The dynamic random subgraph model for the clustering of evolving networks

Abstract

In recent years, many clustering methods have been proposed to extract information from networks. The principle is to look for groups of vertices with homogenous connection profiles. Most of these techniques are suitable for static networks, that is to say, not taking into account the temporal dimension. This work is motivated by the need of analyzing evolving networks where a decomposition of the networks into subgraphs is given. Therefore, in this paper, we consider the random subgraph model (RSM) which was proposed recently to model networks through latent clusters built within known partitions. Using a state space model to characterize the cluster proportions, RSM is then extended in order to deal with dynamic networks. We call the latter the dynamic random subgraph model (dRSM). A variational expectation maximization (VEM) algorithm is proposed to perform inference. We show that the variational approximations lead to an update step which involves a new state space model from which the parameters along with the hidden states can be estimated using the standard Kalman filter and Rauch–Tung–Striebel smoother. Simulated data sets are considered to assess the proposed methodology. Finally, dRSM along with the corresponding VEM algorithm are applied to an original maritime network built from printed Lloyd’s voyage records.

Appendices

Appendix 1: Construction of a tractable lower bound

We rely on a bound introduced in Jordan et al. (1999). Such a general bound can easily been derived by noticing that $C(\cdot )$ is a concave function of $\sum _{l=1}^{K}\exp (\gamma _{s_{i}l}^{(t)})$ and therefore a first order Taylor expansion of the normalizing constant, at any $\xi _{s}^{(t)}\in {\mathbb {R}}^{*+}$, will lead to the inequality:

$$\begin{aligned} \log \left( \sum _{l=1}^{K}\exp \left( \gamma _{sl}^{\left( t\right) }\right) \right) \le \xi _{s}^{-1\left( t\right) }\left( \sum _{l=1}^{K}\exp \left( \gamma _{sl}^{\left( t\right) }\right) \right) -1+\log \left( \xi _{s}^{\left( t\right) }\right) . \end{aligned}$$

(8)

The bounds (8) on the $C(\gamma _{s}^{(t)})$ terms induce a lower bound on the quantity $\log p(Z|\gamma )$:

$$\begin{aligned} \log p(Z|\gamma )= & {} \sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{i=1}^{N}Z_{ik}^{(t)}\log \Big (f_k\Big (\gamma _{s_{i}}^{(t)}\Big )\Big )\\= & {} \sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{i=1}^{N}\sum _{s=1}^{S} y_{is}Z_{ik}^{(t)}\Big (\gamma _{sk}^{(t)}-\log \Big (\sum _{l=1}^{K}\exp \Big (\gamma _{sl}^{(t)}\Big )\Big )\Big )\\\ge & {} \log h(Z,\gamma ,\xi ), \end{aligned}$$

where $\xi$ denotes the set of all variational parameters $(\xi _s^{(t)})_{st}$ and the function $h(\cdot ,\cdot ,\cdot )$ is such that:

$$\begin{aligned}&\log h(Z,\gamma ,\xi )\\&\quad = \sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{i=1}^{N} \sum _{s=1}^{S} y_{is}Z_{ik}^{(t)}\Big (\gamma _{sk}^{(t)}-\big (\xi _{s}^{-1(t)}\sum _{l=1}^{K}\exp \big (\gamma _{sl}^{(t)}\big )-1+\log \big (\xi _{s}^{(t)}\big )\big )\Big ) . \end{aligned}$$

Replacing $\log p(Z|\gamma )$ by $\log h(Z,\gamma ,\xi )$ in ${\mathcal {L}} (q, \theta )$, leads to a new lower bound $\tilde{{\mathcal {L}}}(q,\theta ,\xi )$ for $\log p(X|\theta )$ which satisfies:

$$\begin{aligned} \log p(X|\theta )\geqslant {\mathcal {L}}(q,\theta )\geqslant \tilde{{\mathcal {L}}}(q,\theta ,\xi ), \end{aligned}$$

where

$$\begin{aligned}&\tilde{{\mathcal {L}}}(q,\theta ,\xi )\\&\quad =\sum _{Z}\int _{\gamma }\int _{\nu }q(Z,\gamma ,\nu )\log \dfrac{p(X|Z,\Pi )h(Z,\gamma ,\xi )p(\gamma |B,\nu ,\Sigma )p(\nu |\mu _{0},A,\Phi ,V_{0})}{q(Z,\gamma ,\nu )} d\gamma \, d\nu . \end{aligned}$$

