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Multilayer Modularity Belief Propagation To Assess Detectability Of Community Structure

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Pith's one-line read This paper extends modularity-based belief propagation to multilayer networks and argues that convergence to a non-trivial fixed point is a reliable signal that communities are detectable, while matching a leading modularity optimizer in…

desk verdict A real multilayer extension of modularity belief propagation with a serious sign error in the published update equation that needs to be resolved before the results can be trusted. read the letter →

arxiv 1908.04653 v2 pith:ZDGLL6LE submitted 2019-08-13 cs.SI physics.data-anphysics.soc-ph

William H. Weir , Benjamin Walker , Lenka Zdeborová , Peter J. Mucha This is my paper · ORCID
classification cs.SIphysics.data-anphysics.soc-ph MSC 68Q2568R1068U05
keywords multilayernetworkscommunitydetectionmodularitybeliefpropagationmessagepassingresolutionparameterdetectabilitysoftpartitioning
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

The paper extends message-passing (belief propagation) community detection from single-layer to multilayer networks, starting from the multilayer modularity objective and adding a resolution parameter plus an interlayer coupling parameter. It tries to establish that convergence of the algorithm to a non-trivial fixed point is a trustworthy sign that the network actually has detectable community structure, and that the retrieved partition, measured by adjusted mutual information (AMI), is competitive with a widely used modularity optimizer on synthetic benchmarks. A second claim is that the node-level marginals give a soft, uncertainty-weighted assignment that practitioners can use to choose parameters and interpret real-world structure. If these claims hold, the method gives network analysts a way to ask whether a modularity-optimizing partition reflects real mesoscale structure or just noise fitting.

What carries the argument

The central object is the multilayer message update $\psi_{i\to k}^{t} \propto \exp\left[\gamma \beta \frac{d_i}{2m_{l_i}}\theta_{l_i}^{t} + \sum_{j\in\partial i\setminus k}\log\left(1+\psi_{j\to i}^{t}(e^{\tilde{A}_{ij}\beta}-1)\right)\right]$, with $\tilde{A}_{ij}=A_{ij}\delta(l_i,l_j)+\omega C_{ij}(1-\delta(l_i,l_j))$. The first term is the layer-specific field from the modularity null model, generalized to the multilayer objective; the sum over neighbors carries both intralayer edge weights and interlayer coupling. The companion stability formula, $\langle ((e^{\beta^*\tilde{A}_{ij}}-1)/(e^{\beta^*\tilde{A}_{ij}}+q-1))^2 \rangle_{ij}\hat{c}=1$, selects the inverse temperature at which the symmetric fixed point destabilizes, which is what makes convergence to a non-trivial fixed point a structured signal rather than a generic property of the iteration.

What would settle it

Generate multilayer networks with no planted communities but with degree sequences, edge weights, and layer coupling matched to benchmarks where multimodbp converges; count how often the algorithm converges to a non-trivial fixed point and how high the adjusted mutual information of that partition is against arbitrary labels. If a substantial fraction of such null networks produce converged non-trivial partitions, the convergence signal is not specific to real structure. A complementary test is to measure, on the temporal benchmark, whether non-converged runs' partitions align with ground truth often enough to undermine convergence as a necessary condition.

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Extended reading notes

Core claim

At the paper's core is a belief-propagation update for multilayer modularity in which each node-layer's belief about belonging to community t is proportional to an exponential of a layer-local null-model field, a sum over intralayer neighbors, and a sum over interlayer neighbors weighted by the coupling parameter. The updates are derived from the multilayer modularity Hamiltonian, and their fixed points are stationary points of the associated free energy. Linearizing around the symmetric fixed point gives a stability condition that selects the inverse temperature $\beta^*$ for each assumed number of communities, and the paper reports that running the algorithm at these $\beta^*$ values lands in the retrieval phase for the synthetic and real networks tested. The central empirical discovery is that the algorithm's convergence itself indicates detectability: when the planted structure becomes too weak, multimodbp stops converging to a non-trivial partition, whereas a modularity optimizer continues to return high-modularity partitions. On top of that, the marginals at the fixed point are used as a soft partition whose per-node entropy measures confidence in the community assignment.

