Watts-Strogatz Small-World Networks

The Watts-Strogatz model generates networks that exhibit the small-world property: high local clustering combined with short global path lengths. This topology captures a structural motif found in many biological neural circuits, where neurons form tight local clusters while maintaining efficient long-range communication.

The Small-World Phenomenon

Regular lattices have high clustering but long path lengths. Random graphs (Erdos-Renyi) have short path lengths but low clustering. Many real-world networks — including cortical neural circuits, social networks, and power grids — lie between these extremes, possessing both properties simultaneously.

The Watts-Strogatz model provides a one-parameter interpolation between a regular lattice and a random graph.

Construction Algorithm

The Watts-Strogatz network is constructed in two phases:

Phase 1: Ring Lattice

Start with $n$ neurons arranged on a ring. Each neuron is connected to its $K$ nearest neighbors on each side, for a total degree of $2K$. This produces a regular structure with high clustering but long path lengths (diameter $\sim n / 2K$).

Phase 2: Rewiring

For each edge in the ring lattice, rewire one endpoint to a uniformly random neuron with probability $\beta$:

• $\beta = 0$: No rewiring. The network remains a regular ring lattice.
• $0 \lt \beta \lt 1$: Some edges are rewired, introducing shortcuts that dramatically reduce path lengths while preserving most of the local clustering.
• $\beta = 1$: All edges are rewired. The network becomes approximately random.

The small-world regime occurs for intermediate values of $\beta$, typically $\beta \in [0.01, 0.3]$.

Key Properties

Clustering Coefficient

The clustering coefficient measures the density of triangles in the network. For the ring lattice ($\beta = 0$):

which approaches $3/4$ for large $K$. As $\beta$ increases, the clustering coefficient decays approximately as:

Even for moderate $\beta$ (e.g., 0.1), the clustering remains substantially higher than in a random graph, where $C_{\text{random}} = 2K / n$.

Average Path Length

The average shortest path length drops rapidly with even a small amount of rewiring:

The transition between these regimes is sharp: a rewiring probability of $\beta = 0.01$ is often sufficient to reduce the average path length to near its random-graph minimum while retaining most of the clustering.

The Small-World Regime

The small-world regime is characterized by:

In words: the clustering is much higher than random, but the path length is as short as random.

Implications for Reservoir Computing

Small-world topology provides two distinct advantages for reservoir dynamics:

Local Processing

High clustering creates local subnetworks where groups of neurons are densely interconnected. These clusters can act as local feature detectors or pattern recognizers, performing computation on localized aspects of the input signal before propagating results to the rest of the network.

Efficient Global Communication

Short path lengths (enabled by the rewired long-range connections) ensure that information from any part of the reservoir can reach any other part within a few synaptic steps. This prevents information bottlenecks and enables the reservoir to integrate information from multiple local clusters.

The combination of local processing and global integration is reminiscent of cortical architecture, where dense local connectivity coexists with sparse long-range projections between cortical areas.

SPIRES API

The Watts-Strogatz topology is selected with:

cfg.connectivity_type = SPIRES_CONN_SMALL_WORLD;
cfg.connectivity      = 0.1;  /* rewiring probability beta */

In SPIRES, the connectivity parameter is mapped to the rewiring probability $\beta$. The initial ring lattice degree $K$ is determined internally based on the desired overall connectivity density.

Example Configuration

spires_reservoir_config cfg = {
    .num_neurons       = 500,
    .num_inputs        = 1,
    .num_outputs       = 1,
    .spectral_radius   = 0.95,
    .ei_ratio          = 0.8,
    .input_strength    = 0.1,
    .connectivity      = 0.1,
    .dt                = 1.0,
    .connectivity_type = SPIRES_CONN_SMALL_WORLD,
    .neuron_type       = SPIRES_NEURON_FLIF_GL,
    .neuron_params     = NULL,
};

Choosing the Rewiring Probability

For reservoir computing, $\beta \in [0.05, 0.15]$ is a common starting range. This balances the benefits of clustering (local computation) with sufficient rewiring for global communication.

Small-World Networks and Memory

The interplay between clustering and path length has implications for the reservoir’s memory capacity. The clustered local subnetworks can sustain reverberating activity patterns that persist longer than in a random network, potentially enhancing the fading memory of the reservoir. Meanwhile, the shortcuts prevent the network from becoming too modular, ensuring that information is shared across the entire reservoir.

When combined with fractional-order neurons, the small-world topology provides a second mechanism (structural, in addition to the neuron-level power-law memory) for tuning the temporal dynamics of the reservoir.

When to Use Watts-Strogatz

Choose the Watts-Strogatz topology when:

• Tasks have spatiotemporal structure: Pattern recognition, speech processing, or sensor array inputs where local correlations are important.
• Biological plausibility matters: Cortical microcircuits exhibit small-world properties.
• You want structured diversity: The mix of local and long-range connections produces richer dynamics than a homogeneous random network.
• Baseline performance is insufficient: If Erdos-Renyi underperforms, small-world structure may provide the organizational benefit needed.

For tasks where heterogeneous degree distributions (hub neurons) are beneficial, consider Barabasi-Albert.

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