups, simulate a concrete continuous-time oscillator dynamics (e.g., Kuramoto with heterogeneous natural frequencies) and compare its synchronized clusters with the MPA clusters for the same coupling strength. If nodes that the MPA puts in one cluster systematically fail to synchronize, or if nodes in different MPA clusters do synchronize, the surrogate claim is refuted. A simpler search: find any graph where the MPA predicts a cluster partition that no plausible continuous dynamics reproduces at any coupling.

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Extended reading notes

Core claim

The central claim is that equal-time conditional probabilities of an unknown stochastic dynamics on a graph can be replaced, without bias, by a binary Ising model whose couplings are solved by message passing. Sweeping the uniform coupling J across the critical points J1, J2, ... of this effective model, the local magnetizations split into groups; nodes with equal magnetization (within machine precision) form the synchronized partitions. Topologically equivalent (TE) groups are shown to nucleate clusters, and clusters tend to absorb entire TE groups as J grows. In the noiseless version with random initial conditions, Q and Qsynch grow and then drop discontinuously at each critical point, yie

Load-bearing premise

The load-bearing premise is that equal stationary magnetizations of a binary message-passing Ising model stand in for real synchronization, even though the paper states that this surrogate is an artificial probe with no relation to the actual physical dynamics (as for the power grid) and that the binary reduction is a low-resolution coarse graining.

Editorial extensions

If this is right

  • The approach can partition networks of hundreds of thousands of nodes at near-linear cost, where spectral synchronization methods become infeasible.
  • No specific dynamical model (Laplacian or Kuramoto) has to be chosen, so the detected clusters are unbiased with respect to the unknown dynamics.
  • Abrupt desynchronization does not require higher-order interactions; it can arise from ordinary pair couplings in a simple graph.
  • In the noisy version, the number and height of plateaus in the participation ratio are independent of noise amplitude, giving a stable structural fingerprint.
  • Topologically equivalent groups are expected to be contained in synchronized clusters, linking structural symmetry to cluster synchronization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable refinement would be to run known oscillator dynamics on the same graphs and check whether the MPA clusters are the maximal synchronizing sets; the paper only asserts containment for the power grid, not equality.
  • The binary coarse graining merges distinct TE groups into large clusters; extending the surrogate to multi-state or continuous variables could resolve sub-clusters that the binary version cannot see.
  • The noise-amplitude-independent plateaus suggest the noisy MPA could serve as a universal descriptor of a network's synchronization propensity, comparable across networks of different sizes and topologies.
  • If the surrogate hypothesis holds, the same machinery could be adapted to flag consensus-prone or oscillation-prone modules before any dynamics is simulated, as a preprocessing step for control or prediction studies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a message-passing algorithm (MPA) on a binary Ising model with uniform couplings as an 'unbiased surrogate dynamics' for detecting clusters of synchronized nodes in large networks. Three variants are presented: noiseless with positive initial conditions, noiseless with random initial conditions, and a noisy version with random local fields. The method is validated on a synthetic graph with known topologically equivalent (TE) groups and applied to the US power-grid and WordNet networks. The paper reports abrupt desynchronization events in the noiseless case and smoother, plateau-like behavior with noise, and emphasizes scalability, with code publicly archived.

Significance. If correct, the approach would provide a scalable, dynamics-agnostic method for network cluster detection tied to synchronization-relevant symmetries. The release of Fortran code and the application to networks with up to ~146k nodes are concrete strengths, and the synthetic benchmark gives an instructive demonstration. However, the central claims rest on several assumptions that are either internally inconsistent or insufficiently justified, most notably the noisy variant's clustering threshold and the interpretation of MPA magnetizations as synchronization. The paper's value would improve substantially if these gaps were addressed.

