Random Graphs 2

1
Random Graphs 2

• Ben Byer
• Ara Hayrapetyan
• Jeff Phillips

2
Threshold functions

• Evolutionary reasoning allows us to reason more
  intuitively about graphs
• Properties appear at thresholds
• p cn-k/(k-1)
• p c/n (0 lt c lt 1)
• p c/n (c gt 1)
• p (ln n)/n
• p -gt c

3
p cn-k/(k-1)

• Before this, nodes are largely unconnected
• At this point, G is a forest of trees of size k
• Every tree of k vertices appears No tree of k1
  vertices appears
• No cycles

4
p c/n (0 lt c lt 1)

• Cycles begin to appear
• largest component is log n
• all components are trees or unicyclic

5
p c/n (c gt 1)

• at c1, number of cycles approaches infinity
• size of largest component jumps from log n to
  n2/3
• at cgt1, large component appears -- added edges
  are more likely to fall in the large component
  than outside of it
• isolated vertices still exist
• G is likely to have some cycle with three
  crossing chords and to be nonplanar

6
p (ln n)/n

• isolated vertices disappear
• large component takes over everything
• graph is connected

7
p -gt c

• large component swallows entire graph, and it
  becomes dense.
• Appearance of arbitrary subgraphs
• high chromatic number