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Influence
From Galactic Civilizations III Wiki
Influence is the abstracted measure of the soft power of your civilization and the attractiveness of your culture to others. A highly influential civilization will command respect and keep enemies at a distance, even if its military forces pale in comparison. Conversely, a powerful military but weak influence will affect diplomacy negatively.
In gameplay terms, influence allows for culture flipping - taking over colonies of opposing civilizations within your zone of control.
