Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in stark contrast to prac- tice where data is inherently dynamic and subject to discrete changes over time. A temporal graph is a graph whose edges are assigned a set of integer time labels, indicating at which discrete time steps the edge is active. In this paper we present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and a natural number ∆, we ask for a coloring sequence for each vertex such that (i) in every sliding time window of ∆ consecutive time steps, in which an edge is active, this edge is properly colored (i.e. its endpoints are assigned two different colors) at least once dur- ing that time window, and (ii) the total number of different colors is minimized. This sliding window temporal color- ing problem abstractly captures many realistic graph color- ing scenarios in which the underlying network changes over time, such as dynamically assigning communication chan- nels to moving agents. We present a thorough investigation of the computational complexity of this temporal coloring prob- lem. More specifically, we prove strong computational hard- ness results, complemented by efficient exact and approxi- mation algorithms. Some of our algorithms are linear-time fixed-parameter tractable with respect to appropriate parame- ters, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH).
Motivated by this observation, we formulate this problem as a joint learning problem. We first learn dynamic graph topology via position and motion of the skeleton and then apply them as a prior to the GCN recognition framework. In particular, we propose a novel Spatio-Temporal Graph Routing (STGR) scheme to model the semantic connections among the joints in a disentangled way. Rather than us- ing fixed human skeleton, two sub-networks are responsible to capture both spatial and temporal dependancies between each two nodes, serving as routers for all nodes. As shown in Figure 1, a spatial graph router (SGR) discovers the con- nectivity relationships among the joints based on sub-group clustering along the spatial dimension. A temporal graph router (TGR) explores the structural information by mea- suring the correlation degrees between temporal joint node trajectories. The spatio-temporal skeleton-joint-connectivity graphs are then fed into ST-GCN in multiple routing ways.
Forecasting the traffic flows is a critical issue for researchers and practitioners in the field of transportation. However, it is very challenging since the traffic flows usually show high nonlinearities and complex patterns. Most existing traffic flow prediction methods, lacking abilities of modeling the dy- namic spatial-temporal correlations of traffic data, thus can- not yield satisfactory prediction results. In this paper, we propose a novel attention based spatial-temporal graph con- volutional network (ASTGCN) model to solve traffic flow forecasting problem. ASTGCN mainly consists of three in- dependent components to respectively model three tempo- ral properties of traffic flows, i.e., recent, daily-periodic and weekly-periodic dependencies. More specifically, each com- ponent contains two major parts: 1) the spatial-temporal at- tention mechanism to effectively capture the dynamic spatial- temporal correlations in traffic data; 2) the spatial-temporal convolution which simultaneously employs graph convolu- tions to capture the spatial patterns and common standard convolutions to describe the temporal features. The output of the three components are weighted fused to generate the fi- nal prediction results. Experiments on two real-world datasets from the Caltrans Performance Measurement System (PeMS) demonstrate that the proposed ASTGCN model outperforms the state-of-the-art baselines.
This paper proposed hand gesture graph convolutional network which is modified from spatial temporal graph convolutional networks for skeleton-based dynamic hand gesture recognition. A special adjacent matrix is designed to be multiplied with feature maps to amend the prop- agating directions of joint weights. Also, the dimensions of joint coordinates are expanded to better use domain knowledge for CNNs. Moreover, the coordinates are nor- malized to make every gesture start from the same posi- tion. Hand gesture graph convolutional network achieves high accuracy on both two challenge datasets with very fast speed. This shows the effectiveness of the proposed method. As for future development, more complex link ways of joints can be designed for a larger dataset. Other domain knowledge can also be introduced.
• To learn the Laplacian matrix at a specific time-of-day dy- namically according to the global and local data compo- nents, we design a deep learning-based Laplacian matrix estimator with detailed theoretical derivation and design basis. The Laplacian matrix estimated in real-time will be sent to the graph convolutional layers for forecasting. The rest of this paper is organized as follows. We first sum- marize the related work of GCNN in Section 2, and in- troduce the background knowledge in Section 3. We then present the technical details of our novel DGCNN model in Section 4. After that, we evaluate the performance of our proposed model through experiments on real-world data sets in Section 5, and conclude our work in Section 6.
