Spectral analysis for gene communities in cancer cells

Ayumi Kikkawa

REVIEW 4 major objections 6 minor 29 references

Spectral analysis for gene communities in cancer cells

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Once a hub gene exceeds about 40 links, a cancer gene network localizes on that hub, and its eigenvalue spacing separates into Wigner-Dyson and Poisson regimes by degree discrepancy.

desk verdict A useful empirical application of known spectral tools to cancer gene networks, but the headline Wigner/Poisson contrast depends on a post hoc threshold and needs a null-model check. read the letter →

arxiv 1909.02695 v2 pith:RLVGEKNQ submitted 2019-09-06 q-bio.MN physics.data-an

Ayumi Kikkawa This is my paper · ORCID

classification q-bio.MNphysics.data-an

keywords spectralanalysiseigenvectorcentralitylocalizationhubgenescommunitystructureWigner-DysondistributionPoissondegreecorrelation

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 sets out to show that the eigenvalue statistics of cancer gene interaction networks encode the network's community organization. It observes that when a hub gene's degree passes a critical value $d_c \simeq 40$, the leading eigenvector localizes on that hub and its neighbors, and hubs attach preferentially to low-degree nodes. In 27 strongly localized networks whose hubs have degree $d>125$, the paper splits the graph by the degree discrepancy $\Delta$ between linked nodes and finds two distinct spacing laws: edges with $\Delta < 25$ give Wigner-Dyson nearest-neighbor spacing, while edges with $\Delta \ge 25$ give Poisson spacing. On the paper's interpretation, this spectral dichotomy is a signature of hub-centered modular sub-networks, and it opens a spectral route to extracting gene communities that may determine cancer-cell behavior.

What carries the argument

The key objects are the adjacency matrix $A$ of each gene network, the eigenvector centrality $v_i$ (the $i$-th component of the eigenvector of the largest eigenvalue $\lambda_1$), and the inverse participation ratio $\Psi(\lambda_i)=\sum_j (v_i^{(j)})^4$, which detects localized eigenvectors. The paper uses a theoretical formula for $v_i^2$ in a random graph with an added hub, Eq. (14), to locate the localization transition by fitting $d_c = 2c^\dagger$; it defines the degree discrepancy $\Delta=|k_i-k_j|$ for edges and uses it to partition each network. The central diagnostic is the unfolded nearest-neighbor spacing distribution $P(s)$, compared with Wigner-Dyson and Poisson forms by goodness-of-fit tests. The machinery works by turning community structure into a spectral dichotomy: the $\Delta<25$ sub-network is the delocalized 'hairball' showing correlated eigenvalues, while the $\Delta\ge25$ hub-centered sub-network shows uncorrelated Poisson eigenvalues.

What would settle it

Recompute $P(s)$ for the same 27 networks with the splitting threshold varied over $\Delta = 10, 15, 20, 30, 35, 40$: if the Wigner-Dyson/Poisson contrast does not persist across neighboring thresholds, the reported regimes are an artifact of the $\Delta=25$ choice. Alternatively, rewire each network while preserving the degree sequence; if the contrast disappears under rewiring, the effect is purely structural rather than a signature of biological gene communities.

Watch

Extended reading notes

Core claim

The central discovery is that a single spectral observable, the nearest-neighbor eigenvalue spacing distribution $P(s)$, cleanly separates two regimes in cancer gene interaction networks once the graph is split by the degree discrepancy $\Delta = |k_i - k_j|$ of linked nodes. In the 27 strongly localized networks whose largest hub has degree $d>125$, the sub-network formed by edges with $\Delta < 25$ yields a Wigner-Dyson $P(s)$, with 93% of 906 eigenvalue segments passing a goodness-of-fit test at $\alpha=0.05$, while the sub-network of edges with $\Delta \ge 25$ yields a Poisson $P(s)$, passing in 76% of 502 segments. The paper links this to localization: eigenvector centralities take finite values only on hub nodes and their neighbors once the hub degree exceeds $d_c \simeq 40$, the degree correlation function becomes disassortative with $k_{nn}(k) \propto k^{-0.37}$ for $k \gtrsim 40$, and the $\Delta\ge25$ component contains the hubs and their small-degree partners. The conclusion is that hub-centered gene communities can be read off the spectrum of the adjacency matrix.

