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On tail inference in scale-free inhomogeneous random graphs

Daniel Cirkovic, Tiandong Wang, Daren B. H. Cline

Abstract

Both empirical and theoretical investigations of scale-free network models have found that large degrees in a network exert an outsized impact on its structure. However, the tools used to infer the tail behavior of degree distributions in scale-free networks often lack a strong theoretical foundation. In this paper, we introduce a new framework for analyzing the asymptotic distribution of estimators for degree tail indices in scale-free inhomogeneous random graphs. Our framework leverages the relationship between the large weights and large degrees of Norros-Reittu and Chung-Lu random graphs. In particular, we determine a rate for the number of nodes $k(n) \rightarrow \infty$ such that for all $i = 1, \dots, k(n)$, the node with the $i$-th largest weight will have the $i$-th largest degree with high probability. Such alignment of upper-order statistics is then employed to establish the asymptotic normality of three different tail index estimators based on the upper degrees. These results suggest potential applications of the framework to threshold selection and goodness-of-fit testing in scale-free networks, issues that have long challenged the network science community.

On tail inference in scale-free inhomogeneous random graphs

Abstract

Both empirical and theoretical investigations of scale-free network models have found that large degrees in a network exert an outsized impact on its structure. However, the tools used to infer the tail behavior of degree distributions in scale-free networks often lack a strong theoretical foundation. In this paper, we introduce a new framework for analyzing the asymptotic distribution of estimators for degree tail indices in scale-free inhomogeneous random graphs. Our framework leverages the relationship between the large weights and large degrees of Norros-Reittu and Chung-Lu random graphs. In particular, we determine a rate for the number of nodes such that for all , the node with the -th largest weight will have the -th largest degree with high probability. Such alignment of upper-order statistics is then employed to establish the asymptotic normality of three different tail index estimators based on the upper degrees. These results suggest potential applications of the framework to threshold selection and goodness-of-fit testing in scale-free networks, issues that have long challenged the network science community.

Paper Structure

This paper contains 21 sections, 19 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Scale-free inhomogeneous random network models
  4. Ordered inhomogeneous random network
  5. Scale-free property and order statistics
  6. Alignment of upper order statistics
  7. Application to tail index estimators
  8. Hill and Pickands estimators
  9. Probability weighted moment estimator
  10. Simulations
  11. Empirical alignment of upper order statistics
  12. Sensitivity to the choice of $k(n)$
  13. Conclusion
  14. Proofs
  15. Proof of Lemma \ref{['lem:perm']}
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose $\pi$ is a permutation drawn uniformly at random from $S_n$, the set of all permutations on $[n]$. Define for every $i, j \in [n]$ such that $i \leq j$, $\tilde{A}_{ij}(n) = A_{\pi(i)\pi(j)}(n)$. Then $\{\tilde{A}_{ij}(n) \}_{1 \leq i \leq j \leq n}$ is distributed according to an inhomogene

Figures (4)

  • Figure 1: Histograms of 1000 realizations of $K(n)$ for across sample sizes $n \in \{2^{10}, 2^{11}, \dots, 2^{20}\}$ with weight distributions (a) $1 - F(x) = 2/x$ and (b) $1 - F(x) = (2 + x)/x^2$, $x > 2$. The red line corresponds to the curves $y_a = 0.7430 n^{0.2004}$ and $y_b = 0.7717n^{0.1970}$ where the parameters are estimated by regressing the $\log_2(K(n))$ realizations against $\log_2(n)$ for scenarios (a) and (b), respectively.
  • Figure 2: Histograms of centered and scaled Hill estimators. For each $\alpha \in \{1.25, 1.5, 1.75 \}$ we simulate $1{,}000$ Norros-Reittu graphs and for each $\alpha \in \{2.25, 2.5, 2.75 \}$ we simulate $1{,}000$ Chung-Lu graphs. For each multigraph we compute the Hill estimator for $k(n) \in \{32, 64, 128, 256\}$.
  • Figure 3: Histograms of centered and scaled Pickands estimators. For each $\alpha \in \{1.25, 1.5, 1.75 \}$ we simulate $1{,}000$ Norros-Reittu graphs and for each $\alpha \in \{2.25, 2.5, 2.75 \}$ we simulate $1{,}000$ Chung-Lu graphs. For each multigraph we compute the Pickands estimator for $k(n) \in \{8, 16, 32, 64\}$.
  • Figure 4: Histograms of centered and scaled PWM estimators. For each $\alpha \in \{2.25, 2.5, 2.75 \}$ we simulate $1{,}000$ Chung-Lu graphs. For each multigraph we compute the PWM estimator for $k(n) \in \{32, 64, 128, 256, 512, 1024, 2048\}$.

Theorems & Definitions (36)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Lemma \ref{['lem:perm']}
  • Theorem 4
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 26 more