Appendix 2: E-step of the VEM algorithm

1.1 Distribution q(Z)

The VEM update step for each of the distributions $q(Z_{i})$ in q(Z) is given by:

$$\begin{aligned} \log q(Z_{i})= & {} E_{\gamma ,\nu ,Z^{\backslash i}}[\log p(X|Z,\Pi )+\log h(Z,\gamma ,\xi )]+\mathrm {const}\\= & {} \sum _{t=1}^{T}\sum _{k=1}^{K}Z_{ik}^{(t)}\left( \sum _{l=1}^{K}\sum _{c=0}^{C}\sum _{i\ne j}^{N}\delta (X_{ij}^{(t)}=c)\tau _{jl}^{(t)}\Big [\log (\Pi _{kl}^{c})+\log (\Pi _{lk}^{c})\Big ]\right) \\&+\,\sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{s=1}^{S}y_{is}E_{\gamma }\Big [Z_{ik}^{(t)}\log h(Z^{(t)},\gamma ^{(t)},\xi ^{(t)})\Big ]+\mathrm {const}.\\= & {} \sum _{t=1}^{T}\sum _{k=1}^{K}Z_{ik}^{(t)}\left( \sum _{l=1}^{K}\sum _{c=0}^{C}\sum _{i\ne j}^{N}\delta (X_{ij}^{(t)}=c)\tau _{jl}^{(t)}\Big [\log (\Pi _{kl}^{c})+\log (\Pi _{lk}^{c})\Big ]\right) \\&+\,\sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{s=1}^{S}y_{is}E_{\gamma }\Big [\gamma _{sk}^{(t)}-\Big (\xi _{s}^{-1(t)}\sum _{l=1}^{K}\exp (\gamma _{sk}^{(t)})-1+\log (\xi _{s}^{(t)})\Big )\Big ]\\&+\,\mathrm {const}.\\= & {} \sum _{t=1}^{T}\sum _{k=1}^{K}Z_{ik}^{(t)}\left( \sum _{l=1}^{K}\sum _{c=0}^{C}\sum _{i\ne j}^{N}\delta (X_{ij}^{(t)}=c)\tau _{jl}^{(t)}\Big [\log (\Pi _{kl}^{c})+\log (\Pi _{lk}^{c})\Big ]\right) \\&+\,\sum _{t=1}^{T}\sum _{k=1}^{K} \sum _{s=1}^{S} Z_{ik}^{(t)}y_{is}\left( \hat{\gamma }_{sk}^{(t)}\!-\!\Big [\xi _{s}^{-1(t)}\sum _{l=1}^{K}E(\exp (\gamma _{sl}^{(t)}))\!-\!1\!+\!\log (\xi _{s}^{(t)})\Big ]\right) \\&+\,\mathrm {const}.\\= & {} \sum _{t=1}^{T}\sum _{k=1}^{K}Z_{ik}^{(t)}\Bigg (\sum _{l=1}^{K}\sum _{c=0}^{C}\sum _{i\ne j}^{N}\delta (X_{ij}^{(t)}=c)\tau _{jl}^{(t)}\Big [\log (\Pi _{kl}^{c})+\log (\Pi _{lk}^{c})\Big ]\\&+\, \sum _{s=1}^{S} y_{is} \Big ( \hat{\gamma }_{sk}^{(t)}-\Big (\xi _{s}^{-1(t)}\sum _{l=1}^{K}\exp \Big (\hat{\gamma }_{sl}^{(t)}+\dfrac{\hat{\sigma }_{sl}^{2^{(t)}}}{2}\Big )-1+\log (\xi _{s}^{(t)})\Big ) \Big )\Bigg )\\&+\,\mathrm {const}, \end{aligned}$$

