A stochastic block model for interaction lengths

Abstract

We propose a new stochastic block model that focuses on the analysis of interaction lengths in dynamic networks. The model does not rely on a discretization of the time dimension and may be used to analyze networks that evolve continuously over time. The framework relies on a clustering structure on the nodes, whereby two nodes belonging to the same latent group tend to create interactions and non-interactions of similar lengths. We introduce a variational expectation–maximization algorithm to perform inference, and adapt a widely used clustering criterion to perform model choice. Finally, we validate our methodology using simulated data experiments and showing two illustrative applications concerning face-to-face interaction data and a bike sharing network.

Appendix

1.1 Proof of Proposition 1

The evidence lower bound is defined as follows:

$$\begin{aligned}&\mathcal {F} = \mathbb {E}_q\left[ \log p\left( \varvec{\mathcal {E}}, \mathbf{Z} \vert \varvec{\mu }, \varvec{\nu }, \varvec{\lambda }\right) \right] + Ent(q)\\&\quad = \mathbb {E}_q\left[ \ell _{\varvec{\mathcal {E}}}\left( \varvec{\mu }, \varvec{\nu }, \mathbf{Z} \right) \right] + \mathbb {E}_q\left[ \log p\left( \mathbf{Z} \vert \varvec{\lambda }\right) \right] \\&\qquad - \mathbb {E}_q\left[ \log q\left( \mathbf{Z} \vert \varvec{\tau }\right) \right] \end{aligned}$$

We study the terms on the right hand side separately.

$$\begin{aligned}&\mathbb {E}_q\left[ \ell _{\varvec{\mathcal {E}}}\left( \varvec{\mu }, \varvec{\nu }, \mathbf{Z} \right) \right] \\&\quad = \mathbb {E}_q\left[ \sum _{g=1}^K \sum _{h=1}^K \left\{ L_{\mu _{gh}} \log \left( \mu _{gh} \right) + L_{\nu _{gh}} \log \left( \nu _{gh} \right) - \mu _{gh}\eta _{gh} - \nu _{gh}\zeta _{gh}\right\} \right] \\&\quad = \sum _{g=1}^K \sum _{h=1}^K \left\{ \mathbb {E}_q\left[ L_{\mu _{gh}}\right] \log \left( \mu _{gh} \right) + \mathbb {E}_q\left[ L_{\nu _{gh}}\right] \log \left( \nu _{gh} \right) - \mu _{gh}\mathbb {E}_q\left[ \eta _{gh}\right] - \nu _{gh}\mathbb {E}_q\left[ \zeta _{gh} \right] \right\} \\&\quad = \sum _{g=1}^K \sum _{h=1}^K \left\{ \left( \sum _{i \ne j} \tau _{ig} \tau _{jh} \mathcal {A}_{ij}^{(+)}\right) \log \left( \mu _{gh} \right) + \left( \sum _{i \ne j} \tau _{ig} \tau _{jh} \mathcal {A}_{ij}^{(-)}\right) \log \left( \nu _{gh} \right) \right. \\&\qquad - \left. \left( \sum _{i \ne j} \tau _{ig} \tau _{jh} \mathcal {X}_{ij}^{(+)}\right) \mu _{gh} - \left( \sum _{i \ne j} \tau _{ig} \tau _{jh} \mathcal {X}_{ij}^{(-)}\right) \nu _{gh} \right\} \\&\quad = \sum _{g=1}^K \sum _{h=1}^K \left\{ \bar{L}_{\mu _{gh}} \log \left( \mu _{gh} \right) + \bar{L}_{\nu _{gh}} \log \left( \nu _{gh} \right) - \mu _{gh}\bar{\eta }_{gh} - \nu _{gh}\bar{\zeta }_{gh}\right\} \\&\mathbb {E}_q\left[ \log p\left( \mathbf{Z} \vert \varvec{\lambda }\right) \right] = \mathbb {E}_q\left[ \sum _{k=1}^K\sum _{i=1}^N Z_{ik}\log \lambda _k\right] \\&\quad = \sum _{k=1}^K\sum _{i=1}^N\mathbb {E}_q\left[ Z_{ik}\right] \log \lambda _k = \sum _{k=1}^K\sum _{i=1}^N\tau _{ik}\log \lambda _k \mathbb {E}_q\left[ \log q\left( \mathbf{Z} \vert \varvec{\tau }\right) \right] \\&\quad = \mathbb {E}_q\left[ \sum _{k=1}^K\sum _{i=1}^N Z_{ik}\log \tau _{ik}\right] = \sum _{k=1}^K\sum _{i=1}^N\tau _{ik}\log \tau _{ik} \end{aligned}$$

The three parts combined give (3).

1.2 Proof of Proposition 2

The evidence lower bound can be rewritten as follows:

$$\begin{aligned} \mathcal {F}&= \sum _{g=1}^K \sum _{h=1}^K \left\{ \left( \sum _{i \ne j} \tau _{ig}\tau _{jh} \mathcal {A}_{ij}^{(+)}\right) \log \mu _{gh} + \left( \sum _{i \ne j} \tau _{ig}\tau _{jh} \mathcal {A}_{ij}^{(-)}\right) \log \nu _{gh} \right. \\&\quad \left. - \left( \sum _{i \ne j} \tau _{ig}\tau _{jh} \mathcal {X}_{ij}^{(+)}\right) \mu _{gh} - \left( \sum _{i \ne j} \tau _{ig}\tau _{jh} \mathcal {X}_{ij}^{(-)}\right) \nu _{gh} \right\} \\&\qquad + \sum _{i=1}^N \sum _{k=1}^K \tau _{ik} \log \lambda _k + \sum _{i=1}^N \sum _{k=1}^K \tau _{ik} \log \tau _{ik} \\&= \sum _{g=1}^K \sum _{h=1}^K \sum _{i \ne j} \tau _{ig}\tau _{jh} \omega _{ijgh} + \sum _{i=1}^N \sum _{k=1}^K \tau _{ik} \log \lambda _k + \sum _{i=1}^N \sum _{k=1}^K \tau _{ik} \log \tau _{ik} \end{aligned}$$

