ExpectedCommunities {fake}

Expected community structure

Description

Computes expected metrics related to the community structure of a graph simulated with given parameters.

Usage

ExpectedCommunities(pk, nu_within = 0.1, nu_between = 0, nu_mat = NULL)

Arguments

pk

vector of the number of variables per group in the simulated dataset. The number of nodes in the simulated graph is sum(pk). With multiple groups, the simulated (partial) correlation matrix has a block structure, where blocks arise from the integration of the length(pk) groups. This argument is only used if theta is not provided.

nu_within

probability of having an edge between two nodes belonging to the same group, as defined in pk. If length(pk)=1, this is the expected density of the graph. If implementation=HugeAdjacency, this argument is only used for topology="random" or topology="cluster" (see argument prob in huge.generator). Only used if nu_mat is not provided.

nu_between

probability of having an edge between two nodes belonging to different groups, as defined in pk. By default, the same density is used for within and between blocks (nu_within=nu_between). Only used if length(pk)>1. Only used if nu_mat is not provided.

nu_mat

matrix of probabilities of having an edge between nodes belonging to a given pair of node groups defined in pk.

Details

Given a group of nodes, the within degree d^w_i of node i is defined as the number of nodes from the same group node i is connected to. The between degree d^b_i is the number of nodes from other groups node i is connected to. A weak community in the network is defined as a group of nodes for which the total within degree (sum of the d^w_i for all nodes in the community) is stricly greater than the total between degree (sum of d^b_i for all nodes in the community). For more details, see Network Science by Albert-Laszlo Barabasi.

The expected total within and between degrees for the groups defined in pk in a network simulated using SimulateAdjacency can be computed given the group sizes (stored in pk) and probabilities of having an edge between nodes from a given group pair (defined by nu_within and nu_between or by nu_mat). The expected presence of weak communities can be inferred from these quantities.

The expected modularity, measuring the difference between observed and expected number of within-community edges, is also returned. For more details on this metric, see modularity.

Value

A list with:

total_within_degree_c

total within degree by node group, i.e. sum of expected within degree over all nodes in a given group.

total_between_degree

total between degree by node group, i.e. sum of expected between degree over all nodes in a given group.

weak_community

binary indicator for a given node group to be an expected weak community.

total_number_edges_c

matrix of expected number of edges between nodes from a given node pair.

modularity

expected modularity (see modularity).

See Also

SimulateGraphical, SimulateAdjacency, MinWithinProba

Examples

# Simulation parameters
pk <- rep(20, 4)
nu_within <- 0.8
nu_between <- 0.1

# Expected metrics
expected <- ExpectedCommunities(
  pk = pk,
  nu_within = nu_within,
  nu_between = nu_between
)

# Example of simulated graph
set.seed(1)
theta <- SimulateAdjacency(
  pk = pk,
  nu_within = nu_within,
  nu_between = nu_between
)

# Comparing observed and expected numbers of edges
bigblocks <- BlockMatrix(pk)
BlockStructure(pk)
sum(theta[which(bigblocks == 2)]) / 2
expected$total_number_edges_c[1, 2]

# Comparing observed and expected modularity
igraph::modularity(igraph::graph_from_adjacency_matrix(theta, mode = "undirected"),
  membership = rep.int(1:length(pk), times = pk)
)
expected$modularity