Load-bearing premise

The load-bearing assumption is that the degree-discrepancy cutoff $\Delta=25$ is a real community boundary rather than a value chosen after inspecting the data, so if the cutoff is arbitrary or data-dependent the reported Wigner-Dyson/Poisson contrast could be created by the cut itself.

Editorial extensions

If this is right

If the spectral dichotomy is real, the spacing distribution $P(s)$ of an unlabeled gene sub-network reveals whether it is hub-centered: Poisson spacing flags a disassortative hub community, and Wigner-Dyson spacing flags the rest.
The localization threshold $d_c \simeq 40$ and the disassortative exponent $-0.37$ become empirical constraints that any proposed generative model of cancer gene networks should reproduce.
The $\Delta \ge 25$ sub-networks of the 27 super-hub networks define a concrete candidate list of hub-centered gene communities, so the small-degree mediator genes that connect hubs can be prioritized as potential disease-relevant targets.
Because the method needs only the adjacency matrix, it can be applied without community-detection algorithms or gene annotations, giving a purely spectral community extraction procedure.

Reading between the lines

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

A natural extension is to test whether the $\Delta=25$ cutoff rescales with the network's mean degree or node count; if the qualitative Wigner-Dyson/Poisson contrast survives rescaling, the spectral signature is general, while a fixed cutoff would be an artifact of this particular dataset.
Applying the same splitting to single-cell expression networks or protein interaction networks would show whether the two-regime spacing law is a generic feature of scale-free biological networks or specific to these inferred cancer networks.
A degree-preserving random rewiring of the same networks would isolate whether the Poisson spacing in $\Delta\ge25$ parts comes from hub-localized topology alone or from biologically meaningful community organization.
The fraction of segments that fail the Poisson test in the $\Delta\ge25$ part suggests the boundary may be fuzzy; a generative model with a smooth mixture of Wigner-Dyson and Poisson spacing could estimate the mixing fraction as a function of $\Delta$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The one thing you should know: this paper does something new—it splits inferred cancer gene networks by the degree discrepancy of linked nodes and reports that the Δ<25 part shows Wigner-Dyson level spacing while the Δ≥25 part shows Poisson spacing. That is a concrete, checkable observation, and I don't think it appears in the earlier literature. But the threshold is chosen after looking at Figure 5, and the Poisson claim only holds in 76% of the eigenvalue segments, so the contrast is not yet nailed down.

What the paper does well: it uses a large body of data—254 inferred networks, about nine million edges—and it averages carefully. The disassortativity for hub degrees k>40 is plausible and well documented. The eigenvector centrality fit to Eq. (14) is honest: the authors report that the fitted critical degree dc is larger than the theoretical 2c and that the fit fails for dense networks (c>8). The power-law exponents γ≈3.8 and μ≈6.5 are consistent with the known relation μ=2γ−1. The paper does not oversell the mediator genes; that's a speculation at the end, not a claim.

The soft spots are all in the central contrast. The Δ=25 cutoff is not derived from a null model; it's introduced after inspecting the degree-discrepancy histogram. A spectral statistic that depends on where you cut the graph is exactly the kind of thing that can be an artifact of the filter. Cutting out all similar-degree edges mechanically separates a dense assortative core from a disassortative hub-anchored periphery; any scale-free network with broad degree distribution might show Wigner-Dyson in the core and Poisson in the periphery under that filter. Second, the Poisson support is weak: 76% of 502 segments passing KS at α=0.05 is below the ~95% one would expect from a true Poisson null. That's a meaningful gap. Third, the localization threshold dc is a fit parameter, not a prediction, and the paper says the fit breaks down for c>8, so the model is not universal.

None of these are fatal to the paper as an exploratory analysis. But they mean the title-level claim—that the spacing distribution encodes hub-centered community structure—is not yet established. A serious referee should ask for a null-model comparison (e.g., degree-preserving randomized graphs) and a threshold-sensitivity analysis.

This paper is for network biologists who already work with spectral methods and want a concrete example on real gene networks. It deserves peer review because it is honest, reproducible, and the observation is worth checking. I'd recommend sending it, expecting a major revision.

