Overlapping community detection for count-value networks
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Abstract
Detecting network overlapping community has become a very hot research topic in the literature. However, overlapping community detection for count-value networks that naturally arise and are pervasive in our modern life, has not yet been thoroughly studied. We propose a generative model for count-value networks with overlapping community structure and use the Indian buffet process to model the community assignment matrix Z; thus, provide a flexible nonparametric Bayesian scheme that can allow the number of communities K to increase as more and more data are encountered instead of to be fixed in advance. Both collapsed and uncollapsed Gibbs sampler for the generative model have been derived. We conduct extensive experiments on simulated network data and real network data, and estimate the inference quality on single variable parameters. We find that the proposed model and inference procedure can bring us the desired experimental results.
Keywords
Overlapping community detection Count-value networks Generative network model Nonparametric Bayesian model Indian buffet process Inference quality estimationAbbreviations
- rILFM
relational infinite latent feature model
- MCMC
Markov chain monte carlo
- MMSB
mixed membership stochastic block model
- IBP
Indian buffet process
- SBM
stochastic block model
- IRM
infinite relational model
- DCSBM
degree corrected SBM
- LFRM
latent feature relational model
- IMRM
infinite multiple relational
- IDCSBM
infinite degree corrected SBM
- ILA
infinite latent attribute
- HPD
highest posterior eensity
- GML
graph model language
Introduction
Community detection is a fundamental problem in network analysis, as community structure which almost exists in all networks, is the most widely studied structural properties of networks.
An important challenge in community detection is to specify the number of communities in advance, as we do not have good prior knowledge of how many parameters the model requires to explain the data well. The relational infinite latent feature model (rILFM), in which the number of latent variables is unbounded, is a flexible Bayesian nonparametric approach that is a proper choice for such situation, as its number of parameters can be vary along with the data increasing.
The Indian buffet process (IBP) [2] is often used to develop construction for the overlapping community assignment matrix, in which each object is represented by a sparse subset of an unbounded number of features, thus can lead to a Bayesian nonparametric version of the latent feature model.
As show in Fig. 1b, the set of features possessed by a set of objects can be expressed in the form of a binary matrix Z with infinite columns and exchangeable rows, where the ith row is an object, and the kth column corresponds to a feature, \(z_{ik}\) indicates that object i possesses feature k. The infinite binary matrix Z can describe that each individual is characterized by a set of features, or equivalently to say that each individual belongs to multiple communities simultaneously, which is intuitively named as overlapping community structure.
Most of the existing works represent a network as a symmetric binary adjacent matrix and a Bernoulli distribution (or a logistic Gaussian distribution) is chosen to formulate the generative mechanism, for its simplicity. The symmetric binary adjacent matrix representation has two limitations: (1) when we transform these count-value networks into a symmetric binary adjacent matrix representation, we lose many valuable network information which can help to find overlapping community, e.g., if we use binary network, all nodes play equal roles in one community, as there only have two situations: linked or not linked; but, if we consider the interaction times between nodes, they are no longer play equal roles, the count vale may imply which nodes are at the core of one community, which are at the periphery. (2) The MCMC (Markov chain monte carlo) inference of the generative model with Bernoulli likelihood is difficult to derive.
It is well known to us that count-value networks naturally arise and are pervasive in our modern life. For example, in communicate networks, such as email networks, phone call networks, instant messaging networks, worker recruitment influence networks in mobile crowd sensing (MCS) platforms [3] etc., interactions are often directed and have an associated count value, i.e., person i can send mails (make phone calls or send messages) to person j many times. On online social media service platforms such as Twitter, Facebook, BBS, and MCS [4], people follow (comment, like or reply to) those whom they are interested in, such interactions also have direction and are associated with interaction times.
In this article,we concerned on overlapping community detection for count-value social networks. We propose a generative model for count-value networks with overlapping community structure: the network is modeled as a Poisson point process, after applying Poisson factor analysis on the corresponding count matrix, we obtain \(M=Z\Lambda Z^T\), which is akin to the mixed membership stochastic block model (MMSB) [5] that can express the overlapping community structure. The IBP is used as the prior to model the community assignment matrix Z; thus, allows the number of communities K to be determined at inference time instead of to be predefined. Both a collapsed and an uncollapsed Gibbs sampler for the generative model have been derived. We reinforce the validity of the theoretical results via extensive experiments on simulated network data and real network data.
