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1

Jakubík, Ján. "Direct product decompositions of directed groups." Czechoslovak Mathematical Journal 39, no. 4 (1989): 618–21. http://dx.doi.org/10.21136/cmj.1989.102336.

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2

Halpin-Healy, Timothy. "Directed polymers versus directed percolation." Physical Review E 58, no. 4 (1998): R4096—R4099. http://dx.doi.org/10.1103/physreve.58.r4096.

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3

de Graaf, M., A. Schrijver, and P. D. Seymour. "Directed triangles in directed graphs." Discrete Mathematics 110, no. 1-3 (1992): 279–82. http://dx.doi.org/10.1016/0012-365x(92)90719-v.

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4

de Graaf, Maurits. "Directed triangles in directed graphs." Discrete Mathematics 280, no. 1-3 (2004): 219–23. http://dx.doi.org/10.1016/j.disc.2003.11.002.

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5

MacKay, R. S., S. Johnson, and B. Sansom. "How directed is a directed network?" Royal Society Open Science 7, no. 9 (2020): 201138. http://dx.doi.org/10.1098/rsos.201138.

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The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and percolation. The standard definition of trophic levels, however, borrowed from ecology, suffers from drawbacks such as requiring basal nodes, which limit its applicability. Here we propose simple improved definitions of trophic levels and coherence that can be computed on any directed network. We demonstrate how the method can identify node function in examples including ecosystems, supply chain networks, gene expression and global language networks. We also explore how trophic levels and coherence relate to other topological properties, such as non-normality and cycle structure, and show that our method reveals the extent to which the edges in a directed network are aligned in a global direction.
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6

Colbourn, Charles J., D. G. Hoffman, and C. A. Rodger. "Directed star decompositions of directed multigraphs." Discrete Mathematics 97, no. 1-3 (1991): 139–48. http://dx.doi.org/10.1016/0012-365x(91)90428-5.

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7

Lerman, I.-C., and P. Kuntz. "Directed Binary Hierarchies and Directed Ultrametrics." Journal of Classification 28, no. 3 (2011): 272–96. http://dx.doi.org/10.1007/s00357-011-9091-y.

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8

Wang, Hong. "Disjoint directed cycles in directed graphs." Discrete Mathematics 343, no. 8 (2020): 111927. http://dx.doi.org/10.1016/j.disc.2020.111927.

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9

Perlsman, E., and S. Havlin. "The directed-polymer–directed-percolation transition." Europhysics Letters (EPL) 46, no. 1 (1999): 13–17. http://dx.doi.org/10.1209/epl/i1999-00555-0.

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10

Knezevic, Milan, and Jean Vannimenus. "On directed interacting animals and directed percolation." Journal of Physics A: Mathematical and General 35, no. 12 (2002): 2725–44. http://dx.doi.org/10.1088/0305-4470/35/12/303.

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11

WangRID="*"ID="*" This research was, Hong. "Independent Directed Triangles in a Directed Graph." Graphs and Combinatorics 16, no. 4 (2000): 453–62. http://dx.doi.org/10.1007/pl00007230.

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12

Qi, Liqun. "Directed submodularity, ditroids and directed submodular flows." Mathematical Programming 42, no. 1-3 (1988): 579–99. http://dx.doi.org/10.1007/bf01589420.

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13

Li, Xue. "Directed LPA: Propagating labels in directed networks." Physics Letters A 383, no. 8 (2019): 732–37. http://dx.doi.org/10.1016/j.physleta.2018.11.047.

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14

Robertson, Gill. "How 'self' directed is self-directed learning?" Management Education and Development 18, no. 2 (1987): 75–87. http://dx.doi.org/10.1177/135050768701800201.

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15

Zhang, Danhong, and Hong Wang. "Disjoint directed quadrilaterals in a directed graph." Journal of Graph Theory 50, no. 2 (2005): 91–104. http://dx.doi.org/10.1002/jgt.20096.

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16

Rizzi, Romeo, and Marco Rospocher. "Covering partially directed graphs with directed paths." Discrete Mathematics 306, no. 13 (2006): 1390–404. http://dx.doi.org/10.1016/j.disc.2006.01.024.

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17

Jakubík, Ján. "Atomicity of the Boolean algebra of direct factors of a directed set." Mathematica Bohemica 123, no. 2 (1998): 145–61. http://dx.doi.org/10.21136/mb.1998.126304.