where all terms that do not depend on $Z_{i}$ have been put into the constant terms $\mathrm {const}$. Moreover since $\gamma _{sk}^{(t)}\sim {\mathcal {N}}(\hat{\gamma }_{sk}^{(t)},\hat{\sigma }_{sk}^{2^{(t)}})$ we have used:

$$\begin{aligned} {\mathbb {E}}\left[ \exp \left( \gamma _{sk}^{\left( t\right) }\right) \right] =\exp \left( \hat{\gamma }_{sk}^{\left( t\right) }+\dfrac{\hat{\sigma }_{sk}^{2^{\left( t\right) }}}{2}\right) . \end{aligned}$$

We then recognize the functional form of a multinomial distribution:

$$\begin{aligned} q\left( Z_{i}^{\left( t\right) }\right) \sim {\mathcal {M}}\left( Z_{i}^{\left( t\right) };1,\tau _{i}^{\left( t\right) }\right) ,\,\,\forall i,t. \end{aligned}$$

1.2 Distribution $q(\nu )$

The VEM update step for the distribution $q(\nu )$ is given by:

$$\begin{aligned} \log q(\nu )= & {} E_{Z,\gamma }\Big (\log p(\gamma |\nu ,\Sigma ,B)+\log p(\nu |\mu _{0},V_{0},A,\Phi )\Big )+\mathrm {const}\\= & {} \sum _{t=1}^{T}\sum _{s=1}^{S}\Big (E_{\gamma }\big (\log {\mathcal {N}}(\gamma _{s}^{(t)};B\nu ^{(t)},\Sigma )\big )\Big )+\log p(\nu ^{(1)}|\mu _{0},V_{0})\\&+\, \sum _{t=2}^{T}\log p(\nu ^{(t)}|\nu ^{(t-1)},A,\Phi )+\mathrm {const}\\= & {} \sum _{t=1}^{T}\sum _{s=1}^{S}\Big (E_{\gamma }\big (-\dfrac{1}{2}(\gamma _{s}^{(t)})^{\intercal }\Sigma ^{-1}(\gamma _{s}^{(t)})+(\gamma _{s}^{(t)})^{\intercal }\Sigma ^{-1}B\nu ^{(t)}\\&-\,\dfrac{1}{2}(\nu ^{(t)})^{\intercal }B^{\intercal }\Sigma ^{-1}B\nu ^{(t)}\big )\Big )\\&+\, \log p(\nu ^{(1)}|\mu _{0},V_{0})+\sum _{t=2}^{T}\log p(\nu ^{(t)}|\nu ^{(t-1)},A,\Phi )+\mathrm {const}.\\= & {} \sum _{t=1}^{T}\Big ( \sum _{s=1}^{S}\Big (\hat{\gamma }_{s}^{(t)}\Sigma ^{-1}B\nu ^{(t)}\Big ) -\dfrac{1}{2}(\nu ^{(t)})^{\intercal }B^{\intercal }(S\Sigma ^{-1})B\nu ^{(t)} \Big ) \\&+\, \log p(\nu ^{(1)}|\mu _{0},V_{0})+\sum _{t=2}^{T}\log p(\nu ^{(t)}|\nu ^{(t-1)},A,\Phi )+\mathrm {const}, \end{aligned}$$

where all terms that do not depend on $\nu$ have been put into the constant terms $\mathrm {const}$. We recognize the functional form of the posterior distribution of a linear dynamic system:

$$\begin{aligned} \log q(\nu )= & {} \sum _{t=1}^{T}\Big (\log {\mathcal {N}}\Big (\dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{(t)}}{S};B\nu ^{(t)},\dfrac{\Sigma }{S}\Big )\Big )\\&+\, \log p(\nu ^{(1)}|\mu _{0},V_{0})+\sum _{t=2}^{T}\log p(\nu ^{(t)}|\nu ^{(t-1)},A,\Phi )+\mathrm {const}. \end{aligned}$$

