* Hypergraphs
- links may connect more than 2 vertices: hyperedge
- coauthorships, film actors
- a much powerful representation exists!
* Bipartite graphs
- also known as a 'two-mode network' in sociology
- Two types of vertices with edges running only between different types
of vertices
- usually used to represent group memberships, but other examples exist
- incidence matrix, second index over the vertices
- one mode projection
- union of many cliques
- loses information
- number of groups shared by two vertices
- weighted projections
- improvement over the unweighted version
- total number of groups and the exact membership is lost
- Mathematical representation of projection: B^TB and BB^T
* Multilayer
- several types of vertices and several types of edges
- transportation: airports, train stations, bus stops
- each set of vertices forms a layer
* Multiplex networks
- special case of multilayer networks
- each layer represent the same set of vertices
- social network
* Dynamic networks
- a type of multiplex network, sometimes multilayer if vertices change
* Mathematical representation of multiplex and multilayer networks
- multiplex : a set of several adjacency matrices A^alpha
- multilayer: the above set + interlayer adjacency matrices B^{alpha-beta}
* Walks and Paths
- walks and cycles/loops
- walk is a sequence of vertices such that the consecutive vertices in
the sequence are connected by an edge
- cycle or loop is a walk that starts and ends at the same vertex
- paths
- a self avoiding walk
- length of a walk
- number of walks of length r between two vertices : sum(A^r)_{ij}
- number of cycles as a function of A : sum(lambda_i^r)