- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Reproducing the first and second moments of empirical degree distributions

Mattia Marzi1,*, Francesca Giuffrida2,1,3, Diego Garlaschelli1,3,4, and Tiziano Squartini1,5,4

  • *Contact author: mattia.marzi@imtlucca.it
PDF

Phys. Rev. Research 8, 013141 – Published 9 February, 2026

DOI: https://doi.org/10.1103/3vtj-5nlt

Abstract

The study of probabilistic models for the analysis of complex networks represents a flourishing research field. Among the former, exponential random graphs (ERGs) have gained increasing attention over the years. So far, only linear ERGs have been extensively employed to gain insight into the structural organization of real-world complex networks. None, however, is capable of accounting for the variance of the empirical degree distribution. To this aim, nonlinear ERGs must be considered. After showing that the usual mean-field approximation forces the degree-corrected version of the two-star model to degenerate, we define a fitness-induced variant of it. Such a “softened” model is capable of reproducing the sample variance, while retaining the explanatory power of its linear counterpart, within a purely canonical framework.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (47)

  1. V. Colizza, A. Barrat, M. Barthélemy, and A. Vespignani, The role of the airline transportation network in the prediction and predictability of global epidemics, Proc. Natl. Acad. Sci. USA 103, 2015 (2006).
  2. A. Barrat, M. Barthélemy, and A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, UK, 2008).
  3. M. Newman, Networks: An Introduction (Oxford University Press, Oxford, UK, 2010).
  4. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  5. T. Squartini, I. van Lelyveld, and D. Garlaschelli, Early-warning signals of topological collapse in interbank networks, Sci. Rep. 3, 3357 (2013).
  6. S. Battiston, et al., Complexity theory and financial regulation, Science 351, 818 (2016).
  7. M. Bardoscia, S. Battiston, F. Caccioli, and G. Caldarelli, Pathways towards instability in financial networks, Nat. Commun. 8, 14416 (2017).
  8. V. Macchiati, E. Marchese, P. Mazzarisi, D. Garlaschelli, and T. Squartini, Spectral signatures of structural change in financial networks, Chaos, Solitons Fractals 193, 116065 (2025).
  9. E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).
  10. J. Park and M. E. J. Newman, Statistical mechanics of networks, Phys. Rev. E 70, 066117 (2004).
  11. D. Garlaschelli and M. I. Loffredo, Maximum likelihood: Extracting unbiased information from complex networks, Phys. Rev. E 78, 015101 (2008).
  12. T. Squartini and D. Garlaschelli, Analytical maximum-likelihood method to detect patterns in real networks, New J. Phys. 13, 083001 (2011).
  13. F. Saracco, R. Di Clemente, A. Gabrielli, and T. Squartini, Randomizing bipartite networks: The case of the World Trade Web, Sci. Rep. 5, 10595 (2015).
  14. T. Squartini and D. Garlaschelli, Maximum-Entropy Networks (Springer, Berlin, 2017).
  15. G. Cimini, et al., The statistical physics of real-world networks, Nat. Rev. Phys. 1, 58 (2019).
  16. A. Fronczak, P. Fronczak, and J. A. Hołyst, Statistical mechanics of the international trade network, Phys. Rev. E 73, 016108 (2006).
  17. G. Bianconi, Entropy of network ensembles, Phys. Rev. E 79, 036114 (2009).
  18. A. Fronczak and P. Fronczak, Statistical mechanics of the international trade network: Structural correlations and modeling, Phys. Rev. E 85, 056113 (2012).
  19. T. Squartini, R. Mastrandrea, and D. Garlaschelli, Unbiased sampling of network ensembles, New J. Phys. 17, 023052 (2015).
  20. M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Structures & Algorithms 6, 161 (1995).
  21. Y. Artzy-Randrup and L. Stone, Generating uniformly distributed random networks, Physica A 364, 513 (2006).
  22. C. I. Del Genio, H. Kim, Z. Toroczkai, and K. E. Bassler, Efficient and exact sampling of simple graphs with given arbitrary degree sequence, PLoS One 5, e10012 (2010).
