# Polychrony as Chinampas
## 3 Base and Activation Diagrams
### Activation Diagram  
Given a base diagram, we selected a subset of vertices and called them the _primary vertices_. The _activation diagram_\((B,S)\) is a base diagram \(B\) and a subset \(S\) of the primary vertices of \(V(B)\). A _secondary vertex_ with the coordinates \((r,t)\) is a vertex in the base diagram in which each one of \((r,t-1)\) and \((r-1,t-1)\) is either a primary vertex or a secondary vertex. We associated the _activation graph_ (i.e., the underlying colored graph in which the primary and secondary vertices are black and the remaining vertices are white) to each activation diagram. To simplify the interpretation of the deactivation diagrams, we only drew vertices which were either primary or secondary and avoided the others in the base diagram (see Figure 3). **Example 1**. : _Consider \(P=path(l)\) and its corresponding base diagram \(B=tellis(l)\), as described before. Fix a vertex \(v_{i}\in P\), and consider \(S=\{(v_{i+k},t)\}\subset B\) with \(k=0,1,\cdots,l-1\) consecutive vertices with the same time. The set \(S\) is determined by the initial vertex \(v_{i}\) of the path \(P\), its length  
Figure 3: (**left**) An example of an activation diagram. The activation diagram turns into an activation graph. (**right**) A new black vertex appears in a row of the activation graph due to the two corresponding black vertices in the previous row in the activation diagram. Figure 2: Directed labeled graphs and their correponding base diagrams. (**a**) A path(\(l\)) and its corresponding base diagram \(tellis(l)\). (**b**) A \(cyc(l)\) and its corresponding base diagram cylinder(\(l\)). \(l\), and a fixed time \(t\in\mathbb{N}\). We employ a pyramid in the activation diagram \((B,S)\) and denote it as \(\text{pyramid}(l,i,t)\). In Figure 4, the activation diagram \(\text{pyramid}(3,i,t)\) of a path with a length of three is illustrated._  
**Remark 1**. : _Suppose we have an activation diagram in which there is \(\text{pyramid}(l,i,t)\) with an extra primary vertex \((i-1+s,t+s)\) where \(0\leq s\leq l-1\). The vertex affects \(\text{pyramid}(l,i,t)\) by activating the vertices in the diagonal \(\{(i-1+k,t+k)\mid k=s+1,\cdots,l\})\). On the other hand, if the extra primary vertex is localized in \((i+l+1,t+s)\), where \(0\leq s\leq l-1\), then the vertical line \(\{(i+l+1,t+k)\mid k=s+1,\cdots,l\}\) is activated (see Figure 5)._  
With a path of secondary vertices from \((i,t)\) to \((j,s)\), we define a sequence  
Figure 4: A ring (double circle) indicates the primary vertices. There are three secondary vertices, forming a pyramid. Figure 5: Effect of adding a primary vertex to a pyramid. (**a**) If we add an activated vertex \(v\) to the left side of a pyramid, then the vertices in the same diagonal with times greater than the time of \(v\) are activated until reaching the top of the pyramid. (**b**) If we add an activated vertex \(v\) to the right side of the pyramid, then the vertices in the same column with a time greater than the time of \(v\) are activated until reaching the top of the pyramid. \[(i_{0},t_{0}),(i_{1},t_{1}),\cdots,(i_{k},t_{k})\]  
with the first point \((i_{0},t_{0})=(i,t)\) and final point \((i_{k},t_{k})=(j,s)\) such that there is an edge between two consecutive vertices. Here, the edges are considered while ignoring the direction. The edges go from \((i,t)\) to \((i,t+1)\) and from \((i,t)\) to \((i-1,t-1)\). We restricted our study to activation diagrams where every primary vertex contributed to at least one secondary vertex. A _redundant activation diagram_ occurs when a primary vertex is also a secondary vertex (see Figure 6). An activation diagram \(C\) is _connected_ if the secondary vertices of the corresponding activation graph form a connected, undirected graph. As an example, Figures 3 and 4 are connected activation diagrams. Figure 7 is not a connected activation diagram. Figure 6: In this activation graph, there is a primary vertex which is also a secondary vertex. Figure 7: An example of a non-connected activation diagram.