Hostname: page-component-5db6c4db9b-cfm7h Total loading time: 0 Render date: 2023-03-23T22:13:49.769Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Efficient Laplacian spectral density computations for networks with arbitrary degree distributions

Published online by Cambridge University Press:  07 October 2021

Grover E. C. Guzman
Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010 - São Paulo - SP, 05508-090 Brazil (e-mail: and
Peter F. Stadler
Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for Bioinformatics, Universität Leipzig, Härtelstraße 16-18, Leipzig, D-04107, Germany (e-mail:
André Fujita*
Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010 - São Paulo - SP, 05508-090 Brazil (e-mail: and
*Corresponding author. Email:


The network Laplacian spectral density calculation is critical in many fields, including physics, chemistry, statistics, and mathematics. It is highly computationally intensive, limiting the analysis to small networks. Therefore, we present two efficient alternatives: one based on the network’s edges and another on the degrees. The former gives the exact spectral density of locally tree-like networks but requires iterative edge-based message-passing equations. In contrast, the latter obtains an approximation of the spectral density using only the degree distribution. The computational complexities are 𝒪(|E|log(n)) and 𝒪(n), respectively, in contrast to 𝒪(n3) of the diagonalization method, where n is the number of vertices and |E| is the number of edges.

Research Article
Network Science , Volume 9 , Issue 3 , September 2021 , pp. 312 - 327
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Action Editor: Ulrik Brandes


