Instructions
Form alliances with other empires that are in close proximity with your planets. The presence of a nearby friendly planet comforts your people and increases allegiance.
Build cultural structures, such as mass-media and world wonders, on planets that have lower allegiance scores. This tactic will placate your people, foil another empire's competing propaganda and even influence nearby planets that haven't yet joined you.
Invest in technological advances that increase the strength and breadth of your alliances, which passively allows allegiance to rise faster over time.
Build trade routes from your capital planet to outlying colonies. Ships move from planet to planet on phase lines, known routes that allow for fast interplanetary travel. The number of phase lines separating a planet and its capital determines its maximum allegiance rating, so encouraging trade will increase the maximum value possible. If your people are happy, this extra boost will be filled over time automatically.