Tutorial 83: Network Topology Analysis

Analyse the graph structure of your SNN: small-world coefficient, modularity, centrality, clustering, path lengths. These metrics predict network dynamics and inform architecture choices — small-world networks produce richer dynamics than random graphs.

Why Topology Matters

The connectivity pattern of an SNN determines its computational properties more than individual neuron parameters:

Topology	Dynamics	Computation
Random (Erdős-Rényi)	Homogeneous, rapid synchrony	Poor memory, poor separation
Small-world (Watts-Strogatz)	Rich dynamics, edge-of-chaos	Good memory + separation
Scale-free (Barabási-Albert)	Hub-dominated, fragile	High throughput, low robustness
Modular	Community structure	Task-specific subnetworks

SC-NeuroCore uses small-world connectivity by default — it produces the best balance of local clustering and global integration.

TopologyAnalyzer

import numpy as np
from sc_neurocore.topology import TopologyAnalyzer

rng = np.random.RandomState(42)

# Random sparse connectivity (100 neurons, 10% connection probability)
W = (rng.random((100, 100)) < 0.1).astype(float)
np.fill_diagonal(W, 0)

analyzer = TopologyAnalyzer(W)
report = analyzer.analyze()

print(f"Nodes: {report.n_nodes}")
print(f"Edges: {report.n_edges}")
print(f"Density: {report.density:.3f}")
print(f"Clustering coefficient: {report.clustering:.3f}")
print(f"Mean path length: {report.mean_path_length:.2f}")
print(f"Small-world sigma: {report.small_world_sigma:.2f}")
print(f"Modularity: {report.modularity:.3f}")

Key Metrics

Metric	Formula	What It Means	Optimal for SNN
Clustering (C)	Triangle count / triplet count	Local connectivity density	0.1–0.3
Mean path (L)	Average shortest-path hops	Global integration	2–4 hops
Small-world (σ)	(C/C_rand) / (L/L_rand)	σ > 1 = small-world	1.5–5.0
Modularity (Q)	Community structure strength	Q > 0.3 = modular	0.3–0.6
Degree centrality	Connections / (N-1)	Hub detection	Uniform (no hubs)
Betweenness	Fraction of shortest paths through node	Bottleneck detection	Low max

Small-World Analysis

Networks with high clustering AND short path lengths are "small-world." This is the optimal regime for spiking networks:

# Generate small-world connectivity
from sc_neurocore.network.topology import generate_connectivity

# Watts-Strogatz small-world
W_sw = generate_connectivity(100, topology="small_world", p_rewire=0.1, k=10)
report_sw = TopologyAnalyzer(W_sw).analyze()

# Compare with random
W_rand = generate_connectivity(100, topology="random", p=0.1)
report_rand = TopologyAnalyzer(W_rand).analyze()

print(f"Small-world: C={report_sw.clustering:.3f}, L={report_sw.mean_path_length:.2f}, "
      f"σ={report_sw.small_world_sigma:.2f}")
print(f"Random:      C={report_rand.clustering:.3f}, L={report_rand.mean_path_length:.2f}, "
      f"σ={report_rand.small_world_sigma:.2f}")

Modularity Detection

Find functional communities in your network:

communities = analyzer.detect_communities()
print(f"Communities found: {len(communities)}")
for i, comm in enumerate(communities):
    print(f"  Community {i}: {len(comm)} neurons")

Integration with Studio

The Network Canvas shows populations and projections. The topology analyser operates on the underlying weight matrix — use it to verify that your designed connectivity has the desired graph properties.

References

• Watts & Strogatz (1998). "Collective dynamics of 'small-world' networks." Nature 393:440-442.
• Barabási & Albert (1999). "Emergence of Scaling in Random Networks." Science 286:509-512.
• Sporns (2011). "Networks of the Brain." MIT Press.

Interactive Notebook

Run the hands-on notebook: notebooks/09_topology_and_dynamics.ipynb