Related works
Following the seminal work of Erdos and Renyi [6], various random graph models have been proposed. The celebrated SBM (stochastic block model) [7] and its extensions such as the IRM (infinite relational model, Kemp et al. [8]), MMSB (Airoldi et al. [5]), DCSBM (degree-corrected SBM, Karrer et al. [9]), DSBM (dynamic SBM, Pensky [10]), have a wide variety of applications in network community detection, and form a huge corpus especially in social sciences and machine learning. We do not present an exhaustive review here; for an up-to-date account of various aspects, we direct the reader to Fortunato [11], Xie et al. [12] and Matias et al. [13] for reference.
There already have some pioneering works which composing the ideas of the classical MMSB model and the nonparametric Bayesian approach to increase the flexibility of network generative process by letting each node possess potentially infinite number of features, for example, the celebrated LFRM (latent feature relational model) proposed by Miller et al. [14], which was previously described in Meeds et al. [15]. The IMRM (infinite multiple relational) model proposed by Morup et al. [16] is a variant of the LFRM model, in which a noisy-or likelihood was used instead of the logistic Gaussian likelihood. The ILA [17] (infinite latent attribute) model presented in Palla et al. (2015) generalized the LFRM mode by allowing an explicit representation of the partitioning of each general community into subclasses, thus providing a more structured representation of the data. All these models assume that K is not known a priori and use the IBP to account for the number of latent communities.
Although most of the existing work does not consider count-value networks, some research work provides an exception. For example, Karrer and Newman introduced the DCSBM model [9], they assumed that the links between nodes i and j follow a Poisson distribution and, thus, represented network as a count adjacent matrix. This method is reasonable, as the Poisson distribution is the natural probability distribution for modeling counts. Tue Herlau et al. [18] formulated a nonparametric Bayesian generative model for the DCSBM (they named it IDCSBM), where the number of communities is inferred via the Chinese restaurant process [19]. These two models can be used to detect only nonoverlapping communities.
The celebrated IBP model, originally studied by Ghahramni and Griffiths [2], Thibaux and Jordan [20], connected the IBP to the theory of completely random measure by showing that it could be constructed from an exchangeable sequence of beta-Bernoulli processes. They further showed that the beta-Bernoulli process is the underlying de Finetti mixing measure for the IBP.
The Poisson factor model, which is also named the Gamma-Poisson model, is a probabilistic matrix factorization model that has been widely used in many areas such as image reconstruction, text information retrieval, and collaborative filtering etc.. The first application of Poisson factor analysis to network analysis was presented in Zhou et al. [21].
The proposed model
Apparently, the Poisson factor analysis, is guaranteed by the superposition principle of the Poisson point processes.
Superposition is an additive set operation such the superposition of a k-point configuration in \(X_n\) is a \(kn-point\) configuration in X. Examples of Poisson superposition processes include the compound Poisson, and the negative binomial processes.
Theorem 1
(Poisson Superposition Principle) Give k independent Poisson point processes \(\Pi _1,\Pi _2,\ldots ,\Pi _k\), and the corresponding counting processes are \(N_1,N_2,\ldots ,N_k\), which with intensity measure \(\mu _1,\mu _2,\ldots ,\mu _k\), then \(\Pi =\cup _{i=1}^k\Pi _i\) also is a Poisson point process, the corresponding counting process is \(N=\sum _{i=1}^kN_i\), its intensity is \(\mu =\sum _{i=1}^k\mu _i\) [22].
We apply the restriction that links are directly generated by individual features instead of through complex interactions between features, so that feature and community are the same concepts, i.e., stating that node i possesses feature j is equivalent to stating that node i is affiliated with community j.
As show in Fig. 1b, nodes are assigned to a set of communities can be expressed in the form of a binary matrix with infinite columns and exchangeable rows, where the ith row is the community assignment vector \(Z_i\) of the node i, and the jth column corresponds to a community, \(z_{ij}=1\) indicates that node i affiliated to community j. As \(Z_i\) may has many nonzero element, i.e. there is no assumption of mutual exclusivity and exhaust, thus the community affiliation matrix Z can characterize overlapping community structure in a network.