Adamic, L. A., & Huberman, B. A. (2000). Power-law distribution of the world wide web. Science, 287(5461), 21152115.CrossRefGoogle Scholar
Adamic, Lada A, Lukose, Rajan M, Puniyani, Amit R, & Huberman, Bernardo A. (2001). Search in power-law networks. Physical Review E, 64(4), 046135.CrossRefGoogle Scholar
Adcock, A. B., Sullivan, B. D., & Mahoney, M. W. (2013). Tree-like structure in large social and information networks. In 2013 IEEE 13th international conference on data mining (pp. 110). IEEE.CrossRefGoogle Scholar
Agliari, E., & Tavani, F. (2017). The exact Laplacian spectrum for the Dyson hierarchical network. Scientific Reports, 7(1), 121.CrossRefGoogle ScholarPubMed
Albert, R., & Barabási, A.-L. (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74(1), 47.CrossRefGoogle Scholar
Almendral, J. A., & Daz-Guilera, A. (2007). Dynamical and spectral properties of complex networks. New Journal of Physics, 9(6), 187.CrossRefGoogle Scholar
Barahona, M., & Pecora, L. M. (2002). Synchronization in small-world systems. Physical Review Letters, 89(5), 054101.CrossRefGoogle ScholarPubMed
Bollobás, B. (1998). Random graphs. In Modern graph theory (pp. 215252). Springer.CrossRefGoogle Scholar
Booth, T. E. (2006). Power iteration method for the several largest eigenvalues and eigenfunctions. Nuclear Science and Engineering, 154(1), 4862.CrossRefGoogle Scholar
Cantwell, G. T., & Newman, M. E. J. (2019). Message passing on networks with loops. Proceedings of the National Academy of Sciences, 116(47), 2339823403.CrossRefGoogle Scholar
Chen, H., & Zhang, F. (2007). Resistance distance and the normalized Laplacian spectrum. Discrete Applied Mathematics, 155(5), 654661.CrossRefGoogle Scholar
Chen, H., & Jost, J. (2012). Minimum vertex covers and the spectrum of the normalized Laplacian on trees. Linear Algebra and Its Applications, 437(4), 10891101.CrossRefGoogle Scholar
Chung, F., Lu, L., & Vu, V. (2004). The spectra of random graphs with given expected degrees. Internet Mathematics, 1(3), 257275.CrossRefGoogle Scholar
Chung, F. R. K. (1989). Diameters and eigenvalues. Journal of the American Mathematical Society, 2(2), 187196.CrossRefGoogle Scholar
de Lange, S., de Reus, M., & Van Den Heuvel, M. (2014). The Laplacian spectrum of neural networks. Frontiers in Computational Neuroscience, 7, 189.CrossRefGoogle ScholarPubMed
Ding, Q., Sun, W., & Chen, F. (2014). Applications of Laplacian spectra on a 3-prism graph. Modern Physics Letters B, 28(02), 1450009.CrossRefGoogle Scholar
Gander, W. (1980). Algorithms for the QR decomposition. Research Report, 80(02), 12511268.Google Scholar
Granziol, D., Ru, B., Zohren, S., Dong, X., Osborne, M., & Roberts, S. (2018). Entropic spectral learning for large-scale graphs. arxiv preprint arxiv:1804.06802. Google Scholar
Hakimi-Nezhaad, M., & Ashrafi, A. R. (2014). A note on normalized Laplacian energy of graphs. Journal of Contemporary Mathematical Analysis, 49(5), 207211.CrossRefGoogle Scholar
Hata, S., & Nakao, H. (2017). Localization of Laplacian eigenvectors on random networks. Scientific Reports, 7(1), 111.Google ScholarPubMed
Huang, J., & Li, S. (2018). The normalized Laplacians on both k-triangle graph and k-quadrilateral graph with their applications. Applied Mathematics and Computation, 320, 213225.CrossRefGoogle Scholar
Jamakovic, A., & Van Mieghem, P. (2006). The Laplacian spectrum of complex networks. Proceedings of the european conference on complex systems, Oxford, September 25-29, 2006.Google Scholar
Jiang, T. (2012). Empirical distributions of Laplacian matrices of large dilute random graphs. Random Matrices: Theory and Applications, 1(03), 1250004.CrossRefGoogle Scholar
Julaiti, A., Wu, B., & Zhang, Z. (2013). Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications. The Journal of Chemical Physics, 138(20), 204116.CrossRefGoogle ScholarPubMed
Klein, D. J., & Randić, M. (1993). Resistance distance. Journal of Mathematical Chemistry, 12(1), 8195.CrossRefGoogle Scholar
Koller, D., & Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. MIT Press.Google Scholar
Kowalska, E. (1987). On locally tree-like graphs. Applicationes Mathematicae, 19(3–4), 497503.CrossRefGoogle Scholar
Li, D., & Hou, Y. (2017). The normalized Laplacian spectrum of quadrilateral graphs and its applications. Applied Mathematics and Computation, 297, 180188.CrossRefGoogle Scholar
Li, S., Yan, W., & Tian, T. (2016). The spectrum and Laplacian spectrum of the dice lattice. Journal of Statistical Physics, 164(2), 449462.CrossRefGoogle Scholar
Liu, H., & Zhang, Z. (2013). Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications. The Journal of Chemical Physics, 138(11), 114904.CrossRefGoogle ScholarPubMed
Liu, H., Dolgushev, M., Qi, Y., & Zhang, Z. (2015). Laplacian spectra of a class of small-world networks and their applications. Scientific Reports, 5, 9024.CrossRefGoogle ScholarPubMed
Ma, X., & Bian, H. (2019). The normalized Laplacians, degree-kirchhoff index and the spanning trees of hexagonal möbius graphs. Applied Mathematics and Computation, 355, 3346.CrossRefGoogle Scholar
Metz, F. L., Neri, I., & Bollé, D. (2011). Spectra of sparse regular graphs with loops. Physical Review E, 84(5), 055101.CrossRefGoogle Scholar
Mülken, O., Volta, A., & Blumen, A. (2005). Asymmetries in symmetric quantum walks on two-dimensional networks. Physical Review A, 72(4), 042334.CrossRefGoogle Scholar
Nadakuditi, R. R., & Newman, M. E. J. (2013). Spectra of random graphs with arbitrary expected degrees. Physical Review E, 87(1), 012803.CrossRefGoogle ScholarPubMed
Nadler, B., Srebro, N., & Zhou, X. (2009). Statistical analysis of semi-supervised learning: The limit of infinite unlabelled data. Advances in neural information processing systems (pp. 13301338).Google Scholar
Newman, M. (2018). Networks. Oxford university press.CrossRefGoogle Scholar
Newman, M. E. J. (2019). Spectra of networks containing short loops. Physical Review E, 100(1), 012314.CrossRefGoogle ScholarPubMed
Newman, M. E. J., Zhang, X., & Nadakuditi, R. R. (2019). Spectra of random networks with arbitrary degrees. Physical Review E, 99(4), 042309.CrossRefGoogle ScholarPubMed
Pan, Y., Li, J., Li, S., & Luo, W. (2018). On the normalized Laplacians with some classical parameters involving graph transformations. Linear and Multilinear Algebra, 123.Google Scholar
Peixoto, T. P. (2013). Eigenvalue spectra of modular networks. Physical Review Letters, 111(9), 098701.CrossRefGoogle ScholarPubMed
Rogers, T., Castillo, I. P., Kühn, R., & Takeda, K. (2008). Cavity approach to the spectral density of sparse symmetric random matrices. Physical Review E, 78(3), 031116.CrossRefGoogle ScholarPubMed
Semerjian, G., & Cugliandolo, L. F. (2002). Sparse random matrices: The eigenvalue spectrum revisited. Journal of Physics A: Mathematical and General, 35(23), 4837.CrossRefGoogle Scholar
Sleijpen, G. L. G., & Van der Vorst, H. A. (2000). A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Review, 42(2), 267293.CrossRefGoogle Scholar
Sun, S., & Das, K. Ch. (2020). Normalized Laplacian eigenvalues with chromatic number and independence number of graphs. Linear and Multilinear Algebra, 68(1), 6380.CrossRefGoogle Scholar
Sun, W., Xuan, T., & Qin, S. (2016). Laplacian spectrum of a family of recursive trees and its applications in network coherence. Journal of Statistical Mechanics: Theory and Experiment, 2016(6), 063205.CrossRefGoogle Scholar
Sun, Y., Dai, M., Shao, S., & Su, W. (2017). The entire mean weighted first-passage time on infinite families of weighted tree networks. Modern Physics Letters B, 31(07), 1750049.CrossRefGoogle Scholar
Trinajstic, N., Babic, D., Nikolic, S., Plavsic, D., Amic, D., & Mihalic, Z. (1994). The Laplacian matrix in chemistry. Journal of Chemical Information and Computer Sciences, 34(2), 368376.CrossRefGoogle Scholar
Wegner, A. E., Ospina-Forero, L., Gaunt, R. E., Deane, C. M., & Reinert, G. (2018). Identifying networks with common organizational principles. Journal of Complex Networks, 6(6), 887913.CrossRefGoogle Scholar
Wu, F.-Y. (2004). Theory of resistor networks: The two-point resistance. Journal of Physics A: Mathematical and General, 37(26), 6653.CrossRefGoogle Scholar
Zhang, Z., Zhou, S., Qi, Y., & Guan, J. (2008). Topologies and Laplacian spectra of a deterministic uniform recursive tree. The European Physical Journal B, 63(4), 507513.CrossRefGoogle Scholar
Supplementary material: File

Guzman et al. supplementary material

Guzman et al. supplementary material

Download Guzman et al. supplementary material(File)
File 40 KB