Doubling Dimension in Real-World Graphs

1. Doubling Dimension in Real-World Graphs Melitta Lorraine Geistdoerfer Andersen

2. Recap: Definition • A metric space is a set X together with distance function d that gives a non-negative distance between any 2 points in X and satisfies 3 properties: • d(x,y) = 0 if and only if x = y • d(x,y) = d(y,x) • The triangle inequality holds: d(x,y) + d(y,z) ¸d(x,z) • The doubling dimension of a metric space (X,d) is the least k such that any ball of radius R can be covered by 2k balls of radius R/2. • So the doubling dimension is log2 of the maximum over all centers and all radii of the number of balls of half radius it takes to cover a ball with a specific center and radius.

3. An Example with a Set of Points • In this case, all of the points can be covered by 2k=2 balls of radius R/2. • Each of the balls also have a doubling dimension of 2. • And each of those contain no more than 22 points. • When the doubling dimension is a constant (i.e. bounded) the metric is called a doubling metric.

4. Some Uses of Doubling Dimension • Chan, Gupta, Maggs, and Zhou proved that for any network that has a metric with a bounded doubling dimension, a hierarchical routing structure can be imposed on it. • With this structure, the network can be addressed in such a way as to be able to get routing information from the addresses of the source and the destination. • This routing also achieves minimum or near-minimum path length. • There are also efficient nearest-neighbor algorithms that work with a graph of low doubling dimension.

5. Now We Can Apply It To A Graph • We found a 200,000 node router level graph of the Internet at http://www.caida.org/tools/measurement/skitter/router_topology/. • This was an adjacency graph, so we treated all edges as unit distances. • The doubling dimension was ~14.

6. Average Covering for Each Radius • Plotted on a log scale (because the x axis is also on a log scale), the average number of balls increased nearly linearly until it reached radius 8. • One interpretation of the downturn is the finite nature of the graph. • At R=64, only one ball of radius 32 is required to cover the entire ball. Hence, the diameter of the graph is at most 32.

7. But What About Latencies? • This was all well and good for an adjacency graph, but for routing you actually want to know the fastest route. So we needed a weighted graph. • http://www.cs.cornell.edu/People/egs/meridian/data.php yielded a graph that measured latencies between 2,500 sites. • The doubling dimension of this weighted graph was ~9.

8. Covering for a Weighted Graph • Plotted on a log scale, the average number of balls formed a more symmetric curve than the unweighted graph. • There were few nodes within range for the lower radii, and at the higher radii, we again saw the effects of a finite graph. • One thing of note is the spike of 2 after 1 had already been reached.

9. A Possible Explanation • One thing that could cause the spike is a 2 cluster graph. • Everything within a ball of a certain size can be covered by a ball of half the radius, for both clusters. • But when you double that radius, you run into the other cluster, so 2 balls are required to cover the whole thing.

10. Infinite Graphs? • Another thing to note is that the doubling dimension is finite because the graph is finite. • If this were a section of an infinite doubling metric the doubling dimension would eventually flatten out and become constant. • Though the graph does start to flatten out at the peak, we don’t know if this merely indicates that the finite nature of the graph is affecting it.

11. Other Graphs • We had so much fun with doubling dimension on these graphs, we wanted to find other graphs to play with. But what other interesting graphs are out there? • The Citation Graph connects authors of papers by references. An edge indicates that the author cited a paper by the other author in one of his papers. • People use these graphs to study nearest neighbor algorithms. • The doubling dimension of this graph is ~12.

12. The Citation Graph • This graph looks similar to the router graph. • The Citation Graph also has unit distances for the edges, so this similarity makes sense. • The earlier downward turn could be due to the high degree of each node. Many authors write many papers, and cite a large number of papers in them.

13. More Graphs • Doubling dimension can give us information about many types of graphs. • For instance, using the Internet Movie Database a graph of actors can be created with edges connecting two actors who were in the same movie. • The doubling dimension of this graph is ~14.

14. Yet Another Signature Graph • This graph started it’s downward trend right away. • One possible explanation is that this graph is much denser than the router graph, so the balls of radius 2 cover many points that may not be within 1 hop of each other.

15. The Effects of Scaling • The actor graph had 400,000 nodes. This made it an interesting graph for experimentation with scaling. If we included only a portion of the nodes, what would that do to the dimension?

16. Doubling Dimensions • Plotted on a log scale, the graph increases logarithmically until the maximum doubling dimension is reached.

17. Conclusions • Finite graphs have bounded doubling dimensions. • Different types of graphs have different signature cover graphs. • The number of nodes in a graph has some relation to the doubling dimension. • I like playing with graphs.

18. Future Work • Actually implementing the routing algorithm on a graph. • Measuring latencies of adjacent routers to get a more accurate picture to work with. • Figuring out bounds on how scaling effects doubling dimension, possibly working with some infinite graphs.