In this paper, we introduce tiled graphs as models of learning and maturing processes. We show how tiled graphs can combine graphs of learning spaces or antimatroids (partial hypercubes) and maturity models (total orders) to yield models of learning processes. For the visualization of these processes it is a natural approach to aim for certain optimal drawings. We show for most of the more detailed models that the drawing problems resulting from them are NP-complete. The terse model of a maturing process that ignores the details of learning, however, results in a polynomially solvable graph drawing problem. In addition, this model provides insight into the process by ordering the subjects at each test of their maturity. We investigate extremal and random instances of this problem, and provide exact results and bounds on their optimal crossing number. Graph-theoretic models offer two approaches to the design of optimal maturity models given observed data: (1) minimizing intra-subject inconsistencies, which manifest as regressions of subjects, is modeled as the well-known feedback arc set problem. We study the alternative of (2) finding a maturity model by minimizing the inter-subject inconsistencies, which manifest as crossings in the respective drawing. We show this to be NP-complete.

翻译：本文提出将平铺图作为学习与成熟过程的模型。我们展示了平铺图如何结合学习空间或反拟阵（部分超立方体）的图与成熟度模型（全序）以生成学习过程模型。为可视化这些过程，追求某种最优绘制是自然的方法。我们证明，在大多数更精细的模型中，由此产生的绘制问题均为NP完全的。然而，忽略学习细节的简洁成熟过程模型可归结为多项式可解的图绘制问题。此外，该模型通过在每个成熟度测试中对受试者排序，提供了对过程的洞察。我们研究了该问题的极值与随机实例，并给出了其最优交叉数的精确结果与界。基于图论模型，从观测数据设计最优成熟度模型有两种方法：（1）最小化受试者内部不一致性（表现为受试者的倒退），可建模为经典的反馈弧集问题；我们研究了另一种方法（2）通过最小化受试者间不一致性（表现为相应绘制中的交叉）来寻找成熟度模型，并证明该问题是NP完全的。