Well, a verticle power structure is we have prime minister/president/king on top, then cabinet of ministers, or senate and congress, etc. then it trickles down as smaller and smaller power structures within states, cities: police, DMV, school districts, blah, blah, blah...

The horizontal power structure is essentially none of this, but only the connections between businesses and people. Example of horizontal connections would be E-lance that connects clients in the US with programmers and designers in India, China, Ukraine, Europe, etc.