major comments (4)
  1. [Sec. VI.C, Eq. (18); Figs. 4–6] There is an internal contradiction in the noisy version. The smoothing threshold epsilon in Eq. (18) is set to machine precision (10^-15 or 10^-18), while the random fields h_i are drawn from a continuous uniform distribution. For generic h_i, the local magnetizations from Eq. (10) will be distinct for every node, so Eq. (18) should give Q=N and p=0 for all J. The non-trivial plateaus in Figs. 4–6 therefore require an undocumented, larger tolerance. Because the WordNet application and the plateau conclusions rely entirely on this variant, the reported results cannot be reproduced from the stated algorithm. Please specify the actual clustering tolerance used, justify why it is not arbitrary, and show the sensitivity of the plateaus to that tolerance.
  2. [Sec. VII.A; Sec. III, Eqs. (3) and (10)] The paper's central claim equates clusters of equal MPA magnetizations with 'clusters of synchronized nodes,' but Sec. VII.A explicitly states that the MPA dynamics is 'an artificial probe and bears no relation to the actual physical dynamics.' The target definition in Eq. (3) uses actual state variables x_i, whereas the algorithm operates on the mean-field magnetizations <x_i>_t of a binary model. No argument or evidence is provided that equality of these thermal averages corresponds to synchronization in any continuous-time oscillator system, especially given the acknowledged low-resolution binary reduction. Please either provide a concrete mapping or substantially soften the synchronization interpretation.
  3. [Sec. VI.A, Figs. 1–2] The recovery of TE groups in the noiseless positive-initial-condition variant is to a large extent a built-in consequence of the Ising model and the MPA respecting graph automorphisms: uniform couplings and symmetric initial conditions force identical messages and magnetizations for topologically equivalent nodes. Thus Figs. 1–2 confirm this symmetry, but they do not independently validate that the clusters correspond to synchronization under unspecified dynamics. This circularity weakens the paper's main evidence. Please provide a validation against a genuine oscillator dynamics (e.g., Kuramoto or power-grid swing) or clearly frame the method as detecting symmetry-based structures rather than synchronization.
  4. [Sec. VI, Eq. (17); Sec. VII.B] The algorithm uses local magnetizations at a finite tmax even when the global stationarity condition in Eq. (17) is not satisfied. For the noisy synthetic case only 10–35% of messages are stationary, and for WordNet (tmax=10^3) the paper reports no stationarity fraction at all. If the MPA has not converged, the magnetizations in Eq. (10) are time-dependent, so the resulting partition depends on an arbitrary stopping time. Please report the stationarity fraction for WordNet and demonstrate that the reported Q(J) and p(J) are robust to increasing tmax.
minor comments (5)
  1. [Abstract] Typo: 'desyncrhronization' should be 'desynchronization'.
  2. [Sec. VI] The phrase 'with a lager value of tmax' should read 'a larger value.'
  3. [Fig. 3] The label 'Qynch' in the top plot appears to be a typo for 'Qsynch'.
  4. [Sec. VI.D, Fig. 9] The output table shows two nodes both labeled 24 (node 24 appears twice); this is likely a typo and should be corrected for clarity.
  5. [Sec. II] The claim that the method is 'unbiased' is too strong: Eq. (7) fixes the interaction form to Ising-type couplings, which is already a modeling choice. Please qualify.

Circularity Check

1 steps flagged · score 4.0 of 10

TE-group recovery is built into the automorphism-equivariant MPA, but the central algorithm and real-world applications retain independent content.

  1. self definitional [Sec. VI.A and Sec. IV (Eqs. 4-5 vs Eqs. 9-10, 18)]
    "We have verified this hypothesis by comparing the Q synchronized clusters obtained from the dynamics with the QTE TE groups in synthetically generated networks. An illustrative example with QTE = 207 is reported in in Figs. 1 and 2. These results highlight the role of TE groups as effective nucleation points for synchronization, in agreement with previous findings obtained within Kuramoto-like dynamics [23]."

    With Ji,j = J, hi = 0, and symmetric/positive initial conditions, Eqs. (9)-(10) are equivariant under the graph automorphisms that define TE groups via Eqs. (4)-(5). Every node in a TE group therefore receives identical messages and has identical magnetization at every iteration by construction, so Eq. (18) must place these nodes in the same cluster. The reported 'verification' that TE groups fit inside synchronized clusters and act as nucleation centers is a logical consequence of the update equations plus the TE definition, not an independent numerical prediction of the method.

full rationale

The paper's central proposal—using the message-passing algorithm on a uniform-coupling Ising model to produce a scalable partition—is not itself derived from the TE-group concept and has independent algorithmic content. The real-world applications and the noisy/random-initial-condition phenomenology do not reduce to TE symmetry by construction. However, the noiseless positive-initial-condition validation is partially circular: because the MPA update inherits graph symmetries and uniform couplings, TE nodes must share magnetizations, so the claimed confirmation that TE groups are 'nucleation centers' is a built-in consequence rather than an empirical discovery. This warrants a moderate score. A separate internal-consistency issue (continuous random fields plus machine-precision threshold should yield Q=N for all J in the noisy variant) is a reproducibility/correctness concern and is not scored as circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The central assumptions are modeling choices: the Bethe factorization, the binary Ising surrogate with uniform coupling, the heuristic thresholds (epsilon, tmax), and the assumed sequence of critical points. These choices constitute the real 'cost' of the method and are not independently justified against any physical dynamics.

