Lecture given by Ginestra Bianconi
Higher order networks characterize the interactions between more nodes.
Higher ordr networks => Combinatorial Statistical Properties, Simplicial Topology / Geometry => Higher order dynamics
Today:
Hodge Laplacians
Graph Laplacians
Properties of the Hodge Laplacians
Connection with topology
Topological signals are cochains or vector fields.
The \(m\)-dimensional Hodge Laplacian \(L_m\) is defined as
\[ L_m = L^{up}_m + L_m^{down} \in \mathrm{End}(C^m) \]
where
\[ L_m^{up} = \delta_m^*\delta_m, \quad L_m^{down} = \delta_{m-1}\delta_{m-1}^*.\]
In terms of canonical bases
\[ L_m = B_{m+1} B_{m+1}^T + B_m^T B_m.\]
Semi-positive definite (obvious from the definition)
\[ \langle f, L_m f\rangle = \langle \delta_m f, \delta_m f\rangle + \langle \delta_{m-1}^* f, \delta_{m-1}^* f\rangle \geq 0.\]
Hodge Decomposition
We have \(L^d L^u = 0\) and \(L^u L^d = 0\), so \(\mathrm{Im}(L_m^u) \subset \ker (L_m^d)\) and \(\mathrm{Im}(L_m^d) \subset \ker (L_m^u)\).
\[ \ker(L_m) = \ker(L_m^u) \cap \ker(L_m^d). \]
\(\dim \ker(L_m) = \dim H_m = \beta_m\)
Harmonic eigenvectors are related to the generators of the homology classes and can be chosen in such way that they localise on the \(m\)-holes.
A graph is \((V, E)\) where \(V\) is the set of nodes and \(E\) is the set of edges, for our case it is \(1\)-dimensional simplicial complex. (Undirected, unweighted)
You can represent it by adjecency matrix \(A\) or Laplacian matrix \(L = D - A\).
A network is a graph describing interactions between its constituents of a complex system. Vertices are called nodes and edges are called links.
The degree of a node is the number of links connected to it
\[ d_i = \sum_j A_{ij}.\]