Appendix 3: Derivation of the lower bound

In the following, we denote $x^{(t)}=\dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{(t)}}{S}$ an observed variable. The lower bound is given by:

$$\begin{aligned}&\tilde{{\mathcal {L}}}\left( q,\theta ,\xi \right) \\&\quad = \sum _{Z}\int _{\gamma }\int _{\nu }q\left( Z,\gamma ,\nu \right) \log \dfrac{p\left( X|Z,\Pi \right) h\left( Z,\gamma ,\xi \right) p\left( \gamma |\nu ,\Sigma \right) p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) }{q\left( Z,\gamma ,\nu \right) } d\nu d\gamma \\&\quad = E_{Z,\gamma ,\nu }\left[ \log \dfrac{p\left( X|Z,\Pi \right) h\left( Z,\gamma ,\xi \right) p\left( \gamma |\nu ,\Sigma ,B\right) p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) }{q\left( \gamma \right) q\left( \nu \right) \prod _{i=1}^{N}q\left( Z_{i}\right) }\right] \\&\quad = E_{Z}\left( \log p\left( X|Z,\Pi \right) \right) +E_{Z,\gamma }\left( \log h\left( Z,\gamma ,\xi \right) \right) +E_{\gamma ,\nu }\left( \log p\left( \gamma |\nu ,\Sigma ,B\right) \right) \\&\qquad + E_{\nu }\left( \log p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) \right) -E_{\gamma }\left( \log q\left( \gamma \right) \right) -E_{\nu }\left( \log q\left( \nu \right) \right) \\&\qquad -E_{Z}\left( \log \left( \prod _{i=1}^{N}q\left( Z_{i}\right) \right) \right) . \end{aligned}$$

Note that (see Proposition 3.3),

$$\begin{aligned} q(\nu )\propto p(\nu ^{(1)}|\mu _{0},V_{0})\Big [\prod _{t=2}^{T}p(\nu ^{(t)}|\nu ^{(t-1)},A,\Phi )\Big ]\Big [\prod _{t=1}^{T}{\mathcal {N}}\Big (\dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{(t)}}{S};B\nu ^{(t)},\dfrac{\Sigma }{S}\Big )\Big ]. \end{aligned}$$

As pointed out in this proposition, this corresponds to the form of the posterior distribution associated with a state space model with parameter $\theta ^{'}$ and with observed outputs $x=(x^{(t)})_{t}$. If we denote $p(x|\theta ^{'})$ the likelihood associated with this model, and the joint likelihood $p(x,\nu |\theta ^{'})$, we have

$$\begin{aligned} q(\nu ) = \frac{p(x,\nu |\theta ^{'})}{ p(x|\theta ^{'})}. \end{aligned}$$

Therefore

$$\begin{aligned}&E_{\nu }\left( \log q\left( \nu \right) \right) =E_{\nu }\left( \log p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) \right) \\&\quad +E_{\nu }\left( \log p\left( \dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{\left( t\right) }}{S}|\nu ^{\left( t\right) },\dfrac{\Sigma }{S},B\right) \right) - \log p\left( x|\theta ^{'}\right) . \end{aligned}$$

This leads to,

$$\begin{aligned}&E_{\nu }\left( \log p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) \right) -E_{\nu }\left( \log q\left( \nu \right) \right) \\&\quad =-E_{\nu }\left( \log p\left( \dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{\left( t\right) }}{S}|\nu ^{\left( t\right) },\dfrac{\Sigma }{S},B\right) \right) +\log p\left( x|\theta ^{'}\right) , \end{aligned}$$

and $\tilde{{\mathcal {L}}}(q,\theta ,\xi )$ can be written as follows:

