Modeling sparsity in count-weighted networks

TL;DR

A probabilistic model for generating weighted networks that allows us to control network sparsity and incorporates degree corrections for each node and incorporates degree corrections for each node is proposed.

Abstract

Community detection methods have been extensively studied to recover communities structures in network data. While many models and methods focus on binary data, real-world networks also present the strength of connections, which could be considered in the network analysis. We propose a probabilistic model for generating weighted networks that allows us to control network sparsity and incorporates degree corrections for each node. We propose a community detection method based on the Variational Expectation-Maximization (VEM) algorithm. We show that the proposed method works well in practice for simulated networks. We analyze the Brazilian airport network to compare the community structures before and during the COVID-19 pandemic.

Figures (8)

• Figure 1: Mean of NMI between estimated and true communities membership over 100 simulated networks with $n=100$. The networks are sampled from the ZIP model without degree correction and with mean weights $\lambda_{\text{out}}$ (between groups) and $\lambda_{\text{in}}$ (within groups).
• Figure 2: Mean of NMI between estimated and true communities membership over 100 simulated networks with $n=100$. The networks are sampled from the ZIP model without degree correction and with mean weights $\lambda_{\text{out}}=5$ (between groups) and $\lambda_{\text{in}}=8$ (within groups). The parameter $p$ controls the global sparsity of the networks.
• Figure 3: Mean of NMI between estimated and true communities membership over 100 simulated networks with $n=100$. The networks are sampled from the ZIP model with degree correction and with mean weights $\lambda_{\text{out}}$ (between groups) and $\lambda_{\text{in}}$ (within groups).
• Figure 4: The mean and the one standard deviation error bars for the estimated number of communities over 100 simulated networks with balanced communities. The networks are sampled from the ZIP model with degree correction and with mean weights $\lambda_{\text{out}}$ (between groups) and $\lambda_{\text{in}}$ (within groups).
• Figure 5: Mean of NMI between estimated and true communities membership over 100 simulated networks with $n=100$. The networks are sampled from the ZIP model with degree correction and with mean weights $\lambda_{\text{out}}=5$ (between groups) and $\lambda_{\text{in}}=11$ (within groups). The parameter $p$ controls the global sparsity of the networks.