Parameter inference
The IBP is a distribution over an exchangeable binary matrix, it can be constructed in two ways, restaurant construction and stick-breaking construction. The former easily lends itself to MCMC inference, and the latter easily lends itself to variational inference [23]. Although the execution time required for MCMC inference is cubic due to the number of observations and thus often scales poorly [24], we can only use MCMC to infer the rILFM models if we do not want to predefine K because the stick-breaking construction of the IBP leads to a variational method for inference based on truncating to a finite model. Thus we must predefine the truncating level, which is as difficult as predefining K.
In this paper, we derived both a collapsed and an uncollapsed Gibbs sampler for Z. In "Uncollapsed Gibbs sampler" subsection, we illustrate the uncollapsed Gibbs sampler based MCMC inference algorithm, and in "Collapsed Gibbs sampler" subsection, we depict details about derivation of the collapsed sampler.
Uncollapsed Gibbs sampler
Let \(M_1\) denote the set of observed links, \( (i,j)\in M_1\) means that there is a link between node i and j (in other word, \(m_{ij}>0\)), \(EV=\sum \nolimits _{(i,j)\in M_1}m_{ij}\) denote the total number of links, \(C=\sum \nolimits _{i=1}^n\sum \nolimits _{j=1}^n(Z_iZ_j)=Z\bigodot Z^T\) denote the total number of communities shared by node pairs \( (i,j)\in M\) (\(\bigodot \) denote the Hadamard product operation on matrix), HN denote the harmonic number, \(Z_{-ik}\) denote all community assignments except \(z_{ik}\), \(k_{new}\) denote new sampled features for each object. The inference procedure of our model is as follow:
Collapsed Gibbs sampler
Different from the uncollapsed Gibbs sampler, the collapsed Gibbs sampler use P(M|Z, a, b) as likelihood distribution instead of \(P(M|Z,\lambda )\), and thus we need not to update \(\lambda \), i.e., step 2.2 in the Algorithm 1 can be omitted. As differences between the two samplers are very clear, we have no need to illustrate the collapsed Gibbs sampler based MCMC inference algorithm, we just depict details about derivation of the collapsed sampler here.
First, we derive the likelihood distribution which was used in the uncollapsed Gibbs sampler. Let \(M_0\) denote the set of observed unlinks, \( (i,j)\in M_0\) means that there is no link between node i and j (in other word, \(m_{ij}=0\)).
- 1.
Derive the likelihood in the uncollapsed Gibbs sampler
$$\begin{aligned}&P(M|Z,\lambda )=\prod \limits _{(i,j)\in M_1}\frac{\rho _{ij}^{m_{ij}}}{m_{ij}!}exp(-\rho _{ij})\prod \limits _{(i,j)\in M_0}exp(-\rho _{ij})\\&\quad =\prod \limits _{(i,j)\in M_1}\frac{\rho _{ij}^{m_{ij}}}{m_{ij}!}\prod \limits _{(i,j)\in M_1}exp(-\rho _{ij})\prod \limits _{(i,j)\in M_0}exp(-\rho _{ij})\\&\quad =\prod \limits _{(i,j)\in M_1}\frac{\pi _{ij}^{m_{ij}}}{m_{ij}!}\prod \limits _{(i,j)\in M}exp(-\rho _{ij})\\&\quad =\prod \limits _{(i,j)\in M_1}\frac{(\lambda *\sum Z_iZ_j)^{m_{ij}}}{m_{ij}!}\prod \limits _{(i,j)\in M}exp \left(-\sum (Z_iZ_j)*\lambda \right)\\&\quad =\prod \limits _{(i,j)\in M_1}\frac{(\sum Z_iZ_j)^{m_{ij}}*\lambda ^{m_{ij}}}{m_{ij}!}\prod \limits _{(i,j)\in M}exp \left(-\sum (Z_iZ_j)*\lambda \right)\\&\quad =\prod \frac{\left(\sum Z_iZ_j\right)^{m_{ij}}}{m_{ij}!}*\prod \lambda ^{m_{ij}}\prod \limits _{(i,j)\in M}exp\left(-\sum (Z_iZ_j)*\lambda \right)\\&\quad =\frac{\prod (\sum Z_iZ_j)^{x_{ij}}}{\prod x_{ij}!}*\lambda ^{\sum m_{ij}}exp\left(-\sum \limits _{(i,j)\in M}\sum (Z_iZ_j)*\lambda \right)\\&\quad =\frac{\prod (\sum Z_iZ_j)^{m_{ij}}}{\prod m_{ij}!}*\lambda ^{EV}exp(-C*\lambda) \end{aligned}$$As the likelihood distribution in the uncollapsed sampler is conjugate to the prior of \(\lambda \), we can integrate out \(\lambda \) to obtain the likelihood in the collapsed sampler.