$$\begin{aligned} \tilde{{\mathcal {L}}}\left( q,\theta ,\xi \right)= & {} \sum _{Z}\int _{\gamma }\int _{\nu }q\left( Z,\gamma ,\nu \right) \log \dfrac{p\left( X|Z,\Pi \right) h\left( Z,\gamma ,\xi \right) p\left( \gamma |\nu ,\Sigma \right) p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) }{q\left( Z,\gamma ,\nu \right) }\\= & {} E_{Z,\gamma ,\nu }\left[ \log \dfrac{p\left( X|Z,\Pi \right) h\left( Z,\gamma ,\xi \right) p\left( \gamma |\nu ,\Sigma ,B\right) p\left( \nu |\mu _{0},A,\Phi ,V_{0}\right) }{q\left( \gamma \right) q\left( \nu \right) \prod _{i=1}^{N}q\left( Z_{i}\right) }\right] \\= & {} E_{Z}\left( \log p\left( X|Z,\Pi \right) \right) +E_{Z,\gamma }\left( \log h\left( Z,\gamma ,\xi \right) \right) +E_{\gamma ,\nu }\left( \log p\left( \gamma |\nu ,\Sigma ,B\right) \right) \\&-\, E_{\gamma }\left( \log q\left( \gamma \right) \right) -E_{\nu }\left( \log p\left( \dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{\left( t\right) }}{S}|\nu ^{\left( t\right) },\dfrac{\Sigma }{S},B\right) \right) -E_{Z}\left( \log \left( \prod _{i=1}^{N}q\left( Z_{i}\right) \right) \right) \\&+\, \log p\left( x|\theta ^{'}\right) . \end{aligned}$$

We explicit below each of the terms of the bound $\tilde{{\mathcal {L}}}(q,\theta )$.

• 1. $E_{Z}(\log p(X|Z,\Pi ))$:

  $$\begin{aligned} E_{Z}(\log p(X|Z,\Pi ))= & {} \sum _{t=1}^{T}\sum _{k,l}^{K}\sum _{c=0}^{C}\sum _{i\ne j}^{N}E_{z}(\delta (X_{ij}^{(t)}=c)Z_{ik}^{(t)}Z_{jl}^{(t)}\log (\Pi _{kl}^{c})\\= & {} \sum _{t=1}^{T}\sum _{k,l}^{K}\sum _{c=0}^{C}\sum _{i\ne j}^{N}\delta (X_{ij}^{(t)}=c)\tau _{ik}^{(t)}\tau _{jl}^{(t)}\log (\Pi _{kl}^{c}) \end{aligned}$$

• 2. $E_{Z,\gamma }(\log h(Z,\gamma ,\xi ))$:

  $$\begin{aligned}&E_{Z,\gamma }\Big (\log h(Z,\gamma ,\xi )\Big ) \\&\quad = E_{Z,\gamma }\Big [\sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{i=1}^{N}\sum _{s=1}^{S} y_{is} Z_{ik}^{(t)}\Big (\gamma _{sk}^{(t)}-\big (\xi _{s}^{-1(t)}\sum _{l}\exp (\gamma _{sk}^{(t)})\\&\qquad - 1+\log (\xi _{s}^{(t)})\big )\Big )\Big ]\\&\quad = \sum _{t=1}^{T}\sum _{k=1}^{K}\sum _{i=1}^{N}\sum _{s=1}^{S} y_{is} \Big (\tau _{ik}^{(t)}\hat{\gamma }_{sk}^{(t)}-\tau _{ik}^{(t)}\xi _{s}^{-1(t)}\sum _{l=1}^{K}\exp \Big (\hat{\gamma }_{sl}^{(t)}+\dfrac{\hat{\sigma }_{sl}^{2^{(t)}}}{2}\Big )\\&\qquad + \tau _{ik}^{(t)}-\tau _{ik}^{(t)}\log (\xi _{s}^{(t)})\Big )\\&\quad = \sum _{t=1}^{T}\sum _{s=1}^{S}\Big (r_{s}^{(t)}\hat{\gamma }_{sk}^{(t)}-N_{s}\xi _{s}^{-1(t)}\sum _{l=1}^{K}\exp \Big (\hat{\gamma }_{sl}^{(t)}+\dfrac{\hat{\sigma }_{sl}^{2^{(t)}}}{2}\Big )+N_{s}-N_{s}\log (\xi _{s}^{(t)})\Big ) \end{aligned}$$

  where denote $r_{s}^{(t)}$ is a quantity $\sum _{i=1}^{N}\tau _{ik}^{(t)}y_{is}$.

