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I'm looking for software that will allow me to:

  1. Import an adjacency matrix (ideally one with weights, rather than just binary) and visualize the graph
  2. Manually edit the labels and positions of the nodes by clicking & dragging

Any recommendations?

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I'm pretty sure Gephi can do the editing. I'm not sure about importing an adjacency matrix. Gephi is Java-based and offers an interactive editor, IDE-style, if this is what you want. But interactive editing is feasible only for very small graphs.

I personally would use R and the ggraph package (which is a wrapper for the igraph C library/toolset) for this, and do it programmatically.

The free visjs library also has a network module, see this example and allows to edit a network/small graph inside your browser window.

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