Fit Statistical Distributions to Degree Sequence

Fits one or more statistical distributions to the degree sequence of a network by maximum likelihood and evaluates goodness of fit with Kolmogorov-Smirnov statistics. The comparison table is sorted by AIC.

Usage

fit_degree_distribution(
  x,
  distributions = NULL,
  mode = "all",
  directed = NULL,
  xmin = NULL,
  ...
)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object. Without igraph installed, only a numeric matrix is accepted.
distributions: Character vector of distributions to fit. Options: "power_law", "exponential", "poisson", "geometric". Default NULL fits all four. An unknown name raises an error.
mode: For directed networks: "all", "in", or "out". Determines which degree to extract. Default "all".
directed: Logical or NULL. If NULL (default), auto-detect from matrix symmetry. Set TRUE to force directed, FALSE to force undirected.
xmin: Minimum degree to include in fitting. For the power law, NULL triggers automatic estimation (Clauset et al. 2009 via igraph). For the other distributions, NULL defaults to 1. Each fit requires at least two degrees at or above xmin and raises an error otherwise.
...: Additional arguments (currently unused).

Value

An object of class "cograph_degree_fit", a list containing:

fits: Named list, one entry per distribution, each with distribution, parameters (named list of fitted parameters), loglik, aic, bic, ks_stat and ks_p. The parameters are alpha and xmin for the power law, lambda for the exponential and Poisson fits, and p for the geometric fit.
comparison: Data frame sorted by AIC with columns distribution, aic, bic, ks_stat, ks_p.
best: Name of the best-fitting distribution (lowest AIC).
degree: The named degree vector used for fitting.

Details

The power-law model (Pareto type I) is \(P(k) \sim k^{-\alpha}\). With igraph available and xmin = NULL, it is fitted with igraph::fit_power_law(), which implements the Clauset et al. (2009) method. Otherwise the simple MLE \(\alpha = 1 + n / \sum \log(k / k_{min})\) is computed, with \(k_{min}\) equal to the supplied xmin or, without igraph, to the smallest degree (at least 1).

The exponential model is \(P(k) \sim e^{-\lambda k}\), with MLE \(\lambda = 1 / \bar{k}\).

The Poisson model is \(P(k) \sim \lambda^k e^{-\lambda} / k!\), with MLE \(\lambda = \bar{k}\).

The geometric model is \(P(k) \sim (1-p)^k p\), with MLE \(p = 1 / (1 + \bar{k})\).

ks_p comes from stats::ks.test() for the exponential and Poisson fits and is NA for the power-law and geometric fits. The p-values are approximate because the test assumes a continuous distribution. For the exponential fit stats::ks.test() warns when degrees are tied. A power-law fit to degrees that all equal xmin is degenerate and returns NA for alpha, the likelihood and every statistic.

Each likelihood is computed over the degrees at or above the xmin of that fit. With xmin = NULL, the power-law fit uses the estimated xmin and the other fits use 1, so their AIC values are computed on different subsets of the degrees. AIC and BIC count one free parameter per distribution, so the power-law xmin is not penalized.

Printing and plotting

Printing the result shows the best-fitting distribution, the comparison table and the fitted parameters of each distribution. plot() on the result is documented in plot-results.

References

Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703.

Examples

fit_degree_distribution(regulation_net,
  distributions = c("poisson", "geometric"))
#> Degree Distribution Fit
#> =======================
#> N degrees: 10 
#> Best fit:  poisson 
#> 
#> Comparison (sorted by AIC):
#>  distribution     aic     bic ks_stat   ks_p
#>       poisson 40.4388 40.7414  0.3457 0.1831
#>     geometric 59.4163 59.7189  0.5373     NA
#> 
#> Fitted parameters:
#>   poisson: lambda = 6.0000
#>   geometric: p = 0.1429