In static networks (graphs), the minimization criterion is typi- cally just the number of channels (colors) used. In a mobile net- work (temporal graph), however, there are two parameters of inter- est: the total number of colors used, and the number of times a node has to be re-assigned a color due to conflicts caused by mobility. In most real-life networks, a re-assignment of time slot or frequency requires the exchange of control packets which reduces data capac- ity. More crucially, it may cause packet drops since the distributed algorithms that implement a re-assignment of the channel take time to converge. Note that even the addition of a single edge can cause a network-wide ripple effect of re-coloring.
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This paper presents a graph-based correlated topic model (GCTM) to learn and analyse motion patterns by trajectory clustering in a highly cluttered and crowded en- vironment. Unlike previous works that depend on scenes prior, we extract trajectories and apply a spatio-temporal graph (STG) to uncover the spatial and temporal coherence between the trajectories during the learning process. It ad- vances the CTM by integrating a manifold-based clustering as initialization and iterative statistical inference as opti- mization. The output of GCTM are mid-level features that represent the motion patterns used later to generate trajec- tory clusters. Experiments on two different datasets show the effectiveness of the approach in trajectory clustering and crowd motion modelling.
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triangle assigned different time-labels is a time-triangle. Thus, if we insist on different time-labels, the appearance of two adjacent edges excludes the appearance of some third edge if one wants to keep the temporal graph time-triangle free. On the other hand, if one picks some matching, then those edges can appear any possible number of times and at any possible order. These indicate that if one wants to keep a temporal graph time-triangle free then he/she should sacrifice either dynamicity or connectivity (similar things hold in the more general requirement of keeping the graph time-acyclic). This trade-off needs a precise characterization. Moreover, in the temporal- graph design problem (discussed in Section 6) there is great room for approximation algorithms (or even randomized algorithms) for all combinations of optimization parameters and connectivity constraints, or even exact polynomial-time algorithms for specific graph families. Also, though it has turned out to be a generic lower-bounding technique related to the existence of a large edge- kernel in the underlying graph G, we still do not know whether there are other structural properties of the underlying graph that could cause a growth of the temporality (i.e. the absence of a large edge-kernel does not necessarily imply small temporality). Another thing that we do not know is whether a (3/2)-factor is within reach either for the general TTSP(1,2) or for the special case with lifetime restricted to n. What we are looking for here is a new direct approximation algorithm, some better reduction, or the harder way of improving the known approximations for Maximum Independent Set in k-claw free graphs or for Set Packing . It would also be interesting to know how the generic metric TSP problem behaves in temporal graphs. Is there some temporal analogue of triangle inequality (even a different assumption) that would make the problem approximable? Is there some temporal analogue of symmetry (e.g
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Single-labeled and multi-labeled Temporal Graphs. The model of temporal graphs that we consider in this work has a direct relation with the single-labeled model studied in [KKK00] as well as the multi-labeled model studied in [MMCS13]. The main results of [KKK00] and [MMCS13] have to do mainly with connectivity properties and/or cost minimization parameters for temporal network design. In this work we study temporal graphs from a statistical view and mainly focus on how fast we expect to arrive at a target vertex in a temporal graph. In [KKK00], a temporal path is considered to be a path with non-decreasing labels on its edges. In this work, we follow the assumption of [MMCS13] and consider a temporal path to be a path with strictly increasing labels. This choice is also motivated by recent work on dynamic communication systems, in which if it takes one time unit for the transmition of a data packet over a link, then a packet can only be transmitted over paths with strictly increasing labels.
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(Laokulrat et al., 2013), and NavyTime (Cham- bers, 2013), use better designed rules or more fea- tures such as syntactic tree paths and achieve bet- ter results. However, the decisions made by these (local) models are often globally inconsistent (i.e., the symmetry and/or transitivity constraints are not satisfied for the entire temporal graph). Integer linear programming (ILP) methods (Roth and Yih, 2004) were used in this domain to enforce global consistency by several authors including Bram- sen et al. (2006); Chambers and Jurafsky (2008); Do et al. (2012), which formulated TLINK ex- traction as an ILP and showed that it improves over local methods for densely connected graphs. Since these methods perform inference (“I”) on top of pre-trained local classifiers (“L”), they are often referred to as L+I (Punyakanok et al., 2005). In a state-of-the-art method, CAEVO (Chambers et al., 2014), many hand-crafted rules and machine learned classifiers (called sieves therein) form a pipeline. The global consistency is enforced by inferring all possible relations before passing the graph to the next sieve. This best-first architecture is conceptually similar to L+I but the inference is greedy, similar to Mani et al. (2007); Verhagen and Pustejovsky (2008).