Referee Report

4 major / 6 minor

Summary. This manuscript analyzes 254 gene co-expression networks of 8,000 nodes each from the TCNG database. It reports a power-law degree distribution with exponent gamma ~ 3.87, a spectral tail exponent mu ~ 6.5, eigenvector-centrality localization on high-degree hubs with a critical degree d_c ~ 40 for mean degree c = 6.6, and disassortative degree correlations k_nn(k) ~ k^-0.37 for k >~ 40. The authors then select 27 networks with very large hubs and split each network into two subgraphs according to the degree discrepancy Delta between linked nodes: Delta < 25 and Delta >= 25. They report that the nearest-neighbor eigenvalue spacing P(s) follows the Wigner-Dyson distribution in the Delta < 25 subgraphs (93% of 906 segments pass the KS test) and the Poisson distribution in the Delta >= 25 subgraphs (76% of 502 segments pass the KS test). They conclude that hub-centered communities in cancer gene interaction networks are modular and that spectral statistics can identify them.

Significance. If the reported spectral dichotomy were robust, it would provide a relatively simple spectral signature of hub-centered modular organization in gene networks. The paper's strengths are its use of a public, large dataset, explicit KS pass-rate statistics, and a falsifiable aggregate claim: the Delta-split P(s) contrast. However, the two load-bearing ingredients of the claim—the Delta = 25 cut and the critical degree d_c—are chosen or fitted after inspecting the data rather than derived from a null model, and the Poisson claim is statistically weak at a 76% pass rate. These issues leave the central claim plausible but not yet established.

major comments (4)

[Section 3, Fig. 7 and Eq. (16)] The Delta = 25 threshold is introduced after observing the degree-discrepancy histogram in Fig. 5 and the d >~ 40 discussion, and it is never derived or tested. Because the split deletes every edge whose endpoints have similar degree, it mechanically separates a dense assortative residual graph from a disassortative hub-anchored subgraph. A scale-free network with a broad degree distribution could display the same spectral contrast after such a filter even with no biologically meaningful community structure. Please add a stability sweep over Delta (e.g., 10, 15, 20, 30), a configuration-model null with the same degree sequence subjected to the same Delta-split, and a comparison with an independent community-detection method (e.g., modularity optimization or Louvain).
[Section 3, Fig. 3 and Eq. (14)] The localization transition d_c is not predicted from the data; it is obtained by fitting Eq. (14), a formula derived for a Poisson random graph plus a single hub, to scale-free, disassortative networks, with the mean degree c treated as a free fitting parameter (c-dagger = 13 for c = 4.7, and c-dagger = 20 for c = 6.6). The authors explicitly state that the fit fails for c > 8. Thus d_c ~ 40 is an empirical crossover value, not a theoretically derived critical point, and the abstract's 'd_c ~ 40' should be presented as a fitted value. A direct plot of v_1^2 and the inverse participation ratio against d, without the fitted curve, would support the claimed transition independently of Eq. (14).
[Section 3, Fig. 7(b)] The Poisson claim is not strongly supported by the reported KS statistics. For a true Poisson null and alpha = 0.05, approximately 95% of independent segments should pass the KS test; the reported 76% pass rate (N = 502) is far below that expectation and indicates systematic deviations. The authors should report the histogram of KS p-values and compare it with the uniform distribution expected under the null, and provide confidence intervals on the averaged P(s) curve, rather than reporting only the pass rate.
[Table 1] The fraction of edges with Delta >= 25 varies from 5.7% to 65.7% across the 27 networks, and the fraction of nodes varies from 15.3% to 92.7%. The averaged P(s) in Fig. 7 could therefore be dominated by a few networks with very large hub communities. Please report per-network P(s) or per-network KS pass rates and state the weighting used in the average, to confirm that the Poisson behavior is not driven by a small subset of networks.

minor comments (6)

[Throughout] Typographical errors should be corrected: 'dissasortative' (Sections 2.3 and 3), 'eacn' (Section 2.2), 'statisitics' (Table 1 caption), 'Kormogorov-Smirnov' (Section 3), and the placeholder 'greaterorsimilar'.
[Section 2.4] The unfolding and segment construction are only described by reference to Ref. [17]; since P(s) is the central observable, the segment size, the number of eigenvalues per segment, and the unfolding algorithm should be stated explicitly in this manuscript.
[Section 3, Fig. 3] The axis label 'v2_1' is ambiguous; it should be typeset as v_1^2 or v-squared_1, with a clear definition in the caption.
[Section 3, Fig. 4] The condition 'lambda_2max > 200' is confusing. If lambda_max is the largest eigenvalue, write lambda_max^2 > 200 or lambda_max > sqrt(200), and use consistent notation.
[Eq. (14)] The formula as printed is hard to parse; the hub case and neighbor case should be written as separate lines with clear parentheses, and the condition d > 2c should be stated in relation to the fitted parameter c-dagger.
[Supplementary material] The NDEx network is said to be accessible via a supplementary file, but no accession or stable URL is given; please include the NDEx UUID or a persistent link.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hub-degree threshold is an explicitly labeled empirical fit, and the Δ=25 community split is a graph-topological partition rather than a spectral target.