Load-bearing premise

The method's practical value rests on the unproven premise that when the iterative message-passing updates settle into a non-trivial partition, the network genuinely has community structure rather than the iterations simply getting trapped; the paper says its heuristic provides no guarantees and acknowledges that some non-converged runs still match the planted structure.

Editorial extensions

If this is right

  • On the dynamic stochastic block model benchmark, setting the resolution parameter to $\gamma=0.5$ widens the retrieval phase and matches single-layer detectability limits for low intralayer noise, while larger $\gamma$ and larger coupling $\omega$ pool signal across layers and push detection closer to the layer-aggregation bound.
  • On temporal, uniform-multiplex, and block-multiplex benchmarks, the retrieved partitions have AMI comparable to or better than a widely used modularity optimizer, with the largest gains in temporal networks with strong interlayer persistence.
  • Because non-convergence tracks undetectable structure, the method can be used as a check on greedy modularity optimization: a high-modularity partition on a network where multimodbp never converges should be regarded as potentially spurious.
  • Node-level marginals provide a confidence measure per node-layer that can be averaged over layers; in the voting-network example this reproduces known polarization history and labels bipartisan legislators, and in the law-firm example it maps which parameter regions align with metadata.

Reading between the lines

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

  • Editorial inference: the convergence-versus-non-convergence dichotomy could be calibrated against a formal null model, such as degree-preserved random multilayer networks, to yield a model-free test of community significance; the paper demonstrates the signal but does not quantify its false-positive rate.
  • Editorial inference: the marginals could be thresholded or resampled to produce overlapping or ensemble partitions, since a node with near-uniform beliefs is naturally compatible with multiple communities; the paper only reads them as confidence values.
  • Editorial inference: the strong dependence of multiplex performance on spectral initialization suggests the fixed-point landscape has multiple basins; mapping which initializations converge to which partitions would tell users when a converged result is stable versus initialization-driven.
  • Editorial inference: the same stability analysis could be applied to directed, signed, or edge-heterogeneous multilayer null models, giving analogous inverse-temperature formulas for those settings; the paper does not attempt those cases.
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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

3 major / 5 minor

Summary. The paper develops multimodbp, a belief-propagation algorithm for maximizing multilayer modularity Q(γ,ω)=Σ_ij(A_ij−γP_ij+ωC_ij)δ(c_i,c_j). It extends the Zhang–Moore single-layer modularity belief-propagation approach by adding a resolution parameter γ, layer-specific null-model fields, and interlayer coupling in the update equations, and it derives a Bethe free energy and a β* stability criterion for selecting the inverse temperature. The authors evaluate the method on stochastic block models, LFR benchmarks, and multilayer generative benchmarks against GenLouvain, and they use convergence behavior and node-level marginals on U.S. Senate voting and Lazega lawyer networks to argue that the algorithm can signal the detectability of community structure.

Significance. If the technical issues are resolved, this is a useful contribution: it gives practitioners a multilayer community-detection heuristic with soft output (marginals), a convergence-based detection signal, and an accompanying Python package. The supplement contains detailed derivations of the update equations, the free energy, and β*; the synthetic benchmark comparisons are broad (DSBM, temporal, uniform and block multiplex); and the real-world examples illustrate the claimed extra information. That said, the manuscript currently contains a sign inconsistency in the printed update equations, and the empirical comparisons lack uncertainty quantification, so the results should be treated as provisional.