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CAEVO, a CAscading EVent Ordering archi- tecture (Chambers et al., 2014) use sieve-based ar- chitecture (Lee et al., 2013) for temporal event or- dering for the first time. They mix multiple learn- ers according to their precision based ranks and use transitive closure for maintaining consistency of temporal graph. Mirza and Tonelli (2016) re- cently propose CATENA (CAusal and TEmporal relation extraction from NAtural language texts), the first integrated system for the temporal and causal relations extraction between pre-annotated events and time expressions. They also incorpo- rate sieve-based architecture which outperforms existing methods in temporal relation classifica- tion domain. We make use of CATENA for tem- poral graph construction in our work.
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Skeleton-based Action Recognition. With reliable skele- ton data extracted by robust pose estimation algorithms from depth sensors (Shotton et al. 2011) or a single RGB cam- era (Cao et al. 2017; Xiu et al. 2018), skeleton-based ac- tion recognition draws more and more attention from re- searchers. Deep learning has been widely used in model- ing the spatial and temporal patterns of skeleton sequences in this field. Many methods use RNNs due to its advan- tage for learning long-term sequence data (Song et al. 2017; Zhang et al. 2017). Spatio-temporal graph which models the relationship of human body components (spine, arm and leg) has also been introduced into RNNs (Jain et al. 2016), but it should be more effective to model the hu- man parts in a finer way with every joint into the graph. CNNs has shown its superiority to RNNs owing to the par- allelization over every element in a sequence and simpler training process. Skeleton sequences are manually trans- formed into images (Liu, Liu, and Chen 2017) to feed into CNNs (Li et al. 2017), which obtain promising per- formance in action recognition. Nevertheless, GCNs have shown more promising results (Yan, Xiong, and Lin 2018; Tang et al. 2018), because the images used by CNNs cannot fully describe the topology structure of skeletons. Conven-
Integrated Crisis Early Warning System (ICEWS) is a repository that contains political events with a specific timestamp. These political events relate entities (e.g. countries, presidents...) to a num- ber of other entities via logical predicates (e.g. ’Make a visit’ or ’Express intent to meet or ne- gotiate’). Additional information can be found at http://www.icews.com/ . The repository is organized in dumps that contain the events that occurred each year from 1995 to 2015. We cre- ated two temporal KGs out of this repository, i) a short-range version that contains all events in 2014, and ii) a long-range version that contains all events occurring between 2005-2015. We re- fer to these two data sets as ICEWS 2014 and ICEWS 2005-15, respectively. Due to the large number of entities we selected a subset of the most frequently occurring entities in the graph and all facts where both the subject and object are part of this subset of entities. We split the facts into training, validation and test in a pro- portion of 80%/10%/10%, respectively. The pro- tocol for the creation of these data sets is identi- cal to the onw followed in previous work (Bor- des et al., 2013). To create YAGO15 K , we used F REEBASE 15 K (Bordes et al., 2013) (FB15 K ) as
Although MDCOP Graph Miner algorithm improves the efficiency in mixed-drove spatiotemporal co-occurrence patterns detection, but for large spatial and temporal attack data processing and analysis capabilities, or lack of. In order to solve this problem, on the basic of MDCOP Graph Miner, we presented an improved MDCOP Graph Miner algorithm (the LDMDCOP Graph Miner) which can deal with large spatiotemporal at- tack data sets. LDMDCOP Graph Miner takes MDCOP Graph as the storage structure for close relationships between instances, and storage MDCOP Graph on the hard disk. In the process of calculation, only the necessary attack data is loaded into memory, so as to solve attack data storage problems. Meanwhile, LDMDCOP Graph Miner makes index for MDCOP Graph, which improves the data query efficiency. Experiments show that LD- MDCOP Graph Miner algorithm can not only get the complete and correct mixed-drove spatiotemporal co-occurrence patterns detection, as well as have high computational effi- ciency.
