32. Network Analysis

Network Analysis¶

Overview¶

Network analysis represents the brain as a set of interconnected nodes (brain regions or voxels) and edges (anatomical or functional connections), letting us study brain organization, network properties, and dynamics with the mathematical machinery of graph theory — a field whose roots reach back to Euler in the mid-1700s. In network neuroscience, functional connectomes are used to estimate network topology and higher-order graph-theoretic properties: global properties like efficiency and small-worldness, sub-network structure (“communities” or “modules”), and “hub” nodes that connect many regions or link different communities. All of these properties can vary over time and can be analyzed in relation to cognition, behavior, and disease — for example, comparing network topology between patient and control groups to reveal connectivity abnormalities.

A network is represented mathematically as a graph $G(V, E)$: a set of vertices $V$ and pairwise edges $E$ among them. Numerically, a graph is an $N \times N$ matrix, where $N$ is the number of nodes and each element represents an edge. In a connectivity matrix the elements are continuous (“weighted”) values representing connection strength; in an adjacency matrix they are binarized to 1 (connected) or 0 (unconnected). Graphs may also be undirected (a symmetric matrix — the edge from $i$ to $j$ equals the edge from $j$ to $i$) or directed (asymmetric, with rows conventionally denoting sources and columns targets). Functional connectivity matrices are symmetric, implying undirected networks; effective connectivity methods such as Dynamic Causal Modeling or Granger causality (Chapters 35–36) yield directed ones.

Constructing a brain network means filling in the entries of that matrix. In fMRI the matrix is typically a functional connectome (Chapter 30): correlations between the time series of all node pairs, using Pearson or Spearman correlations, partial correlations, coherence, or mutual information. Because many graph metrics require relatively sparse graphs, the next step is usually to threshold the matrix — removing weak connections — and often to binarize it into an adjacency matrix. Thresholds may be a fixed value, a fixed percentage of retained connections, or a statistical significance level (which raises a multiple comparisons problem across the $N(N-1)/2$ edge tests, analogous to mass-univariate GLM analysis). The connection density — surviving edges divided by possible edges — measures the resulting sparsity. Thresholding removes weak, potentially spurious connections, sharpens topological structure, and eases computation, but it has real disadvantages: the “right” threshold is rarely obvious, multiplicity grows rapidly with $N$, and — critically — subjects who differ in overall connectivity strength end up with different densities at a fixed threshold, which by itself changes every metric computed downstream. The standard fix is to vary the threshold per subject so that the number of edges (or the density) is held fixed.

Once built, a network can be characterized at several topological scales. Local measures describe individual nodes. A node’s degree counts its neighbors,

Degree is the most common index of centrality — a node’s importance for information transfer — and hence of “hub status” (with strength as the weighted analog). A node’s clustering coefficient asks how many of its neighbors are also neighbors of each other: if nodes $i$, $j$, and $h$ are all interconnected they form a closed triangle, and with $t_i$ the number of triangles around node $i$,

The network clustering coefficient is a measure of functional segregation: specialized processing within densely interconnected groups. High clustering implies redundant connections, so losing one node matters less. Betweenness centrality counts the fraction of shortest paths between all node pairs that pass through a node, and the participation coefficient measures how evenly a node’s edges are spread across communities — distinguishing provincial hubs (well connected within their own module) from connector hubs (linking different modules). Global measures summarize the whole network. The shortest path length $d_{ij}$ is the minimum number of edges (or minimum total weight) needed to travel between two nodes, and the characteristic path length

The characteristic path length indexes functional integration: short paths mean information can be combined rapidly across the network. Related global measures include global efficiency, assortativity (the tendency of high-degree nodes to connect to each other, forming densely interconnected “rich clubs”), and resilience — the ability to keep functioning as nodes and edges are removed. In between sit mesoscale properties: communities (modules) detected by algorithms such as Louvain, Girvan–Newman, or Clauset–Newman–Moore modularity optimization, hierarchical or spectral clustering, and stochastic block models; and, in multilayer networks spanning time points or data types, recruitment and integration of nodes across layers.