full rationale

The paper's central quantities are not derived from the same quantities used to test them. The critical hub degree d_c is obtained by fitting Eq. (14), taken from the independent results of Nadakuditi/Newman and Martin et al., to the v_1^2-versus-d data, with c as a fitting parameter, and then converting it as d_c = 2c† (Section 3, Figure 3). The paper does not present this fitted d_c as a first-principles prediction; on the contrary, it explicitly contrasts it with the theoretical prediction d_c = 9.4 and reports that the fit gives larger values (26 and 40). Thus the fitted-value concern is a limitation of the modeling assumption, not a disguised reuse of the target result. The Δ=25 subdivision in Section 3 is introduced after inspecting the degree-discrepancy histogram and the k ≳ 40 disassortative regime, but Δ is defined purely by node degrees via Eq. (16); the subsequent P(s) statistics are measured on the resulting subgraphs, so the Wigner-Dyson/Poisson contrast is not forced by the definition of the partition. The only self-citation, Ref. [17], supplies the unfolding procedure and a previously observed dense-network Wigner result; neither is load-bearing for the new hub-community claim. Weak KS support for the Poisson case and the post hoc choice of Δ=25 are correctness and robustness concerns, not circularity.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claims rest on several fitted parameters and ad hoc modeling choices: the effective mean degree c-star is fit to the eigenvector centrality curve, the power-law exponents are fit to chosen bins, and the degree discrepancy threshold delta=25 is chosen by hand. The theoretical localization formula assumes an uncorrelated Poisson random graph with a single hub, which contradicts the reported scale-free, disassortative structure.

free parameters (5)

c-star (effective mean degree in Eq. 14 fit) = 13 for c=4.7 networks; 20 for c=6.6 networks
Fit parameter in Eq. 14 used to predict the localization transition d_c=2c-star; the fitted values are larger than the empirical mean degrees.
delta threshold (degree discrepancy cutoff) = 25
Chosen by hand in Section 3 to split networks into low/high discrepancy communities; no independent derivation is provided.
Degree distribution exponent gamma = 3.87 +/- 0.01
Power-law fit to P(k) over selected bins (c=1.7, bins 5 to 10), chosen to minimize confidence interval width.
Spectral tail exponent mu = 6.52 +/- 0.05
Power-law fit to rho(lambda) for 7<lambda<13; used to compare with Eq. (2).
Degree correlation exponent eta = -0.37
Power-law fit to k_nn(k) in the disassortative regime k greater than or similar to 40.

assumptions (3)

domain assumption Eigenvector centrality formula Eq. (14) from Martin, Zhang, and Newman (2014) applies to the gene networks.
Invoked in Section 3 to fit v1^2 versus hub degree; it assumes a Poisson random graph with one hub and uncorrelated edge degrees, which the paper itself notes is not valid for dense networks.
ad hoc to paper Degree discrepancy delta between linked nodes defines community boundaries.
Used to split networks into delta<25 and delta>=25 subnetworks in Section 3; no justification is given that delta captures modular structure beyond the observed histogram peaks.
domain assumption TCNG Bayesian-inferred gene networks represent true gene interactions.
The analysis treats inferred edges as ground truth; edges are thresholded by edge factor to fix mean degree, but inference error is not modeled.