  23. J. Blitzstein and P. Diaconis, A sequential importance sampling algorithm for generating random graphs with prescribed degrees, Internet Math. 6, 489 (2011).
  24. J. Kim, Z. Toroczkai, P. L. Erdős, I. Miklós, and I. Kondor, Network-based prediction of unobserved edges in complex networks, New J. Phys. 14, 023012 (2012).
  25. E. S. Roberts and A. C. C. Coolen, Unbiased degree-preserving randomization of directed binary networks, Phys. Rev. E 85, 046103 (2012).
  26. T. Squartini, G. Caldarelli, G. Cimini, A. Gabrielli, and D. Garlaschelli, Reconstruction methods for networks: The case of economic and financial systems, Phys. Rep. 757, 1 (2018).
  27. G. Cimini, T. Squartini, D. Garlaschelli, and A. Gabrielli, Systemic risk analysis on reconstructed economic and financial networks, Sci. Rep. 5, 15758 (2015).
  28. A. Almog, R. Bird, and D. Garlaschelli, Enhanced gravity model of trade: Reconciling macroeconomic and network models, Front. Phys. 7, 55 (2019).
  29. S. Hooijmaaijers and G. Buiten, Estimating the Dutch Interfirm Trade Network with Commodity Breakdown, Statistics Netherlands, Technical Report (2019).
  30. C. E. S. Mattsson, et al., Functional structure in production networks, Front. Big Data 4, 666712 (2021).
  31. L. N. Ialongo, et al., Reconstructing firm-level interactions in the Dutch input–output network, Sci. Rep. 12, 11847 (2022).
  32. G. Cimini, R. Mastrandrea, and T. Squartini, Reconstructing Networks, Elements in the Structure and Dynamics of Complex Networks (Cambridge University Press, Cambridge, 2021).
  33. S. Battiston, M. Puliga, R. Kaushik, P. Tasca, and G. Caldarelli, Debtrank: Too central to fail? Financial networks, the Fed and systemic risk, Sci. Rep. 2, 541 (2012).
  34. R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86, 3200 (2001).
  35. V. Sood and S. Redner, Voter model on heterogeneous graphs, Phys. Rev. Lett. 94, 178701 (2005).
  36. V. Sood, T. Antal, and S. Redner, Voter models on heterogeneous networks, Phys. Rev. E 77, 041121 (2008).
  37. G. Iori, G. D. Masi, O. Precup, G. Gabbi, and G. Caldarelli, A network analysis of the Italian overnight money market, J. Econ. Dyn. Control 32, 259 (2008).
  38. K. Finger, D. Fricke, and T. Lux, Network Analysis of the E-MID Overnight Money Market: The Informational Value of Different Aggregation Levels for Intrinsic Dynamic Processes, Kiel Working Paper No. 1782 (Kiel Institute for the World Economy, 2012).
  39. J. Park and M. E. J. Newman, Solution of the two-star model of a network, Phys. Rev. E 70, 066146 (2004).
  40. The asterisk indicates (the value of the quantities measured on) the observed configuration A*.
  41. Limiting ourselves to the position pij=xyki+kj/(1+xyki+kj) would, in fact, lead to an inconsistency since the information on the degrees is not supposed to be available.
  42. To ease numerical manipulations, we have rescaled the strengths, dividing them by their arithmetic mean.
  43. N. Vallarano, M. J. Straka, G. Cimini, and T. Squartini, Fast and scalable likelihood maximization for exponential random graph models via GPU parallelization, Sci. Rep. 11, 5725 (2021).
  44. Recall that the quantile q of a set of (ordered) values indicates the percentage q of the smallest values. Given the empirical degree sequence and the model-based, expected one, we, thus, compute their quantile functions by sorting both in increasing order and scattering the rth order statistics 〈k〉r versus kr, r=1,⋯,N.
  45. F. Chung and L. Lu, Connected components in random graphs with given expected degree sequences, Ann. Comb. 6, 125 (2002).
  46. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).
  47. https://github.com/mattiamarzi/fit2SM.

Outline

Information

Phys. Rev. Research 8, 013141– Published 9 February, 2026
  • Received 31 July 2025
  • Accepted 20 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation