Uniform Distribution of Weights in Weighted Networks

Uniform Distribution of Weights and Power-Law Distribution of Node Strength in Weighted Networks

In real-world networks, connection weights often exhibit specific distribution characteristics. This article discusses a special weighted network model where edge weights follow a uniform distribution, while node strength (the sum of all adjacent edge weights) displays a power-law distribution. This seemingly contradictory combination reveals profound characteristics of network structures.

Uniform weight distribution means any two edges in the network have equal probability of being assigned weight values. This distribution is typically used to simulate connection strength scenarios without obvious preferences. Notably, uniform weight distribution doesn't equate to all edges having identical weights, but rather indicates that weight values appear randomly and with equal probability within a certain interval.

The power-law distribution of node strength manifests as a few nodes possessing extremely high connection strength while most nodes have relatively low strength. This "rich-get-richer" phenomenon commonly exists in real systems like social networks and the internet. When edge weights are uniformly distributed yet node strength still exhibits power-law characteristics, it demonstrates that the network topology itself (i.e., how nodes connect) plays a decisive role in strength distribution.

When analyzing such networks, researchers typically plot two key graphs: the degree distribution graph showing how node connection quantities are distributed, and the strength distribution graph displaying the distribution pattern of node strength values. Comparing these two graphs can visually reveal the interaction mechanism between weight distribution and topology. This model provides important perspectives for studying emergent properties of weighted networks, particularly having reference value for modeling practical systems like transportation networks and neural networks.

From a programming perspective, implementing such networks typically involves creating an adjacency matrix with uniformly distributed random weights using functions like numpy.random.uniform(). Node strength calculation can be efficiently implemented through matrix operations like row-wise summation of the adjacency matrix. The power-law distribution analysis often requires fitting tools such as powerlaw Python package or logarithmic binning techniques for proper visualization and parameter estimation.