Author: Maximiliano Martino, Daniel Fraiman
Date: 2026-06-04

# Generating and Analyzing Brain Networks with Community Structures

#### Maximiliano Martino, Daniel Fraiman

#### 2026-06-04

## Overview

This vignette shows the full `BrainNetTest` workflow on synthetic populations of brain networks with community (block) structure. We simulate two groups of graphs with different intra- and inter-community connection probabilities, run the global L1-distance ANOVA test, and identify the edges and nodes that drive the observed difference.

    library(BrainNetTest)

## Generating a single community-structured graph

`generate_community_graph()` samples a symmetric binary adjacency matrix with a block-stochastic structure: edges within a community are drawn with probability `intra_prob`, edges between communities with `inter_prob`.

    set.seed(42)
    G <- generate_community_graph(
      n_nodes       = 40,
      n_communities = 4,
      intra_prob    = 0.8,
      inter_prob    = 0.2)

    dim(G)
    #> [1] 40 40
    mean(G[upper.tri(G)])
    #> [1] 0.3358974

### Visualising the network

A single community-structured adjacency matrix is best inspected directly with `igraph`. The package’s own visualisation helper, `plot_critical_edges()`, is designed to summarise the result of an analysis (per-population central graphs plus the identified critical edges) and is illustrated at the end of this vignette.

## Simulating populations of graphs

`generate_category_graphs()` produces a list of graphs that share a common community structure but vary slightly in their edge probabilities across replicates, modelling natural between-subject variability.

    control <- generate_category_graphs(
      n_graphs             = 20,
      n_nodes              = 20,
      n_communities        = 2,
      base_intra_prob      = 0.8,
      base_inter_prob      = 0.2,
      intra_prob_variation = 0.05,
      inter_prob_variation = 0.05,
      seed                 = 1)

    patient <- generate_category_graphs(
      n_graphs             = 20,
      n_nodes              = 20,
      n_communities        = 2,
      base_intra_prob      = 0.6,
      base_inter_prob      = 0.4,
      intra_prob_variation = 0.05,
      inter_prob_variation = 0.05,
      seed                 = 2)

    populations <- list(Control = control, Patient = patient)
    lengths(populations)
    #> Control Patient
    #>      20      20

## Global test statistic

    T_obs <- compute_test_statistic(populations, a = 1)
    T_obs
    #>   Control
    #> -85.03753

## Identifying critical edges

`identify_critical_links()` performs the full pipeline: marginal edge p-values, permutation null for `T`, and iterative edge removal. It returns the edges whose removal eliminates the group-level difference.

    result <- identify_critical_links(
      populations,
      alpha          = 0.05,
      method         = "fisher",
      n_permutations = 500,
      seed           = 42)

    nrow(result$critical_edges)
    #> [1] 50
    head(result$critical_edges)
    #>     node1 node2      p_value
    #> 185    14    20 0.0004359198
    #> 117    12    16 0.0021996412
    #> 50      5    11 0.0033420517
    #> 56      1    12 0.0033420517
    #> 65     10    12 0.0033420517
    #> 153    17    18 0.0033420517

### Critical nodes

`get_critical_nodes()` aggregates the critical edges at the node level, reporting the critical degree (number of critical edges incident on each node):

    get_critical_nodes(result)
    #>    node critical_degree
    #> 1    12              10
    #> 2    16               8
    #> 3    14               7
    #> 4    17               7
    #> 5    20               7
    #> 6     1               6
    #> 7     5               6
    #> 8    19               6
    #> 9     3               5
    #> 10    6               5
    #> 11    9               5
    #> 12    2               4
    #> 13    8               4
    #> 14   11               4
    #> 15   13               4
    #> 16   15               4
    #> 17    4               3
    #> 18   10               2
    #> 19   18               2
    #> 20    7               1

## Visualising the result

`plot_critical_edges()` produces a multi-panel figure that summarises the analysis: one panel per population showing the (weighted) central graph, plus a final panel that highlights the critical edges on the chosen reference central graph. Communities can be passed in to color the vertices consistently across panels.

    plot_critical_edges(
      populations,
      result,
      communities = rep(seq_len(2), each = 10),
      reference   = "Control")

## References

Fraiman, D. and Fraiman, R. (2018) An ANOVA approach for statistical comparisons of brain networks. Scientific Reports, 8, 4746. https://doi.org/10.1038/s41598-018-21688-0