These metrics define characteristic types of networks. In a regular network (a lattice or ring) every node has the same degree: clustering $C$ is high but so is path length $L$ — redundant local communities, inefficient global transmission. In a random (Erdős–Rényi) network every node pair connects with equal probability $p$: both $C$ and $L$ are low — information travels easily, but the structure is vulnerable and unclustered. A small-world network resembles an ordered network with a few randomly rewired links: high $C$ and low $L$, making it simultaneously resilient and efficient. Since Watts and Strogatz’s seminal observation that many social, biological, and technological networks share this property, the brain too has been argued to be small-world — maximizing segregation and integration while minimizing wiring cost. Small-worldness is quantified against a null network:

with $\sigma > 1$ commonly taken as the cutoff. Other overall structures include core–periphery organization and disassortative networks, in which high-degree nodes preferentially connect to low-degree ones.

Finally, networks can be compared — patients versus controls, or network properties against behavior. So long as the graphs share the same set of nodes, any graph metric can serve as a dependent variable in a GLM regressed on group membership or cognitive performance, or as a feature in machine-learning models (Chapters 37–41). Group differences can be tested nonparametrically with permutation tests that shuffle group labels to build a null distribution for the difference in a metric. And to ask whether a network’s organization is non-random — small-world, core–periphery — the benchmark is a population of simulated random graphs with the same connection density and number of connected nodes, yielding a null distribution for each topological metric of interest.

Hands-on tutorial¶

In this tutorial you will build a graph from a simulated modular functional connectome, compute the core metrics from the chapter, and see first-hand why threshold choice matters. We simulate 30 nodes organized into three modules (plus a shared global signal that weakly links them), correlate their time series, threshold, and analyze the result.

Step 1 — From connectivity matrix to graph and metrics. We compute the correlation matrix, binarize it at $r > 0.3$, and compute density, degree, clustering, and characteristic path length.

% Base MATLAB graph objects; CanlabCore on path for the fuller lab
rng(7);                                      % fix the random seed for reproducibility
n = 30; T = 200;                             % n = nodes (regions), T = time points
module  = repelem(1:3, 10)';                 % 3 modules of 10 nodes
g       = randn(1, T);                       % shared global signal
signals = randn(3, T);                       % one latent signal per module
% 0.8 = global-signal weight, 1.2 = noise SD
ts = 0.8 * repmat(g, n, 1) + signals(module, :) + 1.2 * randn(n, T);

R = corr(ts');                               % 30 x 30 functional connectome
A = double(R > 0.3);  A(1:n+1:end) = 0;      % threshold at r > 0.3 + binarize
G = graph(A);

density = numedges(G) / nchoosek(n, 2)       % connection density
deg = degree(G);                             % node degree k_i

t_i = diag(A^3) / 2;  k = sum(A, 2);         % triangles around each node
Ci  = 2 * t_i ./ (k .* (k - 1));  Ci(k < 2) = 0;
C   = mean(Ci)                               % network clustering coefficient

D = distances(G);                            % shortest path lengths d_ij
comp = conncomp(G);  in_cc = comp == mode(comp);
Dcc  = D(in_cc, in_cc);                      % largest connected component
L    = mean(Dcc(Dcc > 0))                    % characteristic path length

import numpy as np, networkx as nx

rng = np.random.default_rng(7)                # fix the random seed for reproducibility
n, T = 30, 200                                # n = nodes (regions), T = time points
module  = np.repeat([0, 1, 2], 10)            # 3 modules of 10 nodes
g       = rng.standard_normal(T)              # shared global signal
signals = rng.standard_normal((3, T))         # one latent signal per module
# 0.8 = global-signal weight, 1.2 = noise SD
ts = 0.8 * g + signals[module] + 1.2 * rng.standard_normal((n, T))

R = np.corrcoef(ts)                           # 30 x 30 functional connectome
A = (R > 0.3).astype(int)                     # threshold at r > 0.3 + binarize
np.fill_diagonal(A, 0)
G = nx.from_numpy_array(A)

density = nx.density(G)                       # connection density
degree  = dict(G.degree())                    # node degree k_i
C       = nx.average_clustering(G)            # network clustering coefficient

Gcc = G.subgraph(max(nx.connected_components(G), key=len))
L   = nx.average_shortest_path_length(Gcc)    # characteristic path length
print(f"density={density:.2f}  C={C:.2f}  L={L:.2f}")

Example output:

density=0.31  C=0.99  L=1.95

At $r > 0.3$, roughly a third of all possible edges survive. Each 10-node module is almost fully interconnected — hence the near-ceiling clustering coefficient — while the sparser between-module links keep the characteristic path length just under 2.

