復(fù)雜網(wǎng)絡(luò)重構(gòu)預(yù)測(cè)和控制

1、會(huì)計(jì)學(xué)1復(fù)雜網(wǎng)絡(luò)重構(gòu)預(yù)測(cè)和控制復(fù)雜網(wǎng)絡(luò)重構(gòu)預(yù)測(cè)和控制重構(gòu)、預(yù)測(cè)和控制重構(gòu)、預(yù)測(cè)和控制Who is the source of a message?Social network + some observersAnswer: Question : Z. Shen, et al. under reviewZ. Shen, et al. under reviewH1N1 in China in 2009Z. Shen, et al. under review陶哲軒，現(xiàn)任教于美國(guó)加州大學(xué)洛杉磯分校（UCLA）數(shù)學(xué)系的華裔數(shù)學(xué)家，澳洲惟一榮獲數(shù)學(xué)最高榮譽(yù)“菲爾茨獎(jiǎng)”的澳籍華人數(shù)學(xué)教授，繼1982年的丘成

2、桐之后獲此殊榮的第二位華人。其于1996年獲普林斯頓大學(xué)博士學(xué)位后任教于UCLA，24歲時(shí)便被UCLA聘為正教授。 Around 2004 Emmanuel Cands, Terence Tao and David Donoho 壓縮感知理論證明對(duì)可壓縮信號(hào)可以通過(guò)遠(yuǎn)低于Nyquist采樣頻率的方式進(jìn)行數(shù)據(jù)采樣，仍然能夠精確地恢復(fù)出原始信號(hào)。該理論目前在圖像處理、醫(yī)學(xué)成像、模式識(shí)別、無(wú)線通訊、天文學(xué)等很多領(lǐng)域受到高度關(guān)注，并被美國(guó)科技評(píng)論評(píng)為當(dāng)年十大科技進(jìn)展之一。 應(yīng)用壓縮傳感理論通過(guò)少量測(cè)量值y恢復(fù)稀疏向量x。是在x稀疏并且滿足約束條件的情況下，M可以遠(yuǎn)小于N，并且x中非零元素的個(gè)數(shù)也小于M

3、。信號(hào)重構(gòu)過(guò)程在此條件下轉(zhuǎn)化為一個(gè)優(yōu)化問(wèn)題，求解方法有最小L1范數(shù)法和迭代閾值法等。 Z. Shen, W.-X. Wang*, Y. Fan, Z. Di and Y.-C. Lai, Nature Communications, to appear in 2014.Z. Shen, W.-X. Wang*, Y. Fan, Z. Di and Y.-C. Lai, Nature Communications, to appear in 2014.Compressive sensing 21xNxxaaaXxNeighbors of xyNmatchingFull networkZ. She

4、n, W.-X. Wang*, Y. Fan, Z. Di and Y.-C. Lai, Nature Communications, to appear in 2014.將不同路段平均速度分段，構(gòu)造符號(hào)序列，計(jì)算路段的熵和可預(yù)測(cè)性Complex networkControlling complex networks is ultimate goal! 復(fù)雜網(wǎng)絡(luò)的控制25Minimum input theory: (Liu et al. Nature 473, 167 (2011) )find minimum number of driver nodes to satisfy full ran

5、k condition NNMNtttNCBABAABBCBuAxxrankconditionrank ility controllab sKalman,matrix ility Controllab)()()(system Dynamical12Controllability卡爾曼滿秩理論26可控性：計(jì)算網(wǎng)絡(luò)最大匹配， 控制未匹配節(jié)點(diǎn)可觀測(cè)性：反向網(wǎng)絡(luò)的最大匹配結(jié)構(gòu)可控性適用范圍：有向網(wǎng)絡(luò)（無(wú)雙相邊），隨機(jī)邊權(quán)重（結(jié)構(gòu)矩陣）結(jié)構(gòu)可控性適用范圍：有向網(wǎng)絡(luò)（無(wú)雙相邊），隨機(jī)邊權(quán)重（結(jié)構(gòu)矩陣）任意網(wǎng)絡(luò)：網(wǎng)絡(luò)矩陣特征值的最大幾何重?cái)?shù)無(wú)向網(wǎng)絡(luò)：特征值的最大代數(shù)重?cái)?shù)（相同特征值數(shù)量）稀疏網(wǎng)絡(luò)：網(wǎng)絡(luò)矩陣的

