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DPGM: Dynamic Poisson Gamma Membership Model

Introduction

DGPM is a probabilistic model, it can describe latent node-group memberships after observing the interactions between nodes. I implemented the gamma process edge partition model for static networks, and hierarchical gamma process edge partition models for large sparse dynamic networks

Requirements

pytorch>=1.8

Code

2.1 Data

The synthetic data is a (6, 60, 60) tensor, it records the interactions between 60 nodes at 6 time snapshots.

2.2 Model

The model is expressed as:

$$\begin{array}{l} \phi^{(t)}_{nk} \sim {\rm Gam}(\phi^{(t-1)}_{nk}/\tau,1/\tau), \quad\textup{for \ t=1,...,T} \\\ \phi^{(0)}_{nk} \sim {\rm Gam}(g_0,1/h_0) \\\ r_k \sim {\rm Gam}(\gamma_0/K, 1/c_0) \\\ \lambda_{kk'} \sim \left\{ \begin{array}{ll} {\rm Gam}(\xi r_k, 1/\beta), & \textup{if } k=k' \\ {\rm Gam}(r_k r_{k'}, 1/\beta), & \textup{otherwise} \end{array}\right. \\\ x^{(t)}_{mn} \sim {\rm Po}(\displaystyle\sum_{k=1}^{K}\sum_{k'=1}^{K}\lambda_{kk'}\phi^{(t)}_{nk}\phi^{(t)}_{mk'}) \\\ b^{(t)}_{mn} = \mathbb{1}(x^{(t)}_{mn}\geqslant1) \end{array}$$

$\beta$, $c_0$, $g_0$, $h_0$ and $\tau$ are hyperparameters, others are model parameters

$\phi^{(t)}_{nk}$ : a positive value, measuring how strongly node $n$ is affiliated with community $k$ at time $t$.

$\phi^{(0)}_{nk}$ is the original state

$r_k$ : a positive value, indicates the popularity of community $k$

$\lambda_{kk'}$ : a positive value, measuring how strongly communities $k$ and $k'$ interact with each other

$b^{(t)}_{mn} = \mathbb{1}$ : there is interaction between node $m$ and node $n$ at time $t$

2.3 Inference

We present a gibbs sampling procedure to update hyperparameters and model parameters

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