• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Spatiotemporal Input Control: Leveraging Temporal Variation in Network Dynamics

    2022-04-15 04:20:16YihanLinJiaweiSunGuoqiLiGaoxiXiaoSenior
    IEEE/CAA Journal of Automatica Sinica 2022年4期

    Yihan Lin, Jiawei Sun, Guoqi Li,, Gaoxi Xiao, Senior,

    Changyun Wen, Fellow, IEEE, Lei Deng, Member, IEEE, and H. Eugene Stanley

    Abstract—The number of available control sources is a limiting factor to many network control tasks. A lack of input sources can result in compromised controllability and/or sub-optimal network performance, as noted in engineering applications such as the smart grids. The mechanism can be explained by a linear timeinvariant model, where structural controllability sets a lower bound on the number of required sources. Inspired by the ubiquity of time-varying topologies in the real world, we propose the strategy of spatiotemporal input control to overcome the source-related limit by exploiting temporal variation of the network topology. We theoretically prove that under this regime,the required number of sources can always be reduced to 2. It is further shown that the cost of control depends on two hyperparameters, the numbers of sources and intervals, in a trade-off fashion. As a demonstration, we achieve controllability over a complex network resembling the nervous system of Caenorhabditis elegans using as few as 6% of the sources predicted by a static control model. This example underlines the potential of utilizing topological variation in complex network control problems.

    I. INTRODUCTION

    NETWORKED structure is a common feature in natural and artificial systems, many of which contain complex topology and higher-order interactions among nodes. The analysis of empirical networks has inspired research fields such as community detection [1], influence maximization [2],and information dissemination, which in return boost applications like epidemic prediction [3]–[5] and recommender systems [6]. Many complex networks are also control systems of practical value [7]–[11], which inspired the study of network control, including theoretical efforts to understand network behaviors under control [12]–[14] and applicationdriven work on specific control solutions or algorithms[15]–[18]. However, part of these efforts are confounded by the complexity originating from network topology, dynamics,and sometimes mere scale [13], [19], [20]. The unprecedented network scale and diversity encountered in modern engineering call for control strategies that are not dependent on specific feature or structure but can handle network complexity in a general way.

    Many networks found in nature and human societies show highly dynamic interaction patterns [21]–[24] and are referred to as temporal networks. Among them, there are systems that function robustly under control and can offer inspiration for new control principles. It is suggested that the time-varying topology is essential to a network’s efficiency and flexibility[25]. Although the majority of control problems are studied using approximated linear systems, e.g., the linear timeinvariant (LTI) model [26], [27], there are fundamental differences between a temporal system and what a linear model can capture. An LTI system is governed by the equationx˙(t)=Ax(t)+Bu(t), which assumes slow dynamics of the network connection specified by matricesAandBand approximates them as constant matrices, whereas in the temporal case, the dynamics of these matrices can have a characteristic time scale comparable tox(t) andu(t).

    Despite the attractive properties, there is yet to be a general framework to exploit the advantages of temporal structures for network control. Here, we propose a control strategy featuring time-dependent input connection (i.e., node-source connection) and by modeling and simulation, evaluate its potential in controlling large complex networks. Specifically,we consider a model that extends from the linear network model but incorporates a piece-wise matrixBthat shows variable connectivityBkin different time periods:x˙(t)=Ax(t)+Bkuk(t)fort∈[tk?1,tk) (Fig. 1). In this model,the time intervals, the input matrices, and the control input vectors are open to optimization, which resembles the realworld cases where input signals can be directed to different sets of nodes during control [28]–[31]. Here, matrixBis assumed to evolve at a much higher rate than the adjacency networkAitself, andAis modeled as a constant. Despite the simplicity, the model captures the fast and slow dynamics of network structures and is expected to offer clues on the general principles of control with temporal input connections.

    Fig. 1. An example comparing the spatiotemporal input control with static network control. (a)–(b) Schematics illustrating an 8-node network (nodes are colored in light blue; directed edges are represented by black arrows) made controllable by 2 sources (dark blue) in 4 time intervals or by 3 sources via static input connection. (c)–(d) Optimized node trajectories of the 2 control strategies. Although the target is reached at 8.4 seconds in both cases, the spatiotemporal input gives much more compact trajectories.

    We term the proposed control strategy instantiated with this model as thespatiotemporal input control, given that it integrates the spatial and temporal patterning of the nodesource connection. It is proved that given a sufficient number of time intervals, controllability can always be achieved with 2 control sources; in many cases, 1 source is sufficient. This lifts the lower bound predicted by the LTI model, which can be computed using the maximum matching algorithm [32].The difference in controllability could be explained by the delivery of control signals to the network: in contrast to the LTI model where all nodes are under effective control [32],[33], the spatiotemporal input controls different subsets of nodes in different time intervals.

    We further optimize the input matrices and signals with a proposed semi-analytic method to generate control solutions.By simulating more than 1000 randomly generated networks,we find that the cost of control, or the control energy, drops with the number of intervals and the number of sources. The dependence of energy on intervals and sources could be summarized into a trade-off relationship, where more intervals may compensate for the lack of sources. Practically, this indicates a promising strategy to tackle large complex networks if dividing intervals can be done with a much lower cost than adding control sources. To demonstrate the potential of the method in handling complex networks, we present a test case using the network topology extracted from the nervous system ofCaenorhabditis elegans[34]–[36], and show that controllability can be achieved with fewer than 6% of the nodes computed by the maximum matching algorithm. A fast construction method is proposed to efficiently generate input connections for large networks like this.

