PSO Neighborhood Topology: Global-Best vs Ring

Neighborhood Topology

PSO Parameters

Swarm size 30

Inertia weight w 0.72

Cognitive c₁ (self) 1.49

Social c₂ (neighbors) 1.49

Live State

Iteration

Diversity (avg |x - x̄|)

0.00

Best fitness found f(x*)

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Best position x*

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How it works

Each particle i keeps a position x_i, velocity v_i, and its own best-ever position p_i (personal best). Every step it also looks at the best position found among its neighbors — the neighborhood best g_i — and updates:

v_i ← w·v_i + c1·r1·(p_i − x_i) + c2·r2·(g_i − x_i)
x_i ← x_i + v_i          (r1, r2 ~ Uniform(0,1), resampled every step)

The topology defines who counts as a "neighbor" and therefore what g_i is:

Global-best (star): every particle's neighborhood is the whole swarm, so g_i is the single best position anyone has ever found. Information spreads instantly — the swarm converges fast, but if the whole population rushes toward one promising-looking dip it can get stuck there, missing the true global minimum (premature convergence).
Local-best (ring): particles are arranged on a ring and each only sees its 2 immediate ring neighbors, so g_i is the best position among just those 3 particles. Good solutions propagate around the ring gradually instead of everywhere at once, which slows convergence but preserves diversity and makes premature convergence to a bad local minimum much less likely.

The terrain here is the Rastrigin function, a standard optimization benchmark full of regularly-spaced local minima surrounding one global minimum at the origin:

f(x, z) = 20 + x² + z² − 10·cos(2πx) − 10·cos(2πz)     global min: f(0,0) = 0

Height on screen is drawn as (peak − f), so the swarm visibly climbs toward the tallest peak, which sits exactly at the true optimum. Watch the Diversity readout: global-best usually collapses it toward zero quickly (everyone rushing one direction), while ring topology keeps it higher for longer as sub-groups explore independently.

Particles (cyan trail = personal best) Global incumbent best

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