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A series of real networks invariants

Mikhail Tuzhilin

TL;DR

The paper tackles distinguishing real-world networks from artificial models by introducing a series of Laplacian-based invariants, the $j$-neighborhood centralities, which generalize degree ($j=0$) and ksi-centrality ($j=1$ via $\xi^j_i = \frac{(L\chi^j_i,\chi^j_i)}{(\chi^j_i,\chi^j_i)}$ with $L=(\partial^0)^*\partial^0$). These centralities are tied to the Laplacian spectrum and Cheeger number, and a practical computation scheme is provided. Empirically, the centralities exhibit exponential-like distributions for real networks and different (more centered) distributions for artificial networks, with some exceptions. This suggests the invariants can be used for real-vs-artificial discrimination and as informative features in graph-based learning, while opening avenues for further theoretical and empirical study.

Abstract

In this article we propose a generalization of two known invariants of real networks: degree and ksi-centrality. More precisely, we found a series of centralities based on Laplacian matrix, that have exponential distributions (power-law for the case $j = 0$) for real networks and different distributions for artificial ones.

A series of real networks invariants

TL;DR

The paper tackles distinguishing real-world networks from artificial models by introducing a series of Laplacian-based invariants, the -neighborhood centralities, which generalize degree () and ksi-centrality ( via with ). These centralities are tied to the Laplacian spectrum and Cheeger number, and a practical computation scheme is provided. Empirically, the centralities exhibit exponential-like distributions for real networks and different (more centered) distributions for artificial networks, with some exceptions. This suggests the invariants can be used for real-vs-artificial discrimination and as informative features in graph-based learning, while opening avenues for further theoretical and empirical study.

Abstract

In this article we propose a generalization of two known invariants of real networks: degree and ksi-centrality. More precisely, we found a series of centralities based on Laplacian matrix, that have exponential distributions (power-law for the case ) for real networks and different distributions for artificial ones.
Paper Structure (6 sections, 3 theorems, 5 equations, 17 figures)

This paper contains 6 sections, 3 theorems, 5 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Generalization of degree and ksi centralities
  3. Theory
  4. Calculations
  5. Comparison of distributions for real networks and artificial
  6. Discussion

Key Result

Theorem 1

For any graph $j$-neighborhood centrality where $\mathop{\mathrm{out}}\nolimits(H) = E(H, V(G)\setminus H)$ the set of outer connections from the set $H$ to other vertices.

Figures (17)

  • Figure 3: Distributions of $\xi^j$ centralities for $t=j=1,2,3,4$ for Erdos-Renyi $(n,p) = (4000, 0.02)$ and Barabasi-Albert $(n,m) = (4000, 43)$ networks.
  • Figure 4: Distributions of $\xi^j$ centralities for $t=j=1,2,3,4$ for Watts-Strogatz $(n, k, p) = (4000, 21, 0.3)$ and Boccaletti-Hwang-Latora $(n, m, n_0) = (4000, 20, 100)$ networks.
  • Figure 5: Distributions of $\xi^j$ centralities for $t=j=1,2,3,4$ for real networks structures data from Tuz2.
  • Figure 6: Distributions of $\xi^j$ centralities for $t=j=1,2,3,4$ for real networks structures data from Tuz2.
  • Figure 7: Distributions of $\xi^j$ centralities for $t=j=1,2,3,4$ for real networks structures data from Tuz2.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3