[Paper Review] Preferential compactness of networks

Mikko J. Alava, S. N. Dorogovt︠s︡ev|arXiv (Cornell University)|Jul 23, 2004

Complex Network Analysis Techniques20 references3 citations

TL;DR

This paper introduces a preferential compactness model for growing networks where new vertices attach preferentially to central nodes, favoring compact, low-diameter structures. The model links optimization and self-organization by using distance-dependent attachment probabilities; while finite networks exhibit complex, multi-part degree distributions, infinite networks converge to a rapidly decreasing degree distribution due to the dominance of compactness over randomness in the thermodynamic limit.

ABSTRACT

We introduce evolving networks where new vertices preferentially connect to the more central parts of a network. This makes such networks compact. Finite networks grown under the preferential compactness mechanism have complex architectures, but infinite ones tend towards the opposite, having rapidly decreasing distributions of connections. We present an analytical solution of the problem for tree-like networks. Our approach links a collective self-optimization mechanism of the emergence of complex network architectures to self-organization mechanisms.

Motivation & Objective

• To explore how collective self-optimization mechanisms, particularly preferential attachment based on centrality, shape complex network architectures.
• To investigate the interplay between compactness (central attachment) and randomness (geographic or distance-based selection) in network growth.
• To explain why finite networks under this mechanism display complex degree distributions, while infinite networks exhibit trivial, rapidly decreasing distributions.
• To provide an analytical solution for tree-like networks under the preferential compactness mechanism, bridging optimization and self-organization frameworks.

Proposed method

• New vertices are added sequentially to a growing tree network, attaching to existing nodes with probability proportional to a decreasing function r(ℓ) of their distance ℓ from the root.
• The preference function r(ℓ) = x^ℓ is used to model decreasing attachment likelihood with distance, favoring central nodes and promoting compactness.
• Analytical solutions are derived for the degree distribution and the mean number of nodes at each distance ℓ from the root, using evolution equations for average degrees.
• The model is extended by modifying attachment rules to include a finite probability of attaching to the root or its neighbors, preventing edge condensation and enabling larger observable plateaus in degree distributions.
• The time evolution is reparameterized using τ(t) ≈ √(2pt) as a new time variable to analyze the growth of the central region and degree distribution over time.
• A modified evolution equation is used to track the average number of nodes with in-degree k in the central region, showing a kink-like plateau of width pτ.

Experimental results

Research questions

• RQ1 How does preferential attachment based on distance from the root affect the degree distribution in growing tree networks?
• RQ2 Why do finite networks under preferential compactness exhibit complex, multi-part degree distributions, while infinite networks show only rapidly decreasing distributions?
• RQ3 What mechanisms can stabilize complex degree distributions in large but finite networks under distance-based attachment?
• RQ4 To what extent does the use of inter-vertex distance in attachment rules lead to structural trivialization in the thermodynamic limit?
• RQ5 Can the model be modified to observe complex network structures in larger networks without edge condensation on the root?

Key findings

• Finite networks grown under preferential compactness exhibit a complex degree distribution with a plateau-like structure, where the width of the plateau is xτ and the height is 1/x, with τ ≈ √(2pt) in modified models.
• In the infinite network limit, the degree distribution becomes trivial and rapidly decreasing, as the compactness mechanism dominates over randomness.
• The model shows that the thermodynamic limit is approached slowly, allowing complex structures to persist in reasonably large finite networks.
• The introduction of a finite attachment probability to the root or its neighbors prevents edge condensation and enables a broader observable range of complex degree distributions.
• The analytical solution for the degree distribution in the modified model shows a plateau of width pτ and height p, with τ growing as √t, allowing for larger-scale observation of complex connectivity patterns.
• The degree distribution in the original model is non-power-law and multi-part, with a plateau region that becomes increasingly narrow in the infinite limit, indicating structural trivialization.