[Paper Review] Node-node distance distribution for growing networks

Krzysztof Malarz, JUSTYNA KARPIŃSKA+2 more|Sep 10, 2003

Complex Network Analysis Techniques1 references4 citations

TL;DR

This paper presents a simulation-based method for tracking node-node distance (NND) distributions in growing networks, using iterative updates of the distance matrix to model exponential and scale-free trees (m=1,2). It derives analytical formulas for the n-th moments of NND distributions in exponential trees, showing logarithmic growth of average distance and dispersion with network size, consistent with the small-world effect.

ABSTRACT

We present the simulation of the time evolution of the distance matrix. The result is the node-node distance distribution for various kinds of networks. For the exponential trees, analytical formulas are derived for the moments of the distance distribution.

Motivation & Objective

• To model the time evolution of node-node distance (NND) distributions in growing networks using distance matrix updates.
• To analyze how attachment rules (random or preferential) affect NND characteristics in exponential and scale-free networks.
• To derive analytical expressions for the n-th moments of the NND distribution in exponential trees.
• To quantify the small-world effect through the logarithmic growth of average NND with network size.
• To provide a computationally efficient algorithm for simulating NND distributions in growing graphs.

Proposed method

• The distance matrix S is dynamically updated as new nodes are added, with new row/column entries computed based on shortest path rules.
• For m=1 (trees), the new node's distances are set as s_{N+1}(i) = s_p(i) + 1, where p is the attachment node.
• For m=2 (graphs), the new node's distance is s_{N+1}(i) = min(s_p(i), s_q(i)) + 1, with p and q as attachment nodes.
• A reevaluation step ensures shortest paths are preserved: s_{N+1}(i,j) = min(s_N(i,j), s_N(i,p)+2+s_N(q,j), ...).
• Preferential attachment is implemented via a Kertész-style algorithm using node degrees as selection weights.
• Iterative formulas are derived for the n-th moments of the NND distribution, enabling analytical prediction of statistical properties.

Figure 1: The NND distribution for different types trees and graphs.

Experimental results

Research questions

• RQ1 How does the node-node distance distribution evolve in growing exponential and scale-free networks?
• RQ2 What are the analytical expressions for the n-th moments of the NND distribution in exponential trees?
• RQ3 How do the average distance and dispersion of NND grow with network size N?
• RQ4 To what extent do the NND distributions in exponential and scale-free networks differ in shape and concentration?
• RQ5 Do the higher-order moments (skewness, kurtosis) of the NND distribution converge to normal-like values as N increases?

Key findings

• The average node-node distance d_N^1 grows logarithmically with network size N, confirming the small-world effect.
• For exponential trees with m=1, the average distance follows d(N) = 2.00 ln N - 2.84, and dispersion σ² = 2.00 ln N - 1.44.
• For scale-free trees with m=1, the average distance is d(N) = 1.00 ln N - 0.08, and dispersion σ² = 1.00 ln N - 1.64.
• The iterative formula (9) accurately predicts the n-th moments d_N^n for exponential trees, matching simulation results over 10^4 realizations.
• Skewness and kurtosis of the NND distribution in exponential trees do not vanish even for large N, indicating non-normal, asymmetric behavior.
• NND distributions in graphs are more condensed than in trees, and scale-free networks are more condensed than exponential networks.

Figure 2: The main moments $d^{k}_{N}$ ( $k=1,\cdots,4$ ) for the exponential trees given by Eq. ( 9 ) (lines) and from the direct simulations (symbols). The latter are averaged over $N_{\text{run}}=10^{4}$ independent evolution process realizations.