major comments (3)
  1. [Eq. (2.4), Eq. (2.5), Supplement S.1 (S.6–S.7, S.10) and S.23] In Eq. (2.4) the null-model field enters with a plus sign: the update is proportional to exp(γβd_iθ_{l_i}^t/(2m_{l_i}) + ...), where θ_{l_i}^t = Σ_{j∈V_{l_i}} d_jψ_j^t. In the derivation in Supplement S.1 the same field is derived with a minus sign (Eqs. S.6–S.7 and S.10): exp(−βd_iθ_{l_i}^t/(2m_{l_i})). The plus sign makes the null model attractive for high-degree nodes, which is the opposite of the penalty in Eq. (1.1), so a program implementing Eq. (2.4) would not be solving the stated multilayer modularity objective. The Bethe free energy in Eq. (2.5), whose final term has a plus sign, is consistent with the minus-sign update, as is the S.1 derivation; the same plus sign appears in Eq. (S.23). The authors must reconcile these equations and, ideally, confirm from the released code that the implemented update uses the minus sign.
  2. [Figs. 5 and 7] Figures 5 and 7 report point averages over 50 and 100 realizations, respectively, but no error bars, confidence intervals, or significance tests are given. Several claims of comparable or superior performance (e.g., the statement in §3.2.2 that GenLouvain has "a slight edge overall" in the uniform multiplex case, Figure 7B) depend on differences whose magnitude relative to run-to-run variability cannot be assessed. The authors should add error bars or otherwise quantify uncertainty.
  3. [§2.4, Fig. 8, Fig. S.24] The paper's title and contributions center on using convergence to non-trivial fixed points as a signal of detectable community structure, but this premise is explicitly heuristic: §2.4 notes that the β* heuristic "provides no guarantees," and Figure S.24 shows that some non-converged runs still align with ground truth. To support the detectability claim, the authors should quantify the behavior of the convergence signal on structureless null networks (e.g., Erdős–Rényi or configuration-model graphs with matched degree sequences) and report the overlap of the converged and non-converged AMI distributions over the parameter grid; Figure 8 alone covers one topology and one value of p.
minor comments (5)
  1. [§3.2.2 vs Fig. 7 caption] In Section 3.2.2 the temporal benchmark is described as having "150 node-layers in each layer with 100 layers," while the caption of Figure 7A says "100 node-layers in each layer with 150 layers"; both give 15,000 node-layers, but the text should agree.
  2. [Eq. (S.10)] Equation (S.10) contains a garbled symbol in the definition of θ_{l_i}^t ("V↕⟩"), which should be V_{l_i}.
  3. [Reference [63]] The reference for the released software [63] points to a GitHub repository, while the text states that the tool is available on PyPI; the reference should be updated or clarified.
  4. [Section 2.4] In Section 2.4, "Schülkeet al." should read "Schülke et al."
  5. [Fig. 1 caption] The Figure 1 caption phrase "From left to right, the plots show the retrieval modularity, number of iterations to convergence, and the AMI ... and the effective number of communities" describes four panels but is awkwardly worded; please revise for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multilayer BP updates are derived from the modularity Hamiltonian and validated against external ground-truth benchmarks.

full rationale

The paper's central derivation is self-contained: Section S.1 of the supplement starts from the multilayer modularity objective (Eq. 1.1), splits the interaction into the edge term and the null-model field term, applies the standard sparse-graph 'field trick,' and obtains the update equations (Eq. 2.4). The null-model derivation is shown explicitly in Eqs. S.6-S.7, and the Bethe free energy is derived in Sections S.2-S.3 rather than being assumed. The stability heuristic for beta* (Eq. 2.9) is imported from prior work (Shi et al., and ultimately Zhang-Moore and Decelle et al.), but it is not fitted to the output being predicted; it is a parameter-free formula in the edge weights, degree distribution, q, and coupling, and the paper tests it against known synthetic ground truth (e.g., Figures S.13-S.14). The performance claims are benchmarked externally: AMI against planted partitions in DSBM and Bazzi et al. generative models, and comparison with GenLouvain on the same objective. Self-citations to Mucha et al. for multilayer modularity and to Weir et al. for domain post-processing are background building blocks, not load-bearing circular premises. The one notable internal issue is a sign inconsistency between main-text Eq. (2.4), which has a '+' on the null-model field term, and the supplement's derivation (S.6-S.7), which yields '−'. This is a correctness/typo concern about whether the published equation matches the stated objective, not a circularity, because the derivation proceeds from the objective rather than assuming the conclusion. Likewise, the paper's definition of 'significant structure' in terms of convergence is an explicit convention that is then empirically checked against ground truth, and Section 2.4 honestly states that the beta* heuristic 'provides no guarantees.' No prediction is equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central method introduces no new physical entities. The algorithm depends on user-chosen or heuristically selected parameters gamma, omega, beta, qmax, and a spectral initialization scale. The theoretical derivation rests on sparsity and random-perturbation assumptions, and the detectability interpretation is an empirical heuristic rather than a theorem.