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18

Tirabassi, Giulio, Linda Sommerlade, and Cristina Masoller. "Inferring directed climatic interactions with renormalized partial directed coherence and directed partial correlation." Chaos: An Interdisciplinary Journal of Nonlinear Science 27, no. 3 (2017): 035815. http://dx.doi.org/10.1063/1.4978548.

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19

May, Simon Căbulea. "Directed Duties." Philosophy Compass 10, no. 8 (2015): 523–32. http://dx.doi.org/10.1111/phc3.12239.

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20

Cairns, J. "Directed mutation." Science 260, no. 5112 (1993): 1221–24. http://dx.doi.org/10.1126/science.8493560.

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21

van der Hoeven, Joris, and Grégoire Lecerf. "Directed evaluation." Journal of Complexity 60 (October 2020): 101498. http://dx.doi.org/10.1016/j.jco.2020.101498.

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22

McCarthy, Nicola. "Directed therapy." Nature Reviews Cancer 13, no. 8 (2013): 520. http://dx.doi.org/10.1038/nrc3575.

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23

McCarthy, Nicola. "As directed." Nature Reviews Cancer 13, no. 12 (2013): 824–25. http://dx.doi.org/10.1038/nrc3629.

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24

McCarthy, Nicola. "Directed development." Nature Reviews Cancer 14, no. 2 (2014): 74–75. http://dx.doi.org/10.1038/nrc3673.

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25

Hecht, Jeff. "Directed energy." Nature 440, no. 7086 (2006): 968. http://dx.doi.org/10.1038/440968a.

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26

Hilling, Lana, Eric Bakow, James Fink, Chris Kelly, and Peter A. Southorn. "Directed Cough." International Anesthesiology Clinics 34, no. 1 (1996): 87–94. http://dx.doi.org/10.1097/00004311-199603410-00012.

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27

VAN EYKEN, ALBERT. "Directed Evolution." Journal of Applied Philosophy 13, no. 3 (1996): 251–58. http://dx.doi.org/10.1111/j.1468-5930.1996.tb00170.x.

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28

Fahrenberg, Ulrich. "Directed Homology." Electronic Notes in Theoretical Computer Science 100 (October 2004): 111–25. http://dx.doi.org/10.1016/j.entcs.2004.09.005.

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29

Grohmann, Davide, and Marino Miculan. "Directed Bigraphs." Electronic Notes in Theoretical Computer Science 173 (April 2007): 121–37. http://dx.doi.org/10.1016/j.entcs.2007.02.031.

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30

Corsi, Giovanna, and Silvio Ghilardi. "Directed frames." Archive for Mathematical Logic 29, no. 1 (1989): 53–67. http://dx.doi.org/10.1007/bf01630811.

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31

Burkey, Shelby H. "Directed Parathyroidectomy." Archives of Surgery 138, no. 6 (2003): 604. http://dx.doi.org/10.1001/archsurg.138.6.604.

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32

MA, XIAOYAN, XIAOHUI HUANG, and ZHAO ZHANG. "DIAMETER VULNERABILITY OF DIRECTED CYCLES AND DIRECTED TORI." Discrete Mathematics, Algorithms and Applications 04, no. 01 (2012): 1250004. http://dx.doi.org/10.1142/s1793830912500048.

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The diameter of a graph is important for the network as it measures the maximum communication delay between any pair of processors. In this paper, we study the diameter vulnerability of the directed cycle [Formula: see text] and the directed torus [Formula: see text] with respect to arc addition, that is, what is the minimum number of arcs the addition of which reduces the diameter by at least k.
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33

Mendes, J. F. F., R. Dickman, and H. Hermann. "Crossover from directed percolation to compact directed percolation." Physical Review E 54, no. 4 (1996): R3071—R3074. http://dx.doi.org/10.1103/physreve.54.r3071.

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34

Rustagi, Tarun, and Michael Golioto. "Endoscopist-Directed Versus Anesthesiologist-Directed Sedation for ERCP." Gastroenterology 140, no. 5 (2011): S—552. http://dx.doi.org/10.1016/s0016-5085(11)62287-7.

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35

Alon, N. "Monochromatic directed walks in arc-colored directed graphs." Acta Mathematica Hungarica 49, no. 1-2 (1987): 163–67. http://dx.doi.org/10.1007/bf01956320.

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36

DiMasi, Elaine, Matthew J. Olszta, Vishal M. Patel, and Laurie B. Gower. "When is template directed mineralization really template directed?" CrystEngComm 5, no. 61 (2003): 346. http://dx.doi.org/10.1039/b309800a.