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Cite this review

Pith. "Pith review of Spectral analysis for gene communities in cancer cells." pith.science (2026). https://pith.science/paper/RLVGEKNQ

@misc{pith2026190902695,
  author       = {Pith},
  title        = {Pith review of: Spectral analysis for gene communities in cancer cells},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLVGEKNQ}},
  note         = {Machine review of arXiv:1909.02695}
}

abstract

We investigate gene interaction networks in various cancer cells by spectral analysis of the adjacency matrices. We observe localization of the networks on hub genes which have extraordinarily many links. The eigenvector centralities take finite values only on special nodes when the hub degree exceeds a critical value $d_c \simeq 40$. The degree correlation function shows the disassortative behavior in the large degrees, and the nodes whose degrees $d \gtrsim 40$ have tendencies to link to small degree nodes. The communities of the gene networks centered at the hub genes are extracted by the amount of node degree discrepancies between linked nodes. We verify the Wigner-Dyson distribution of the nearest neighbor eigenvalues spacing distribution $P(s)$ in the small degree discrepancy communities, and the Poisson $P(s)$ in the communities of large degree discrepancies including the hubs.

Figures

Figures reproduced from arXiv: 1909.02695 by the authors.

**Figure 2.** (a) The distribution of node degree P(k) in the log-log scale. The logarithmic bins are used and the n-th bin edge is k (n) = (1.7 n − 1)/0.7. In this figure 1 ≤ n ≤ 11 bins are shown. The curve fitting by a power function for 5th to 10th bins are performed. The result P(k) ∝ k −3.8 is plotted by the dotted line. (b) The eigenvalue density ρ(λ) in the tail region λ > 5. The bin width is 0.25. The power-law behavior … view at source ↗

**Figure 3.** (a) v 2 1 v.s. the hub degrees are plotted for 112 gene interaction networks where the mean degree c = 4.7 . The dashed line is the result of a curve fitting by the function in eq.(14). The fitting parameter is c † = 13. The localization transition at dc = 2c † = 26 is larger than the theoretical prediction dc = 9.4. (b) The same plot for the networks where c = 6.6 and we obtain dc = 40 from the fitting (the dashed … view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The scatter plot of the inverse participation ratio Ψ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Histogram of the degree discrepancies ∆ between two nod [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: The correlation function of the node degrees averaged o [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: (a) The distribution of the nearest neighbor eigenvalues s [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Mehta, M. L. (1991) RANDOM MATRICES. Academic Press, Inc. 13

work page 1991
[2]

& Weidenmueller, H

Guhr, T., Mueller-Groeling, A. & Weidenmueller, H. A. (1998) Random Matrix Theories in Quantum Physics: Common Concepts. Phys. Rep., 299, 189–425

work page 1998
[3]

(2008) Spectra of sparse random matrices

Kuhn, R. (2008) Spectra of sparse random matrices. J. Phys. A Math. Theor. 41, 295002

work page 2008
[4]

Mirlin, A. D. & Fyodorov, Y. V. (1991) Universality of Level Correlation-Function of Sparse Random Matrices. J. Phys. A. Math. Gen. 24, 2273–2286

work page 1991
[5]

Rodgers, G. J. & Bray, A. J. (1988) Density of states of a sparse random matrix. Phys. Rev. B 37, 3557–3562

work page 1988
[6]

& Rodgers, G

Nagao, T. & Rodgers, G. J. (2008) Spectral density of complex net- works with a ﬁnite mean degree. J. Phys. A Math. Theor. 41, 265002

work page 2008
[7]

N., Goltsev, A

Dorogovtsev, S. N., Goltsev, A. V., Mendes, J. F. F. & Samukhin, A. N. (2003) Spectra of complex networks. Phys. Rev. E 68, 046109

work page 2003
[8]

(2002) Statistical mechanics of complex networks

Bar´abasi, A.-L. (2002) Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97

work page 2002

Show all 29 references

[9]

(2016) NETWORK SCIENCE

Bar´abasi, A.-L. (2016) NETWORK SCIENCE . Cambridge University Press

2016
[10]

& Golinelli, O

Bauer, M. & Golinelli, O. (2001) Random incidence matrices: Mo- ments of the spectral density. J. Stat. Phys. 103, 301–337

2001
[11]

A., Simonsen, I., Maslov, S

Eriksen, K. A., Simonsen, I., Maslov, S. & Sneppen, K. (2003) Modularity and Extreme Edges of the Internet. Phys. Rev. Lett. 90, 148701

2003
[12]

& Sneppen, K

Simonsen, I., Astrup Eriksen, K., Maslov, S. & Sneppen, K. (2004) Diﬀusion on complex networks: a way to probe their large-scale topo logical structures. Phys. A Stat. Mech. its Appl. 336, 163–173

2004
[13]

