Uncovering mechanisms of network formation

We provide a Python package called mechanistic_net_abc that contains implementations of mechanistic models to describe the formation of networks. We focus on models of mixture of mechanisms that combine different node-level mechanisms. Models are fitted using Approximate Bayesian Computation (ABC) techniques. To improve ABC inference we propose a summary statistic selection method using multi-output random forests. This package also provides codes for reproducibility of the analysis carried in our manuscript Uncovering mechanisms of network formation by L. Raynal and J.-P. Onnela.

Table of contents

• Description
• Installation
• Package outline
• Simple examples
• Household network study
• Authors

Description

This project focuses on models of mixture of mechanisms to understand network formation from the observation of a single graph. Inference is performed using ABC methods.

Our objectives consists in:

1. identifying the mechanisms explaining the formation of a network, and in which proportion each mechanism intervene;
2. proposing a summary statistic selection technique for improve ABC parameter inference.

Installation

Install with pip from a terminal

pip install git+https://github.com/onnela-lab/mechanistic-net-abc.git

Important: Before running the different examples please edit settings.py to indicate a valid working directory for each studied example. Figures, simulated data and results will be stored in these directories.

Package outline

The mechanistic_net_abc folder contains the package, made of the following modules:

• abc.py contains an implementation of the replenishment sequential Monte Carlo ABC (SMC-ABC) algorithm proposed by Drovandi and Pettitt, (2011).
• data_generation.py to generate individual network data (summarized) from a model of mixture of mechanisms (data_indiv_simulation) or a reference table (data_ref_table_simulation).
• data_reading.py: read real data.
• mechanisms.py provides various mechanisms for network formation. Examples are the functions node_addition, preferential_attachment_growth, neg_preferential_attachment_growth, random_attachment_growth, random_node_deletion, random_edge_deletion, random_edge_rewiring, triadic_closure, triadic_closure_without_rewiring or triangle_formation_node_addition.
• models.py implements the model of mixture of mechanisms to generate networks (see function mixture_model_simulation).
• summaries.py to compute summary statistics of undirected networks using networkx.
• summary_selection.py implements our recursive selection technique based on multi-output random forests.
• utility.py: various utility functions.

The folder 1-PA_one_noise focuses on the first simulated example of our paper where we identify preferential attachment from one noise mechanism that is random attachment.

The folder 2-PA_two_noises considers the previous example where we add an additional noise mechanism, triangle formation.

The folder 3-PA_RA_TF_mixture contains a simulated example where three mechanisms are relevant and their weights and parameters are inferred.

The folder 4-household_analysis is dedicated to the study of a real household network.

Simulated examples

We provide three simulated studies that can show you how to formulate a model of mixture of mechanisms, generate data from this model, select summaries with our random forest-based proposal and infer model parameters with ABC. Codes are ready to launch in the folders 1-PA_one_noise, 2-PA_two_noises and 3-PA_RA_TF_mixture.

Household network study

An empirical household network is also studied. The data contains 710 nodes that are households with edges representing contacts between them. We propose a mixture model of four mechanisms to understand its formation and the impact of these mechanisms. Full analysis can be replicated from codes in the 4-household_analysis folder.

Authors

• Louis Raynal - Harvard University
• Jukka-Pekka Onnela - Harvard University