- 2.
Integrate out \(\lambda \) to obtain the likelihood in the collapsed sampler
$$\begin{aligned}&P(M|Z,a,b)=\int _\lambda P(M|Z,\lambda )P(\lambda |a,b)d\lambda \\ \\&\quad =\int _\lambda \frac{\prod (\sum Z_iZ_j)^{m_{ij}}}{\prod m_{ij}!}*\lambda ^{EV}exp(-C*\lambda )\frac{b^a}{\Gamma (a)}\lambda ^{a-1}exp(-b\lambda )d\lambda \\&\quad =\frac{\prod (\sum Z_iZ_j)^{m_{ij}}}{\prod m_{ij}!}\frac{b^a}{\Gamma (a)}\int _\lambda \lambda ^{a+EV-1}exp(-(b+C)\lambda )d\lambda \\&\quad =\frac{\prod (\sum Z_iZ_j)^{m_{ij}}}{\prod m_{ij}!}\frac{b^a}{\Gamma (a)}\frac{\Gamma (a+EV)}{(b+C)^{a+EV}}\int _\lambda \frac{(b+C)^{a+EV}}{\Gamma (a+EV)}\lambda ^{a+EV-1}exp(-(b+C)\lambda )d\lambda \\&\quad =\frac{\prod (\sum Z_iZ_j)^{m_{ij}}}{\prod m_{ij}!}\frac{b^a}{\Gamma (a)}\frac{\Gamma (a+EV)}{(b+C)^{a+EV}}\\&\quad =\frac{\prod (\sum Z_iZ_j)^{m_{ij}}}{\prod m_{ij}!}\frac{b^a\prod _{k=1}^ {EV} (k+a)}{(b+C)^{a+EV}} \end{aligned}$$
Inference tricks
In order to derive a feasible MCMC inference procedure, we make the following assumptions for our model:
- 1.
We assume that \(\Lambda \) is a diagonal matrix, links only exist between nodes in the same community, i.e., there’s no link from a node in community \(k_1\) to a node in community \(k_2\) when \(k_1!=k_2\);
- 2.
We restrict all link probability \(\lambda _{k_1k_2}\) to take the same value \(\lambda \), this means nodes within each community have same opportunity to form a link.
These two assumptions can bring us two benefits, one is that we don’t need to change the shape of \(\lambda \) along with the changes of K, the other is that we can obtain the conjugacy between the likelihood and the Gamma prior for \(\lambda \). Under this circumstance, \(\lambda \) can be integrated away and a collapsed Gibbs sampler for Z can be derived.
The IBP has a major weakness: the generated Z is determined only by N and \(\alpha \), regardless of the characteristics of the observations. For example, if node i is an isolated node, its community assignment vector should be an all-zero vector, but the IBP ignores this fact and assigns node i to some communities. Some steps are taken to correct this clear mistake and to avoid unnecessarily updating of the all-zero rows in Z. And accordingly make the MCMC inference accelerated.
- 1.
Assign a flag to isolated node
We maintain a flag vector with all-zero initial values. First, we check each node in the graph. If its in-degree and out-degree both are zero, we set its flag to one to indicate that the node is not affiliated with any community;
- 2.
Skip unnecessary update steps
After the initial Z has been generated, according to the flag, we change the corresponding row in Z to an all-zero vector. In the process of each MCMC iteration, when we update Z, if a node’s flag is one, we don’t update the corresponding row.
After we perform posterior inference on Z, based on the assumption that a community should contain at least three nodes, we will cancel those columns in the inferred Z which have less than three non-zero values.
Per-iteration running times
For both the uncollapsed Gibbs sampler and the collapsed Gibbs sampler, when analysis algorithm complexity, we only consider the number of the Hadamard product operates on Z (i.e., element-wise matrix multiplication \(Z\bigodot Z^T\)) for one sweep through a \(N*K\) community assignment matrix Z under a compound Poisson likelihood model.
The running time of both two Gibbs samplers are dominated by the computation of the likelihood. When we change one element of Z, the likelihood need to be calculated twice, thus Z may be updated in \(O(N^3K)\) time.