Now consider the following Lagrangian:

$$\begin{aligned} \mathcal {H} = \mathcal {F} + \sum _{i=1}^N \xi _{i} \left( \sum _{k=1}^K \tau _{ik}-1 \right) \end{aligned}$$

with multipliers $\xi _1,\ldots ,\xi _N$. The derivative is equal to the following:

$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial \tau _{\ell k}} = \sum _{j=1}^N \sum _{h=1}^K \tau _{jh} \omega _{\ell jkh} + \sum _{i=1}^N \sum _{g=1}^K \tau _{ig} \omega _{i\ell gk} + \log \lambda _k - \log \tau _{\ell k} - 1 + \xi _{\ell } \end{aligned}$$

with root:

$$\begin{aligned} \tau _{\ell k} = \exp \left\{ \sum _{j=1}^N \sum _{h=1}^K \tau _{jh} \omega _{\ell jkh} + \sum _{i=1}^N \sum _{g=1}^K \tau _{ig} \omega _{i\ell gk} + \log \lambda _k - 1 + \xi _{\ell }\right\} \end{aligned}$$

(4)

Regarding the constraints:

$$\begin{aligned} 1 = \sum _{k=1}^K \tau _{\ell k} = \exp \left\{ \xi _{\ell }\right\} \sum _{k=1}^K \exp \left\{ \sum _{j=1}^N \sum _{h=1}^K \tau _{jh} \omega _{\ell jkh} + \sum _{i=1}^N \sum _{g=1}^K \tau _{ig} \omega _{i\ell gk} + \log \lambda _k - 1\right\} \end{aligned}$$

This yields the following:

$$\begin{aligned} \xi _{\ell } = \log \sum _{k=1}^K \exp \left\{ \sum _{j=1}^N \sum _{h=1}^K \tau _{jh} \omega _{\ell jkh} + \sum _{i=1}^N \sum _{g=1}^K \tau _{ig} \omega _{i\ell gk} + \log \lambda _k - 1\right\} \end{aligned}$$

This critical point is a maximum. Using this result in (4) finishes the proof.

1.3 Proof of Proposition 3

Consider the following Lagrangian:

$$\begin{aligned} \mathcal {H} = \sum _{g=1}^K \left( \sum _{i=1}^N \tau _{ig}\right) \log \lambda _g + \xi \left( \sum _{g=1}^K \lambda _g -1\right) \end{aligned}$$

and its derivative:

$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial \lambda _k} = \frac{\sum _{i=1}^N \tau _{ik}}{\lambda _k} + \xi \end{aligned}$$

This gives the root $\lambda _k = - \sum _{i=1}^N \tau _{ik} / \xi$ and in turn:

$$\begin{aligned} \xi = -\sum _{i=1}^N \sum _{g=1}^K \tau _{ig} = -N \end{aligned}$$

which leads to the result of the proposition. This critical point is a maximum.

1.4 Proof of Proposition 4

From (3):

$$\begin{aligned} \frac{\partial \mathcal {F}}{\partial \mu _{gh}} = \frac{ \bar{L}_{\mu _{gh}} }{ \mu _{gh} } - \bar{\eta }_{gh} \end{aligned}$$

has root $\mu _{gh} = \frac{ \bar{L}_{\mu _{gh}} }{ \bar{\eta }_{gh} }$ which corresponds to a maximum. The formula for $\nu _{gh}$ is obtained analogously.

1.5 Proof of Proposition 5

The proof of this Proposition follows closely the proof of Proposition 8 in Daudin et al. (2008). Our model selection criterion is the exact integrated completed log-likelihood, which is defined as:

$$\begin{aligned} \log p(\varvec{\mathcal {E}},\mathbf{Z} |K) = \log p(\varvec{\mathcal {E}}|\mathbf{Z} ,K) + \log p(\mathbf{Z} |K) \end{aligned}$$

(5)

The first term on the right hand side can be calculated using a BIC-like approximation, as follows:

$$\begin{aligned} \log p(\varvec{\mathcal {E}}|\mathbf{Z} ,K)&\approx \max _{\varvec{\mu },\varvec{\nu }} \log p\left( \varvec{\mathcal {E}}\vert \mathbf{Z} , \varvec{\mu }, \varvec{\nu }\right) - \frac{1}{2}\left( 2K^2\right) \log \left( \sum _{i\ne j}W_{ij}\right) \\&\approx \log p\left( \varvec{\mathcal {E}}\vert \hat{\mathbf{Z }}, \hat{\varvec{\mu }}, \hat{\varvec{\nu }} \right) - K^2\log \left( \sum _{i\ne j}W_{ij}\right) \end{aligned}$$

where $2K^2$ is the number of components’ parameters and $\sum _{i\ne j}W_{ij}$ is the number of data points. The second term on the right hand side of (5) can be calculated using the same approximation proposed by Daudin et al. (2008):

$$\begin{aligned} \log p(\mathbf{Z} |K)&\approx \max _{\varvec{\lambda }} \log p\left( \hat{\mathbf{Z }} \vert \lambda \right) - \frac{K-1}{2}\log N \\&\approx \log p\left( \hat{\mathbf{Z }} \vert \hat{\lambda } \right) - \frac{K-1}{2}\log N \end{aligned}$$

Combining the two formulas gives the result in Proposition 5.