A Poisson Gamma Probabilistic Model for Latent Node-group Memberships in Dynamic Networks

A Poisson Gamma Probabilistic Model for Latent Node-group Memberships in Dynamic Networks

Sikun Yang, Heinz Koeppl
Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt
64283 Darmstadt, Germany
{sikun.yang, heinz.koeppl}@bcs.tu-darmstadt.de

We present a probabilistic model for learning from dynamic relational data, wherein the observed interactions among networked nodes are modeled via the Bernoulli Poisson link function, and the underlying network structure are characterized by nonnegative latent node-group memberships, which are assumed to be gamma distributed. The latent memberships evolve according to Markov processes. The optimal number of latent groups can be determined by data itself. The computational complexity of our method scales with the number of non-zero links, which makes it scalable to large sparse dynamic relational data. We present batch and online Gibbs sampling algorithms to perform model inference. Finally, we demonstrate the model’s performance on both synthetic and real-world datasets compared to state-of-the-art methods.



A Poisson Gamma Probabilistic Model for Latent Node-group Memberships in Dynamic Networks

Sikun Yang, Heinz Koeppl Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt 64283 Darmstadt, Germany {sikun.yang, heinz.koeppl}@bcs.tu-darmstadt.de

Copyright © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.


Considerable work has been done on the analysis of static networks in terms of community detection or link prediction (???). However, due to the temporal evolution of nodes (e.g. individuals), their role within a network can change and hence observed links among nodes may appear or disappear over time (??). Given such dynamic network data, one may be interested in understanding the temporal evolution of groups in terms of their size and node-group memberships and in predicting missing or future unobserved links based on historical records.

A dynamic network of nodes can be represented as a sequence of adjacency matrices , , where indicates the presence of a link between node and at time point and otherwise. For the sake of clarity we focus on undirected and unweighted networks but the presented method can be extended to weighted networks via the compound Poisson distribution (?) and to multi-relational networks (??).

Many of the current probabilistic methods for dynamic networks map observed binary edge-variables (either links or non-links) to latent Gaussian random variables via the logistic or probit function (?). The time-complexity of these approaches often scales quadratically with the number of nodes, i.e., . This will become infeasible for large networks and will become especially inefficient for sparse networks. In this work we leverage the Bernoulli-Poisson link function (???) to map an observed binary edge to a latent Poisson count random variable, which leads to a computational cost that only scales with the number of non-zero edges. As large, real-world networks are usually very sparse, the proposed method yields significant speed-up and enables the analysis of larger dynamic networks. We allow for a time-varying nonnegative degree of membership of a node to a group. We realize this by constructing a gamma Markov chain to capture the time evolution of those latent membership variables. Inspired by recent advances in data augmentation and marginalization techniques (?), we develop an easy-to-implement efficient Gibbs sampling algorithm for model inference. We also present an online Gibbs sampling algorithm that can process data in mini-batches and thus readily scales to massive sparse dynamic networks. The algorithms performs favorably on standard datasets when compared to state-of-the-art methods (???).

Dynamic Poisson Gamma Membership Model

In the proposed model, each node is characterized by a time-dependent latent membership variable that determines its interactions or involvement in group at the snapshot of the dynamic networks. This latent node-group membership is modeled by a gamma random variable and is, thus, naturally nonnegative real-valued. This is contrast to multi-group memberships models (or latent feature models) (???) where each node-group membership is represented by a binary latent feature vector. These models assume that each node either associates to one group or not – simply by a binary feature. The proposed model on the other hand can characterize how strongly each node associates with multiple groups.

Dynamics of latent node-group memberships. For dynamic networks, the latent node-group membership can evolve over time to interpret the interaction dynamics among the nodes. For example, latent group could mean “play soccer” and could mean how frequently person plays soccer or how strongly person likes playing soccer. The person’s degree of association to this group could be increasing over time due to, for instance, increased interaction with professional soccer players, or decreasing over time as a consequence of sickness. Hence, in order to model the temporal evolution of the latent node-group memberships, we assume the individual memberships to form a gamma Markov chain. More specifically, is drawn from a gamma distribution, whose shape parameter is the latent membership at the previous time

where the parameter controls the variance without affecting the mean, i.e., .