free parameters (5)
  • gamma (resolution parameter) = 0.5, 1.0, 1.5, 3.0 used in experiments
    Controls the scale of detected communities; user-chosen and scanned. No automatic selection is provided.
  • omega (interlayer coupling) = e.g., 0, 1, 4 in DSBM experiments
    Weights interlayer edges; user-chosen and scanned in experiments.
  • beta (inverse temperature) = computed from Eq. 2.9, e.g., 0.5-4.5 scans
    Selected via the stability heuristic of Eq. 2.9 based on user-supplied q and qmax; not fitted to ground truth but a free algorithmic parameter.
  • qmax (maximum number of communities) = 4 for DSBM; unspecified for Bazzi benchmarks
    User-chosen upper bound on number of communities; affects the beta* scan range and runtime.
  • spectral initialization factor = 5
    Beliefs for the spectral partition are set to a factor 5 times the other beliefs (Section 3.2.2).
assumptions (6)
  • domain assumption Sparse, tree-like network assumption (d_i << sqrt(2m_l/gamma))
    Used in the field trick derivation in Supplement S.1 and in the beta* stability analysis; stated in Section 2.4.
  • domain assumption Uncorrelated random perturbations around the trivial fixed point
    Basis for the linear stability analysis in Supplement S.4 that leads to Eq. 2.9.
  • domain assumption Newman-Girvan null model within each layer
    Equation 1.2 defines P_ij; the BP field term is derived from this null model.
  • domain assumption Mucha et al. multilayer modularity as the objective
    Equation 1.1 is taken as the Hamiltonian for the Boltzmann ensemble; the method inherits the assumptions of that objective.
  • domain assumption Loopy belief propagation provides good approximations on loopy graphs
    Invoked in Section 1 with citations [45,34,64].
  • ad hoc to paper Convergence to non-trivial fixed point indicates detectable structure
    This is the paper's empirical detectability heuristic, asserted in Sections 1 and 2.4 and tested in synthetic benchmarks.

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Pith. "Pith review of Multilayer Modularity Belief Propagation To Assess Detectability Of Community Structure." pith.science (2026). https://pith.science/paper/ZDGLL6LE

@misc{pith2026190804653,
  author       = {Pith},
  title        = {Pith review of: Multilayer Modularity Belief Propagation To Assess Detectability Of Community Structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDGLL6LE}},
  note         = {Machine review of arXiv:1908.04653}
}
read the original abstract

Modularity based community detection encompasses a number of widely used, efficient heuristics for identification of structure in networks. Recently, a belief propagation approach to modularity optimization provided a useful guide for identifying non-trivial structure in single-layer networks in a way that other optimization heuristics have not. In this paper, we extend modularity belief propagation to multilayer networks. As part of this development, we also directly incorporate a resolution parameter. We show that adjusting the resolution parameter affects the convergence properties of the algorithm and yields different community structures than the baseline. We compare our approach with a widely used community detection tool, GenLouvain, across a range of synthetic, multilayer benchmark networks, demonstrating that our method performs comparably to the state of the art. Finally, we demonstrate the practical advantages of the additional information provided by our tool by way of two real-world network examples. We show how the convergence properties of the algorithm can be used in selecting the appropriate resolution and coupling parameters and how the node-level marginals provide an interpretation for the strength of attachment to the identified communities. We have released our tool as a Python package for convenient use.

Figures

Figures reproduced from arXiv: 1908.04653 by the authors.

