典型的網路分析大多基於最短路徑進行分析。然而這些分析方法無法展現出神經網路兩種特性性：
第一，神經網路有特定的資訊傳遞方向。第二，在功能上，經由多個突觸的路徑可能比最短路徑來得
重要。為了考慮到此兩個特性，我們量化兩個全新的特性：第一，縱向傳遞：所有輸入到輸出間的主
要資訊傳遞路徑多快會建立完成。第二：橫向傳遞：訊息進入網路後多快散佈到眾多的輸出節點。我
們分析了線蟲的神經網路、果蠅的前腦橋網路，並用人工產生的規則網路、小世界網路以及隨機網路
作為比較。結果顯示比起小世界網路，線蟲及前腦橋神經網路在縱向及橫向傳遞上更有效率。更多的
分析顯示在小世界網路中，不同的核心樞紐可以改善不同的傳遞效率：區域型核心樞紐可以提昇縱向
傳遞、無親型核心樞紐可以改善橫向傳遞、連接型核心樞紐可以同時提昇縱向及橫向的傳遞效率。此
外，在破壞線蟲神經網路的核心樞紐之實驗，也同樣支持此結果。我們的實驗結果顯示在神經網路的
訊息傳遞中，不同的核心樞紐可能扮演不同的重要功能及角色，而這也啟發我們，對於初階感知系統
中聯絡神經元之功能的想像。

Typical analyses of the network architecture focus on the shortest path. However, the approach
may not fully characterize the features of neural networks in at least two ways: a) A neural
network has a specific direction of information flow. b) The neural pathways via multiple synaptic
connections may be functionally more important than the shortest pathways. To address
the issues, we measures two novel quantities: a) Vertical propagation is how quickly the main information
pathway are established between input and output nodes. b) Horizontal propagation
is how quickly the information from input neurons could propagate to multiple output neurons.
We analysed the C. Elegans neural network, protocerebral bridge network in Drosophila, and, as
comparison, artificially generated regular, small-world and random networks. Our results show
that the C. Elegans and PCB neural networks are more efficient in both vertical and horizontal
propagation than the small-world networks. Further analysis show that different hubs could improve
the different propagation efficiency in small-world networks: provincial hubs enhance vertical
propagation, kinless hubs improve horizontal propagation, and connector hubs increase the
efficiency of the both propagations. In addition, this result could be supported by lesioning hubs
in the C. Elegans neural network. Our results suggest that the various hubs may play different
important roles in information propagations of the neural networks, and our works may deliver
insight into the functions of the interneurons in primary sensory systems.

L. a. N. Amaral, A. Scala, M. Barthélémy, and H. E. Stanley. Classes of Small-World
networks. Proceedings of the National Academy of Sciences, 97(21):11149–11152, October
2000. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.200327197. URL http:
//www.pnas.org/content/97/21/11149.
Albert-László Barabási and Réka Albert. Emergence of scaling in random networks.
Science, 286(5439):509–512, October 1999. ISSN 0036-8075, 1095-9203. doi: 10.1126/
science.286.5439.509. URL http://www.sciencemag.org/content/286/5439/509.
Norman Biggs. Algebraic Graph Theory. Cambridge University Press, 1993. ISBN
9780521458979.
Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre.
Fast unfolding of communities in large networks. Journal of Statistical Mechanics:
Theory and Experiment, 2008(10):P10008, October 2008. ISSN 1742-5468. doi: 10.
1088/1742-5468/2008/10/P10008. URL http://iopscience.iop.org/1742-5468/2008/
10/P10008.
Davi D. Bock, Wei-Chung Allen Lee, Aaron M. Kerlin, Mark L. Andermann, Greg Hood,
Arthur W. Wetzel, Sergey Yurgenson, Edward R. Soucy, Hyon Suk Kim, and R. Clay
Reid. Network anatomy and in vivo physiology of visual cortical neurons. Nature, 471
(7337):177–182, March 2011. ISSN 0028-0836. doi: 10.1038/nature09802. URL http:
//www.nature.com/nature/journal/v471/n7337/full/nature09802.html.
Kevin L. Briggman, Moritz Helmstaedter, and Winfried Denk. Wiring specificity in the
direction-selectivity circuit of the retina. Nature, 471(7337):183–188, March 2011. ISSN
0028-0836. doi: 10.1038/nature09818. URL http://www.nature.com/nature/journal/
v471/n7337/full/nature09818.html.
