* Graph theory
*
- Vertices and edges
- Nodes and links
- Sites and bonds
- Actors and ties
- n and m
* Multiedge, self-edge/self-loop, simple network, multigraph
* Edge list
- undirected graphs on n vertices, label ordering doesn't matter
- list of all edges
- sometimes used to store networks on computers, but are cumbersome
otherwise
* The adjacency matrix
- simple undirected graph
- diagonal elements are zero, and the matrix is symmetric
- multiedges and self-loops
- multiedge : multiplicity
- self-loops: 2 (two ends, each is connected to the other),
general mathematical results, non-self edge appears twice
- multi-self-loops: twice the multiplicity of self-loops
* Weighted networks
- for many networks, edges have on-off structure
- in other cases, it is useful to represent edges as having a strength,
value or a weight to them
- examples: strengths in social networks, amount of data flowing along
edges in the Internet, distances in transport networks etc
- adjacency matrix
- multigraphs can be thought as weighted networks, sometimes handy
- weights could be negative: social enemity
* Directed networks
- WWW, citations, food webs
- adjacency matrix is asymmetric
- undirected => directed in both the directions, same adjacency matrix!
- self-edges => 1 on the diagonal
* Degree
- undirected network
- total edges and degree
- mean degree
- directed networks (digraphs and arcs)
- in and out degrees
- degree and number of edges
- mean in and out degrees
- density/connectance, sparse and dense graphs
- extremely sparse: rho -> 1/n
- most networks are sparse
- k-regular graphs
* Convert directed networks to undirected
- analysis might become easier
- ignore the directions => discards useful information
- Cocitation: the number of vertices that have outgoing edges pointing
both to i and j
- C = AA^T and C is symmetric
- ignore diagonal elements
- Bibliographic coupling: the number of other vertices to which both i and
j point to
- B = A^TA
- Cocitation is useful only for influential papers, bibliographic coupling
is more uniform. Also, it can be computed as soon as the paper is
published. Cocitation changes with time.