Joe Barton and Leopold Eddington are hypothetical characters, so I'll provide a general framework on how characters' actions can influence a plot. In any narrative, characters like Joe Barton and Leopold Eddington can influence the plot in several ways:

1. Driving the Conflict: One or both characters might be central to the story's main conflict. For example, if Joe Barton is a protagonist with a specific goal, Leopold Eddington might be an antagonist whose actions create obstacles for Joe, thus driving the narrative forward.

2. Character Development: Their interactions and decisions can lead to significant character development. If Joe is faced with moral dilemmas due to Leopold’s actions, it might lead to growth or change in Joe’s character, which can be a crucial element of the plot.

3. Plot Twists: Either character could be involved in a plot twist. For instance, Leopold could reveal a hidden motive or identity that changes the direction of the story, impacting Joe and other characters in unexpected ways.

4. Theme Exploration: Through their actions and interactions, themes can be explored. If Joe and Leopold have opposing worldviews, their conflict can highlight themes such as justice, power, or redemption.

5. Resolution: Their actions can lead to the climax and resolution of the story. The decisions they make might resolve the central conflict, bringing the story to its conclusion.

In essence, characters like Joe Barton and Leopold Eddington can shape the plot through their motivations, conflicts, relationships, and growth, ultimately driving the story’s progression and resolution.