    II. MODEL

    A. Temporal Network Model

    Consider the following piece-wise linear model with temporal input connection. Specifically, it contains a constant directed network ofNnodes controlled byMindependent sources. The time duration is split intoKintervals, i.e.,[tk?1,tk),k=1,...,K. For simplicity, we only consider intervals of equal length, giving Δt=tK/Kfor all intervals.The dynamics is determined by the following system of ordinary differential equations:

    B. Controllability

    A prerequisite for successful control is the system’s controllability [32]. Given proper input, a controllable system can access any target in the RNspace starting from any initial state. In contrast to the LTI system of which controllability is determined by the Kalman rank condition [37], networks with temporal input structure have a slightly different condition of controllability. Assuming that the control sources are homogeneous and can be connected to any node at any time,the following lemmas give the controllability condition of the model in (1).

    C. Control Energy as an Objective Function

    D. Optimal Control Problem

    III. METHODS

    A. Constructing Input Connections for Controllability

    For a classical LTI network, a graph-based method,maximum matching, gives a node-source mapping that uses the fewest control sources to ensure controllability. However,this lower bound may not hold true for temporal input connections given that a control source can deliver signals to multiple sets of nodes by switching connections across intervals. In this section, the lower bound of control sources is recalculated by constructing input matrices. The following definitions and lemmas are presented for the proof of Theorem 1.

    Algorithm 1 Construct Input Matrices If the Adjacency Matrix A Has Complex Eigenvalues{Bk}As{tk}Kk=1 Require: ,{Bk}Kk=1 Ensure:function MATRIXGENERATOR ( , )K ≥γ+「ζ As{tk}Assign 2 ■,γ= β 2 As Jt=P?1AsP Real Jordan canonical form transform of : .Use j as the row number as the Jordan block.k=1 →K kth for do k ≤「ζ if then Cbk=zeros(2,j)2■Cbk[1:2,j]=ones(2,2)Ok=zeros(2,j)Ck=[O1 ··· Cbk ··· Oχ]T end if「ζ 2■+γ ≥k>「ζ if then Cbk=zeros(1,j)2■Cbk[1,j]=1 Ok=zeros(1,j)Ck=[O1 ··· Cbk ··· Oχ]T end if k>「ζ if then Ck=O 2■+γ end if{Bk}Kk=1=P×{Ck}Kk=1 Calculate .Note: the computed should have no more than 2 columns end for end function Bk(k=1,...,K)

    Algorithm 2 Construct Input Matrices If the Adjacency Matrix A Has Only Real Eigenvalues{Bk}As{tk}K Require: ,{Bk}K k=1 Ensure:k=1 function MATRIXGENERATORR K ≥ζ(As,{tk})Assign As Js=P?1AP Real Jordan canonical form transform of : .Use j as the row number as the Jordan block.k=1 →K kth for do Cbk=zeros(1,j)Cbk[1,j]=1 Ok=zeros(1,j)if then Ck=[O1 ··· Cbk ··· Oζ]T 1 ≤k ≤ζ end if k>ζ if then Ck=0.end if end for{Bk}K k=1=P×{Ck}K Calculate .Bk(k=1,...,K)k=1 Note: the computed should have only 1 column.end function

    B. Optimization

    The functionalEin (6) is introduced both as a metric for evaluating the control input and as an objective for problem(7). In this subsection, problem (7) is solved, and in the following sections, the minimized energy is calculated under various conditions to investigate the system’s hyperparameter dependence. This is of practical interest since energy is often an analogy to real-life control effort.

    After solving (7), we seek to find a general relationship between energyE, total number of intervalsK, and total number of sourcesM. In order to compare these cases of different hyperparameters, the factors of specific control input, namely {Bk} and {uk(t)}, need to be eliminated by optimization. In other words, it is much less meaningful to compare the energy of systems that are far away from their optima. Therefore, a series of optimal control problems in the form of (7) with differentKs andMs are solved to obtain the minimized energy. Note that the case ofK=1 reduces to an LTI model.

    To start with, we have the following theorems and methods for optimization.

    Theorem 2:Givenx˙(t)=Akx(t)+Bkuk(t), the input vector minimizingEkof time interval [tk?1,tk) is given by

    Hence we have

    Proof:Plugging (8) and (10) into (6) yields,

    update , = ControlEnergy(A, , , ).^El^Γl{Bl+1 k } tkΛl?Λ =^ΓlΛl compute .?E wlk compute projected gradient using (49) and obtain unit vector .ˉwlk apply Adam algorithm to determine the step size and update .Δ=∑Λl+1=Λl ?μlkˉwlkμlk k ?Blk‖+‖Λl+1 ?Λl‖compute .‖Bl+1 end while end function k

    C. Coordinate Descent Algorithm and Simulation

    In order to combine the optimization of the input matrices{Bk}and the vector Λ that are related tox(t) and {uk(t)}, a coordinate descent algorithm is proposed based on the methods described in the last subsection to solve the optimization problem in (38), as shown in Algorithm 3.Specifically, {Bk} and Λ will be updated in an alternating fashion in each iteration. Note that this algorithm also applies to more general cases where bothAandBare piece-wise constant matrices.

