25,249 research outputs found
Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs
In this paper we study weighted distances in scale-free spatial network
models: hyperbolic random graphs (HRG), geometric inhomogeneous random graphs
(GIRG) and scale-free percolation (SFP). In HRGs,
vertices are sampled independently from the hyperbolic disk with radius and
two vertices are connected either when they are within hyperbolic distance ,
or independently with a probability depending on the hyperbolic distance. In
GIRGs and SFP, each vertex is given an independent weight and location from an
underlying measured metric space and , respectively, and two
vertices are connected independently with a probability that is a function of
their distance and weights. We assign i.i.d. weights to the edges of the random
graphs and study the weighted distance between two uniformly chosen vertices.
In SFP, we study the weighted distance from the origin of vertex-sequences with
norm tending to infinity. In particular, we study the case when the parameters
are so that the degree distribution in the graph follows a power law with
exponent (infinite variance), and the edge-weight distribution
is such that it produces an explosive age-dependent branching process with
power-law offspring distribution. We show that in all three models, typical
distances within the giant/infinite component converge in distribution, solving
an open question in [Explosion and distances in scale-free percolation (2017)].
The main tools of our proof are to couple the models to infinite versions, to
follow the shortest paths to infinity and to connect these paths using
weight-dependent percolation on the graphs: delete edges attached to vertices
with higher weight with higher probability. We realise this using the
edge-weights: only short edges connected to high weight vertices will stay,
yielding arbitrarily short upper bounds for the connections.Comment: 49 pages, 4 figure
The trace-reinforced ants process does not find shortest paths
In this paper, we study a probabilistic reinforcement-learning model for ants
searching for the shortest path(s) between their nest and a source of food. In
this model, the nest and the source of food are two distinguished nodes and
in a finite graph . The ants perform a sequence of random walks
on this graph, starting from the nest and stopped when first hitting the source
of food. At each step of its random walk, the -th ant chooses to cross a
neighbouring edge with probability proportional to the number of preceding ants
that crossed that edge at least once. We say that {\it the ants find the
shortest path} if, almost surely as the number of ants grow to infinity, almost
all the ants go from the nest to the source of food through one of the shortest
paths, without loosing time on other edges of the graph.
Our contribution is three-fold: (1) We prove that, if is a tree
rooted at whose leaves have been merged into node , and with one edge
between and , then the ants indeed find the shortest path. (2) In
contrast, we provide three examples of graphs on which the ants do not find the
shortest path, suggesting that in this model and in most graphs, ants do not
find the shortest path. (3) In all these cases, we show that the sequence of
normalised edge-weights converge to a {\it deterministic} limit, despite a
linear-reinforcement mechanism, and we conjecture that this is a general fact
which is valid on all finite graphs. To prove these results, we use stochastic
approximation methods, and in particular the ODE method. One difficulty comes
from the fact that this method relies on understanding the behaviour at large
times of the solution of a non-linear, multi-dimensional ODE
The trace-reinforced ants process does not find shortest paths
In this paper, we study a probabilistic reinforcement-learning model for ants
searching for the shortest path(s) between their nest and a source of food. In
this model, the nest and the source of food are two distinguished nodes and
in a finite graph . The ants perform a sequence of random walks
on this graph, starting from the nest and stopped when first hitting the source
of food. At each step of its random walk, the -th ant chooses to cross a
neighbouring edge with probability proportional to the number of preceding ants
that crossed that edge at least once. We say that {\it the ants find the
shortest path} if, almost surely as the number of ants grow to infinity, almost
all the ants go from the nest to the source of food through one of the shortest
paths, without loosing time on other edges of the graph.
Our contribution is three-fold: (1) We prove that, if is a tree
rooted at whose leaves have been merged into node , and with one edge
between and , then the ants indeed find the shortest path. (2) In
contrast, we provide three examples of graphs on which the ants do not find the
shortest path, suggesting that in this model and in most graphs, ants do not
find the shortest path. (3) In all these cases, we show that the sequence of
normalised edge-weights converge to a {\it deterministic} limit, despite a
linear-reinforcement mechanism, and we conjecture that this is a general fact
which is valid on all finite graphs. To prove these results, we use stochastic
approximation methods, and in particular the ODE method. One difficulty comes
from the fact that this method relies on understanding the behaviour at large
times of the solution of a non-linear, multi-dimensional ODE
Faster Replacement Paths
The replacement paths problem for directed graphs is to find for given nodes
s and t and every edge e on the shortest path between them, the shortest path
between s and t which avoids e. For unweighted directed graphs on n vertices,
the best known algorithm runtime was \tilde{O}(n^{2.5}) by Roditty and Zwick.
For graphs with integer weights in {-M,...,M}, Weimann and Yuster recently
showed that one can use fast matrix multiplication and solve the problem in
O(Mn^{2.584}) time, a runtime which would be O(Mn^{2.33}) if the exponent
\omega of matrix multiplication is 2.
We improve both of these algorithms. Our new algorithm also relies on fast
matrix multiplication and runs in O(M n^{\omega} polylog(n)) time if \omega>2
and O(n^{2+\eps}) for any \eps>0 if \omega=2. Our result shows that, at least
for small integer weights, the replacement paths problem in directed graphs may
be easier than the related all pairs shortest paths problem in directed graphs,
as the current best runtime for the latter is \Omega(n^{2.5}) time even if
\omega=2.Comment: the current version contains an improved resul
Fully dynamic all-pairs shortest paths with worst-case update-time revisited
We revisit the classic problem of dynamically maintaining shortest paths
between all pairs of nodes of a directed weighted graph. The allowed updates
are insertions and deletions of nodes and their incident edges. We give
worst-case guarantees on the time needed to process a single update (in
contrast to related results, the update time is not amortized over a sequence
of updates).
Our main result is a simple randomized algorithm that for any parameter
has a worst-case update time of and answers
distance queries correctly with probability , against an adaptive
online adversary if the graph contains no negative cycle. The best
deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time
of and assumes non-negative weights. This is the first
improvement for this problem for more than a decade. Conceptually, our
algorithm shows that randomization along with a more direct approach can
provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201
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