nips nips2010 nips2010-67 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Katsuhiko Ishiguro, Tomoharu Iwata, Naonori Ueda, Joshua B. Tenenbaum
Abstract: We propose a new probabilistic model for analyzing dynamic evolutions of relational data, such as additions, deletions and split & merge, of relation clusters like communities in social networks. Our proposed model abstracts observed timevarying object-object relationships into relationships between object clusters. We extend the infinite Hidden Markov model to follow dynamic and time-sensitive changes in the structure of the relational data and to estimate a number of clusters simultaneously. We show the usefulness of the model through experiments with synthetic and real-world data sets.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract We propose a new probabilistic model for analyzing dynamic evolutions of relational data, such as additions, deletions and split & merge, of relation clusters like communities in social networks. [sent-7, score-0.797]
2 Our proposed model abstracts observed timevarying object-object relationships into relationships between object clusters. [sent-8, score-0.12]
3 We extend the infinite Hidden Markov model to follow dynamic and time-sensitive changes in the structure of the relational data and to estimate a number of clusters simultaneously. [sent-9, score-0.644]
4 Many statistical models for relational data have been presented [10, 1, 18]. [sent-12, score-0.35]
5 The stochastic block model (SBM) [11] and the infinite relational model (IRM) [8] partition objects into clusters so that the relations between clusters abstract the relations between objects well. [sent-13, score-0.878]
6 SBM requires specifying the number of clusters in advance, while IRM automatically estimates the number of clusters. [sent-14, score-0.19]
7 Similarly, the mixed membership model [2] associates each object with multiple clusters (roles) rather than a single cluster. [sent-15, score-0.286]
8 However, a large amount of relational data in the real world is time-varying. [sent-17, score-0.35]
9 Recently some researchers have investigated the dynamics in relational data. [sent-23, score-0.35]
10 They assumed a transition probability matrix like HMM, which governs all the cluster assignments of objects for all time steps. [sent-28, score-0.437]
11 Thus, it cannot represent more complicated time variations such as split & merge of clusters that only occur temporarily. [sent-30, score-0.324]
12 This model is very general for time series relational data modeling, and is good for tracking gradual and continuous changes of the relationships. [sent-34, score-0.421]
13 In addition, previous models assume the number of clusters is fixed and known, which is difficult to determine a priori. [sent-37, score-0.19]
14 1 In this paper we propose yet another time-varying relational data model that deals with temporal and dynamic changes of cluster structures such as additions, deletions and split & merge of clusters. [sent-38, score-0.851]
15 Instead of the continuous world view of [4], we assume a discrete structure: distinct clusters with discrete transitions over time, allowing for birth, death and split & merge dynamics. [sent-39, score-0.33]
16 More specifically, we extend IRM for time-varying relational data by using a variant of the infinite HMM (iHMM) [15, 3]. [sent-40, score-0.35]
17 By incorporating the idea of iHMM, our model is able to infer clusters of objects without specifying a number of clusters in advance. [sent-41, score-0.454]
18 This specific form of iHMM enables the model to represent time-sensitive dynamic properties such as split & merge of clusters. [sent-43, score-0.167]
19 2 Infinite Relational Model We first explain the infinite relational model (IRM) [8], which can estimate the number of hidden clusters from a relational data. [sent-45, score-0.918]
20 In IRM, Dirichlet process (DP) is used as a prior for clusters of an unknown number, and is denoted as DP(γ, G0 ) where γ > 0 is a parameter and G0 is a base measure. [sent-46, score-0.19]
21 The IRM is an application of the DP for relational data. [sent-54, score-0.35]
22 The IRM divides the set of N objects into multiple clusters based on the observed relational data X = {xi, j ∈ {0, 1}; 1 ≤ i, j ≤ N}. [sent-60, score-0.614]
23 The IRM is able to infer the number of clusters at the same time because it uses DP as a prior distribution of the cluster partition. [sent-61, score-0.473]
24 We k,l=1 sample a cluster index of the object i, zi = k, k ∈ {1, 2, . [sent-74, score-0.355]
25 (4) ηk,l is the strength of a relation between the objects in clusters k and l. [sent-80, score-0.295]
26 Generating the observed relational data xi, j follows Eq. [sent-81, score-0.35]
27 (5) conditioned by the cluster assignments Z and the strengths H. [sent-82, score-0.289]