    To implement the algorithm, {Bk} can be initialized using the constructive methods described in Algorithms 1 and 2 to ensure controllability. In order to avoid converging to suboptimal local minima, we launch gradient descent at multiple initial states with different step sizes and take the case with the lowest energy as an approximation of the optimal solution.As a test case, control energy is optimized for an example network given different numbers of intervals, and a significant decrease ofEwith an increasingKis observed (Fig. 2).

    To better illustrate the effect of optimization, a video is provided showing examples of randomly generated networks containing 8 nodes and 10% of all possible connections among nodes (see demo1.mp4). The dynamics of the system is visualized using 4 two-dimensional trajectories representing the state of the 8 nodes. The first and the last 4 nodes are visualized as thexandycoordinates, respectively.

    D. Control by Network Partition

    It is challenging to solve control problems of large networks. In particular, the computational complexity of the proposed Algorithms 1 to 3 scale with the network size. In addition, the split of time into several intervals also brings about the challenge of temporal complexity to the finite computational resources one can possibly utilize. In this subsection, we discuss the complexity of the proposed algorithms and present a method to circumvent this challenge while constructing and optimizing input connections.

    In Algorithm 1, the time complexity of calculating a Jordan canonical form isO(N4) [43]. Other operations in Algorithm 1 are linear except for matrix multiplication, of which the time complexity isO(N3). Regarding space complexity, the most costly operation is also calculating the Jordan canonical form,whose complexity is bounded byO(N2).

    Fig. 2. More time intervals result in lower control energy in randomly generated 8-node networks. (a) Histograms of control energy before and after optimization using the proposed coordinate descent algorithm. Networks are randomly generated and are driven from the origin to the same target state within K = 4 time intervals. Although the specific network topology has a substantial impact on the energy, the algorithm overall reduces energy by multiple orders of magnitude. (b) Histograms of optimized energy of more than 1000 randomly generated 8-node networks given different number of time intervals. The total time is divided into K = 4, 5, 7 or 8 intervals. The increase in the number of intervals significantly reduces control energy.

    Fig. 3. An example illustrating the method of control by network partition. A 16-node temporal network is split into two 8-node subnetworks (shown in green and pink), with each controlled by one input source (s1 or s2). The 2 subnetworks, with the internal edges (blank arrows), are controllable in 5 time intervals.The combined 16-node network is also controllable regardless of the edges connecting the 2 subnetworks (grey allows), which is consistent with Theorem 5.

    The time complexity of Algorithm 3 is hard to determine due to the nonlinear operations such as numerical integration and matrix exponential, which is also dependent on numerical precision. When ignoring the unpredictable iterative steps of gradient descent, the theoretical time complexity is bounded by the calculation of the energy gradient, which isO((NK)3×N×M×K)=O(N4K4M). The space complexity isO(N2K2), which is due to the storage of the gradient.

    Remark 1:Note that the above theorems only guarantee structural controllability [45], [46], which is a slightly weaker condition than controllability. However, it is practically useful for controlling a temporal network model, since the probability of having a set of weights that make a network with structural controllability uncontrollable is often 0. Thus,without solving the original optimal control problem, which can be costly, we may instead solve a series of simpler optimal control problems based on the partition of the large network, although the performance of the combined input signals and connections may vary according to the partition.Nonetheless, this indicates the possibility of controlling a large-scale network by controlling parts of the network, and the general methodology is not limited to temporal networks.

    IV. NUMERICAL EXAMPLES

    A. The Trade-Off Between Energy, Time Intervals and Input Sources

    So far, the results regarding controllability and energy in the above sections suggest two pairs of trade-offs, namely betweenKandMand betweenEandK. Here, by simulating an example network, we investigate the relations between energy and the two important hyperparameters, in search of general principles governing temporal network control. By solving the optimal control problems with various combinations ofMandK, we are able to approximately determine the minimal energy in each case, based on which we provide a visualization of the hyperparameter space by fitting a 2D surface from the discrete data points (Fig. 4).Although the results are based on a simple network model, it may explain the power of temporal input connections, which can be partly boiled down to the trade-off among energy,sources and intervals. Compared with static input connections,the spatiotemporal input introduces an extra dimension ofKto the hyper-parameter space, allowing new possibilities of reducing energy or the total number of sources.

    B. A Test Using the Neural Network of Caenorhabditis Elegans

    Complex networks, a common structure studied across fields, typically have a large size and a non-trivial topology.Here, we consider the applicability of the spatiotemporal input as a general strategy for controlling such networks.Particularly, based on the above-mentioned trade-off relations,we focus on the choice ofMandKin search of a proper balance.

    A biological network from the nervous system ofC. elegansis used as an example. Its complexity and the extensive research on its cellular connections make it a perfect test case[46]. It is known to be a sparse network of 2177 nodes and 4007 edges (multiple edges are counted as 1 if they are between the same 2 neurons). In the simulation, the network is treated as a directed graph, while the biological context is neglected. First, we demonstrate the feasibility of achieving controllability and optimizing control over a small 30-node subnetwork. Then, we show how the entire network can be controlled by a small number of control sources. The partitioning method introduced in the previous section is applied to the full network.

    Fig. 4. (a) Visualization of the trade-off relationship among energy, the number of sources, and the number of time intervals. The solid dots represent simulation results from an 8-node example network, while the circles are obtained by spline interpolation from the simulated data points due to limited computational power. (b) Projection of (a) on the M-K plane. The dashed line in the lower-left corner shows the boundary of controllability. (c) Projection of (a)on the E-K plane. (d) Projection of (a) on the E-M plane.