Model of latent groups. We characterize the interactions or correlations among latent groups by a matrix of size , where relates to the probability of there being a link between node affiliated to group and node affiliated to group . Specifically, we assume the latent groups to be generated by the following hierarchical process: we first generate a separate weight for each group as


and then generate the inter-group interaction weight and intra-group weight as


where and . The reasonable number of latent groups can be inferred from dynamic relational data itself by the shrinkage mechanism of our model. More specifically, for fixed , the redundant groups will effectively be shrunk as many of the groups weights tend to be small for increasing . Thus, the interaction weights between the redundant group and all the other groups , and all the node-memberships to group will be shrunk accordingly. In practice, the intra-group weight would tend to almost zero if for small , and the corresponding groups will disappear inevitably. Hence, we use a separate variable to avoid overly shrinking of small groups with less interactions with other groups. As has a large effect on the number of the latent groups, we do not treat it as a fixed parameter but place a gamma prior over it, i.e., . Given the latent node-group membership and the interaction weights among groups, the probability of there being a link between node and is given by


Interestingly, we can also generate by truncating a latent count random variable at , where can be seen as the integer-valued weight for node and and can be interpreted as the number of times the two nodes interacted. More specifically, can be drawn as


where indicates the Poisson distribution. We can obtain Eq. (3) by marginalizing out the latent count from the above expression. The conditional distribution of the latent count can then be written as

where is the zero-truncated Poisson distribution with probabilisty mass function (PMF) and and denotes the set of all node-group membership variables. The usefulness of this construction for will become clear in the inference section. We note that the latent count only needs to be sampled for , using rejection sampling detailed in (?). To this end the proposed hierarchical generative model is as follows.

Figure 1: The graphical model of the Poisson gamma latent membership model; auxillary variables introduced for inference are not shown.

For our model’s hyperparameters, we draw and from . The graphical model is shown in Fig. 1.

Related Work

Approaches to analyze dynamic networks range from non-Bayesian methods such as the exponential random graph models (?) or matrix and tensor factorization based methods (?) to Bayesian latent variable models (?????). Our work falls into the latter class and we hence confine ourselves to discuss it’s relation to this class. Dynamic extensions of mixed membership models, where each node is assigned to a set of latent groups represented by multinomial distribution, have been developed (??). One limitation of mixed membership models is that if the probability that node associates to group is increased, the probability that node associates to group has to be decreased. The multi-group memberships models use a binary latent feature vector to characterize each node’s multi-group memberships. In multi-group memberships models, a node’s membership to one group does not limit its memberships to other groups. However, differences in the degree associations of a node to different groups cannot be captured by such models (???). One possible extension is to introduce a Gaussian distributed random variables to characterize how strongly each node is associated to different groups as previously done for latent factor models (?). Such approaches where membership variables evolve according to linear dynamical systems (?) can exploit the rich and efficient toolset for inference, such as Kalman filtering. However, the resulting signed-valued latent features lack an intuitive interpretation, e.g., in terms of degree of membership to a group. In contrast to these approaches, our model is based on a bilinear Poisson factor model (?), where each node’s memberships to groups are represented by a nonnegative real-valued memberships variable. The model does not only allow each node to be associated with multiple groups but also captures the degree at which each node is associated to a group. It means that our model combines the advantages of both mixed membership and multi-group memberships models. We exploit recent data augmentation technique (?), to construct a sampling scheme for the time evolution of the nonnegative latent features. Related to our work is the dynamic gamma process Poisson factorization model (D-GPPF) (?) where the underlying groups’ structure can evolve over time but each node-group membership is static. This is in constrast to our approach where the node’s memberships evolve over time. We note that the gamma Markov chain used by our method and by D-GPPF is motivated by the augmentation techniques in  (?). In the experiment section we compare our model to (1) the hierarchical gamma process edge partition model (HGP-EPM) (?), which is the static counterpart of our model, (2) the dynamic relational infinite feature model (DRIFT) (?) which uses binary latent features to represent the node-group memberships, and characterizes the temporal dependences of latent features via a hidden Markov process and (3) the D-GPPF model.