Experiments
We implemented our model and the inference algorithm using python. After we finished Bayesian analysis, the posterior which contains all the information about model parameters according to the observed data and the model, was need to be summarized [25].
For single variable parameters such as \(\alpha \) and \(\lambda \), it is easy to communicate the result, as the most probable posterior value is given by the mode of the posterior distribution (i.e., the peak of the distribution). It is also a good choice to report the mean (or median) of the distribution and some other measure, such as standard deviation or HPD (highest posterior density) interval, to have an idea of the dispersion and hence the uncertainty in our estimate [25].
Experiment on synthetic data
Occurring times of all the \(K_s\) values sampled from six chains
4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
U | chain1 | 0 | 5 | 439 | 2660 | 2672 | 959 | 187 | 16 | 2 | 0 | 0 | 0 |
chain2 | 0 | 0 | 184 | 1393 | 2396 | 1952 | 836 | 199 | 35 | 4 | 1 | 0 | |
chain3 | 2 | 66 | 406 | 1311 | 1951 | 1784 | 954 | 393 | 104 | 27 | 2 | 0 | |
C | chain4 | 9 | 181 | 742 | 1628 | 1754 | 1106 | 427 | 127 | 21 | 4 | 1 | 0 |
chain5 | 8 | 100 | 581 | 1418 | 2004 | 1707 | 827 | 268 | 77 | 9 | 1 | 0 | |
chain6 | 3 | 62 | 338 | 1137 | 1842 | 1825 | 1127 | 490 | 125 | 42 | 8 | 1 |
The true communities and the inferred communities
Communities | Size | |
---|---|---|
Ground truth | C1 = \(\{2,3,4,5,7,8,9, 10, 11, 14, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29\}\) | 23 |
C2 = \(\{1, 3, 4,6, 7,9, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30\}\) | 23 | |
C3 = \(\{1,2,5,6,13,17,26,30\}\) | 8 | |
C4 = \(\{1,8,15,16,28\}\) | 5 | |
C5 = \(\{5,12,17,24,28\}\) | 5 | |
C6 = \(\{3,7,10,13,18,30\}\) | 6 | |
C7 = \(\{4,12,20,25\}\) | 4 | |
C8 = \(\{13,14,23\}\) | 3 | |
Inferred result, the 6997th sample | C1 = \(\{2,3,4,5,7,8,9, 10, 11, 14, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29\}\) | 23 |
C2 = \(\{1, 3, 4,6, 7,9, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30\}\) | 23 | |
C3 = \(\{5,12,13,17,24,25,28\}\) | 7 | |
C4 = \(\{1,2,5,6,17,26,30\}\) | 7 | |
C5 = \(\{1,2,8,15,16,28\}\) | 6 | |
C6 = \(\{6,13,18,23,30\}\) | 5 | |
C7 = \(\{3,10,21\}\) | 3 | |
C8 = \(\{10,13,21\}\) | 3 | |
The 1000th sample | C1 = \(V_t\) | 30 |
C2 = \(\{3,4,7,10,11,14,16,20,21,22,23,25,29\}\) | 13 | |
C3 = \(\{5,20,23,24,25,27\}\) | 6 | |
C4 = \(\{2,5,8,17,26\}\) | 5 | |
C5 = \(\{1,2,6,30\}\) | 4 | |
C6 = \(\{6,21,26,27,28\}\) | 5 | |
C7 = \(\{9,12,14,20,27\}\) | 5 | |
C8 = \(\{9,10,30\}\) | 3 |
From Table 2, we can see that the biggest two communities C1 and C2 have the same objects in both the ground truth and the inferred results, but those small communities are different from each other. Only 6 objects \(v_{11},v_{18},v_{19},v_{22},v_{27},v_{29}\) have the same community affiliation, imply that for an unsupervised learning task, such as overlapping community detection, even if we known the ground truth, it is hard to obtain accuracy results via statistical machine learning method. Let us see Z sampled from chain2 in the 1000th MCMC iteration, it is very far from the ground truth, so it’s necessary to throw the burning samples away.