Figure 1
Figure 1. Demonstration of multimodbp on two realizations of a (non-degree corrected) stochastic block model (SBM). From left to right, the plots show the retrieval mod￾ularity, number of iterations to convergence, and the AMI of the retrieval partition with known community assignments and the effective number of communities. (a) A 4-community planted partition SBM with n = 1000,  = pout pin = .1, mean degree cavg = 4, and e… view at source ↗
Figure 2
Figure 2. Performance of multimodbp and sbmbp over many LFR benchmark realiza￾tions with a range of values for the mixing parameter µ. Each point represents an average over 100 realizations of LFR with 1000 nodes, an average degree of 3 (with a max of 10), and other parameters set to default values. best community partition in a range γ ∈ [1.4, 4] [61, 62]. To investigate how the value of γ affects the retrieval phase, we ran… view at source ↗
Figure 3
Figure 3. Testing multimodbp on the 2000-2001 Division I-A College Football network [11, 16]. A) The average number of iterations until convergence in the retrieval phase across a range of γ values. B) The average number of communities detected in the retrieval phase as γ increases and the corresponding adjusted mutual information (AMI) of those partitions. C) ForceAtlas2 [21] layout of the football network with each node col… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: We show the performance characteristics of the algorithm for 3 differ [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Accuracy of multimodbp on a multilayer SBM (assessed by hAMIi averaged over 50 independently sampled networks at each point) for different values of model parameters , and η (horizontal and vertical axes respectively) and parameters γ and ω (moving horizontally and ve…
Figure 6
Figure 6. Figure 6: Graphical representation of the community structures for networks sampled [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: (Caption on next page.) layer assignments with probability p for layers within a given block. Within each block the structure is the same as the uniform multiplex, however there is a complete discontinuity in node-layer community assignments from block to block. Note t…
Figure 7
Figure 7. Figure 7: (Previous page.) We compare the performance of [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Detectability of communities in the multilayer temporal benchmarking network [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: We ran multimodbp on the US Senate voting similarity network comprised of 1884 Senators across the first 110 Congresses [37, 60]. A) The relationship between Bethe free energy and retrieval modularity is given by equation Eq. 2.5. We see that the Bethe free energy corr…
Figure 10
Figure 10. Figure 10: Several visualizations of the Lazega Lawyer network [ [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

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Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    Arenas, A

    A. Arenas, A. Fern ´andez, and S. G ´omez, Analysis of the structure of complex networks at different resolution levels, New Journal of Physics, 10 (2008), p. 053039

    work page 2008

  2. [2]

    J. P. Bagrow, Communities and bottlenecks: trees and treelike networks have high modularity, Physical Review E, 85 (2012), p. 066118

    work page 2012

  3. [3]

    Bazzi, L

    M. Bazzi, L. G. S. Jeub, A. Arenas, S. Howison, and M. Porter , A framework for the construction of generative models for mesoscale structure in multilayer networks , Physical Review Research, (2020), https://journals.aps.org/prresearch/abstract/10.1103/ PhysRevResearch.2.023100

    work page 2020

  4. [4]

    Bazzi, M

    M. Bazzi, M. Porter, S. Williams, M. McDonald, D. Fenn, and S. Howison , Community Detection in Temporal Multilayer Networks, with an Application to Correlation Networks , Multiscale Modeling & Simulation, 14 (2016), pp. 1–41

    work page 2016

  5. [5]

    Bennett, A

    L. Bennett, A. Kittas, S. Liu, L. G. Papageorgiou, and S. Tsoka , Community Struc- ture Detection for Overlapping Modules through Mathematical Programming in Protein Interaction Networks, PLOS ONE, 9 (2014), p. e112821

    work page 2014

  6. [6]

    V. D. Blondel, J.-L. Guillaume, R. Lambiotte, and E. Lefebvre , Fast unfolding of com- munities in large networks, Journal of Statistical Mechanics: Theory and Experiment, 2008 (2008), p. P10008

    work page 2008

  7. [7]

    Brandes, D

    U. Brandes, D. Delling, M. Gaertler, R. Gorke, M. Hoefer, Z. Nikoloski, and D. Wag- ner, On modularity clustering , IEEE Transactions on Knowledge and Data Engineering, 20 (2008), pp. 172–188

    work page 2008

  8. [8]

    de Montgolfier, M

    F. de Montgolfier, M. Soto, and L. Viennot, Asymptotic modularity of some graph classes, International Symposium on Algorithms and Computation, (2011), pp. 435–444

    work page 2011

Show all 67 references
  1. [9]

    Decelle, F

    A. Decelle, F. Krzakala, C. Moore, and L. Zdeborov ´a, Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications , Physical Review E, 84 (2011), p. 066106

    2011

  2. [10]

    Decelle, F

    A. Decelle, F. Krzakala, C. Moore, and L. Zdeborov ´a, Inference and Phase Transitions in the Detection of Modules in Sparse Networks , Physical Review Letters, 107 (2011), p. 065701