We present a Gibbs sampling procedure to draw samples of from their posterior distribution given the observed dynamic relational data and the hyper-parameters . In order to circumvent the technical challenges of drawing samples from the gamma Markov chain which does not yield closed-form posterior, we make use of the idea of data augmentation and marginalization technique and of the gamma-Poisson conjugacy to derive a closed-form update.

Notation. When expressing the full conditionals for Gibbs sampling we will use the shorthand “–” to denote all other variables or equivalently those in the Markov blanket for the respective variable according to Fig. 1. We use “” as a index summation shorthand, e.g., .

We repeatedly exploit the following three results (???) to derive the conditional distributions used in our sampling algorithm.

Result 1. A negative binomially (NB) distributed random variable can be generated from a gamma mixed Poisson distribution as, i.e., and , as seen by marginalizing over .

Result 2. The Poisson-logarithmic bivariate distributed variable  (?) with and a Chinese restaurant table (CRT) (?) distributed variables , can equivalently be expressed as a sum-logarithmic (SumLog) and Poisson variable, i.e., with and .

Result 3. Let , where are independently drawn from a Poisson distribution with rate , then according to the Poisson-multinomial equivalence (?), we have and .

Gibbs sampling

Sampling latent counts . We sample a latent count for each time dependent observed edge as


Sampling individual counts . We can partition the latent count
using the Poisson additive property as
, where . Then, via the Poisson-multinomial equivalence, we sample the latent count as


Sampling group weights . Via the Poisson additive property, we have


where we defined and . We can marginalize out from Eq. (8) and (2) using the gamma-Poisson conjugacy, which gives

where and denotes the Kronecker delta. According to Result 2, we introduce the auxiliary variables as


We then re-express the bivariate distribution over and as


Using Eq. (1) and (10), via the gamma-Poisson conjugacy, we obtain the conditional distribution of as


Sampling intra-group weight . We resample the auxiliary variables using Eq. (9), and then exploit the gamma-Poisson conjugacy to sample as


Sampling inter-group weights . We sample from its conditional obtained from Eq. (2) and (8) via the gamma-Poisson conjugacy as


Sampling hyperparameter . Using Eq. (10) and the Poisson additive property, we have as

Marginalizing out using the gamma-Poisson conjugacy, we have

where . We introduce the auxiliary variables and re-express the bivariate distribution over and as


Using Eq. (14), we can then sample via the gamma-Poisson conjugacy as


Sampling latent memberships . Since the latent memberships evolve over time according to our Markovian construction, the backward and forward information need to be incorporated into the updates of . We start from time slice ,


Via the gamma-Poisson conjugacy, we have


Marginalizing out yields


where . According to Result 2, the NB distribution can be augmented with an auxiliary variable as


We re-express the bivariate distribution over and as




Given , via the Poisson additive property, we have


Combing the likelihood in Eq. (21) with the gamma prior placed on , we immediately have the conditional distribution of via the gamma-Poisson conjugacy as


Here, can be considered as the backward information passed from to . Recursively, we augment at each time slice with an auxiliary variable as


where the NB distribution over is obtained via the Poisson additive property and gamma-Poisson conjugacy with . Repeatedly using Result 2, we have

By repeatedly exploiting the Poisson additive property and gamma-Poisson conjugacy, we obtain


We sample the auxiliary variables and update recursively from to , which can be considered as the backward filtering step. Then, in the forward pass we sample from to .
Sampling hyperparameters. Via the gamma-gamma conjugacy, we sample and as


Algorithm 1 summarizes the full procedure.

input : dynamic relational data
Initialize the maximum number of groups , hyperparameters , and parameters repeat
       Sample for non-zero links (Eq. 6) Sample (Eq. 7) and update   Sample (Eq. 9) and calculate the quantities:  ,  Sample (Eq. 11), (Eq. 12), and (Eq. 13) for  to  do
             Sample (Eq. 23) and update (Eq. 20)
       end for
      for  to  do
             Sample (Eq. 24)
       end for
      Sample , (Eq. 25) and  (Eq. 15)
until convergenceoutput : posterior mean
Algorithm 1 Batch Gibbs Sampling