    Fig. 5. Optimized trajectories of a 30-node sub-network extracted from the nervous system of C. elegans. (a)–(b) Trajectories of all 30 nodes controlled by 1 or 3 sources, respectively. The right panels provide zoom-in views near the target states, to which the relative error of each node is contained within 0:1%. (c) A graph of the sub-network. The nodes are labeled by their cellular code in [47]. Each node is colored differently according to its degree.

    1) Test on a Subnetwork:A small subnetwork of 30 cells with sparse connection is extracted from the large neural network of an adultC. eleganshermaphrodite. The cells include ADELa1~ADELb5, AS01a1~AS01b5, and MDL05a1~MDL05b5, which together form 23 edges in total,with an average in-degree of 0.756 (Fig. 5(c)). According to Theorem 1, the network is controllable by 1 source. A set of input matrices {Bk} are generated that spans acrossK=11 intervals. The optimal control problem is solved, giving the trajectories in Fig. 5(a). Another set of parametersM=3 andK=4 yield a much lower energy after optimization (Fig. 5(b)).In contrast to these cases, the maximum matching algorithm predicts a lower bound of 11 sources connected to cells ADELa1, ADELa2, ADELa3, ADELa4, ADELa5, MDL5a5,AS01a1, AS01a2, AD01a3, AD01a4 and AD01a5,respectively.

    Fig. 6. Graphical demonstration of the node-node and node-source connections in the controlled neural network of C. elegans. The first and last 3 intervals out of 34 are shown here. The nodes and the sources are colored in blue and black, respectively. Each interval is of 1 second. See connection.mov in the supplementary materials for more details.

    It is noted that given 3 sources, ‖x(t)‖ is smaller by roughly an order of magnitude on overage than the 1 source case,while the overall “complexity” of control input, which can be thought of asM×K, is similar. It is possible that a non-trivial interplay exists between the numbers of sources and intervals in terms of energy scaling, which requires further studies.

    2) Test on the Entire Network:It is important to understand how nature exploits limited resources and energy for efficient manipulation of complex systems. In living matter, it is reasonable to speculate that systems deliver or respond to control signals in a time-dependent fashion with only a small number of sources being used or activated at the same time,given examples like the sparse and temporal firing patterns in cortical activities [48]–[51]. Here we show that in agreement with this intuition, the spatiotemporal input can achieve controllability over the entire network of 2177 nodes by frequently switching the connections from a limited number of sources.

    According to Fig. 4, we can balance the numbers of sources and time intervals given a certain level of energy. For small networks, this can be done by generating random instances of{Bk}until the controllability condition (3) is satisfied. Here,given a large network, we construct input connections by the method of network partition, as introduced in the previous section. For each subnetwork, the input connections are generated by Algorithm 1 or 2.

    The full network is split intoR=44 subnetworks, each of which is controllable by 1 source, givingM=R=44. This results inK=34 time intervals. The node-node and nodesource connections are illustrated in Fig. 6. Controllability is tested directly using equation (3). As a comparison, the same network requiresM≥754 nodes if controlled by static input.Therefore, the spatiotemporal input strategy saves more than 94%of the control sources. It is worth noting that there are numerous ways of partitioning the network, and the decision will likely be reflected in the control energy.

    To accelerate the simulation, we increaseRto 223 and still assumeM=R. The network is controllable withinK=5 intervals. Although we have successfully achieved controllability, the trajectories are not optimized due to the incapability of the optimization algorithm to overcome network scaling, which is a typical phenomenon for gradient descent algorithms, known as the curse of dimension. Due to the limited computational resources, it is difficult to achieve an optimal result within an acceptable amount of time.Instead, a few iterations of gradient descent are carried out for partial optimization (Fig. 7(a)). A short animation(connection.mov) is provided in the supplementary materials to show the temporal sequence of the node-node and nodesource connections for this case.

    Fig. 7. Simulated dynamics of the C. elegans neural network controlled by 223 sources in 5 intervals. The target state is randomly generated. Trajectories for 2177 nodes are plotted in different colors. A zoom-in view near the target state is provided in the right panel, underlining the accuracy of the control process.

    V. CONCLUSIONS

    This work explores the idea of using temporal input connections in complex network control. Specifically, a piecewise linear model is studied, which yields a novel control strategy termed the spatiotemporal input control. Regarding controllability, we prove that as few as 2 input sources are sufficient for an arbitrary network given sufficient intervals.Regarding the energy cost of control, we show by simulation that having more intervals can effectively reduce control energy after optimizing the input. Further simulation suggests that the advantage of temporal topology may be attributed to the trade-offs between energy, control sources and time intervals. By making the input matrix temporal, we are essentially introducing an extra dimension of intervals to the hyperparameter space, expanding the space of possible control solutions. It is worth noting that although the above properties are revealed with a relatively simple model, the trade-off relationships may generally apply to different forms of temporal topology.

    The example of theC. elegansnervous system underlines the potential of temporal input connections in controlling large networks. In real-world applications where the time intervals can be engineered, this strategy can prevent the number of input sources from scaling proportionally with the network size and thus becoming unaffordable. However, solving the optimal control problem can pose challenges on computational resources due to the time complexity and the gradient descent method. It remains an interesting problem to develop a faster algorithm for optimizing the temporal topology in general.