    2011

  3. [11]

    T. S. Evans , Clique graphs and overlapping communities , Journal of Statistical Mechanics: Theory and Experiment, 2010 (2010), p. P12037

    2010

  4. [12]

    Fortunato, Community detection in graphs , Physics Reports, 486 (2010), pp

    S. Fortunato, Community detection in graphs , Physics Reports, 486 (2010), pp. 75–174

    2010

  5. [13]

    Fortunato and M

    S. Fortunato and M. Barth´elemy, Resolution limit in community detection , Proceedings of the National Academy of Sciences, 104 (2007), pp. 36–41

    2007

  6. [14]

    Fortunato and D

    S. Fortunato and D. Hric, Community detection in networks: A user guide , Physics Reports, 659 (2016), pp. 1–44

    2016

  7. [15]

    Ghasemian, P

    A. Ghasemian, P. Zhang, A. Clauset, C. Moore, and L. Peel , Detectability Thresholds and Optimal Algorithms for Community Structure in Dynamic Networks , Physical Review X, 6 (2016), p. 031005

    2016

  8. [16]

    Girvan and M

    M. Girvan and M. E. J. Newman , Community structure in social and biological networks. , Proceedings of the National Academy of Sciences, 99 (2002), pp. 7821–7826

    2002

  9. [17]

    Q. Han, K. S. X. 0001, and E. M. Airoldi, Consistent estimation of dynamic and multi-layer block models., ICML, (2015)

    2015

  10. [18]

    M. B. Hastings, Community detection as an inference problem, Physical Review E, 74 (2006), p. 035102

    2006

  11. [19]

    T. C. Havens, J. C. Bezdek, C. Leckie, K. Ramamohanarao, and M. Palaniswami , A Soft Modularity Function For Detecting Fuzzy Communities in Social Networks , IEEE Transactions on Fuzzy Systems, 21 (2013), pp. 1170–1175

    2013

  12. [20]

    P. W. Holland, K. B. Laskey, and S. Leinhardt, Stochastic blockmodels: First steps, Social Networks, 5 (1983), pp. 109–137

    1983

  13. [21]

    Jacomy, T

    M. Jacomy, T. Venturini, S. Heymann, and M. Bastian , ForceAtlas2, a Continuous Graph Layout Algorithm for Handy Network Visualization Designed for the Gephi Software , PLOS ONE, 9 (2014), pp. 1–12

    2014

  14. [22]

    L. G. S. Jeub, A Python framework for generating multilayer networks with planted mesoscale structure., (2019)

    2019

  15. [23]

    L. G. S. Jeub and M. Bazzi, A generative model for mesoscale structure in multilayer networks implemented in MATLAB, tech. report, 2019

    2019

  16. [24]

    L. G. S. Jeub, M. Bazzi, J. S. Inderjit, and P. J. Mucha , A general- ized Louvain method for community detection implemented in MATLAB , URL http://github.com/GenLouvain/GenLouvain, (2011)

    2011

  17. [25]

    Karrer and M

    B. Karrer and M. E. Newman, Stochastic blockmodels and community structure in networks, MULTILAYER MODULARITY BELIEF PROPAGATION 45 Physical Review E, 83 (2011), p. 016107

    2011

  18. [26]

    Kivel¨a, A

    M. Kivel¨a, A. Arenas, M. Barth ´elemy, J. P. Gleeson, Y. Moreno, and M. A. Porter , Multilayer networks, Journal of Complex Networks, 2 (2014), pp. 203–271

    2014

  19. [27]

    H. W. Kuhn , The Hungarian method for the assignment problem , Naval Research Logistics Quarterly, 2 (1955), pp. 83–97

    1955

  20. [28]

    D. Lai, X. Shu, and C. Nardini , Correlation enhanced modularity-based belief propagation method for community detection in networks , Journal of Statistical Mechanics: Theory and Experiment, 05 (2016), pp. 053301–

    2016

  21. [29]

    Lambiotte, J.-C

    R. Lambiotte, J.-C. Delvenne, and M. Barahona, Laplacian dynamics and Multiscale Mod- ular structure in Networks , arxiv.org, 1 (2009), pp. 76–90