free parameters (5)
  • global coupling J = scanned over [Jmin, Jmax], e.g., [0, 15]
    The partition output depends on J; no principled criterion is given for choosing a particular J, so the method returns a family of partitions.
  • noise amplitude h* = 0, 0.1, 0.5, 1, 2
    Chosen ad hoc; the plateaus are claimed to be independent of h*, but the coupling cost changes.
  • clustering threshold epsilon = machine precision (1e-15 to 1e-18)
    Set to machine precision in the text, but this is inconsistent with the reported clusters in the noisy version; a practical threshold is not disclosed.
  • tmax = 1e3 to 5e4 depending on graph and variant
    Chosen heuristically via a stationarity-percentage criterion; only partial stationarity is achieved in the noisy version.
  • inverse temperature beta = 1
    Set to 1 throughout; a scale choice that absorbs into J.
assumptions (4)
  • domain assumption Bethe factorization (Eq. 6): conditional probability of a node's state given its neighbors factorizes over edges.
    Exact only on tree-like graphs; for loopy graphs it is an approximation that defines the surrogate dynamics.
  • ad hoc to paper Binary Ising states and pairwise couplings encode synchronization (Eqs. 6-7).
    The reduction to xi in {-1,1} is a coarse-graining introduced to make the problem tractable; its connection to real synchronized dynamics is assumed, not derived.
  • ad hoc to paper The MPA dynamics is the actual surrogate dynamics even when the graph has dense loops and MPA does not converge (Sec. V).
    The paper states 'we still assume that our dynamics is defined via the MPA' even if its connection to the Ising model is poor, making the algorithm itself the model.
  • domain assumption A sequence of critical couplings J1, J2, ... exists with non-analytic behavior of dm0/dJ for general graphs (Sec. VI.A).
    Derived explicitly only for regular random graphs (Eq. 13); the extension to arbitrary graphs and the existence of multiple transitions are assumed.

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Pith. "Pith review of Partitioning networks into clusters of synchronized nodes via the message-passing algorithm: an unbiased scalable approach." pith.science (2026). https://pith.science/paper/3HZBKKD6

@misc{pith2026260115944,
  author       = {Pith},
  title        = {Pith review of: Partitioning networks into clusters of synchronized nodes via the message-passing algorithm: an unbiased scalable approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3HZBKKD6}},
  note         = {Machine review of arXiv:2601.15944}
}
read the original abstract

Partitioning large networks into stable clusters of synchronized nodes is a challenging task. Recent approaches based on spectral analysis can provide exact results on specific dynamics but remain unfeasible for very large networks. Moreover, within a stochastic framework, it is unclear which dynamics should be chosen to study synchronization. Here we propose an unbiased and scalable method based on the message-passing algorithm. By exploiting the collective behavior emerging across critical points of an effective Ising-like model, we identify dynamically coherent clusters of synchronized nodes and illustrate the approach on some large real-world networks. We find that, unlike continuous-time dynamics, abrupt desyncrhronization occurs even in simple graphs, without the need to invoke higher order interactions. However, when noise is included, the transition to synchronization becomes smoother and proceeds through the formation of plateaus, albeit at the cost of requiring larger coupling strengths.

Figures

Figures reproduced from arXiv: 2601.15944 by the authors.

Figure 1
Figure 1. FIG. 1: Number of synchronized clusters [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Spontaneous magnetization [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 6
Figure 6. FIG. 6: Comparing the participation ratio [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5: The participation ratio [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: A toy graph with [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The number [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The number [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: The number [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The number [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

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Reference graph

Works this paper leans on

53 extracted references

  1. [1]

    Barab´ asi,Network Science, Cambridge University Press, Cambridge, (2016)

    A.-L. Barab´ asi,Network Science, Cambridge University Press, Cambridge, (2016)

    2016

  2. [2]

    Newman, Networks: An Introduction , Oxford Univer- sity Press, Oxford, (2018)

    M. Newman, Networks: An Introduction , Oxford Univer- sity Press, Oxford, (2018)

    2018

  3. [3]

    S. N. Dorogovtsev and J. F. F. Mendes, The Nature of Complex Networks , Oxford University Press, Oxford, (2022)

    2022

  4. [4]

