Dear,
I have made a script tool that generates the spider network(as you needed)- see shot attached
Usage: just locate the tool script in arc catalog. It needs four parameters
1. Input point feature
2.Output folder/work space
3.Output shapefile name
4.Search radius (for your case get the diagonal distance between two neighbor points using measure tool)
Tool link https://www.dropbox.com/s/4a7ayysvul6h4ot/tool.rar?dl=0
Enoy..