Tracing Network Evolution Using the PARAFAC2 Model
Table of Contents
We generated two types of simulated data sets: (i) data sets following Equation (2) from the paper as well as the Parafac2 constraint, and (ii) data sets following Equation (2) from the paper, having time-evolving patterns Bₖ that do not necessarily follow the Parafac2 constraints.
First mode: Suppose that the first mode of the tensor represents subjects. In this mode, we introduce a clustering structure since our goal with the analysis of time-evolving data is often to identify patterns that can separate different groups of subjects. The factor vectors, i.e., columns of A, in this mode can be regarded as a measure of how much each underlying pattern (network) contributes to the signal of each subject. In order to generate the clustering structure, two vectors, representing the mean of each group, m₁ and m₂ were drawn randomly from {0, 1}ᴿ. Then, for each subject, i, a group, gᵢ ∈ {1, 2}, was chosen at random and a row vector aᵢ ∈ ℝ¹ˣᴿ was drawn from a normal distribution with μ = m₁ (mean) if gᵢ = 1 and μ = m₂ and σ=0.5 (standard deviation). These row vectors were stacked to generate an I×R matrix A. (See Figure 2 in the paper).
Second mode (dynamic networks): B matrices, i.e., Bₖ, are simulated with column vectors representing temporally evolving networks, where each element, [bₖ]ⱼᵣ, can be considered as a node while each factor vector [bₖ]ᵣ corresponds to a network. If node j is present in network r at time step k, then [bₖ]ⱼᵣ is high, i.e., node j is an active node; otherwise, [bₖ]ⱼᵣ is low, and node j is a passive node.
First, networks are initialized by dividing J nodes into R networks randomly. Once initialized, networks may evolve in three different ways at each time step: (i) Growing: With pᵢ probability, a random passive node becomes active. (ii) Shrinking: With pₘ probability, a random active node becomes deactivated. Deactivated nodes are not re-added to the network. (iii) Shifting: With probability pₛ, node indices increase by 1. We also generate a baseline matrix, Ꞵₖ with entries [ꞵₖ]ⱼᵣ randomly drawn from a normal distribution with μ=0.2 and σ=0.1. Finally, in order to generate Bₖ, we iterate through all nodes in network r, and if node j is active in network r at time k, then [bₖ]ⱼᵣ = [ꞵₖ]ⱼᵣ + [ρₖ]ⱼᵣ, where [ρₖ]ⱼᵣ is drawn from a normal distribution (μ=0.8 and σ=0.1). Otherwise, [bₖ]ⱼᵣ = [ꞵₖ]ⱼᵣ. These steps are repeated for a simulation that lasts for K time steps, forming Bₖ, for k=1, 2,..., K.
The evolving network set-up contains a network generated using pₘ = pᵢ = 0 and pₛ=1 (shifting), two networks using pᵢ > pₘ , one of which with pₛ=0.5 (growing & shifting) and the other with pₛ=0 (growing), and one network with pᵢ < pₘ and pₛ=0 (shrinking) (See Figure 3 in the paper). We also generated random Bₖs that follow the Parafac2 constraints. Note that evolving networks do not often satisfy this constraint.
Third mode (time): Factor vectors in this mode are simulated using two different set-ups: (i) all factor vectors drawn from a uniform distribution, 𝓤(1.1, 2.1) (ii) one factor vector follows a uniform distribution, 𝓤(1.1, 2.1), and the remaining three follow an exponential function, a sigmoidal function and a sine wave (See Figure 2 in the paper).
These high-quality gifs were generated using the OpenSource tool
gifski 
Three toolboxes were used during this project, TensorKit, a Python library for tensor decompositions. SynTen, the simulation library developed for this paper and its dependency, TensorKit Tools, an (undocumented) library to run large scale experiments with TensorKit. Additionally, the code used to generate the figures is available here.
Commits: The following commits can be used to replicate the findings in this paper
- TensorKit:
82462e2bdf2add038ec6489cd1c493c0a44215c5 - TensorKit Tools:
10b5581e501982263c1aaa8739befdbc106fe683 - SynTen:
93dfe8be3d4fa0e38bb0008422ee182c1c10bfea
React
Typescript
Django
Laravel
Facebook
Microsoft
Google
Alibaba
D3
Tencent