150,013 research outputs found
Ollivier curvature, betweenness centrality and average distance
We give a new upper bound for the average graph distance in terms of the
average Ollivier curvature. Here, the average Ollivier curvature is weighted
with the edge betweenness centrality. Moreover, we prove that equality is
attained precisely for the reflective graphs which have been classified as
Cartesian products of cocktail party graphs, Johnson graphs, halved cubes,
Schl\"afli graphs, and Gosset graphs
Shortest Distances as Enumeration Problem
We investigate the single source shortest distance (SSSD) and all pairs
shortest distance (APSD) problems as enumeration problems (on unweighted and
integer weighted graphs), meaning that the elements -- where
and are vertices with shortest distance -- are produced and
listed one by one without repetition. The performance is measured in the RAM
model of computation with respect to preprocessing time and delay, i.e., the
maximum time that elapses between two consecutive outputs. This point of view
reveals that specific types of output (e.g., excluding the non-reachable pairs
, or excluding the self-distances ) and the order of
enumeration (e.g., sorted by distance, sorted row-wise with respect to the
distance matrix) have a huge impact on the complexity of APSD while they appear
to have no effect on SSSD.
In particular, we show for APSD that enumeration without output restrictions
is possible with delay in the order of the average degree. Excluding
non-reachable pairs, or requesting the output to be sorted by distance,
increases this delay to the order of the maximum degree. Further, for weighted
graphs, a delay in the order of the average degree is also not possible without
preprocessing or considering self-distances as output. In contrast, for SSSD we
find that a delay in the order of the maximum degree without preprocessing is
attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit
Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees
The average distance from a node to all other nodes in a graph, or from a
query point in a metric space to a set of points, is a fundamental quantity in
data analysis. The inverse of the average distance, known as the (classic)
closeness centrality of a node, is a popular importance measure in the study of
social networks. We develop novel structural insights on the sparsifiability of
the distance relation via weighted sampling. Based on that, we present highly
practical algorithms with strong statistical guarantees for fundamental
problems. We show that the average distance (and hence the centrality) for all
nodes in a graph can be estimated using single-source
distance computations. For a set of points in a metric space, we show
that after preprocessing which uses distance computations we can compute
a weighted sample of size such that the average
distance from any query point to can be estimated from the distances
from to . Finally, we show that for a set of points in a metric
space, we can estimate the average pairwise distance using
distance computations. The estimate is based on a weighted sample of
pairs of points, which is computed using distance
computations. Our estimates are unbiased with normalized mean square error
(NRMSE) of at most . Increasing the sample size by a
factor ensures that the probability that the relative error exceeds
is polynomially small.Comment: 21 pages, will appear in the Proceedings of RANDOM 201
Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs
We consider the problem of computing shortest paths in weighted unit-disk
graphs in constant dimension . Although the single-source and all-pairs
variants of this problem are well-studied in the plane case, no non-trivial
exact distance oracles for unit-disk graphs have been known to date, even for
.
The classical result of Sedgewick and Vitter [Algorithmica '86] shows that
for weighted unit-disk graphs in the plane the search has average-case
performance superior to that of a standard shortest path algorithm, e.g.,
Dijkstra's algorithm. Specifically, if the corresponding points of a
weighted unit-disk graph are picked from a unit square uniformly at random,
and the connectivity radius is , finds a shortest path in
in expected time when , even though has
edges in expectation. In other words, the work done by the
algorithm is in expectation proportional to the number of vertices and not the
number of edges.
In this paper, we break this natural barrier and show even stronger sublinear
time results. We propose a new heuristic approach to computing point-to-point
exact shortest paths in unit-disk graphs. We analyze the average-case behavior
of our heuristic using the same random graph model as used by Sedgewick and
Vitter and prove it superior to . Specifically, we show that, if we are
able to report the set of all points of from an arbitrary rectangular
region of the plane in time, then a shortest path between
arbitrary two points of such a random graph on the plane can be found in
expected time. In particular, the state-of-the-art range
reporting data structures imply a sublinear expected bound for all
and expected bound for
after only near-linear preprocessing of the point set.Comment: Full version of a SoCG'21 paper. Abstract truncated to meet arxiv
requirement
An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
We prove the following Alon-Boppana type theorem for general (not necessarily
regular) weighted graphs: if is an -node weighted undirected graph of
average combinatorial degree (that is, has edges) and girth , and if are the
eigenvalues of the (non-normalized) Laplacian of , then (The Alon-Boppana theorem implies that if is unweighted and
-regular, then if the diameter is at least .)
Our result implies a lower bound for spectral sparsifiers. A graph is a
spectral -sparsifier of a graph if where is the Laplacian matrix of and is
the Laplacian matrix of . Batson, Spielman and Srivastava proved that for
every there is an -sparsifier of average degree where
and the edges of are a
(weighted) subset of the edges of . Batson, Spielman and Srivastava also
show that the bound on cannot be reduced below when is a clique; our Alon-Boppana-type result implies that
cannot be reduced below when comes
from a family of expanders of super-constant degree and super-constant girth.
The method of Batson, Spielman and Srivastava proves a more general result,
about sparsifying sums of rank-one matrices, and their method applies to an
"online" setting. We show that for the online matrix setting the bound is tight, up to lower order terms
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