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I want to visualise a network in a way that the nodes are along a line (and the edges are not on that line). I have included a simple example below.

Network on a line

Which network visualisation programs (Cytoscape, Gephi, R igraph etc) do have this layout option?

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That plot is called an Arc diagram you can google that and use the package or program that best suits you.

Here are 2 examples of the use of Arc diagrams in R:
Example 1: enter image description here

Example 2: enter image description here

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