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We used the arc length function and its inverse in order to introduce a new way to graph functions by curving the axes as shown in Figure 10 and Figure 12. Using this, we found a more conveniently method to solve the temporal equation and then we changed the fourth step of the algorithm given in the first part in order to graph each component of the solution of Equation (1) as given in the mentioned figures. Following this algorithm, we solved the harmonic oscillator, the pendulum, the particle under the action of two elastic springs and Kepler’s problems. We were able to solve the pendulum and the particle under the action of two elastic springs problems without using elliptical integrals and to see the periods of both problems as the length of a curve given in Equa- tions (43) and (58) respectively. Then, using this fact, we approximate the periods of both problems in Equations (46) and (61) with a relative error less than 0.17. Finally, in Kepler’s problem, we solved the trajectory equation and we proved that the solution describes an ellipse with focus in the origin (for some values of the energy). We could obtain the semi-major and semi-minor axes, the center of the ellipse and the orbital period by using this formalism.
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In conclusion, the relationship between apparent diffusion coefficient values and memory perfor- mance is complex. MR diffusion abnormalities af- fect not only the mesial temporal lobe structures but possibly the whole brain in patients with tem- poral lobe epilepsy. Our preliminary findings sug- gest that diffusion abnormalities within the brain are related to a patient’s cognitive function and that a negative correlation exists between hip- pocampal apparent diffusion coefficients and both verbal and visuospatial memory scores, particularly on the left. Further study is needed to elucidate the mechanism by which diffusion abnormalities occur in these patients and to determine whether tissue water diffusivity in the hippocampus can be used as an indicator of cognitive function in patients with temporal lobe epilepsy.
Moreover, we evaluated TCDF on two simulated financial time series datasets with complex underlying causal structures that include confounders, feedback loops and self-causation. The learnt graphs by TCDF were close to the ground truth, with F1-scores that varied from 0.86 to 0.59. TCDF outperformed tsFCI and TiMINo in discovering causal relations, and had comparable or better accuracy than PCMCI for both stationary and non-stationary data. Our experiments also showed that TCDF discovered 82% to 100% of the delays correctly, which was comparable with the delay discovery results of existing methods. However, we found that the accuracy of TCDF decreases drastically for non-linear data. Although TCDF still per- formed comparable or better than existing methods when applied to non-linear financial data, we think it is worthwhile to test other activation functions in the AD-DSTCN architecture which might increase accuracy. Lastly, TCDF includes a novel algorithm to detect the presence of hidden confounders. Our experiments show that, in contrast to existing temporal causal discovery methods, TCDF can successfully discover the presence of a hidden confounder when the time delays to the confounder’s e↵ects are equal.
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c = 100 |E| . Also note the noise in temporal Katz centrality rank distance curves due to the effect of the most recently selected edges, as described in Section Convergence properties. To summarize our experiments in Fig. 9, we considered the behavior of temporal Katz centrality with different parameters as well as temporal PageRank after the two changes in sampling distribution marked by vertical bars in the Figure. We observed that tempo- ral PageRank forgets the old distribution very slow, while temporal Katz centrality very quickly becomes similar to the new static distribution. The best parameter for temporal Katz centrality is a weak decay c = |E| 1 , which is still sufficient to forget the old distribution but gives less fluctuation compared to the very highly adaptive, stronger decay versions with larger values of c.
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The inner point number i(G) of a planar graph G is the minimum number of points not belonging to the boundary of the exterior region in any embedding of G in the plane. A graph G is said to be k-minimally nonouterplanar if i(G) = k, k ≥ 1. This concept was introduced by Kulli in . A graph is outerplanar if i(G) = 0. A 1-minimally nonouterplanar graph is called minimally nonouterplanar, see . The concepts of outerplanar and minimally nonouterplanar were studied, for example, in [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].
D(G) adjacent if u ∈ V and v is a minimal dominating set in G containing u. This concept was introduced by Kulli et al in . Several graph valued functions in graph theory were studied, for example, in [4, 5, 6, 7, 8, 9, 10, 11, 12,13, 14, 15, 16] and also several graph valued functions in domination theory were studied, for example, in [17, 18, 19, 20, 21, 22, 23, 24, 25].