28 1 Dynamic Infinite Relational Model (dIRM) Time-varying relational data First, we define the time-varying relational data considered in }this paper. [sent-84, score-0.7]
29 Time-varying relational { data X have three subscripts t, i, and j: X = xt,i, j ∈ {0, 1} , where i, j ∈ {1, 2, . [sent-85, score-0.35]
30 We assume that there is no relation between objects belonging to a different time step t and t . [sent-94, score-0.161]
31 The time-varying relational data X is a set of T (static) relational data for T time steps. [sent-95, score-0.729]
32 It is natural to assume that every object transits between different clusters along with the time evolution. [sent-101, score-0.286]
33 Observing several real world time-varying relational data, we assume there are several properties of transitions, as follows: • P1. [sent-102, score-0.35]
34 Time evolutions of clusters are not stationary nor uniform. [sent-105, score-0.258]
35 The number of clusters is time-varying and unknown a priori. [sent-107, score-0.19]
36 P1 is a common assumption for many kinds of time series data, not limited to relational data. [sent-108, score-0.379]
37 P2 tries to model occasional and drastic changes from frequent and minor modifications in relational networks. [sent-111, score-0.417]
38 This will cause an addition and deletion of a user cluster (community). [sent-115, score-0.254]
39 We first consider several straightforward solutions based on the IRM for analyzing time-varying relational data. [sent-119, score-0.35]
40 ˜ The simplest way is to convert time-varying relational data X into “static” relational data X = { xi, j } ˜ ˜ ˜ and apply the IRM to X. [sent-120, score-0.7]
41 This solution cannot represent the time changes of clustering because it assume the same clustering results for all the time steps (z1,i = z2,i = · · · = zT,i ). [sent-122, score-0.196]
42 We may separate the time-varying relational data X into a series of time step-wise relational data Xt and apply the IRM for each Xt . [sent-123, score-0.729]
43 Since β is shared over all time steps, we may expect that the clustering results between time steps will have higher correlations. [sent-129, score-0.12]
44 This implies that the tIRM is not suitable for modeling time evolutions since the order of time steps are ignored in the model. [sent-131, score-0.154]
45 3 dynamic IRM To address three conditions P1∼3 above, we propose a new probabilistic model called the dynamic infinite relational model (dIRM). [sent-133, score-0.474]
46 (12) is a transition probability that an object remaining in the cluster k ∈ {1, 2, . [sent-150, score-0.339]
47 } at time t − 1 will move to the cluster l ∈ {1, 2, . [sent-153, score-0.283]
48 This implies that this DP encourages the self-transitions of objects, and we can achieve the property P1 for time-varying relational data. [sent-164, score-0.35]
49 πt,k is sampled for every time step t, thus, we can model time-varying patterns of transitions, including additions, deletions and split & merge of clusters as extreme cases. [sent-166, score-0.362]
50 These changes happen only temporarily, therefore, time-dependent transition probabilities are indispensable for our purpose. [sent-167, score-0.114]
51 Note that the transition probability is also dependent on the cluster index k, as in conventional iHMMs. [sent-168, score-0.326]
52 Also the dIRM can automatically determine the number of clusters thanks to DP: this enables us to hold P3. [sent-169, score-0.19]
53 Equation (13) generates a cluster assignment for the object i at time t, based on the cluster, where the object was previously (zt−1,i ) and its transition probability π. [sent-170, score-0.408]
54 Equation (14) generates a strength parameter η for the pair of clusters k and l, then we obtain the observed sample xt,i, j in Eq. [sent-171, score-0.19]
55 Thus, we may interpret the dIRM as an extension of the iHMM, which has N (= a number of objects) hidden sequences to handle relational data. [sent-176, score-0.378]
56 Given U, the number of clusters can be reduced to a finite number during the inference, and it enables us an efficient sampling of variables. [sent-179, score-0.221]
57 (20) Because of I(u < π), values of cluster indices k are limited within a finite set. [sent-197, score-0.277]
58 First β is assumed as a K + 1-dimensional vector (mixing ratios ∑K of unrepresented clusters are aggregated in βK+1 = 1 − k=1 βk ). [sent-201, score-0.19]
59 To apply the IRM to time-varying relational data, we use Eq. [sent-222, score-0.35]
60 To synthesize datasets, we first determined the number of time steps T , the number of clusters K, and the number of objects N. [sent-230, score-0.321]
61 Next, we manually assigned zt,i in order to obtain cluster split & merge, additions, and deletions. [sent-231, score-0.285]
62 After obtaining Z, we defined the connection strengths between clusters H = {ηk,l }. [sent-232, score-0.19]
63 IOtables summarize the transactions of goods and services between industrial sectors. [sent-240, score-0.141]