    Dean and S

    J. Dean and S. Ghemawat, MapReduce: Simplified Data Processing on Large Clusters, Communications of the ACM, 51, 107 (2008)

    2008

  5. [5]

    Manyika et al., Big Data: The Next Frontier for Inno- vation, Competition, and Productivity, McKinsey Global Institute Report (2011)

    J. Manyika et al., Big Data: The Next Frontier for Inno- vation, Competition, and Productivity, McKinsey Global Institute Report (2011)

    2011

  6. [6]

    Gandomi and M

    A. Gandomi and M. Haider, Beyond the Hype: Big Data Concepts, Methods, and Analytics, International Jour- nal of Information Management , 35, 137 (2015)

    2015

  7. [7]

    Dorogovtsev, A.V

    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Critical phenomena in complex networks , Rev. Mod. Phys. 80, 1275 (2008)

    2008

  8. [8]

    Kelling et al., Data-Intensive Science: A New Paradigm for Biodiversity Studies, BioScience, 59, 7, 613 (2009)

    S. Kelling et al., Data-Intensive Science: A New Paradigm for Biodiversity Studies, BioScience, 59, 7, 613 (2009)

    2009

Show all 53 references
  1. [9]

    O. J. Reichman, M. B. Jones, and M. P. Schildhauer, Challenges and Opportunities of Open Data in Ecology, Science, 331, 703 (2011)

    2011

  2. [10]

    Marx, Biology: The Big Challenges of Big Data, Na- ture, 498, 255 (2013)

    V. Marx, Biology: The Big Challenges of Big Data, Na- ture, 498, 255 (2013)

    2013

  3. [11]

    Z. D. Stephens et al., Big Data: Astronomical or Ge- nomical?, PLoS Biology, 13, no. 7, e1002195 (2015)

    2015

  4. [12]

    Ravasz and A.-L

    E. Ravasz and A.-L. Barab´ asi,Hierarchical organization in complex networks , Phys. Rev. E 67, 026112 (2003)

    2003

  5. [13]

    Fortunato, Community detection in graphs, Phys

    S. Fortunato, Community detection in graphs, Phys. Rep. 486, 75 (2010)

    2010

  6. [14]

    Fortunato, Community detection in networks: A user guide, Phys

    S. Fortunato, Community detection in networks: A user guide, Phys. Rep. 659, 1 (2016)

    2016

  7. [15]

    local matrices

    B. D. MacArthur, R. J. S´ anchez-Garc ´ ıa, J. W. Anderson, Symmetry in complex networks , Discrete Applied Math- ematics, 156, 3525 (2008). 12 TABLE I: A synthetic description of our algorithm for partitioning a given graph G into Q clusters of stable synchronized nodes. Note...

    2008

  8. [16]

    edges.txt

    Read G = (V, E) via file named “edges.txt”

  9. [17]

    For J ∈ {Jmin, . . . , Jmax}

  10. [18]

    Initialize N random fields from uniform distributions in [ −h∗, h∗]

  11. [19]

    Initialize E incoming messages from uniform distribution in [ −1, 1]

  12. [20]

    For t ∈ {0, . . . , tmax}

  13. [21]

    Compute |E| outgoing messages via Eq. (9)

  14. [22]

    Read the new |E| incoming messages from the |E| outgoing messages

  15. [23]

    Check stationarity of all incoming and outgoing messages via Eq. (17)

  16. [24]

    If stationarity holds for all messages GOTO 12

  17. [25]

    non stationarity

    Return “non stationarity”

  18. [26]

    Compute local magnetization of each node via Eq. (11)

  19. [27]

    Order values of local magnetizations

  20. [28]

    Find Q clusters via Eqs. (3)

  21. [29]

    Compute metrics such as mean and variance of N (l), k(l), k(IN;l), k(OUT;l)

  22. [30]

    Return full cluster decomposition and its metrics

  23. [31]

    L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Cluster synchronization and iso- lated desynchronization in complex networks with sym- metries, Nat. Commun. 5, 4079 (2014)

    2014

  24. [32]

    W. Lin, H. Fan, Y. Wang, H. Ying, and X. Wang, Con- trolling synchronous patterns in complex networks , Phys. Rev. E 93, 042209 (2016)

    2016

  25. [33]

    M. T. Schaub, N. O’Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and, M. Barahona Graph partitions and cluster synchronization in networks of oscillators , Chaos 26, 094821 (2016)

    2016

  26. [34]