Newman, M. E. J. (2004) Detecting community structure in networks. Eur. Phys. J. B 38, 321–330

2004
[14]

& Higham, D

Estrada, E. & Higham, D. J. (2010) Network Properties Revealed through Matrix Functions. SIAM Rev. 52, 696–714

2010
[15]

Newman, M. E. J. (2006) Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74, 036104

2006
[16]

Tamada, Y. et al. (2011) Estimating genome-wide gene networks using nonparametric bayesian network models on massively parallel compu ters. IEEE/ACM Trans. Comput. Biol. Bioinforma. 8, 683–697. 14

2011
[17]

(2018) Random Matrix Analysis for Gene Interaction Net- works in Cancer Cells

Kikkawa, A. (2018) Random Matrix Analysis for Gene Interaction Net- works in Cancer Cells. Sci. Rep. 8, 10607

2018
[18]

Barrett, T. et al. (2012) NCBI GEO: archive for functional genomics data setsupdate. Nucleic Acids Res. 41, D991–D995

2012
[19]

Nadakuditi, R. R. & Newman, M. E. J. (2013) Spectra of random graphs with arbitrary expected degrees. Phys. Rev. E 87, 012803

2013
[20]

& Newman, M

Martin, T., Zhang, X. & Newman, M. E. J. (2014) Localization and centrality in networks. Phys. Rev. E 90, 052808

2014
[21]

& Vespignani, A

Pastor-Satorras, R., V ´azquez, A. & Vespignani, A. (2001) Dynam- ical and correlation properties of the internet. Phys. Rev. Lett. 87, 258701

2001
[22]

da F., Rodrigues, F

Costa, L. da F., Rodrigues, F. A., Travieso, G. & Boas, P. R. V. (2007) Characterization of complex networks: A survey of measu rements. Adv. Phys. 56, 167–242

2007
[23]

Newman, M. E. J. (2002) Assortative Mixing in Networks. Phys. Rev. Lett. 89, 208701

2002
[24]

Newman, M. E. J. (2003) Mixing patterns in networks. Phys. Rev. E 67, 026126

2003
[25]

& Yadav, A

Jalan, S. & Yadav, A. (2015) Assortative and disassortative mixing investigated using the spectra of graphs. Phys. Rev. E 91, 012813

2015
[26]

(2011) Oxford University Press

The Oxford handbook of random matrix theory. (2011) Oxford University Press

2011
[27]

MATLAB 2018a (The MathWorks, Inc., Natick, Massachusetts, U. S.)
[28]

Evangelou, S. N. (1992) A numerical study of sparse random matrices. J. Stat. Phys. 69, 361–383

1992
[29]

Pratt, D. et al. (2015) NDEx, the Network Data Exchange. Cell Syst. 1, 302–305. 15

2015

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Mehta, M. L. (1991) RANDOM MATRICES. Academic Press, Inc. 13

work page 1991

[2] [2]

& Weidenmueller, H

Guhr, T., Mueller-Groeling, A. & Weidenmueller, H. A. (1998) Random Matrix Theories in Quantum Physics: Common Concepts. Phys. Rep., 299, 189–425

work page 1998

[3] [3]

(2008) Spectra of sparse random matrices

Kuhn, R. (2008) Spectra of sparse random matrices. J. Phys. A Math. Theor. 41, 295002

work page 2008

[4] [4]

Mirlin, A. D. & Fyodorov, Y. V. (1991) Universality of Level Correlation-Function of Sparse Random Matrices. J. Phys. A. Math. Gen. 24, 2273–2286

work page 1991

[5] [5]

Rodgers, G. J. & Bray, A. J. (1988) Density of states of a sparse random matrix. Phys. Rev. B 37, 3557–3562

work page 1988

[6] [6]

& Rodgers, G

Nagao, T. & Rodgers, G. J. (2008) Spectral density of complex net- works with a ﬁnite mean degree. J. Phys. A Math. Theor. 41, 265002

work page 2008

[7] [7]

N., Goltsev, A

Dorogovtsev, S. N., Goltsev, A. V., Mendes, J. F. F. & Samukhin, A. N. (2003) Spectra of complex networks. Phys. Rev. E 68, 046109

work page 2003

[8] [8]

(2002) Statistical mechanics of complex networks

Bar´abasi, A.-L. (2002) Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97

work page 2002