Visualizing and exploring modular networks based on a probabilistic model

We propose a method to investigate modular structure in networks based on fitted probabilistic model, where the connection probability between nodes is related to a set of introduced local attributes. The attributes, as parameters of the empirical model, can be estimated by maximizing the likelihood function of the observed network. We demonstrate that the distribution of attributes provides an informative visulization of modular networks on low-dimensional space, and suggest the attribute space can be served as a better platform for further network analysis.

💡 Research Summary

The paper introduces a probabilistic framework for uncovering and visualizing modular structure in complex networks. Instead of assigning discrete community labels as in traditional community‑detection or stochastic block‑model (SBM) approaches, each node i is endowed with a continuous latent attribute vector x_i ∈ ℝ^d (typically d = 2 or 3). The probability of an edge between nodes i and j is modeled as a decreasing function of the Euclidean distance between their attribute vectors:

 p_{ij} = σ(−‖x_i − x_j‖²),

where σ is a sigmoid (or logistic) function mapping the distance‑based similarity to the interval (0,1). This formulation yields a smooth, distance‑driven connectivity landscape: nodes that are close in attribute space are more likely to be linked, while distant nodes have low connection probabilities.

Parameter estimation proceeds by maximizing the log‑likelihood of the observed adjacency matrix A:

 L = Σ_{i<j}