ExpectedCommunities: 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)

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).

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.

Examples

Run this code

# 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