    2009

  22. [30]

    Lambiotte, J.-C

    R. Lambiotte, J.-C. Delvenne, and M. Barahona , Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks , IEEE transactions on network science and engineering, 1 (2014), pp. 76–90

    2014

  23. [31]

    Lancichinetti, S

    A. Lancichinetti, S. Fortunato, and F. Radicchi, Benchmark graphs for testing community detection algorithms, Physical Review E, 78 (2008), p. 046110

    2008

  24. [32]

    Lazega and others , The collegial phenomenon: The social mechanisms of cooperation among peers in a corporate law partnership , Oxford University Press on Demand, 2001

    E. Lazega and others , The collegial phenomenon: The social mechanisms of cooperation among peers in a corporate law partnership , Oxford University Press on Demand, 2001

    2001

  25. [33]

    Liu, Fuzzy modularity and fuzzy community structure in networks , 77 (2010), pp

    J. Liu, Fuzzy modularity and fuzzy community structure in networks , 77 (2010), pp. 547–557

    2010

  26. [34]

    M´ezard and A

    M. M´ezard and A. Montanari , Information, physics, and computation , Oxford University Press, 2009

    2009

  27. [35]

    Moody and P

    J. Moody and P. J. Mucha, Portrait of political party polarization, Network Science, 1 (2013), pp. 119–121

    2013

  28. [36]

    Mossel, J

    E. Mossel, J. Neeman, and A. Sly , Belief propagation, robust reconstruction and optimal recovery of block models , in International Conference on Machine Learning, May 2014, pp. 356–370

    2014

  29. [37]

    P. J. Mucha and M. A. Porter , Communities in multislice voting networks , Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), p. 041108

    2010

  30. [38]

    P. J. Mucha, T. Richardson, K. Macon, M. A. Porter, and J.-P. Onnela , Commu- nity structure in time-dependent, multiscale, and multiplex networks. , Science, 328 (2010), pp. 876–878

    2010

  31. [39]

    R. R. Nadakuditi and M. E. J. Newman, Graph Spectra and the Detectability of Community Structure in Networks , Physical Review Letters, 108 (2012), p. 188701

    2012

  32. [40]

    Newman and M

    M. Newman and M. Girvan, Finding and evaluating community structure in networks , Phys- ical Review E, 69 (2004), p. 268

    2004

  33. [41]

    M. E. J. Newman , Analysis of weighted networks , Physical Review E, 70 (2004), p. 056131

    2004

  34. [42]

    M. E. J. Newman, Equivalence between modularity optimization and maximum likelihood meth- ods for community detection , Physical Review E, 94 (2016), p. 052315

    2016

  35. [43]

    A. R. Pamfil, S. D. Howison, R. Lambiotte, and M. A. Porter , Relating Modularity Max- imization and Stochastic Block Models in Multilayer Networks , SIAM Journal on Mathe- matics of Data Science, 1 (2019), pp. 667–698

    2019

  36. [44]

    Papadopoulos, F

    A. Papadopoulos, F. Pachet, P. Roy, and J. Sakellariou , Exact Sampling for Regular and Markov Constraints with Belief Propagation , in Research in Computational Molecular Biology, Springer International Publishing, Cham, Aug. 2015, pp. 341–350

    2015

  37. [45]

    Pearl , Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference , (1988)

    J. Pearl , Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference , (1988)

    1988

  38. [46]

    L. Peel, D. B. Larremore, and A. Clauset , The ground truth about metadata and commu- nity detection in networks , Science Advances, 3 (2017), p. e1602548

    2017

  39. [47]

    T. P. Peixoto , Nonparametric Bayesian inference of the microcanonical stochastic block model, Physical Review E, 95 (2017), p. 012317

    2017

  40. [48]

    M. A. Porter, J. P. Onnela, and P. J. Mucha , Communities in networks , Notices of the American Mathematical Society, 56 (2009), pp. 1082–1097

    2009

  41. [49]

    Reichardt and S

    J. Reichardt and S. Bornholdt , Statistical mechanics of community detection , Physical Review E, (2006)

    2006

  42. [50]