    Fiore, G

    D. Fiore, G. Russo, and M. di Bernardo, Exploiting nodes symmetries to control synchronization and consensus pat- terns in multiagent systems , IEEE Control Syst. Lett., 1, 364 (2017)

    2017

  27. [35]

    L. V. Gambuzza, M. Frasca and V. Latora, Distributed Control of Synchronization of a Group of Network Nodes, IEEE Transactions on Automatic Control64, 365 (2019)

    2019

  28. [36]

    Della Rossa, L

    F. Della Rossa, L. Pecora, K. Blaha, A. Shirin, I. Klick- stein, and F. Sorrentino, Symmetries and cluster synchro- nization in multilayer networks , Nat Commun 11, 3179 (2020)

    2020

  29. [37]

    R. J. S´ anchez-Garc ´ ıa,Exploiting symmetry in network analysis, Commun. Phys. 3, 87 (2020)

    2020

  30. [38]

    Ostilli, Topologically protected synchronization in net- works, Phys

    M. Ostilli, Topologically protected synchronization in net- works, Phys. Rev. E 112, L042202 (2025)

    2025

  31. [39]

    Bayani, F

    A. Bayani, F. Nazarimehr, S. Jafari, K. Kovalenko, G. Contreras-Aso, K. Alfaro-Bittner, R. J. S´ anchez-Garc ´ ıa, and S. Boccaletti, The transition to synchronization of networked systems, Nat. Comm. 15, 4955 (2024)

    2024

  32. [40]

    J. S. Yedidia, W. T. Freeman, and Y. Weiss, General- ized belief propagation , in Advances in Neural Informa- tion Processing Systems, NIPS, pp. 689–695. MIT Press, Cambridge, MA. (2001)

    2001

  33. [41]

    S., Freeman, W

    Yedidia, J. S., Freeman, W. T., and Weiss, J. S. Yedida, W. T. Freeman, and Y. Weiss, Constructing free energy approximations and generalized belief propagation algo- rithms, IEEE Trans. Inf. Theory, 51, 2282 (2005)

    2005

  34. [42]

    M´ ezard and G

    M. M´ ezard and G. Parisi, The Bethe lattice spin glass revisited, Eur. Phys. J. B, 20, 217 (2001)

    2001

  35. [43]

    M´ ezard and A

    M. M´ ezard and A. Montanari,Information, Physics, and Computation, Oxford University Press (2009). 13

    2009

  36. [44]

    S.Skardal and A

    P. S.Skardal and A. Arenas, Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching, Comm. Phys. 3, 218 (2020)

    2020

  37. [45]

    S. H. Strogatz, Nonlinear Dynamics and Chaos, with Ap- plications to Physics, Biology, Chemistry, and Engineer- ing, (Addison-Wesley, Massachusetts, 1994)

    1994

  38. [46]

    Pikovsky, M

    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- tion, A Universal Concept in Nonlinear Sciences , (Cam- bridge, 2001)

    2001

  39. [47]

    Arenas, A

    A. Arenas, A. D ´ ıaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks , Physics Re- ports 469, 93 (2008)

    2008

  40. [48]

    Krzakala, F

    F. Krzakala, F. Ricci-Tersenghi, and L. Zdeborov´ a,Elu- sive Spin-Glass Phase in the Random Field Ising Model , Phys. Rev. Lett. 104, 207208 (2010)

    2010

  41. [49]

    S.-W. Son, H. Jeong, J.D. Noh, Random field Ising model and community structure in complex networks , Eur. Phys. J. B 50, 431–437 (2006)

    2006

  42. [50]

    Peixoto, Network Reconstruction and Community De- tection from Dynamics , Phys

    T. Peixoto, Network Reconstruction and Community De- tection from Dynamics , Phys. Rev. Lett. 123, 128301 (2019)

    2019

  43. [51]

    J Duncan and S

    W. J Duncan and S. H. Strogatz, Collective dynamics of small-world networks , Nature, 393, 440 (1998)

    1998

  44. [52]

    R. A. Rossi and N. K. Ahmed, The Network Data Repos- itory with Interactive Graph Analytics and Visualization , AAAI, url, https://networkrepository.com (2015)

    2015

  45. [53]

    G. A. Miller, WordNet: A Lexical Database for English ; Comm. of the ACM 38, 39 (1995); Princeton University About WordNet, WordNet. Princeton University (2010). C. Fellbaum, WordNet: An Electronic Lexical Database , Cambridge, MA, MIT Press. (1998)

    1995

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