netproperties: Structural Properties of an Unweighted Undirected Network
In SyNet: Inference and Analysis of Sympatry Networks

Description Usage Arguments Details Value Note Author(s) References Examples

Calculates i) density of the network, ii) node degrees, iii) geodesic distances between pairs of vertices. It also finds components and enumerates maximal cliques in the network.

1	netproperties (mt, cutoff = 0, dichotomization = ">")

`mt`	Any valued symmetric matrix.
`cutoff`	Numeric. It is the threshold that dichotomizes the input matrix.
`dichotomization`	Character. Dichotomization rule expressed as a comparison operator.

The input matrix is transformed into a binary adjacency matrix applying the dichotomization rule over the cutoff. Then, several connecting properties of the resulting simple graph are studied.

A brief summary of graph concepts follows. The density is the proportion of dyadic connections (links) which are actually present. The degree of a node is the number of neighbors directly tied to that node. Cliques are fully connected groups of nodes, that is complete subgraphs in a graph. A clique is maximal if it cannot be extended to a larger clique. In graph theory, the distance between two vertices is the number of edges in a shortest path connecting them. A connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. If there is no path connecting the two vertices, i.e. if they belong to different connected components, then conventionally the geodesic distance can be defined as NA or infinite.

An object of class cleavogram, which is a list with components:

`Adjacency`	Binary matrix resulting from dichotomization.
`Components`	A vector of integers indicating the component to which each vertex (or node) is allocated.
`Degree`	A vector of integers indicating the number of neighbors directly linked to each vertex (or node).
`Geodesic`	Geodesic distance matrix between vertices. If two vertices belong to different components, the respective entry is set NA.
`Cliques`	Data frame. Columns are maximal cliques whereas rows are vertices of the network. Membership of each node to maximal cliques are indicated through 1 (present) and 0 (absent).

Geodesic distances between vertices was calculated following Seidel (1995). Maximal cliques have been enumerated following the recursive algorithm of Bron and Kerbosch (1973).

Daniel A. Dos Santos <dadossantos@csnat.unt.edu.ar>

Bron C., Kerbosch J. 1973. Algorithm 457: Finding All Cliques of an Undirected Graph. Commun. ACM 16(9): 575-577.

Seidel R. 1995. On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs.In Proceedings of J. Comput. Syst. Sci.: 400-403.

  # "A" corresponds to the 0-1 adjacency matrix
  # associated to a simple graph of 10 nodes
  # The main diagonal has zeroes.
  A <- matrix(0, 10, 10)
  A[lower.tri(A)] <- ifelse(runif(5*9) < 0.5, 1, 0)
  pmin(A + t(A), 1) -> A
  netproperties(A)