6、秩致密網(wǎng)絡(luò)：?jiǎn)挝魂嚰泳W(wǎng)絡(luò)矩陣的秩29Identifying driver nodesUniversal Symmetry in Controlling Complex NetworksC. Zhao, W.-X. Wang*, Y.-Y. Liu* and J.-J. Slotine*First-order nodal dynamicsA mixture of nodal dynamics with different orders無(wú)標(biāo)度網(wǎng)絡(luò)網(wǎng)絡(luò)結(jié)構(gòu)動(dòng)力學(xué)小世界網(wǎng)絡(luò)社團(tuán)網(wǎng)絡(luò)權(quán)重網(wǎng)絡(luò)同步博弈交通抗毀性調(diào)控功能復(fù)雜網(wǎng)絡(luò)三元素正問(wèn)題: (1)網(wǎng)絡(luò)結(jié)構(gòu)如何演化 (2) 網(wǎng)絡(luò)結(jié)構(gòu)如何影響動(dòng)力學(xué)和功能反問(wèn)

7、題：從動(dòng)力學(xué)反推網(wǎng)絡(luò)結(jié)構(gòu)螢火蟲(chóng)發(fā)光，鼓掌合作涌現(xiàn)，經(jīng)濟(jì)行為物種多樣性互聯(lián)網(wǎng)，輸運(yùn)網(wǎng)絡(luò)中的擁塞金融危機(jī)，大停電基因調(diào)控網(wǎng)，神經(jīng)網(wǎng)絡(luò)控制復(fù)雜網(wǎng)絡(luò)研究的核心問(wèn)題：三元素的關(guān)系最終目標(biāo)Full social network structureCompressive sensingTime series ofagents (Detectable)(1) payoffs (2) strategiesEvolutionary gamesEvolutionary gamesyTxxyyxxyxyxyaMyxayxaarrSGbPDDCPSSPPSS :agent from agent of Payoffnet

8、work connection no if 0 connects if 1 :.matrix Adjacency (SG) gameSnowdrift 0111)( (PDG) game dilemma sPrisoner 001)( :matrix Payoff 10)( defector ;01)( cooperator :StrategiesPrisoners dilemma 21xNxxaaaXxNeighbors of xyNFull network structurematchingCompressive sensingseries timefrom obtainable : ;d

9、etectable : by )()()()()()()()()()()()()()()()()()(ted)reconstruc be (to agent of neighbors : )()()( )()()()()()()( : at time of payoff21222222121112111121212211XYXYAPSSPSSPSSPSSPSSPSSPSSPSSPSSXXYPSSPSSPSSNmttttttttttttttttttxaaatMtMtMttattattatMtxmNmTxmmTxmmTxNTxTxTxNTxTxTxxNxxmxxxNTxxNTxxTxxx00.10

10、.0.91DataSuccessful rateSocial network detection, #N=100 edgenoedge01DataSuccessful rateSocial network detection, #N=100 edgenoedge01DataSuc

11、cessful rateSocial network detection, #N=100 edgenoedgePDG for network with 100 nodesScale-freeSmall-worldRandom 01DataSuccessful rateSocial network detection, #N=100 edgenoedge01DataSucce

12、ssful rateSocial network detection, #N=100 edgenoedge01DataSuccessful rateSocial network detection, #N=100 edgenoedgeSG for network with 100 nodesScale-freeSmall-worldRandom Relationship network of 22 students0100.10.

13、0.91DataSuccessful rateSocial network detection, #N=20 edgenoedgeSuccess rate of prediction22 students play PDG together andwrite down their payoffs and strategiesPayoff vs number of neighbors12345670.450.500.550.600.650.700.750.800.85 Normailized payoffnode degreeWinner: Lingchao

14、 JiReconstructing traffic networks and local routing strategy,其中Measurable data: incoming and out going flux of nodesAccording to the flux conservation Z. Shen, et al. under reviewZ. Shen, W.-X. Wang*, Y. Fan, Z. Di and Y.-C. Lai, Nature Communications, to appear in 2014.Compressive sensing 21xNxxaaaXxNeighbors of xyNmatchingFull netw