The graph provides an easy visualisation for the audience to draw connections between the different characters in the short story. The author, being the narrator of the story, has connections with the most number of characters, since the story is told in their point of view. What the network graph is unable to provide, is information about what these connections or relationships are. We are unable to draw meaningful deductions from the network graph alone, as the lines only tell us that these two nodes (people) are connected in some way, but do not tell us how they are connected.
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