Introduction to Homology and Cohomology
Simplicial complexes
Algebraic Topology
Higher-order networks
Hodge Laplacian and Introduction to Synchronization
Topological Kuramoto model
Weighted homology and Global topological synchronization
dirac operator and dynamics
Note course change on Nov 11.
A \(d\)-dimensional simplex is formed by a set of \((d+1)\) ordered nodes \[\alpha = [v_0,\dots,v_d]\]
A face of a \(d\)-dim simplex \(\alpha\) is a simplex \(\alpha'\) formed by a proper subset of nodes \(\alpha'\subset\alpha\).
A simplicial complex \(\mathcal{K}\) is formed by a set of simplices that is closed under the inclusion of the faces of each simplex, whose dimension is the maximal dimension of its simplexes.
\[\alpha\in \mathcal{K}\Rightarrow \mathrm{Faces}(\alpha)\subset \mathcal{K}\]
The facets of a simplicial complex os a face that is not the face of any other simplex.
a pure simplisial compex is formed by a set of \(d\) dim simplexes and their faces (no other dimensions) It is fully determined by an (tensored) adjacency matrix.
generalized degree \(k_{m',m}(\alpha)\) of a \(m\)-face \(\alpha\) is the number of \(m'\)-dim simplices incident to \(\alpha\).
Simplicial complex \(\to\) its skeleton (\(0\) and \(1\) simplices)
A reverse (not really) process: clique complex, adds as much as possible simplices (but should not generate extra \(1\) morphisms) by assuming that each clique is a complex.
Euler characteristic
\[ \chi = \sum_{m=0}^d (-1)^m \beta_m \]
Boundary Operators \(\partial\)
can we use these differentials to compute gradient descend.. lol
We consider a \(d\)-dimensional simplicial complex \(K\) having \(N_m\) positively oriented simplexes \(\alpha_r^m\), whose set is \(Q_m(K)\).
The orientations for each simplex is determined by the indexes on nodes, i.e. an order on \(Q_0(K)\), we require
\[ [v_0,\dots,v_m] = (-1)^{\mathrm{sgn}(\sigma)} [v_{\sigma(0)},\dots,v_{\sigma(m)}].\]
\(m\)-chains \(C_m=C_m(K)\) is the free \(\mathbb{Z}\)-module generated by all \(m\)-simplexes of the simplicial complex. Its elements are
\[ c_m = \sum_{\alpha_r\in Q_m(K)} c_m^r \alpha_r^m.\]
The boundary operator has type \(\partial_m : C_m\to C_{m-1}\) given by
\[ \partial_m [v_0,\dots, v_m] := \sum_{p=0}^m (-1)^p [v_0,\dots,\hat{v_p},\dots,v_m]\]
where the hat means to skip.
The most important fact about boundary operator is, consecutive boundary always gives zero
\[ \partial_m \circ \partial_{m+1} = 0. \]
topological signals are numbers or functions associated to the nodes, edges and faces.
\(m\)-cochains are just functions in \(C^m = C_m^\vee = \mathrm{Hom}(C_m, k)\), with induced differentials \(\partial^m = (\circ \partial_{m+1})\).
You can have an inner product on \(\mathrm{Hom}(C_m, k)\) by using the dual basis of \(C^m\) induced by the basis of \(C_m\)
\[ \langle f, g\rangle = \sum_{\alpha\in Q_m(K)} f(\alpha)g(\alpha).\]
But this might not be the natural choice, we can define using any matrix.