Online Gibbs sampling

To make our model applicable to large-scale dynamic networks, we propose an online Gibbs sampling algorithm based on the recent developed Bayesian conditional density filtering (BCDF) (?), which has been adapted for Poisson tensor factorization (?) recently. The main idea of BCDF is to partition the data into small mini-batches, and then to perform inference by updating the sufficient statistics using each mini-batch in each iteration. Specifically, the sufficient statistics used in our model are the latent count numbers. We use and to denote the indices of the entire data and the mini-batch in iteration respectively. We define the quantities updated with the mini-batch in iteration as:

The main procedure of our online Gibbs sampler is then as follows. We first update the sufficient statistics used to sample model parameters as

where , where and is the decay factor commonly used for online methods (?). We calculate the sufficient statistics for each mini-batch and then resample the model parameters using the procedure in batch Gibbs sampling algorithm outlined in Algorithm 1.


We evaluate our model by performing experiments on both synthetic and real-world datasets. First, we generate a synthetic data with the true underlying network structure evolving over time to test our model on dynamic community detection. For quantitive evaluation, we determine the model’s ability to predict held-out missing links. Our baseline methods include DRIFT, D-GPPF and HGP-EPM as we discussed before. For DRIFT, we use default settings as the code released online. 111http://jfoulds.informationsystems.umbc.edu/code/DRIFT.tar.gz. We implemented D-GPPF by ourselves and set the hyperparameters and initialize the model parameters with the values provided in (?). For HGP-EPM, we used the code released for (?). 222https://github.com/mingyuanzhou/EPM. In the following, we refer to our model as DPGM (Dynamic Poisson Gamma Membership model). For DPGM, we set and use , where is the number of nodes, for initilization. We obtain similar results when instead setting in a sensitivity analysis. For online Gibbs sampling, we set , and mini-batch size . All our experiments were performed on a standard desktop with 2.7 GHz CPU and 24 GB RAM. Following (?), we generate a set of small-scale dynamic networks from real-world relational data while we use held-out relational data to evaluate our model. The following three real-world datasets are used in our experiments, the detail of which are summarized in Table 1.

NIPS. The dataset records the co-authorship information among 5722 authors on publications in NIPS conferences over the past ten years (?). We first take the 70 authors who are most connected across all years to evaluate all methods (NIPS 70). We also use the whole dataset (NIPS 5K) for evaluation.

Enron. The dataset contains 136776 emails among 2359 persons over 28 months () (?). We generate a binary symmetric matrix for each monthly snapshot. The presence or absence of an email between each pair of persons during one month is described by the binary link at that time. We first select 61 persons by taking a 7-core of the aggregated network for the entire time and filter out the authors with email records less than 5 snapshots (Enron 61). We also use the whole dataset for evaluation (Enron 2K).

DBLP. The DBLP dynamic networks dataset (?) are generated from the co-authorship recordings among 347013 authors over 25 years, which is a subset of data contained in the DBLP database. We first choose 7750 authors by taking a 7-core of the aggregated network for the entire time (DBLP 7K) and subsequently filter out authors with less than 10 years of consecutive publication activity to generate a small dataset (DBLP 96). The proposed method and the two baselines are applied to all six datasets, except for DRIFT. DRIFT could not be applied to NIPS 5K, Enron 2K and DBLP 7K due to its unfavorable computational complexity. Most of these datasets exhibit strong sparsity that the proposed algorithm can exploit through its Bernoulli-Poisson link function.

Datasets NIPS 70 DBLP 96 Enron 61
Nodes 70 96 61
Time slices 10 25 28
Non-zero links 528 1392 1386
NIPS 5K DBLP 7K Enron 2K
Nodes 5722 7750 2359
Time slices 10 10 28
Non-zero links 5514 108980 76828
Table 1: Details of the dataset used in our experiments.