    In summary, the paper contributes a) conceptually by suggesting a shift from static to temporal input connections,which is tested with a piece-wise linear model; b) theoretically by proving that the lower bound of the number of input sources can be relaxed to 1 or 2, and presenting the trade-off relationship among energy, sources and intervals; c)methodologically by solving the optimal control problem for temporal networks and presenting the methods for constructing temporal input matrices. The results suggest a promising control strategy for complex networks and a possible explanation for the highly efficient control systems observed in nature. The methods proposed with our model may also be of practical value in future applications.

    APPENDIX

    And according to Lemmas 1 and 2, the controllable space of the system in (1) is

    国产精品乱码一区二三区的特点| 天堂动漫精品| 久久香蕉精品热| 国产亚洲精品综合一区在线观看 | 香蕉av资源在线| 婷婷六月久久综合丁香| 久久狼人影院| 男女做爰动态图高潮gif福利片| 1024香蕉在线观看| 免费高清视频大片| 国产一区二区三区在线臀色熟女| 日本免费a在线| 亚洲五月天丁香| 国产欧美日韩一区二区精品| x7x7x7水蜜桃| 日本在线视频免费播放| av片东京热男人的天堂| 成人三级做爰电影| 精品一区二区三区四区五区乱码| 99久久国产精品久久久| xxxwww97欧美| 丰满人妻熟妇乱又伦精品不卡| 精品国产乱子伦一区二区三区| 男男h啪啪无遮挡| 不卡av一区二区三区| 黄网站色视频无遮挡免费观看| 国产精品久久久久久亚洲av鲁大| 亚洲第一av免费看| 国产精品,欧美在线| 久久中文看片网| 亚洲久久久国产精品| 老汉色av国产亚洲站长工具| 俄罗斯特黄特色一大片| 亚洲精品久久国产高清桃花| 无人区码免费观看不卡| 狂野欧美激情性xxxx| 午夜久久久在线观看| 免费在线观看日本一区| 9191精品国产免费久久| 老司机在亚洲福利影院| 欧美亚洲日本最大视频资源| 久久婷婷成人综合色麻豆| 黑人巨大精品欧美一区二区mp4| 一进一出抽搐gif免费好疼| 亚洲国产精品sss在线观看| 99热只有精品国产| 欧美激情极品国产一区二区三区| 亚洲七黄色美女视频| 一级作爱视频免费观看| 欧美黄色淫秽网站| 国产aⅴ精品一区二区三区波| 99国产精品99久久久久| 久热爱精品视频在线9| av天堂在线播放| 亚洲无线在线观看| 欧美丝袜亚洲另类 | 狂野欧美激情性xxxx| 香蕉久久夜色| 国产三级在线视频| 日日干狠狠操夜夜爽| 精品欧美一区二区三区在线| 欧美精品啪啪一区二区三区| 久久久精品欧美日韩精品| 两性夫妻黄色片| 50天的宝宝边吃奶边哭怎么回事| 成人国产一区最新在线观看| 一夜夜www| 亚洲第一欧美日韩一区二区三区| 黄色成人免费大全| 久久99热这里只有精品18| 一级片免费观看大全| 精品日产1卡2卡| 搡老熟女国产l中国老女人| 99国产极品粉嫩在线观看| 99国产综合亚洲精品| 最新在线观看一区二区三区| 免费无遮挡裸体视频| 日韩精品免费视频一区二区三区| cao死你这个sao货| 精品第一国产精品| 亚洲国产欧美日韩在线播放| 黑丝袜美女国产一区| 欧美色欧美亚洲另类二区| 香蕉久久夜色| 国产又色又爽无遮挡免费看| 一区二区三区精品91| 日本一本二区三区精品| 亚洲av成人不卡在线观看播放网| 最近最新中文字幕大全免费视频| 正在播放国产对白刺激| 欧美一区二区精品小视频在线| 亚洲精品国产精品久久久不卡| 三级毛片av免费| 在线国产一区二区在线| 日韩精品免费视频一区二区三区| 麻豆av在线久日| 国产在线精品亚洲第一网站| 神马国产精品三级电影在线观看 | 99久久精品国产亚洲精品| av电影中文网址| 亚洲国产精品sss在线观看| 