Soft community detection in networks with nmfkc.net"
In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5
)

Introduction

nmfkc.net() is a symmetric NMF for network (adjacency) matrices. With type = "bi" it factorises a symmetric \eqn{N \times N} matrix as \eqn{Y \approx X X^\top} with \eqn{X \ge 0}. Each row of \eqn{X} is a node's graded membership across communities, so this is soft community detection: unlike a hard method (e.g.\ Louvain), boundary nodes are revealed by a mixed membership rather than being forced into one group.

This vignette uses a self-contained synthetic network with a known community structure, so we can check that nmfkc.net() recovers it and that nmfkc.net.rank() finds the right number of communities.

library(nmfkc)

A synthetic network (stochastic block model, 3 communities)

We draw an undirected graph on 60 nodes from a stochastic block model: three communities of 20 nodes, with a high within-community edge probability and a low between-community one. We then deliberately add three "bridge" nodes (one per community), each also wired to a second community, so that the network contains a few genuinely mixed nodes. The true community of each node is known.

set.seed(1)
K <- 3; n_each <- 20; N <- K * n_each       # 60 nodes, 3 communities
block <- rep(1:K, each = n_each)            # true community labels

p_in <- 0.6; p_out <- 0.05
Prob <- matrix(p_out, N, N)
for (k in 1:K) Prob[block == k, block == k] <- p_in
Y <- matrix(rbinom(N * N, 1, Prob), N, N)   # 0/1 adjacency
Y[lower.tri(Y)] <- t(Y)[lower.tri(Y)]        # make symmetric
diag(Y) <- 0

# add three bridge nodes, each also linked to a second community
bridge <- c(20, 40, 60); into <- c(2, 3, 1)
for (i in seq_along(bridge)) {
  b <- bridge[i]; tgt <- which(block == into[i])
  e <- rbinom(length(tgt), 1, 0.45)
  Y[b, tgt] <- pmax(Y[b, tgt], e); Y[tgt, b] <- Y[b, tgt]
}

isSymmetric(Y); sum(Y) / 2                   # symmetric, number of edges

# adjacency matrix in the original (random) node order -- structure is hidden
image(1:N, 1:N, Y[, N:1], col = c("white", "steelblue"),
      xlab = "node", ylab = "node", main = "Adjacency (original order)")

Soft community detection with `nmfkc.net`

res <- nmfkc.net(Y, rank = 3, type = "bi", nstart = 10, maxit = 500)
res$r.squared                 # fit
head(round(res$X.prob, 2))    # soft membership: rows = nodes, columns = communities
head(res$X.cluster)           # hard label = argmax membership

Did it recover the known communities?

table(true = block, estimated = res$X.cluster)

# the three bridge nodes have genuinely split memberships
round(res$X.prob[bridge, ] * 100, 1)

The cross-tabulation is essentially block-diagonal: apart from the bridge nodes, every node is assigned to its true community. The bridge nodes (rows above) are roughly 50/50 between two communities, so their hard label simply tips to whichever side is marginally stronger --- exactly the situation where a soft membership is more informative than a forced assignment.

Visualising the result

Re-ordering the nodes by the recovered community turns the adjacency matrix into a clean block-diagonal, and the membership matrix shows three crisp blocks:

ord <- order(res$X.cluster)
op <- par(mfrow = c(1, 2), mar = c(4, 4, 3, 1))
image(1:N, 1:N, Y[ord, rev(ord)], col = c("white", "steelblue"),
      xlab = "node (reordered)", ylab = "", main = "Adjacency by community")
image(1:N, 1:K, res$X.prob[ord, , drop = FALSE],
      col = hcl.colors(20, "Blues", rev = TRUE),
      xlab = "node (reordered)", ylab = "community", axes = FALSE,
      main = "Soft membership")
axis(1); axis(2, at = 1:K)
par(op)

Network diagram (Graphviz DOT): soft vs. hard

nmfkc.net.DOT() turns the fit into a Graphviz DOT graph: each network node is linked to the community (basis) hubs by edges whose thickness encodes the membership strength. We raise threshold to 0.25 so that only substantial memberships are drawn --- each pure node then links to a single hub, which lets the neato layout pull the three communities apart, while the bridge nodes keep both links and settle in between. Rendered with the suggested package DiagrammeR, it embeds directly in the HTML output. The only difference between the two views below is the node colour:

Y.cluster = "soft" (default) blends the community colours in proportion to the membership, so the three bridge nodes appear in a mixed colour.
Y.cluster = "hard" paints each node in the single colour of its dominant community.

has_dg <- requireNamespace("DiagrammeR", quietly = TRUE)

Soft -- the three bridge nodes added earlier (bridge <- c(20, 40, 60), each wired to a second community) show a blended colour:

dot_soft <- nmfkc.net.DOT(res,
  Y.label = as.character(1:N), X.label = paste("Community", 1:K),
  Y.title = "Network nodes",   X.title = "Communities",
  layout  = "neato", threshold = 0.25, Y.cluster = "soft")
DiagrammeR::grViz(as.character(dot_soft))

Hard -- those same bridge nodes (20, 40, 60) are forced into one community colour:

dot_hard <- nmfkc.net.DOT(res,
  Y.label = as.character(1:N), X.label = paste("Community", 1:K),
  Y.title = "Network nodes",   X.title = "Communities",
  layout  = "neato", threshold = 0.25, Y.cluster = "hard")
DiagrammeR::grViz(as.character(dot_hard))

(If DiagrammeR is not installed, both diagrams are skipped.)

How many communities? `nmfkc.net.rank`

The same rank-selection diagnostics as nmfkc.rank are available for the symmetric model. The recommended rank is the cross-validation (ECV) minimum; the effective rank and R-squared elbow corroborate it.

rk <- nmfkc.net.rank(Y, rank = 1:6, type = "bi", nstart = 5)
rk$rank.best
round(rk$criteria, 3)

Remarks

nmfkc.net() recovers the three planted communities for the core nodes, and nmfkc.net.rank() selects three. Because the membership is soft (res$X.prob), the three bridge nodes that sit between communities show a split membership instead of being forced into one group --- the practical advantage of NMF over hard community detection, made visible by the soft-vs-hard DOT graphs above.

type = "bi" (\eqn{Y \approx X X^\top}) is the usual choice for an undirected, non-negative network; type = "tri" (\eqn{Y \approx X C X^\top}) adds a community-interaction matrix \eqn{C}, and type = "signed" handles networks with negative weights. For choosing the number of communities, the same advice as in the rank-selection vignette applies: use the ECV minimum as the primary criterion and corroborate it with the effective rank.

See ?nmfkc.net, ?nmfkc.net.ecv, ?nmfkc.net.rank, and the companion vignette "Choosing the NMF rank on data with a known true rank".