Dynamic community detection

          (a)           (b)           (c)           (d)           (e)
Figure 2: Dynamic community detection on synthetic data. We generate a dynamic network with five time snapshots as shown in column (a) ordered from top to bottom. The link probabilities estimated by D-GPPF and DPGM are shown in column (b) and (d). The association weights of each node to the latent groups can be calculated by for D-GPPF and for DPGM as shown in column (c) and (e), respectively. The pixel values are displayed on scale.
NIPS 70 DBLP 96 Enron 61
DPGM (batch)
NIPS 5K DBLP 7K Enron 2K
DPGM (batch)
DPGM (online)
Table 2: Missing links prediction. We highlight the performance of the best scoring model in bold.
NIPS 70 DBLP 96 Enron 61 NIPS 5K DBLP 7K Enron 2K
D-GPPF 0.0388 0.1350 0.2161 D-GPPF 7.6440 7.9160 7.8227
DRIFT 11.7047 42.1853 24.7505 DPGM (batch) 10.6240 15.6576 15.8584
DPGM (batch) 0.1283 0.6302 0.7364 DPGM (online) 8.9501 10.8152 10.4521
Table 3: Comparison of per-iteration computation time (seconds).

We adapt the synthetic example used in (?) to generate a dynamic network of size that evolve over five time slices as shown in Fig. 2. More specifically, we generate three groups at , and split the second group at . From to , the second and third group merge into one group. In Fig. 2, column (b) and (d) show the discovered latent groups over time by D-GPPF and DPGM, respectively. D-GPPF captures the evolution of the discovered groups but has difficulties to characterize the changes of node-group memberships over time. Our model (DPGM) can detect the dynamic groups quite distinctively. We also show the associations of the nodes to the inferred latent groups by D-GPPF and DPGM in column (c) and (e), respectively. In particular, we calculate the association weights of each node to the latent groups as for D-GPPF and for DPGM. For both models, most of the redundant groups can effectively be shrunk even though we initialize both algorithms with . The node-group association weights estimated by DPGM vary smoothly over time and capture the evolution of the node-group memberships, which is consistent to the ground truth shown in column (a).

Missing link prediction

For the task of missing link prediction, we randomly hold out of the observed interactions (either links or non-links) at each time as test data. The remaining data is used for training. HGP-EPM, DRIFT and D-GPPF are considered as the baseline methods. We train a HGP-EPM model on the training entries for each time slice separately. For each method, we use 2000 burn-in iterations, and collect 1000 samples of the model posterior. We estimate the posterior mean of the link probability for each held-out edge in the test data by averaging over the collected Gibbs samples. We then use these link probabilities to evaluate the predictive performance of each model by calculating the area under the curve of the receiver operating characteristic (AUC-ROC) and of the precision-recall (AUC-PR). Table 2 shows the average evaluation metrics for each model over 10 runs. Overall, our model (DPGM) shows the best performance. We observe that both DRIFT and DPGM outperform D-GPPF because the evolution of individual node-group memberships are explicitly captured in these two models. D-GPPF essentially assumes that the nodes’ memberships are static over time and thus has difficulties to fully capture the dynamics of each node’s interactions caused by the same node’s memberships’ evolution. We see that DPGM outperforms its static counterpart, HGP-EPM, via capturing the evolution of nodes’ memberships over time. We also compare per-iteration computation time of each model (all models are implemented in Matlab), as shown in Table 3. The computational cost of DRIFT scales quadratically with the number of nodes. Both D-GPPF and DPGM are much faster than DRIFT because the former two models scale only with the number of non-zero edges. We also report per-iteration computation time of GPPF and DPGM with Matlab/MEX/C implementation on medium-scale data in Table 3.


We have presented a probabilistic model for learning from dynamic relational data. The evolution of the underlying structure is characterized by the Markovian construction of latent memberships. We also proposed efficient batch and online Gibbs algorithms that make use of the data augmentation technique. Experimental results on synthetic and real datasets illustrate our model’s interpretable latent representations and competitive performance. Our model is dedicated to dynamic networks modeling but can be considered for other related problems such as dynamic multi-relational graph model (?). Another interesting direction is to scale up the model inference algorithm via stochastic gradient variational Bayes (?).


The authors thank Adrian Šošić and the reviewers for constructive comments. This work is funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement 668858.


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