Step 2 — The threshold changes everything. Sweep the threshold and watch density, clustering, and path length co-vary — none of these metrics is interpretable without knowing the density it was computed at.

thr = 0.10:0.05:0.50;                    % thresholds to sweep, lenient -> strict
[dens, Cs, Ls] = deal(zeros(size(thr)));
for i = 1:numel(thr)
    A = double(R > thr(i));  A(1:n+1:end) = 0;
    G = graph(A);
    dens(i) = numedges(G) / nchoosek(n, 2);
    t_i = diag(A^3)/2;  k = sum(A, 2);
    Ci = 2*t_i ./ (k .* (k-1));  Ci(k < 2) = 0;  Cs(i) = mean(Ci);
    D = distances(G);  comp = conncomp(G);
    Dcc = D(comp == mode(comp), comp == mode(comp));
    Ls(i) = mean(Dcc(Dcc > 0));
end
plot(thr, dens, '-o', thr, Cs, '-s', thr, Ls / max(Ls), '-^');
legend({'Density', 'Clustering C', 'Path length L (scaled)'});
xlabel('Correlation threshold');

import matplotlib.pyplot as plt

thresholds = np.arange(0.10, 0.51, 0.05)      # thresholds to sweep, lenient -> strict
dens, Cs, Ls = [], [], []
for thr in thresholds:
    A = (R > thr).astype(int); np.fill_diagonal(A, 0)
    G = nx.from_numpy_array(A)
    dens.append(nx.density(G))
    Cs.append(nx.average_clustering(G))
    Gcc = G.subgraph(max(nx.connected_components(G), key=len))
    Ls.append(nx.average_shortest_path_length(Gcc))

plt.plot(thresholds, dens, "-o", label="Density")
plt.plot(thresholds, Cs, "-s", label="Clustering C")
plt.plot(thresholds, np.array(Ls) / max(Ls), "-^", label="Path length L (scaled)")
plt.xlabel("Correlation threshold"); plt.legend()

Example output:

Every curve moves together: a “group difference in clustering” at a fixed threshold may be nothing more than a group difference in density. The full labs push further: they plant a connector hub in the network and find it with degree and betweenness centrality, detect communities with greedy modularity optimization and score them against the ground-truth modules, test small-worldness ($\sigma$) against density-matched random graphs, and demonstrate that comparing two “subjects” with different overall connectivity strength is confounded at a fixed threshold but fair after density matching.

Thought questions¶

1. Node degree, betweenness centrality, and participation coefficient can all be used to call a region a “hub” — yet they can disagree sharply for the same node. Construct (or sketch) a network in which a node has modest degree but very high betweenness, and explain what functional role such a region might play in the brain. Why does the provincial/connector distinction matter for predicting the consequences of a lesion?

2. A study reports that patients have lower clustering coefficients than controls, using a fixed correlation threshold of $r > 0.25$ applied to every subject. The patients also show globally weaker functional connectivity (e.g., due to greater head motion or vascular differences). Walk through the chain of consequences from weaker correlations to the reported group difference. What analyses would convince you the topological difference is real?

3. Small-worldness $\sigma$ compares a network’s clustering and path length to those of random graphs. What properties should the null networks be matched on (density? degree sequence? connectedness?), and how could a poor choice of null model manufacture — or hide — small-world structure?

4. Functional connectomes based on correlation are undirected, so all the metrics in this chapter ignore the direction of influence. For which network properties (hubs, communities, efficiency) do you think ignoring directionality is most misleading, and how might metrics computed on a directed effective-connectivity graph (Chapters 35–36) change the picture?

5. Community-detection algorithms will happily partition any network — including a random one — into “modules.” Drawing on the network-comparison logic of the chapter, design a test for whether the modular structure detected in a resting-state connectome is statistically meaningful rather than an artifact of the algorithm.

Quiz yourself¶

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