An algorithm to draw graphs without checking all pairs of vertices?

Question

Graph drawing algorithms, such as those described here, check all the vertices two-by-two and apply additional forces if two vertices are connected by an edge. If we have a very large graph, checking all pairs of vertices would be costly. Is there any graph drawing algorithm that draws a large graph using only existing edges, not by verifying all possible pairs?

EDIT
By drawing algorithm, I meant an algorithm that assigns a 2D or 3D position to each vertex such that rendering spheres or circles (or any other shape) as vertices at their assigned positions lead to a plausible visual representation of the whole graph.

Answer 1

Check this Spring-Electrical Embedding It is in O(nlog n).

Answer 2

If you have sparse matrix you can consider about creating graph as list of neighbours or simplier like pair of vertices (e.g. (1, 3) and 1 and 3 are the numbers of vertices).