Behind complex networks, there are networks that describe the interaction between components.
Basics
A network is a set of nodes interconnected by a set of links. The adjacency matrix of a network is the matrix which contains non-zero element
if there exists an edge between node
and
. The resulting graph is undirected if the adjacency matrix
is symmetric
or directed otherwise.
A cycle exists in a graph if there exists an edge in the graph that can be removed without dividing it into two. A graph without cycles is considered a tree.
A bipartite graph connects two different kinds of nodes. The bipartite graph can be projected onto either type of node by matrix multiplication and
respectively.
Centrality Measures
To assess a network the centrality of each nodes needs to be analysed. Several options are available:
- Degree: The number of incoming edges is easy to measure, but does not centrality over a larger neighbourhood of nodes.
- Eigenvector: For a vertex
its importance can be defined as a vector
which can be computed as
.
- Closeness: A geodesic path connects two vertices with the lowest amount of edges. The length of a path between
and
measured in edges is denoted
. Closeness then can be expressed as the harmonic mean between
and all other nodes
.
- Betweenness: The number of geodesic paths between
and
is denoted as
. The number of geodesic paths through a node
is denoted as
. Then betweenness is defined as
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