Karl Emanuel Busch, Patrick Laurent, Zoltan Soltesz, Robin Joseph Murphy, Olivier
Faivre, Berthold Hedwig, Martin Thomas, Heather L Smith, and Mario de Bono. Tonic
signaling from o2 sensors sets neural circuit activity and behavioral state. Nature
Neuroscience, 15(4):581– 591, 2012. ISSN 1097-6256. doi: 10.1038/nn.3061. URL
http://www.nature.com/neuro/journal/v15/n4/full/nn.3061.html.
Benny H.H. Cheung, Merav Cohen, Candida Rogers, Onder Albayram, and Mario
de Bono. Experience-Dependent modulation of c. elegans behavior by ambient oxygen.
Current Biology, 15(10):905– 917, May 2005. ISSN 0960-9822. doi: 10.
1016/j.cub.2005.04.017. URL http://www.sciencedirect.com/science/article/pii/
S0960982205003891.
Ann-Shyn Chiang, Chih-Yung Lin, Chao-Chun Chuang, Hsiu-Ming Chang, Chang-Huain
Hsieh, Chang-Wei Yeh, Chi-Tin Shih, Jian-Jheng Wu, Guo-Tzau Wang, Yung-Chang
Chen, Cheng-Chi Wu, Guan-Yu Chen, Yu-Tai Ching, Ping-Chang Lee, Chih-Yang Lin,
Hui-Hao Lin, Chia-Chou Wu, Hao-Wei Hsu, Yun-Ann Huang, Jing-Yi Chen, Hsin-Jung
Chiang, Chun-Fang Lu, Ru-Fen Ni, Chao-Yuan Yeh, and Jenn-Kang Hwang. Three-
Dimensional reconstruction of brain-wide wiring networks in drosophila at Single-Cell
resolution. Current Biology, 21(1):1–11, January 2011. ISSN 0960-9822. doi: 10.
1016/j.cub.2010.11.056. URL http://www.sciencedirect.com/science/article/pii/
S0960982210015228.
Ya-Hui Chou, Maria L Spletter, Emre Yaksi, Jonathan C S Leong, Rachel I Wilson, and
Liqun Luo. Diversity and wiring variability of olfactory local interneurons in the drosophila
antennal lobe. Nature Neuroscience, 13(4):439–449, 2010. ISSN 1097-6256.doi: 10.1038/nn.2489. URL http://www.nature.com/neuro/journal/v13/n4/full/nn.
2489.html.
Juliet C. Coates and Mario de Bono. Antagonistic pathways in neurons exposed to body
fluid regulate social feeding in caenorhabditis elegans. Nature, 419(6910):925–929, October
2002. ISSN 0028-0836. doi: 10.1038/nature01170. URL http://www.nature.com/
nature/journal/v419/n6910/full/nature01170.html#B1.
Richard Michael Durbin. Studies on the development and organisation of the nervous
system of Caenorhabditis elegans. PhD thesis, University of Cambridge, 1987. URL
http://www.wormatlas.org/ver1/durbinv1.2/durbinindex.html.
P. Erdös and A. Rényi. On the evolution of random graphs. In PUBLICATION OF THE
MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES,
page 17–61, 1960.
Ernesto Estrada and Naomichi Hatano. Communicability in complex networks. Physical
Review E, 77(3):036111, March 2008. doi: 10.1103/PhysRevE.77.036111. URL http:
//link.aps.org/doi/10.1103/PhysRevE.77.036111.
Giorgio Fagiolo. Clustering in complex directed networks. Physical Review E, 76(2):
026107, 2007. doi: 10.1103/PhysRevE.76.026107. URL http://link.aps.org/doi/10.
1103/PhysRevE.76.026107.
L Freeman. Centrality in social networks: Conceptual clarification. Social Networks, 1
(3):215–239, 1979. ISSN 03788733. URL http://dx.doi.org/10.1016/0378-8733(78)
90021-7.
M. Girvan and M. E. J. Newman. Community structure in social and biological networks.
Proceedings of the National Academy of Sciences, 99(12):7821–7826, June 2002.
ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.122653799. URL http://www.pnas.org/
content/99/12/7821.
Roger Guimerà and Luís A. Nunes Amaral. Functional cartography of complex metabolic
networks. Nature, 433(7028):895–900, February 2005. ISSN 0028-0836. doi: 10.
1038/nature03288. URL http://www.nature.com/nature/journal/v433/n7028/full/
nature03288.html.
D. H. Hall and R. L. Russell. The posterior nervous system of the nematode caenorhabditis
elegans: Serial reconstruction of identified neurons and complete pattern of synaptic
interactions. The Journal of Neuroscience, 11(1):1–22, January 1991. ISSN 0270-6474,
1529-2401. URL http://www.jneurosci.org/content/11/1/1.