    Rosvall and C

    M. Rosvall and C. T. Bergstrom, Maps of random walks on complex networks reveal com- munity structure, Proceedings of the National Academy of Sciences, 105 (2008), pp. 1118– 1123

    2008

  43. [51]

    M. T. Schaub, J.-C. Delvenne, M. Rosvall, and R. Lambiotte , The many facets of com- munity detection in complex networks , Applied Network Science, 2 (2017), p. 4

    2017

  44. [52]

    Sch¨ulke and F

    C. Sch¨ulke and F. Ricci-Tersenghi , Multiple phases in modularity-based community detec- tion, Physical Review E, 92 (2015), p. 042804

    2015

  45. [53]

    S. Shai, N. Stanley, C. Granell, D. Taylor, and P. J. Mucha , Case studies in network community detection, arXiv, (2017). 46 WEIR, WALKER, ZDEBOROV´A AND MUCHA

    2017

  46. [54]

    C. Shi, Y. Liu, and P. Zhang , Weighted community detection and data clustering using message passing, Journal of Statistical Mechanics: Theory and Experiment, 2018 (2018), p. 033405

    2018

  47. [55]

    Stanley, S

    N. Stanley, S. Shai, D. Taylor, and P. J. Mucha, Clustering network layers with the strata multilayer stochastic block model. , IEEE transactions on network science and engineering, 3 (2016), pp. 95–105

    2016

  48. [56]

    Taylor, S

    D. Taylor, S. Shai, N. Stanley, and P. J. Mucha , Enhanced Detectability of Community Structure in Multilayer Networks through Layer Aggregation, Physical Review Letters, 116 (2016), p. 228301

    2016

  49. [57]

    V. A. Traag, G. Krings, and P. Van Dooren , Significant scales in community structure , Scientific reports, 3 (2013), p. 75

    2013

  50. [58]

    N. X. Vinh, J. Epps, and J. Bailey , Information theoretic measures for clusterings compar- ison: is a correction for chance necessary? , the 26th Annual International Conference, (2009), pp. 1073–1080

    2009

  51. [59]

    N. X. Vinh, J. Epps, and J. Bailey , Information theoretic measures for clusterings compar- ison: Variants, properties, normalization and correction for chance , Journal of Machine Learning Research, 11 (2010), pp. 2837–2854

    2010

  52. [60]

    A. S. Waugh, L. Pei, J. H. Fowler, P. J. Mucha, and M. A. Porter , Party polarization in congress: A network science approach , arXiv.org, physics.soc-ph (2009)

    2009

  53. [61]

    W. H. Weir, S. Emmons, R. Gibson, D. Taylor, and P. Mucha , Post-Processing Partitions to Identify Domains of Modularity Optimization , Algorithms, 10 (2017), p. 93

    2017

  54. [62]

    W. H. Weir, R. Gibson, and P. J. Mucha, CHAMP package: Convex Hull of Admissible Mod- ularity Partitions in Python and MATLAB, 2017, https://github.com/wweir827/CHAMP

    2017

  55. [63]

    W. H. Weir and B. Walker , multimodbp, 2018-2020, https://github.com/bwalker1/ ModularityBP Cpp

    2018

  56. [64]

    Zdeborov´a and F

    L. Zdeborov´a and F. Krzakala , Statistical physics of inference: thresholds and algorithms , Advances in Physics, 65 (2016), pp. 453–552

    2016

  57. [65]

    Zhang, F

    P. Zhang, F. Krzakala, J. Reichardt, and L. Zdeborov´a, Comparative study for inference of hidden classes in stochastic block models , Journal of Statistical Mechanics: Theory and Experiment, 2012 (2012), p. P12021

    2012

  58. [66]

    Zhang and C

    P. Zhang and C. Moore , Scalable detection of statistically significant communities and hi- erarchies, using message passing for modularity , Proceedings of the National Academy of Sciences, 111 (2014), pp. 18144–18149

    2014

  59. [67]

    Zhang, R.-S

    S. Zhang, R.-S. Wang, and X.-S. Zhang , Identification of overlapping community structure in complex networks using fuzzy c-means clustering , Physica A: Statistical Mechanics and its Applications, 374 (2007), pp. 483–490

    2007

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