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Does anyone know about the differences in accuracy between the three different pagerank functions in Networkx?

I have a graph of 1000 nodes and 139732 edges, and the "plain" pagerank function didn't seem to work at all -- all but two of the nodes had the same PG, so I'm assuming this function doesn't work quite as well for large graphs?

pagerank_numpy's values also seemed to be a little bit more spread out than pagerank_scipy's values. The documentation for this function says that "This will be the fastest and most accurate for small graphs." What is meant by "small" graphs?

Also, why doesn't pagerank_numpy allow for max_iter and tol arguments?

Saullo G. P. Castro
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wrongusername
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1 Answers1

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Each of the three functions uses a different approach to solving the same problem:

networkx.pagerank() is a pure-Python implementation of the power-method to compute the largest eigenvalue/eigenvector or the Google matrix. It has two parameters that control the accuracy - tol and max_iter.

networkx.pagerank_scipy() is a SciPy sparse-matrix implementation of the power-method. It has the same two accuracy parameters.

networkx.pagerank_numpy() is a NumPy (full) matrix implementation that calls the numpy.linalg.eig() function to compute the largest eigenvalue and eigenvector. That function is an interface to the LAPACK dgeev function which is uses a matrix decomposition (direct) method with no tunable parameters.

All three should produce the same answer (within numerical roundoff) for well-behaved graphs if the tol parameter is small enough and the max_iter parameter is large enough. Which one is faster depends on the size of your graph and how well the power method works on your graph. 

In [12]: import networkx as nx

In [13]: G=nx.gnp_random_graph(1000,0.01,directed=True)

In [14]: %timeit nx.pagerank(G,tol=1e-10)
10 loops, best of 3: 157 ms per loop

In [15]: %timeit nx.pagerank_scipy(G,tol=1e-10)
100 loops, best of 3: 14 ms per loop

In [16]: %timeit nx.pagerank(G)
10 loops, best of 3: 137 ms per loop
user
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Aric
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    Awesome explanation, thanks! By the way, how does pagerank_numpy know when to stop without those tunable parameters? – wrongusername Oct 24 '12 at 00:18
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    The algorithm that computes the eigenvalues in pagerank_numpy() (LAPACK's dgeev) does a fixed number of operations that depends only on the matrix size. I think it should be roughly n^3 where n is the number of nodes. See http://stackoverflow.com/questions/713878/how-expensive-is-it-to-compute-the-eigenvalues-of-a-matrix for a longer discussion of this in the context of PageRank. – Aric Oct 24 '12 at 01:06
  • By the way, what do you mean by "well-behaved graphs"? Graphs that are connected and have no dead ends? Because I wonder if the disprecancies in the results I observe could be due to that... – wrongusername Jan 08 '13 at 19:38