Dissertation: hypergraphs + clique algorithms

dissertation/dissertation.tex

@@ -138,6 +138,10 @@ structure of the metric space.
 
 %% TODO figure with examples of simplicial complexes
 
+The notion of simplicial complex is closely related to that of a
+hypergraph. The important distinction lies in the fact that a subset
+of a hyperedge is not necessarily a hyperedge itself.
+
 Using these definitions, we can define homology on simplicial
 complexes. %% TODO add reference for more details/do it myself?
 
@@ -364,6 +368,14 @@ then apply WRCF on each static graph in the sequence, obtaining a
 filtered complex for each window, to which we can then apply
 persistent homology.
 
+This method can quickly become very computationally expensive, as
+finding all maximal cliques (using the Bron-Kerbosch algorithm for
+example) is a complicated problem in itself. In practice, we often
+restrict the search to cliques of dimension lower than a certain bound
+$d_M$. With this restriction, the new simplicial complex is
+homologically equivalent to the original: they have the same homology
+groups up to $H_{d_M-1}$.
+
 This method is sensitive to the choice of sliding windows on the time
 scale. The width and the overlap of the windows can completely change
 the networks created and their topological features. Too small a