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New exponential law for real networks

Mikhail Tuzhilin

TL;DR

The paper defines ksi-centrality and the exponential-ksi law as an invariant of real networks, showing across 40 real networks that ksi distributions follow an exponential form, in contrast to random graphs which are bell-shaped. It develops a fast algorithm for computing ksi and normalized ksi-centralities and compares real networks to Barabasi-Albert and Watts-Strogatz models, finding exponential-ksi occurs only for limited parameter ranges. The results highlight the bow-tie structure as a potential structural reason for the observed exponential distribution and challenge common generative models to capture this aspect of real networks. The findings offer a new, quantitative invariant for distinguishing real network structure from typical random or generative models, with implications for network analysis and modeling.

Abstract

In this article we have shown that the distributions of ksi satisfy an exponential law for real networks while the distributions of ksi for random networks are bell-shaped and closer to the normal distribution. The ksi distributions for Barabasi-Albert and Watts-Strogatz networks are similar to the ksi distributions for random networks (bell-shaped) for most parameters, but when these parameters become small enough, the Barabasi-Albert and Watts-Strogatz networks become more realistic with respect to the ksi distributions.

New exponential law for real networks

TL;DR

The paper defines ksi-centrality and the exponential-ksi law as an invariant of real networks, showing across 40 real networks that ksi distributions follow an exponential form, in contrast to random graphs which are bell-shaped. It develops a fast algorithm for computing ksi and normalized ksi-centralities and compares real networks to Barabasi-Albert and Watts-Strogatz models, finding exponential-ksi occurs only for limited parameter ranges. The results highlight the bow-tie structure as a potential structural reason for the observed exponential distribution and challenge common generative models to capture this aspect of real networks. The findings offer a new, quantitative invariant for distinguishing real network structure from typical random or generative models, with implications for network analysis and modeling.

Abstract

In this article we have shown that the distributions of ksi satisfy an exponential law for real networks while the distributions of ksi for random networks are bell-shaped and closer to the normal distribution. The ksi distributions for Barabasi-Albert and Watts-Strogatz networks are similar to the ksi distributions for random networks (bell-shaped) for most parameters, but when these parameters become small enough, the Barabasi-Albert and Watts-Strogatz networks become more realistic with respect to the ksi distributions.

Paper Structure

This paper contains 9 sections, 2 theorems, 7 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Calculations
  4. Results
  5. Exponential law
  6. Normalized ksi coefficient
  7. Comparison with Watts-Strogatz and Barabasi-Albert networks
  8. Discussion
  9. Acknowledgments

Key Result

Lemma 1

where $\overline a_{ki} = 1-a_{ki}.$

Figures (13)

  • Figure 1: Linear approximation of ksi distribution on log-scale $y$ and comparison of fitted exponential function with it for Facebook and Collaboration networks.
  • Figure 2: Ksi distribution for Erdos-Renyi graphs.
  • Figure 3: The root mean squared difference between the fitted (exponential) distributions and the original for fixed parameters $(k, p)$ of the 2000-node Watts-Strogatz network. The X-axis corresponds to $k$, and the Y-axis corresponds $p$.
  • Figure 4: Initial ksi distributions (right) and in logarithmic scale (left) for the case where Barabasi-Albert and Watts-Strogatz networks are exponential-ksi networks. The number of nodes is 2000.
  • Figure 5: Ksi-distributions (left) and corresponding log-scale for y axes (right) for networks from table \ref{['tab:net']}. The X-axis corresponds to the ksi value, and the Y-axis corresponds to the number of vertices with that ksi value.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Corollary 1
  • Definition 3
  • Definition 4