Computing a pagerank on a weighted graph with absolute weights - graph

Question

I am facing the same issue as expressed in this link (Networkx PageRank - Equal Ranks with Different Weights).
Essentially, I am using networkx to compute the pagerank on a graph. Since, pagerank computation first converts the graph to a right stochastic matrix (all out-going edges are normalised to one).
What I need is a way to not normalise the edge weights. So, If one node as only one outgoing edge with weight 0.1 and another one has only one outgoing edge with weight 0.05, I want this information to be used in the computation of pagerank (rather than being normalized to 1 each).
Does anyone know what could be the right way of modifying pagerank to achieve this.
thanks in advance,
Amit

Answer 1

Maybe you are thinking of what Larry and Sergey called "Personalized PageRank"? You can adjust the weighting of the nodes in the random jump part of the algorithm to create a bias. E.g.
In [1]: import networkx as nx
In [2]: G = nx.DiGraph()
In [3]: G.add_path([1,2,3,4])
In [4]: nx.pagerank_numpy(G)
Out[4]:
{1: 0.11615582303660349,
2: 0.2148882726177166,
3: 0.29881085476166286,
4: 0.370145049584017}
In [5]: nx.pagerank_numpy(G,personalization={1:1,2:10,3:1,4:1})
Out[5]:
{1: 0.031484535189871404,
2: 0.341607206810105,
3: 0.3218506609784609,
4: 0.3050575970215628}
See, for example, the discussion here http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf

Answer 2

I have a graph which is not right stochastic (i.e. edge weights are absolute and consistent across nodes). I changed pagerank implementation of networkx to avoid converting initial matrix to a right stochastic matrix thus giving me the right answer. However, this implies that pagerank doesn't converge as sometimes total sum of edges > 1, but usually rankings are fairly consistent after 30-40 iterations.
In essence, removing this line from the networkx code (algorithms/link_analysis/pagerank_alg.py) did the job:-
W = x.stochastic_graph(D, weight=weight)