亚洲aⅴ乱码一区二区在线播放 | 国产99白浆流出| www.999成人在线观看| 真人一进一出gif抽搐免费| 亚洲 欧美一区二区三区| 久久精品人妻少妇| 久久久久久久精品吃奶| 国产精品一区二区免费欧美| 日本五十路高清| 免费女性裸体啪啪无遮挡网站| 免费看日本二区| 国产黄片美女视频| 欧美激情极品国产一区二区三区| 人妻久久中文字幕网| 少妇 在线观看| 丰满的人妻完整版| av福利片在线| 搡老熟女国产l中国老女人| 啦啦啦免费观看视频1| 免费搜索国产男女视频| 精品一区二区三区av网在线观看| 亚洲美女黄片视频| 桃红色精品国产亚洲av| 色播在线永久视频| 在线永久观看黄色视频| 国产极品粉嫩免费观看在线| 亚洲国产欧美网| 啦啦啦免费观看视频1| 欧美成人一区二区免费高清观看 | 亚洲国产精品久久男人天堂| 精品电影一区二区在线| 99国产综合亚洲精品| 成熟少妇高潮喷水视频| 亚洲成人久久性| a级毛片a级免费在线| 免费看日本二区| 国产精品98久久久久久宅男小说| 国产三级黄色录像| 欧美一级毛片孕妇| 亚洲成人久久性| 国产精品av久久久久免费| 色综合欧美亚洲国产小说| 国产精品免费视频内射| 99精品欧美一区二区三区四区| 久99久视频精品免费| 91在线观看av| 国产av一区二区精品久久| 好男人在线观看高清免费视频 | 亚洲七黄色美女视频| 正在播放国产对白刺激| 男女午夜视频在线观看| 99riav亚洲国产免费| 啦啦啦韩国在线观看视频| av天堂在线播放| 国产精品久久久久久人妻精品电影| 观看免费一级毛片| 亚洲国产毛片av蜜桃av| 制服诱惑二区| 亚洲国产日韩欧美精品在线观看 | 啦啦啦韩国在线观看视频| 国产av在哪里看| 国产精品久久久久久人妻精品电影| 亚洲人成网站高清观看| 亚洲国产毛片av蜜桃av| 中文字幕人妻熟女乱码| 欧美不卡视频在线免费观看 | 中文字幕人妻熟女乱码| 国产精品一区二区精品视频观看| 久久久国产精品麻豆| 久久久国产成人精品二区| 国产精品1区2区在线观看.| 亚洲国产看品久久| 国产黄a三级三级三级人| 91av网站免费观看| 久久久国产成人免费| 精品第一国产精品| 一二三四在线观看免费中文在| 国产在线观看jvid| 叶爱在线成人免费视频播放| bbb黄色大片| 国产99久久九九免费精品| 国产亚洲av高清不卡| 午夜福利一区二区在线看| 夜夜爽天天搞| 亚洲国产中文字幕在线视频| 又黄又爽又免费观看的视频| av中文乱码字幕在线| av欧美777| 亚洲九九香蕉| 最近在线观看免费完整版| 久久热在线av| 亚洲av五月六月丁香网| 精品人妻1区二区| 国产午夜精品久久久久久| 一本一本综合久久| 亚洲午夜精品一区,二区,三区| 成人国产综合亚洲| 搡老岳熟女国产| 51午夜福利影视在线观看| x7x7x7水蜜桃| 精品高清国产在线一区| 熟女电影av网| 亚洲激情在线av| 日韩大码丰满熟妇| 亚洲avbb在线观看| 亚洲午夜精品一区,二区,三区| 中文字幕人成人乱码亚洲影| 叶爱在线成人免费视频播放| 国内精品久久久久久久电影| 国产人伦9x9x在线观看| 搞女人的毛片| 午夜福利在线在线| 男人舔女人下体高潮全视频| 欧美日韩中文字幕国产精品一区二区三区| 给我免费播放毛片高清在线观看| 在线国产一区二区在线| 国产精品久久久人人做人人爽| 十八禁人妻一区二区| 中出人妻视频一区二区| 成人一区二区视频在线观看| 久久精品aⅴ一区二区三区四区| 亚洲一卡2卡3卡4卡5卡精品中文| 国产精品免费视频内射| 91成年电影在线观看| 免费看日本二区| 亚洲av中文字字幕乱码综合 | 夜夜爽天天搞| 91在线观看av| 99精品欧美一区二区三区四区| 国产97色在线日韩免费| 一区二区三区高清视频在线| 男人的好看免费观看在线视频 | 亚洲国产精品sss在线观看| 一本一本综合久久| 黑人巨大精品欧美一区二区mp4| 国产精品av久久久久免费| 亚洲人成网站高清观看| 亚洲国产看品久久| 性欧美人与动物交配| 妹子高潮喷水视频| 亚洲国产欧美一区二区综合| 国产免费男女视频| 久久久久久久午夜电影| 91老司机精品| 欧美黑人精品巨大| 国产单亲对白刺激| 日韩欧美国产一区二区入口| 一本一本综合久久| 亚洲三区欧美一区| 色精品久久人妻99蜜桃| 婷婷亚洲欧美| 日韩视频一区二区在线观看| 中文字幕久久专区| 国产私拍福利视频在线观看| 亚洲精品国产一区二区精华液| 日韩欧美国产一区二区入口| 欧美日韩福利视频一区二区| 一边摸一边做爽爽视频免费| 最近最新免费中文字幕在线| 美女国产高潮福利片在线看| 日本a在线网址| 成人一区二区视频在线观看| 欧美黑人欧美精品刺激| 国产片内射在线| 最好的美女福利视频网| 18禁黄网站禁片午夜丰满| 午夜两性在线视频| 欧美在线一区亚洲| 国产免费男女视频| 欧美日韩亚洲综合一区二区三区_| 精品国内亚洲2022精品成人| 