Network Structure Change Point Detection by Posterior Predictive Discrepancy

Abstract

Detecting changes in network structure is important for research into systems as diverse as financial trading networks, social networks and brain connectivity. Here we present novel Bayesian methods for detecting network structure change points. We use the stochastic block model to quantify the likelihood of a network structure and develop a score we call posterior predictive discrepancy based on sliding windows to evaluate the model fitness to the data. The parameter space for this model includes unknown latent label vectors assigning network nodes to interacting communities. Monte Carlo techniques based on Gibbs sampling are used to efficiently sample the posterior distributions over this parameter space.

Appendix: Derivation of the Collapsing Procedure

We now create a new parameter vector $\varvec{\eta }=\lbrace \alpha _{1}+m_{1},\ldots ,\alpha _{K}+m_{K}\rbrace$. We can collapse the integral $\int p(\mathbf{z} ,\mathbf{r} \vert K)d\mathbf{r}$ as the following procedure.

$$\begin{aligned} \int p(\mathbf{z} ,\mathbf{r} \vert K)d\mathbf{r}= & {} \int p(\mathbf{r} \vert K)p(\mathbf{z} \vert \mathbf{r} , K)d\mathbf{r} \\= & {} \int N(\varvec{\alpha })\prod _{k=1}^{K}r_{k}^{\alpha _{k}-1} \prod _{k=1}^{K}r_{k}^{m_k}d\mathbf{r} \\= & {} \int N(\varvec{\alpha })\prod _{k=1}^{K}r_{k}^{\alpha _{k}+m_{k}-1} d\mathbf{r} \\= & {} \frac{N(\varvec{\alpha })}{N(\varvec{\eta })}\int N(\varvec{\eta })\prod _{k=1}^{K}r_{k}^{\alpha _{k}+m_{k}-1} d\mathbf{r} \\= & {} \frac{\varGamma (\sum _{k=1}^{K}\alpha _{k})}{\varGamma (\sum _{k=1}^{K}(\alpha _{k}+m_{k})}\prod _{k=1}^{K}\frac{\varGamma (\alpha _{k}+m_{k})}{\varGamma (\alpha _{k})}. \end{aligned}$$

Integral of the form $\int p(\mathbf{x} _{kl},\pi _{kl}\vert \mathbf{z} )d\pi _{kl}$ can be calculated as

$$\begin{aligned} \int _{0}^{1}p(\mathbf{x} _{kl},\pi _{kl}\vert \mathbf{z} )d\pi _{kl}= & {} \int _{0}^{1}p(\pi _{kl})p(\mathbf{x} _{kl}\vert \pi _{kl}, \mathbf{z} )d\pi _{kl} \\= & {} \int _{0}^{1}\frac{\pi _{kl}^{a-1}(1-\pi _{kl})^{b-1}}{B(a,b)}\pi _{kl}^{n_{kl}}(1-\pi _{kl})^{w_{kl}-n_{kl}}d\pi _{kl}\\= & {} \int _{0}^{1}\frac{\pi _{kl}^{n_{kl}+a-1}(1-\pi _{kl})^{w_{kl}-n_{kl}+b-1}}{B(a,b)}d\pi _{kl}\\= & {} \frac{B(n_{kl}+a,w_{kl}-n_{kl}+b)}{B(a,b)}\\&\times \int _{0}^{1}\frac{\pi _{kl}^{n_{kl}+a-1}(1-\pi _{kl})^{w_{kl}-n_{kl}+b-1}}{B(n_{kl}+a,w_{kl}-n_{kl}+b)}d\pi _{kl}\\= & {} \frac{B(n_{kl}+a,w_{kl}-n_{kl}+b)}{B(a,b)}. \end{aligned}$$