Fit Statistical Distributions to Degree Sequence

Fits one or more statistical distributions to the degree sequence of a network via maximum likelihood estimation and evaluates goodness-of-fit using Kolmogorov-Smirnov tests. Returns a comparison table 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.
distributions: Character vector of distributions to fit. Options: "power_law", "exponential", "poisson", "geometric". Default NULL fits all four.
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 power-law, NULL triggers automatic estimation (Clauset et al. 2009 via igraph). For other distributions, NULL defaults to 1.
...: Additional arguments (currently unused).

Value

An object of class "cograph_degree_fit" containing:

fits: Named list, one entry per distribution, each with: distribution, parameters (named list of fitted params), loglik, aic, bic, ks_stat, ks_p.
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 degree vector used for fitting.

Details

Power-law (Pareto Type I): \(P(k) \sim k^{-\alpha}\). When igraph is available, uses igraph::fit_power_law() implementing the Clauset et al. (2009) method. Otherwise, computes the simple MLE: \(\alpha = 1 + n / \sum \log(k / k_{min})\).

Exponential: \(P(k) \sim e^{-\lambda k}\). MLE: \(\lambda = 1 / \bar{k}\).

Poisson: \(P(k) \sim \lambda^k e^{-\lambda} / k!\). MLE: \(\lambda = \bar{k}\). Note: the KS test uses a continuous approximation for a discrete distribution; p-values are approximate.

Geometric: \(P(k) \sim (1-p)^k p\). MLE: \(p = 1 / (1 + \bar{k})\).

References

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

Examples

adj <- matrix(c(0, 1, 1, 0, 0,
                1, 0, 1, 1, 0,
                1, 1, 0, 1, 1,
                0, 1, 1, 0, 1,
                0, 0, 1, 1, 0), 5, 5, byrow = TRUE)
rownames(adj) <- colnames(adj) <- LETTERS[1:5]
fit <- cograph::fit_degree_distribution(adj,
  distributions = c("exponential", "poisson"))
#> Warning: ties should not be present for the one-sample Kolmogorov-Smirnov test
print(fit)
#> Degree Distribution Fit
#> =======================
#> N degrees: 5 
#> Best fit:  poisson 
#> 
#> Comparison (sorted by AIC):
#>  distribution     aic     bic ks_stat   ks_p
#>       poisson 17.4664 17.0758  0.4695 0.2205
#>   exponential 22.2962 21.9056  0.5105 0.1476
#> 
#> Fitted parameters:
#>   exponential: lambda = 0.3571
#>   poisson: lambda = 2.8000