中文字幕人妻丝袜一区二区| xxx96com| 久久精品国产亚洲av香蕉五月| 在线av久久热| 男人舔女人的私密视频| 在线观看日韩欧美| 欧美日韩黄片免| 色播亚洲综合网| 国产亚洲精品一区二区www| 欧美黑人欧美精品刺激| 观看免费一级毛片| 久久精品国产清高在天天线| 亚洲avbb在线观看| 国产又色又爽无遮挡免费看| 亚洲欧美精品综合久久99| 18禁观看日本| 国产色视频综合| 曰老女人黄片| 亚洲九九香蕉| 亚洲av片天天在线观看| 90打野战视频偷拍视频| 男男h啪啪无遮挡| 日本精品一区二区三区蜜桃| xxx96com| 国产91精品成人一区二区三区| 特大巨黑吊av在线直播 | 国产精品久久电影中文字幕| 一a级毛片在线观看| aaaaa片日本免费| 黄色 视频免费看| 中国美女看黄片| 欧美性长视频在线观看| 男女之事视频高清在线观看| 女人爽到高潮嗷嗷叫在线视频| 国产欧美日韩精品亚洲av| 中文在线观看免费www的网站 | 亚洲黑人精品在线| 在线十欧美十亚洲十日本专区| 亚洲无线在线观看| 亚洲国产欧美网| 亚洲精品粉嫩美女一区| 中文字幕av电影在线播放| 国产一卡二卡三卡精品| 中文字幕最新亚洲高清| 少妇被粗大的猛进出69影院| 国产高清videossex| avwww免费| 国产成人欧美在线观看| 午夜福利18| 大香蕉久久成人网| 国产99久久九九免费精品| 午夜福利18| 午夜老司机福利片| 国产精品久久久久久亚洲av鲁大| 日本一区二区免费在线视频| 久久久久久九九精品二区国产 | 国产99久久九九免费精品| 欧美黄色片欧美黄色片| √禁漫天堂资源中文www| 亚洲国产高清在线一区二区三 | 亚洲欧美精品综合久久99| 男女午夜视频在线观看| 久久久久久亚洲精品国产蜜桃av| 神马国产精品三级电影在线观看 | 亚洲av成人一区二区三| 亚洲人成77777在线视频| 成人一区二区视频在线观看| 国产一区二区激情短视频| 久久青草综合色| 黄片小视频在线播放| 国产精品永久免费网站| av欧美777| 老司机福利观看| 丁香欧美五月| 色综合欧美亚洲国产小说| 成人三级黄色视频| 国产亚洲欧美精品永久| 午夜a级毛片| 亚洲av电影不卡..在线观看| 国产主播在线观看一区二区| 国产片内射在线| 少妇被粗大的猛进出69影院| 成人18禁高潮啪啪吃奶动态图| 一区二区日韩欧美中文字幕| 日本三级黄在线观看| 国产高清激情床上av| 老司机福利观看| 18禁黄网站禁片免费观看直播| 色老头精品视频在线观看| 欧美性猛交╳xxx乱大交人| 久久久久久久久中文| av有码第一页| 真人做人爱边吃奶动态| 天堂√8在线中文| 免费在线观看日本一区| 欧美性猛交黑人性爽| 成人精品一区二区免费| 俄罗斯特黄特色一大片| 国产精品乱码一区二三区的特点| 一个人观看的视频www高清免费观看 | 美女高潮到喷水免费观看| 啦啦啦韩国在线观看视频| 国产欧美日韩精品亚洲av| 免费在线观看日本一区| 午夜福利免费观看在线| 黑人操中国人逼视频| 露出奶头的视频| 窝窝影院91人妻| 99在线人妻在线中文字幕| 国产激情欧美一区二区| 亚洲狠狠婷婷综合久久图片| 久久久久免费精品人妻一区二区 | 国产精品野战在线观看| 欧美性猛交黑人性爽| 99久久99久久久精品蜜桃| 亚洲狠狠婷婷综合久久图片| 国产伦在线观看视频一区| 国内毛片毛片毛片毛片毛片| 国产精品国产高清国产av| 人成视频在线观看免费观看| 黄色成人免费大全| 欧美乱妇无乱码| 麻豆久久精品国产亚洲av| 青草久久国产| 亚洲成人久久性| 日本一区二区免费在线视频| 久久久国产欧美日韩av| 精品久久久久久成人av| 婷婷亚洲欧美| 女性被躁到高潮视频| 黄网站色视频无遮挡免费观看| 长腿黑丝高跟| 久久久久久亚洲精品国产蜜桃av| 女人爽到高潮嗷嗷叫在线视频| 久久婷婷人人爽人人干人人爱| 麻豆国产av国片精品| 成在线人永久免费视频| 高清在线国产一区| 淫妇啪啪啪对白视频| 国产亚洲精品第一综合不卡| 国产亚洲av嫩草精品影院| 不卡av一区二区三区| 亚洲av日韩精品久久久久久密| 国产片内射在线| 亚洲国产毛片av蜜桃av| 国产极品粉嫩免费观看在线| 精品少妇一区二区三区视频日本电影| 91av网站免费观看| 国产1区2区3区精品| 国产一卡二卡三卡精品| 校园春色视频在线观看| av中文乱码字幕在线| 国产极品粉嫩免费观看在线| 在线观看www视频免费| 日本 欧美在线| av免费在线观看网站| 在线观看免费日韩欧美大片| 男人舔女人的私密视频| 亚洲免费av在线视频| 亚洲国产看品久久| 两个人看的免费小视频| 90打野战视频偷拍视频| 女性被躁到高潮视频| 在线观看午夜福利视频| 亚洲一区二区三区色噜噜| 在线观看免费午夜福利视频| 亚洲欧美日韩无卡精品| 正在播放国产对白刺激| 国产一区二区激情短视频| 欧美乱码精品一区二区三区| 亚洲av熟女| 一二三四在线观看免费中文在| 