ETH

CGSS: Complex Networks

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 A of a network is the matrix which contains non-zero element A_{ij} if there exists an edge between node i and j. The resulting graph is undirected if the adjacency matrix A is symmetric A_{ij} = A_{ji} 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 AA^T and A^TA 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 i its importance can be defined as a vector x_i which can be computed as x_i \sum_k A_{ik}x_k.
  • Closeness: A geodesic path connects two vertices with the lowest amount of edges. The length of a path between i and j measured in edges is denoted d_{ij}. Closeness then can be expressed as the harmonic mean between i and all other nodes C_i = \frac{1}{n-1}\sum_{j(\neq i)}\frac{1}{d_{ij}}.
  • Betweenness: The number of geodesic paths between s and t is denoted as \sigma{st}. The number of geodesic paths through a node i is denoted as \sigma_{st}(v_i). Then betweenness is defined as C(v_i) = \sum_{s,t} = \frac{\sigma_{st}(v_i)}{\sigma_{st}}

 <k^2> = \sum_i P(k=i ) i^2= \sum_{i=0}^\infty i - \gamma i^2   <k> = \sum_i P(k=i ) i  = \sum_{i=0}^\infty i - \gamma i

Standard