Mark D. Humphries and Kevin Gurney. Network ‘Small-World-Ness’: a quantitative
method for determining canonical network equivalence. PLoS ONE, 3(4):e0002051,
April 2008. doi: 10.1371/journal.pone.0002051. URL http://dx.plos.org/10.1371/
journal.pone.0002051.
Alex C. Keene and Scott Waddell. Drosophila olfactory memory: single genes to complex
neural circuits. Nature Reviews Neuroscience, 8(5):341–354, May 2007. ISSN 1471-
003X. doi: 10.1038/nrn2098. URL http://www.nature.com/nrn/journal/v8/n5/full/
nrn2098.html.
Vito Latora and Massimo Marchiori. Efficient behavior of Small-World networks. Physical
Review Letters, 87(19):198701, October 2001. doi: 10.1103/PhysRevLett.87.198701. URL
http://link.aps.org/doi/10.1103/PhysRevLett.87.198701.
E. A. Leicht and M. E. J. Newman. Community structure in directed networks. Physical
Review Letters, 100(11):118703, March 2008. doi: 10.1103/PhysRevLett.100.118703. URL
http://link.aps.org/doi/10.1103/PhysRevLett.100.118703.
Richard H. Masland. The fundamental plan of the retina. Nature Neuroscience, 4(9):
877–886, September 2001. ISSN ERROR! NO ISSN. doi: 10.1038/nn0901-877. URL
http://www.nature.com/neuro/journal/v4/n9/abs/nn0901-877.html.
Dharmendra S Modha and Raghavendra Singh. Network architecture of the Long-
Distance pathways in the macaque brain. Proceedings of the National Academy of
Sciences, 107(30):13485–13490, July 2010. ISSN 0027-8424, 1091-6490. doi: 10.1073/
pnas.1008054107. URL http://www.pnas.org/content/107/30/13485.
M. E. J. Newman. Modularity and community structure in networks. Proceedings of
the National Academy of Sciences, 103(23):8577–8582, June 2006. ISSN 0027-8424,
1091-6490. doi: 10.1073/pnas.0601602103. URL http://www.pnas.org/content/103/
23/8577.
Shawn R. Olsen and Rachel I. Wilson. Cracking neural circuits in a tiny brain: new approaches
for understanding the neural circuitry of drosophila. Trends in Neurosciences,
31(10):512–520, October 2008. ISSN 0166-2236. doi: 10.1016/j.tins.2008.07.006. URL
http://www.sciencedirect.com/science/article/pii/S0166223608001823.
Mikail Rubinov and Olaf Sporns. Complex network measures of brain connectivity:
Uses and interpretations. NeuroImage, 52(3):1059–1069, September 2010. ISSN 1053-
8119. doi: 10.1016/j.neuroimage.2009.10.003. URL http://www.sciencedirect.com/
science/article/pii/S105381190901074X.
Olaf Sporns. Networks of the Brain. The MIT Press, 1 edition, October 2010. ISBN
0262014696.
Mark Stopfer, Seetha Bhagavan, Brian H. Smith, and Gilles Laurent. Impaired odour
discrimination on desynchronization of odour-encoding neural assemblies. Nature, 390
(6655):70–74, November 1997. ISSN 0028-0836. doi: 10.1038/36335. URL http://www.
nature.com/nature/journal/v390/n6655/full/390070a0.html.
Jeffrey Travers and Stanley Milgram. An experimental study of the small world problem.
Sociometry, 32:425–443, 1969.
Lav R. Varshney, Beth L. Chen, Eric Paniagua, David H. Hall, and Dmitri B. Chklovskii.
Structural properties of the caenorhabditis elegans neuronal network. PLoS Comput
Biol, 7(2):e1001066, February 2011. doi: 10.1371/journal.pcbi.1001066. URL http://dx.
doi.org/10.1371/journal.pcbi.1001066.
Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’ networks.
Nature, 393(6684):440–442, June 1998. ISSN 0028-0836. doi: 10.1038/30918. URL http:
//www.nature.com/nature/journal/v393/n6684/full/393440a0.html.
J. G White, E. Southgate, J. N Thomson, and S. Brenner. The structure of the
nervous system of the nematode caenorhabditis elegans. Philosophical Transactions
of the Royal Society of London. B, Biological Sciences, 314(1165):1–340, November
1986. ISSN 0962-8436, 1471-2970. doi: 10.1098/rstb.1986.0056. URL http://rstb.
royalsocietypublishing.org/content/314/1165/1.