[Paper Review] Scale-free networks with self-growing weight

Takuma Tanaka, Toshio Aoyagi|ArXiv.org|Jan 15, 2007

Modular Robots and Swarm Intelligence3 citations

TL;DR

This paper proposes a novel analytically tractable model of weighted scale-free networks where link weights grow independently of node attachment via a strength-driven, self-growing mechanism. The model generates power-law distributions for degree, weight, and strength, with the control parameter $ c $ determining network structure—small $ c $ fits coauthorship networks, large $ c $ fits e-mail networks—enabling theoretical derivation of statistical properties in complex systems.

ABSTRACT

We present a novel type of weighted scale-free network model, in which the weight grows independently of the attachment of new nodes. The evolution of this network is thus determined not only by the preferential attachment of new nodes to existing nodes but also by self-growing weight of existing links based on a simple weight-driven rule. This model is analytically tractable, so that the various statistical properties, such as the distribution of weight, can be derived. Finally, we found that some type of social networks is well described by this model.

Motivation & Objective

• To develop a model of weighted complex networks where link weights grow independently of node addition, capturing real-world dynamics such as repeated collaborations.
• To explain the emergence of power-law distributions in link weights, degrees, and node strengths through a single, analytically tractable mechanism.
• To demonstrate that the model captures statistical properties of real social networks, including coauthorship and e-mail networks, by tuning a single parameter $ c $.
• To provide a theoretical framework for studying dynamical processes on weighted networks, such as epidemics and oscillators, by deriving exact statistical properties.

Proposed method

• The network grows by adding one node at a time, with new nodes attaching preferentially to existing nodes based on their strength $ s_i $, using probability $ s_i / ar{s} $.
• At each time step, $ ct $ pairs of nodes are selected with probability proportional to the product of their strengths $ s_i s_j / ar{s}^2 $, and their link weight increases by one if already connected.
• The weight of a link between nodes $ i $ and $ j $ evolves according to the differential equation $ \partial w / \partial t = 2ct \cdot s_i s_j / \bar{s}^2 $, with $ \bar{s} = \int_0^t s(v,t) dv $.
• The model uses a continuous approximation for node age $ u $, leading to a closed-form solution for the weight matrix: $ w(u_1,u_2,t) = (2+ct)^2 / \sqrt{u_1 u_2 (2+cu_1)^3 (2+cu_2)^3} $.
• The distribution of link weights is derived as $ P(w) \sim w^{-3/2} $ for $ cu \gg 1 $ and $ P(w) \sim w^{-3} $ for $ cu \ll 1 $, with normalization constant $ C $.
• The strength $ s(u,t) $ is obtained by integrating $ w(u,v,t) $ over all other nodes, satisfying $ s(u,t) = \int_0^t w(u,v,t) dv $.

Experimental results

Research questions

• RQ1 How can a weighted scale-free network be constructed such that link weights grow independently of node attachment?
• RQ2 What analytical conditions lead to power-law distributions in degree, weight, and strength in such networks?
• RQ3 Can the model reproduce the statistical properties of real-world social networks like coauthorship and e-mail networks?
• RQ4 How does the control parameter $ c $, governing the rate of self-growing weight, affect the network's scaling exponents?
• RQ5 What is the relationship between node strength and link weight distribution in a network where weights grow via strength-driven preferential attachment?

Key findings

• The model produces power-law distributions for link weights with exponents $ \sim w^{-3/2} $ when $ cu \gg 1 $, and $ \sim w^{-3} $ when $ cu \ll 1 $, derived analytically.
• The degree distribution follows a power law with exponent $ \gamma \approx 3 $, consistent with empirical scale-free networks.
• The strength of a node scales with its degree as $ k \propto s^{1/2} $ for large $ c $, indicating a sublinear but nontrivial relationship.
• For the coauthorship network, the estimated parameter $ c = 1.5 \times 10^{-4} $ yields a network with statistical properties matching real data, including total strength $ 1.75 \times 10^6 $.
• The e-mail network, with large $ c $, exhibits a degree exponent of approximately 1.8, falling between the model’s theoretical bounds of 1.5 and 2, confirming empirical fit.
• The model’s analytical tractability allows exact derivation of statistical properties, making it suitable for studying dynamical processes such as epidemics and oscillators on weighted networks.