I assume that you have no header in your network file indicating the names of the column and that you are using Cytoscape 2. When you import your network (File -> Import network as a table), right click on the third column header, give it an appropriate attribute name (e.g. "Weight"), then left click on the column (it should turn blue) to indicate that you want to import this attribute.

Once the network is built, select an edge, go to the data panel (Menu: View -> Show data panel if not visible), click on the "Edge attribute browser", click on the button with two checkboxes and two tickmarks (its tooltip indicate "Select all Attributes") Check that the weight column is there

Then go to the control panel (Left panel of cytoscape window), go to VizMapper, in the visual mapping browser, double click on the row "Edge line width". This should create an "Edge line width" entry under the "Edge visual mapping" table. In this entry, select "Weight" as the value; then "Continuous Mapper" as the Mapping type. Double click on the "Graphical View" of this mapping to fine tune how weight values are mapped to edge width.

You can have a look at these two screenshots detailing the setup:

• http://dx.doi.org/10.6084/m9.figshare.155875
• http://dx.doi.org/10.6084/m9.figshare.155874

On Cytoscape 3.8.0, you can import WGCNA network by File -> Import -> Network from File and selecting the module edge file. On the import dialogue box, you will typically select the fromNode column as the Source Node and the toNode column as the Target Node. The weight column should be left as an Edge Attribute. The direction column should be changed to interaction type. fromAltName is a Source Node Attribute while the toAltName is a Target Node Attribute.