Approximating Single-Source Personalized PageRank with Absolute Error Guarantees
CoRR(2024)
摘要
Personalized PageRank (PPR) is an extensively studied and applied node
proximity measure in graphs. For a pair of nodes s and t on a graph
G=(V,E), the PPR value π(s,t) is defined as the probability that an
α-discounted random walk from s terminates at t, where the walk
terminates with probability α at each step. We study the classic
Single-Source PPR query, which asks for PPR approximations from a given source
node s to all nodes in the graph. Specifically, we aim to provide
approximations with absolute error guarantees, ensuring that the resultant PPR
estimates π̂(s,t) satisfy max_t∈
V|π̂(s,t)-π(s,t)|≤ε for a given error bound
ε. We propose an algorithm that achieves this with high
probability, with an expected running time of
- O(√(m)/ε) for directed graphs, where
m=|E|;
- O(√(d_max)/ε) for undirected
graphs, where d_max is the maximum node degree in the graph;
- O(n^γ-1/2/ε) for power-law
graphs, where n=|V| and γ∈(1/2,1) is the extent
of the power law.
These sublinear bounds improve upon existing results. We also study the case
when degree-normalized absolute error guarantees are desired, requiring
max_t∈ V|π̂(s,t)/d(t)-π(s,t)/d(t)|≤ε_d for
a given error bound ε_d, where the graph is undirected and d(t)
is the degree of node t. We give an algorithm that provides this error
guarantee with high probability, achieving an expected complexity of
O(√(∑_t∈
Vπ(s,t)/d(t))/ε_d). This improves over the previously
known O(1/ε_d) complexity.
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