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37

Yang, Boting, and Yi Cao. "Digraph searching, directed vertex separation and directed pathwidth." Discrete Applied Mathematics 156, no. 10 (2008): 1822–37. http://dx.doi.org/10.1016/j.dam.2007.08.045.

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38

Colbourn, Charles J., D. G. Hoffman, and C. A. Rodger. "Directed star decompositions of the complete directed graph." Journal of Graph Theory 16, no. 5 (1992): 517–28. http://dx.doi.org/10.1002/jgt.3190160511.

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39

Zelinka, Bohdan. "Distances between directed graphs." Časopis pro pěstování matematiky 112, no. 4 (1987): 359–67. http://dx.doi.org/10.21136/cpm.1987.108565.

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40

Zelinka, Bohdan. "Centers of directed cacti." Časopis pro pěstování matematiky 114, no. 3 (1989): 225–29. http://dx.doi.org/10.21136/cpm.1989.118370.

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41

Tran, Loc. "Directed Hypergraph Neural Network." Journal of Advanced Research in Dynamical and Control Systems 12, SP4 (2020): 1434–41. http://dx.doi.org/10.5373/jardcs/v12sp4/20201622.

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42

Jakubík, Ján. "On directed interpolation groups." Czechoslovak Mathematical Journal 40, no. 4 (1990): 648–58. http://dx.doi.org/10.21136/cmj.1990.102418.

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43

Chavan, Sameer, Deepak Kumar Pradhan, and Shailesh Trivedi. "Multishifts on directed Cartesian products of rooted directed trees." Dissertationes Mathematicae 527 (2017): 1–102. http://dx.doi.org/10.4064/dm758-6-2017.

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44

Xu, Taihua, Jie Yang, and Guanglei Gou. "A Force-Directed Algorithm for Drawing Directed Graphs Symmetrically." Mathematical Problems in Engineering 2018 (November 19, 2018): 1–24. http://dx.doi.org/10.1155/2018/6208509.

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Symmetry is one of the most important aesthetic criteria on graph drawing. It is quite necessary to measure the extent to which the drawings can be considered symmetric. For this purpose, a symmetric metric based on vertex coordinate calculation is proposed in this paper. It is proven theoretically and experimentally that the proposed metric is robust to contraction, expansion, and rotation of drawings. This robustness conforms to human perception of symmetry. Star-subgraphs and cycles are two common structures in digraphs. Both of them have inherent symmetry which should be displayed in drawings. For this purpose, a force-directed algorithm named FDS is proposed which can draw star-subgraphs and cycles as symmetrically as possible. FDS algorithm draws cycles as circles whose positions are fixed to provide a scaffolding for overall layout, renders non-leaf vertices by a standard force-directed layout, and places leaf vertices on concentric circles via a deterministic strategy. A series of experiments are carried out to test FDS algorithm. The results show that FDS algorithm draws digraphs more symmetrically than the existing state-of-the-art algorithms and performs efficiency comparable to O(nlog⁡n) YFHu algorithm.
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45

Coles, Simon J., Jeremy G. Frey, Philip A. Gale, et al. "Anion-directed assembly: the first fluoride-directed double helix." Chemical Communications, no. 5 (February 4, 2003): 568–69. http://dx.doi.org/10.1039/b210847j.

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46

Perlsman, E., and S. Havlin. "Directed polymer ? directed percolation transition: the strong disorder case." European Physical Journal B 43, no. 4 (2005): 517–20. http://dx.doi.org/10.1140/epjb/e2005-00085-7.

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47

Duan, Jun, Xiaokui Tang, Shicong Huang, Jinwei Jia, and Shuliang Guo. "Protocol-directed versus physician-directed weaning from noninvasive ventilation." Journal of Trauma and Acute Care Surgery 72, no. 5 (2012): 1271–75. http://dx.doi.org/10.1097/ta.0b013e318249a0d5.

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48

Lebedev, N. I., and Y. C. Zhang. "On the connection between directed percolation and directed polymers." Journal of Physics A: Mathematical and General 28, no. 1 (1995): L1—L6. http://dx.doi.org/10.1088/0305-4470/28/1/001.

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49

Zhao, Star X., and Fred Y. Ye. "Exploring the directed h-degree in directed weighted networks." Journal of Informetrics 6, no. 4 (2012): 619–30. http://dx.doi.org/10.1016/j.joi.2012.06.007.

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50

Williams, M., and G. F. Woodman. "Directed forgetting versus directed remembering in visual working memory." Journal of Vision 8, no. 6 (2010): 201. http://dx.doi.org/10.1167/8.6.201.

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