生活中不難發現網路的存在，例如社交網路、代謝網路、生物網絡和引文網路等，而相關的議題在各領域中也被廣泛地討論。在實驗設計領域中，實驗單位被視為網路中的節點，有關聯的實驗單位被稱為彼此的鄰居。當一個處理被應用於一個實驗單位上時，它會同時影響自身及其鄰居，因此產生了兩種對反應變數的影響：處理效應和網路效應。Parker et al. (2017) 提出一線性模型來處理具備網路結構的實驗，並且假設處理效應和網路效應都是固定效應。Chang et al. (2020) 採用了類似的模型，但他們將網路效應假設為隨機效應。本文將此模型擴展至有方向與加權之網路結構，並且尋找特定網路結構下的最佳設計。最後我們歸納出相關的理論結果，並提供對應的最佳設計。
關鍵字: 處理效應、網路效應、非結構化處理、二分圖/路徑圖/循環圖、D最佳化準則、(M.S)最佳化準則

Connected experimental units can be characterized by a network structure, where related examples are abundant in the world. If there exists a connection between two units, then they are neighbors of each other. In this case, a treatment affects both the unit to which it applies and the neighbors of that unit, simultaneously causing a treatment effect and a network effect. Parker, Gilmour, and Schormans (2017) launched a numerical study for designing experiments on general network structures. They proposed a linear model with unstructured treatments and assumed both treatment effects and network effects were fixed effects. A relevant work is Chang, Phoa, and Huang (2020), which argued using random network effects. In this work, we adopt the model in Chang, Phoa, and Huang (2020) and extend the network structure to directed/weighted graphs. We study optimal designs on specific graphs, devote to obtaining theoretical results, and provide templates of the corresponding optimal designs.
Keywords: Treatment Effect, Network Effect, Unstructured Treatment, Bipartite/Path/Cycle Graph, D-optimality, (M.S)-optimality.

Besag, J. and Kempton, R. (1986). Statistical analysis of field experiments using
neighbouring plots, Biometrics, 42(2), 231-251.

Chang, M. C., Phoa, F. K. H., and Huang, J. W. (2020). Optimal designs for
network experimentations with unstructured treatments, manuscript.

Croes, D., Couche, F., Wodak, S. J., and Helden, J. (2006). Inferring meaningful
pathways in weighted metabolic networks. J. Mol. Biol., 356(1), 222-236.

Druilhet, P. (1999). Optimality of neighbour balanced designs, J. Statist. Plann. Inference, 81(1), 141-152.

Gallager, R. G. (1963). Low-Density Parity-Check Codes. Cambridge, MA: MIT Press.
Hosmer Jr., D. W., Lemeshow, S., and Sturdivant, R. X. (2013). Applied Logistic
Regression. Hoboken, New Jersey: John Wiley and Sons, Inc.

Lewis, D. D., Yang, Y., Rose, T. G., and Li, F. (2004). RCV1: A new benchmark
collection for text categorization research. J. Mach. Learn. Res., 5, 361-397.

Lindquist, M. and Mckeague, I. (2012). Logistic regression with brownian-like pre-
dictors. J. Amer. Statist. Assoc., 104(488), 1575-1585.

Ma, Y., Pan, R., Zou, T., and Wang, H. (2020). A naive least squares method for
spatial autoregression with covariates. Statist. Sinica, to appear.

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Model (Second Edition). London: Chapman and Hall.

Parker, B. M., Gilmour, S. G., and Schormans, J. (2017). Optimal design of
experiments on connected units with application to social networks. J. R. Stat. Soc.
Ser. C. Appl. Stat., 66(3), 455-480.

Polson, N., Scott, J., and Windle, J. (2013). Bayesian inference for logistic models
using polya-gamma latent variables. J. Amer. Statist. Assoc., 108(504), 1339-1349.

Rosvall, M. and Bergstrom, C. T. (2008). Maps of random walks on complex
networks reveal community structure. Proc. Natl. Acad. Sci. USA, 105(4), 1118-1123.

Tanner, R. M. (1981). A recursive approach to low complexity codes. IEEE Trans. Inf. Theory, 27(2), 533-547.

Weng, L., Menczer, F., and Ahn, Y. Y. (2013). Virality prediction and community
structure in social networks. Sci. Rep., 3:2522.

Yan, T., Jiang, B., Fienberg, S. E., and Leng, C. (2019). Statistical inference in a
directed network model with covariates. J. Amer. Statist. Assoc., 114(526), 857-868.

Zhang, X., Pan, R., Guan, G., Zhu, X., and Wang, H. (2019). Logistic regression
with network structure. Statist. Sinica, to appear.