In Fig. 7, the black curve describes the posterior using a kernel density estimation, mode, ROPE means lower and upper values of the region of practical equivalence. When we say that the \(95\%\) HPD for \(\alpha \) is 1.33, 4.78, we mean that according to our data and model we think \(\alpha \) in question is between 1.33 and 4.78 with a 0.95 probability. \(95\%\)HPD of retained \(\alpha \), \(\lambda \) which were drawn from chain1 and chain3 were depicted in Fig. 8.
Summarization about mode and 95% HPD of retained \(\alpha \), \(\lambda \)
chain1 | chain2 | chain3 | chain1 | chain2 | chain3 | ||
---|---|---|---|---|---|---|---|
Mode of \(\alpha \) | 2.46 | 2.75 | 2.97 | \(95\%\) HPD of \(\alpha \) | 1.11, 4.47 | 1.33, 4.78 | 1.37, 4.94 |
Mode of \(\lambda \) | 1.24 | 1.25 | 1.25 | \(95\%\) HPD of \(\lambda \) | 1.2, 1.27 | 1.21, 1.28 | 1.22, 1.29 |
Experiment on the LESMIS network
We obtain a data file in GML format, we convert it into a CSV file. The file contains an upper triangular matrix, with all diagonal elements as 0. Note that we have no ground truth about Z and K. For greater reliability, we ran two chains: chain1, which starts with \(a = b = 1,\,e = 24,\,f = 1/HN\); and chain2, which starts with \(a = b = 1,\,e = 44,\,f = 1/HN\). We ran each chain for \(maxIter=10000\) MCMC iterations, with \(burnin=4000\) and collected the last 6000 samples.
Occurring times of all the \(K_s\) values sampled from the two chains
\(K_s=13\) | \(K_s=14\) | \(K_s=15\) | \(K_s=16\) | \(K_s=17\) | \(K_s=18\) | \(K_s=19\) | |
---|---|---|---|---|---|---|---|
chain1 | 183 | 2284 | 2878 | 626 | 25 | 4 | 0 |
chain2 | 377 | 1776 | 2020 | 1246 | 507 | 70 | 4 |
Conclusion
The paper makes the following contributions: (1) we propose a generative model for count-value networks with overlapping community structure; (2) we use the IBP to model the community assignment matrix Z, so the number of communities K is not required to be fixed in advance, it is able to increase as more and more data are encountered; (3) both uncollapsed Gibbs sampler and collapsed Gibbs sampler for the generative model have been derived; (4) we analysis the inference quality on single variable parameters; (5) we conduct extensive experiments on simulated network data and real network data, we find that the proposed model and inference procedure can bring us the desired experimental results.
Most count value networks are overdisperse, the negative binomial likelihood is more suitable for these overdisperse count value data. But inference of the rILFM model with negative binomial likelihood requires great care in selecting appropriate starting point, we aim it as one of our future work.
For single variable parameters, the posterior inference result is easy to communicate. But for structured parameters such as \(Z_{ik}\hbox {s}\), how to summarize the posterior inference results and estimate the inference quality, is a considerable challenge, we aim it as another one of our future work.
Notes
Acknowledgements
We thanks Mikkel N. Schmidt etc. for their great research work and helpful open-source code. Thanks anonymous reviewers for their comments.
Authors' contributions
QY and ZWY are responsible for model design; QCY and ZW are responsible for model inference; QCY and XFW are responsible for experiment design and experiment implementation; QCY and YZ Wang are responsible for data analysis; QCY, ZW and ZWY are responsible for writing. All authors read and approved the final manuscript.
Funding
This work was partially supported by the National Basic Research Program of China (973) (No. 2015CB352401), the National Natural Science Foundation of China (No. 61332005, 61725205), the Research Project of the North Minzu University (No.2019XYZJK02, 2019XYZJK05, 2017KJ24, 2017KJ25, 2019MS002).
Competing interests
The authors declare that they have no competing interests.