64 Each element in the matrix ei, j represents that one unit of demand in the jth sector invokes ei, j productions in the ith sector. [sent-242, score-0.113]
65 Thus we obtain a time-varying relational data of N = 32 and T = 5. [sent-245, score-0.35]
66 Differences in the number of realized clusters ˆ were computed between Zt and Zt , and we calculated the average of these errors for T steps. [sent-262, score-0.19]
67 Particularly, dIRM showed good results in the Synth2 and Enron datasets, where the changes in relationships are highly dynamic and unstable. [sent-300, score-0.144]
68 Thus we can say that the dIRM is superior in modeling time-varying relational data, especially for dynamic ones. [sent-302, score-0.412]
69 The panel (a) illustrates the estimated ηk,l using the dIRM, and the panel (b) presents the time evolution of cluster assignments, respectively. [sent-305, score-0.283]
70 For example, dIRM groups the machine industries into cluster 5, and infrastructure related industries are grouped into cluster 13. [sent-308, score-0.602]
71 For example, demands for machine industries (cluster 5) will cause large productions for “iron and steel” sector (cluster 7). [sent-312, score-0.16]
72 However, the sector transits to cluster 1 afterwards, which does not connect strongly with clusters 5 and 7. [sent-317, score-0.553]
73 Next, the “transport” sector enlarges its roll in the market by moving to cluster 14, and it causes the deletion of cluster 8. [sent-319, score-0.59]
74 From 1985 to 2000, this sector is in the cluster 9 which is rather independent from other clusters. [sent-321, score-0.336]
75 However, in 2005 the cluster separated, and telecom industry merged with cluster 1, which is a influential cluster. [sent-322, score-0.604]
76 Figure 4 (a) tells us that clusters 1 ∼ 7 are relatively separated communities. [sent-326, score-0.19]
77 For example, members in cluster 4 belong to a restricted domain business such energy, gas, or pipeline businesses. [sent-327, score-0.254]
78 Cluster 5 is a community of financial and monetary departments, and cluster 7 is a community of managers such as vice presidents, and CFOs. [sent-328, score-0.338]
79 One interesting result from the dIRM is finding cluster 9. [sent-329, score-0.254]
80 This cluster notably sends many messages to other clusters, especially for management cluster 7. [sent-330, score-0.508]
81 The number of objects belonging to this cluster is only three throughout the time steps, but these members are the key-persons at that time. [sent-331, score-0.384]
82 6 Cluster 7 iron and steel iron and steel iron and steel iron and steel iron and steel 0. [sent-333, score-0.875]
83 1 14 1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 l (b) (a) Figure 3: (a) Example of estimated ηk,l (strength of relationship between clusters k, l) for IOtable data by dIRM. [sent-341, score-0.19]
84 (d) Time-varying clustering assignments for selected clusters by dIRM. [sent-342, score-0.259]
85 dIRM: Learned ηkl for Enron data “Inactive” object cluster CEO of Enron America The founder COO (a) (b) Figure 4: (a): Example of estimated ηk,l for Enron dataset using dIRM. [sent-343, score-0.369]
86 (b): Number of items belonging to clusters at each time step for Enron dataset using dIRM. [sent-344, score-0.28]
87 First, the CEO of Enron America stayed at cluster 9 in May (t = 5). [sent-345, score-0.254]
88 Next, the founder of Enron was a member of the cluster in August t = 8. [sent-346, score-0.295]
89 Finally, the COO belongs to the cluster in October t = 10. [sent-348, score-0.254]
90 4 (b) presents the time evolutions of the cluster memberships; i. [sent-351, score-0.351]
91 the number of objects belonging to each cluster at each time step. [sent-353, score-0.384]
92 For example, the volume of cluster 6 (inactive cluster) decreases as time evolves. [sent-356, score-0.283]
93 On the contrary, cluster 4 is stable in membership. [sent-358, score-0.277]
94 6 Conclusions We proposed a new time-varying relational data model that is able to represent dynamic changes of cluster structures. [sent-361, score-0.708]
95 The dynamic IRM (dIRM) model incorporates a variant of the iHMM model and represents time-sensitive dynamic properties such as split & merge of clusters. [sent-362, score-0.229]
96 Experiments with synthetic and real-world time series datasets showed that the proposed model improves the precision of time-varying relational data analysis. [sent-364, score-0.445]
97 Learning systems of concepts with an infinite relational model. [sent-426, score-0.35]
98 The enron corpus: A new dataset for email classification research. [sent-431, score-0.31]
99 A Bayesian approach toward finding communities and their evolutions in dynamic social networks. [sent-482, score-0.157]
100 Stochastic relational models for large-scale dyadic data using mcmc. [sent-494, score-0.35]
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