In this paper, we introduce a new family of combinatorial cubical complexes \(\{X_n\}_{n\ge 1}\) . Each such complex \(X_n\) models the process of updating spanning tree networks on n vertices over time. The vertices of these complexes model the networks themselves, whereas the higher-dimensional cubes model the network realignments. Such realignments may become necessary in order to maintain the connectivity of the networks by local changes, in case the connections to the leaves become weak. Our goal is to study the topology of \(X_n\) . Our main result states that for any n, there exists a strong deformation retraction from the complex \(X_n\) to the complete graph \(K_n\) , the latter viewed as a topological space. In particular, the homology vanishes in dimensions 2, and above, and, in fact, these complexes are homotopy equivalent to wedges of circles.
