  Knowledge is a network of interconnected concepts. Yet, precisely how the
topological structure of knowledge constrains its acquisition remains unknown,
hampering the development of learning enhancement strategies. Here we study the
topological structure of semantic networks reflecting mathematical concepts and
their relations in college-level linear algebra texts. We hypothesize that
these networks will exhibit structural order, reflecting the logical sequence
of topics that ensures accessibility. We find that the networks exhibit strong
core-periphery architecture, where a dense core of concepts presented early is
complemented with a sparse periphery presented evenly throughout the
exposition; the latter is composed of many small modules each reflecting more
narrow domains. Using tools from applied topology, we find that the
expositional evolution of the semantic networks produces and subsequently fills
knowledge gaps, and that the density of these gaps tracks negatively with
community ratings of each textbook. Broadly, our study lays the groundwork for
future efforts developing optimal design principles for textbook exposition and
teaching in a classroom setting.