• 3. $E_{\gamma ,\nu }(\log p(\gamma |\nu ,\Sigma ,B))$:

  $$\begin{aligned} E_{\gamma ,\nu }(\log p(\gamma |\nu ,\Sigma ,B))= & {} E_{\gamma ,\nu }\Big (\log \prod _{t=1}^{T}\prod _{s=1}^{S}{\mathcal {N}}(\gamma _{s}^{(t)};B\nu _{s}^{(t)},\Sigma )\Big )\\= & {} \sum _{t=1}^{T}\sum _{s=1}^{S}\Big (\log {\mathcal {N}}(\hat{\gamma }_{s}^{(t)},B\hat{\nu }_{s}^{(t)},\Sigma )-\dfrac{1}{2}tr(\Sigma ^{-1}B^{T}\hat{V}^{(t)}B)\\&-\,\dfrac{1}{2}tr\Big (\Sigma ^{-1}\hat{\sigma }_{s}^{(t)^{2}}\Big )\Big ) \end{aligned}$$

• 4. $E_{\gamma }(\log q(\gamma ))$:

  $$\begin{aligned} E_{\gamma }(\log q(\gamma ))= & {} E_{\gamma }\Big (\prod _{t=1}^{T}\prod _{s=1}^{S}\prod _{k=1}^{K}{\mathcal {N}}(\gamma _{sk}^{(t)};\hat{\gamma }_{sk}^{(t)},\hat{\sigma }_{sk}^{2^{(t)}})\Big )\\= & {} \sum _{t=1}^{T}\sum _{s=1}^{S}\sum _{k=1}^{K} -\log \Big ( (2\pi )^{\frac{1}{2}}\hat{\sigma }_{sk}^{(t)}\Big )-\dfrac{TKS}{2}. \end{aligned}$$

• 5. $E_{\nu }\Big (\log p\Big (\dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{(t)}}{S}|\nu ^{(t)},\dfrac{\Sigma }{S},B\Big )\Big )$ :

  $$\begin{aligned}&E_{\nu }\Big (\log p\Big (\dfrac{\sum _{s=1}^{S}\hat{\gamma }_{s}^{(t)}}{S}|\nu ^{(t)},\dfrac{\Sigma }{S},B\Big )\Big )\\&\quad = \sum _{t=1}^{T}\Big (\log {\mathcal {N}}(x^{(t)};B\hat{\nu }^{(t)},\Sigma /S)-\dfrac{1}{2}tr(\Sigma ^{-1}SB^{T}\hat{V}^{(t)}B)\Big ). \end{aligned}$$

• 6. $E_{Z}(\log (\prod _{i=1}^{T}q(Z_{i})))$:

$$\begin{aligned} E_{Z}\left( \log \left( \prod _{i=1}^{T}q\left( Z_{i}\right) \right) \right)= & {} \sum _{i=1}^{N}E_{Z}\left( \log q\left( Z_{i}\right) \right) \\= & {} \sum _{i=1}^{N}E_{Z}\left( \sum _{t=1}^{T}\sum _{k=1}^{K}Z_{ik}^{\left( t\right) }\log \left( \tau _{ik}\right) \right) \\= & {} \sum _{i=1}^{N}\sum _{t=1}^{T}\sum _{k=1}^{K}\tau _{ik}^{\left( t\right) }\log \left( \tau _{ik}^{\left( t\right) }\right) . \end{aligned}$$