Doubling Dimension in Real-World Graphs

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Doubling Dimension in Real-World Graphs

• Melitta Lorraine Geistdoerfer Andersen

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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.

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An Example with a Set of Points

• In this case, all of the points can be covered by
  2k2 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.

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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.

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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.

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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 R64, only one ball of radius 32 is required
  to cover the entire ball. Hence, the diameter of
  the graph is at most 32.

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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.

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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.

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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.

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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 dont know if this merely indicates that
  the finite nature of the graph is affecting it.

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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.

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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.

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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.

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Yet Another Signature Graph

• This graph started its 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.

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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?

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Doubling Dimensions

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

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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.

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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.