成人手机av| ponron亚洲| 在线av久久热| 精品国产美女av久久久久小说| 精品国产乱子伦一区二区三区| 成人亚洲精品一区在线观看| 757午夜福利合集在线观看| www.www免费av| 久久久久久人人人人人| 成年女人毛片免费观看观看9| 亚洲精品中文字幕一二三四区| 特大巨黑吊av在线直播 | 精品国产乱码久久久久久男人| 99精品在免费线老司机午夜| 在线免费观看的www视频| 丰满人妻熟妇乱又伦精品不卡| av在线播放免费不卡| 国产区一区二久久| 日韩欧美 国产精品| tocl精华| 欧美成人午夜精品| 精品久久久久久久末码| 精品福利观看| 成人午夜高清在线视频 | 精品免费久久久久久久清纯| 国产精品亚洲美女久久久| 欧美激情高清一区二区三区| 精品不卡国产一区二区三区| 国产国语露脸激情在线看| 村上凉子中文字幕在线| 日本三级黄在线观看| 一进一出好大好爽视频| 亚洲国产精品久久男人天堂| 成人三级黄色视频| 国产日本99.免费观看| www日本黄色视频网| 久久热在线av| av在线天堂中文字幕| 亚洲色图av天堂| 制服诱惑二区| av中文乱码字幕在线| 两人在一起打扑克的视频| 国产精品久久久久久亚洲av鲁大| 国产成人一区二区三区免费视频网站| 久久婷婷成人综合色麻豆| 禁无遮挡网站| 国产99白浆流出| 欧美色视频一区免费| 亚洲真实伦在线观看| 日本在线视频免费播放| av有码第一页| 老熟妇乱子伦视频在线观看| 亚洲色图 男人天堂 中文字幕| 国产亚洲精品第一综合不卡| 国产精品永久免费网站| 亚洲av日韩精品久久久久久密| 久久久久国产精品人妻aⅴ院| 两性夫妻黄色片| 人人妻人人澡欧美一区二区| 久久伊人香网站| 久久精品影院6| 欧美 亚洲 国产 日韩一| 久久精品成人免费网站| 久久久久亚洲av毛片大全| 欧美国产日韩亚洲一区| 亚洲无线在线观看| 黄色成人免费大全| 午夜激情福利司机影院| 午夜a级毛片| 可以在线观看毛片的网站| 日日干狠狠操夜夜爽| 999久久久精品免费观看国产| 久久这里只有精品19| 久久狼人影院| 免费观看精品视频网站| 最近最新中文字幕大全免费视频| 黄色 视频免费看| 久久久久国产一级毛片高清牌| 久久中文看片网| 免费一级毛片在线播放高清视频| 啪啪无遮挡十八禁网站| 久久精品人妻少妇| 成人18禁在线播放| 亚洲人成77777在线视频| 久久国产精品影院| 老司机午夜十八禁免费视频| 国产欧美日韩一区二区三| 国产成人精品久久二区二区免费| 亚洲五月天丁香| 日韩精品青青久久久久久| 国产精品永久免费网站| or卡值多少钱| 在线观看66精品国产| 亚洲人成伊人成综合网2020| 精品国产美女av久久久久小说| 亚洲av成人一区二区三| 俺也久久电影网| 久久人人精品亚洲av| 国产精品一区二区免费欧美| 欧美日韩乱码在线| 妹子高潮喷水视频| 精品欧美一区二区三区在线| 日韩三级视频一区二区三区| 免费无遮挡裸体视频| 黄色片一级片一级黄色片| 国产蜜桃级精品一区二区三区| 亚洲av成人一区二区三| 桃红色精品国产亚洲av| 成人三级黄色视频| 精品国内亚洲2022精品成人| 国产精品久久久久久人妻精品电影| 18禁观看日本| 亚洲欧洲精品一区二区精品久久久| 淫秽高清视频在线观看| 精品少妇一区二区三区视频日本电影| 国产亚洲欧美98| 亚洲男人天堂网一区| 亚洲最大成人中文| 在线观看免费午夜福利视频| 黄色女人牲交| 欧美乱码精品一区二区三区| 亚洲成av人片免费观看| 国产又色又爽无遮挡免费看| 中文字幕高清在线视频| 免费看十八禁软件| 无遮挡黄片免费观看| 欧美一级a爱片免费观看看 | 成熟少妇高潮喷水视频| 色综合站精品国产| 熟女电影av网| 亚洲第一欧美日韩一区二区三区| 久久香蕉国产精品| 少妇 在线观看| 欧美丝袜亚洲另类 | 国产精品免费视频内射| 久99久视频精品免费| 少妇被粗大的猛进出69影院| 一区二区三区国产精品乱码| 亚洲av第一区精品v没综合| 黑人巨大精品欧美一区二区mp4| 美女 人体艺术 gogo| 国产欧美日韩一区二区三| 国产免费男女视频| 操出白浆在线播放| 国产私拍福利视频在线观看| 国产精品乱码一区二三区的特点| 一进一出抽搐动态| 久久中文看片网| 757午夜福利合集在线观看| 三级毛片av免费| 精品乱码久久久久久99久播| 岛国在线观看网站| 午夜激情福利司机影院| 亚洲精品色激情综合| 欧美不卡视频在线免费观看 | 欧美乱色亚洲激情| 中文字幕人妻熟女乱码| 亚洲激情在线av| www日本在线高清视频| 可以在线观看的亚洲视频| 精华霜和精华液先用哪个| 亚洲欧洲精品一区二区精品久久久| 欧美黄色片欧美黄色片| 国内精品久久久久精免费| 露出奶头的视频| 久久久久久免费高清国产稀缺| 国产精品久久电影中文字幕| 俄罗斯特黄特色一大片| 成人av一区二区三区在线看| 日本三级黄在线观看| 国产亚洲精品久久久久5区|