Mapping matrix of origins and destinations in QGIS?

If you are just looking to connect orgin/destination points and not needing the curved "great circle" lines, take a look at the QGIS plugin called "mmqgis". It has a Hub Lines tool that I think will create the visual that you are looking for.

"The hub lines tool creates hub and spoke diagrams with lines drawn from points on the "Spoke Point" layer to matching points in the "Hub Point" layer. Determination of which hub goes with each point is based on a match between the "Hub ID Attribute" on the hub points and the "Spoke Hub ID Attribute" on the spoke points."

For this to work, you will need a working PostGIS installation. Shapefiles can be loaded into PostGIS using the QGIS PostGIS Manager tool.

SQL queries can be executed in multiple tools from within QGIS (if it has connection to a PostGIS database), eg. RT SQL Layer plugin.

If you have only one destination point but multiple source points (table "source", the SQL query in http://underdark.wordpress.com/2011/08/20/visualizing-global-connections/ can be simplified to:

CREATE TABLE connections AS (
  source.id,
  SELECT ST_Transform(ST_Segmentize(ST_MakeLine(
    ST_Transform(ST_SetSRID(ST_Makepoint(--your lon lat here--),4326), 953027),
    ST_Transform(source.the_geom, 953027)
  ), 100000 ), 4326 ) AS the_geom
FROM source
);

Don't forget to first insert the CRS #953027 first.

You can depict flow maps via node-2-node flow mapping extension which is available from QGIS plugin repo;

http://plugins.qgis.org/plugins/FlowMapper/

there is also sample data and documentation within the zip file.

input node coordinates and interaction matrix should be in plane text format, white space delimited.