References
- 1.Zhu W, Zhang D, Zhou X, Yang D, Zhiwen Y (2017) Discovering and profiling overlapping communities in location-based social networks. IEEE Trans Syst Man Cybern Syst 44(4):499–509Google Scholar
- 2.Griffiths T, Ghahramani Z (2005) Infinite latent feature models and the Indian buffet process. In: International conference on neural information processing systemsGoogle Scholar
- 3.Wang J, Feng W, Wang Y, Zhang D, Qiu Z (2018) Social-network-assisted worker recruitment in mobile crowd sensing. IEEE Trans Mob Comput 99:1–1Google Scholar
- 4.Wang Z, Guo B, Yu Z, Zhou X (2018) Wi-Fi CSI-based behavior recognition: from signals and actions to activities. IEEE Commun Mag 56(5):109–119CrossRefGoogle Scholar
- 5.Airoldi EM, Blei DM, Fienberg SE, Xing EP (2008) Mixed membership stochastic blockmodels. J Mach Learn Res 9(5):1981zbMATHGoogle Scholar
- 6.Erdos P, Renyi A (1959) On random graphs. Publicationes Mathematicae 6(4):3286–3291zbMATHGoogle Scholar
- 7.Brian K, Newman MEJ (2011) Stochastic blockmodels and community structure in networks. Phys Rev E Stat Nonlinear Soft Matter Phys 83(2):016107MathSciNetGoogle Scholar
- 8.Kemp C, Tenenbaum JB, Griffiths TL (2006) Learning systems of concepts with an infinite relational model. Cogn Sci 21(1):61Google Scholar
- 9.Karrer B, Newman ME (2011) Stochastic blockmodels and community structure in networks. Phys Rev E Stat Nonlinear Soft Matter Phys 83(2):016–107MathSciNetGoogle Scholar
- 10.Pensky M (2016) Dynamic network models and graphon estimation. arXiv preprint arXiv:1607.00673
- 11.Fortunato S (2009) Community detection in graphs. Phys Rep 486(3):75–174MathSciNetGoogle Scholar
- 12.Xie J, Kelley S, Szymanski BK (2011) Overlapping community detection in networks: the state-of-the-art and comparative study. ACM Comput Surv 45(4):1–35zbMATHCrossRefGoogle Scholar
- 13.Matias C, Robin S (2014) Modeling heterogeneity in random graphs through latent space models: a selective review. ESAIM Proc Surv 47:55–74MathSciNetzbMATHCrossRefGoogle Scholar
- 14.Miller KT (2011) Bayesian nonparametric latent feature models. Dissertations and Theses—Gradworks, pp 201–226Google Scholar
- 15.Meeds E, Ghahramani Z, Neal RM, Roweis ST (2006) Modeling dyadic data with binary latent factors. In: International conference on neural information processing systemsGoogle Scholar
- 16.Morup M, Schmidt MN, Hansen LK (2011) Infinite multiple membership relational modeling for complex networks. Comput Sci 19(5):1–6Google Scholar
- 17.Konstantina Palla, Knowles David A, Zoubin Ghahramani (2015) Relational learning and network modelling using infinite latent attribute models. IEEE Trans Pattern Anal Mach Intell 37(2):462–474CrossRefGoogle Scholar
- 18.Herlau T, Schmidt MN, Morup M (2014) Infinite-degree-corrected stochastic block model. Phys Rev E Stat Nonlinear Soft Matter Phys 90(3):032819CrossRefGoogle Scholar
- 19.Aldous David J (1985) Exchangeability and related topics. Springer, BerlinzbMATHCrossRefGoogle Scholar
- 20.Thibaux R, Jordan MI (2007) Hierarchical beta processes and the Indian buffet process. In: Proceedings of the 11th international conference on artificial intelligence and statistics, pp 1135–1143Google Scholar
- 21.Zhou M (2015) Infinite edge partition models for overlapping community detection and link prediction. In: In AISTATS2015, vol 38, pp 1135–1143Google Scholar
- 22.De Blasi P, Favaro S, Lijoi A, Mena RH, Prunster I, Ruggiero M (2015) Are gibbs-type priors the most natural generalization of the dirichlet process? IEEE Trans Pattern Anal Mach Intell 37(2):212–229CrossRefGoogle Scholar
- 23.Doshi F, Miller KT, Van Gael J, Teh YW (2008) Variational inference for the Indian buffet process. J Mach Learn Res 5:137–144Google Scholar
- 24.Gershman SJ, Blei DM (2012) A tutorial on bayesian nonparametric models. J Math Psychol 56(1):1–12MathSciNetzbMATHCrossRefGoogle Scholar
- 25.Martin O (2016) Bayesian analysis with python. Packt PublishingGoogle Scholar
- 26.Rossi RA, Ahmed NK (2015) The network data repository with interactive graph analytics and visualization. In: Proceedings of the twenty-ninth AAAI conference on artificial intelligenceGoogle Scholar
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