Algorithms for Visualizing Large Networks

Graphs are often used to encapsulate relationship between objects. Graph drawing enables visualization of such relationships. The usefulness of this visual representation is dependent on whether the drawing is aesthetic. While there are no strict criteria for aesthetics of a drawing, it is generally agreed, for example, that such a drawing has minimal edge crossing, with vertices evenly distributed in the space, connected vertices close to each other, and with symmetry that may exist in the graph preserved. One of the earliest example of graph drawing is probably the Mill (also known as Morris) game boards, found in the 13th century The Book of Games, produced under the direction of Alfonso X (1221-1284), King of Castile (Spain). Other examples of early graph drawing are drawings of family trees, and trees of virtues and vices. In 1735, Euler presented and later published his famous paper [13] on the Seven Bridges of Königsberg. The problem was to find a walk through the city of Königsberg, which include two islands connected to the mainland by seven bridges (Figure 1). The walk must cross each bridge once and only once. Euler proved that such a walk is not possible. Euler’s paper laid the foundation of graph theory. Interestingly, however, Euler did not provide a drawing of the graph representation of the problem itself in the paper. In the 18th century, the Irish physicist, astronomer, and mathematician, William Hamilton, invented the Iconsian game. To play the game, one has to visit every node of a dodecahedral graph once, and back to the starting node. Every edge can be passed at most once. In 1857 he sold the game to a London game dealer for 25 Pounds (about 3000 US dollars in today’s money after taking into account of inflation). The dealer made two versions; unfortunately, neither sold well, probably because the game was so simple that even a child could succeed. Around the same time, British mathematician Arthur Cayley wrote a paper on the enumeration of tree diagrams. In the paper he pioneered the