Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mean first-passage time for random walks in general graphs with a deep
  trap   We provide an explicit formula for the global mean first-passage time (GMFPT)
for random walks in a general graph with a perfect trap fixed at an arbitrary
node, where GMFPT is the average of mean first-passage time to the trap over
all starting nodes in the whole graph. The formula is expressed in terms of
eigenvalues and eigenvectors of Laplacian matrix for the graph. We then use the
formula to deduce a tight lower bound for the GMFPT in terms of only the
numbers of nodes and edges, as well as the degree of the trap, which can be
achieved in both complete graphs and star graphs. We show that for a large
sparse graph the leading scaling for this lower bound is proportional to the
system size and the reciprocal of the degree for the trap node. Particularly,
we demonstrate that for a scale-free graph of size $N$ with a degree
distribution $P(d)\sim d^{-\gamma}$ characterized by $\gamma$, when the trap is
placed on a most connected node, the dominating scaling of the lower bound
becomes $N^{1-1/\gamma}$, which can be reached in some scale-free graphs.
Finally, we prove that the leading behavior of upper bounds for GMFPT on any
graph is at most $N^{3}$ that can be reached in the bar-bell graphs. This work
provides a comprehensive understanding of previous results about trapping in
various special graphs with a trap located at a specific location.

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The categories of this paper are: quant-ph
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The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: math
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th