2,329 research outputs found
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Robust Block Coordinate Descent
In this paper we present a novel randomized block coordinate descent method
for the minimization of a convex composite objective function. The method uses
(approximate) partial second-order (curvature) information, so that the
algorithm performance is more robust when applied to highly nonseparable or ill
conditioned problems. We call the method Robust Coordinate Descent (RCD). At
each iteration of RCD, a block of coordinates is sampled randomly, a quadratic
model is formed about that block and the model is minimized
approximately/inexactly to determine the search direction. An inexpensive line
search is then employed to ensure a monotonic decrease in the objective
function and acceptance of large step sizes. We prove global convergence of the
RCD algorithm, and we also present several results on the local convergence of
RCD for strongly convex functions. Finally, we present numerical results on
large-scale problems to demonstrate the practical performance of the method.Comment: 23 pages, 6 figure
Rumor Spreading on Random Regular Graphs and Expanders
Broadcasting algorithms are important building blocks of distributed systems.
In this work we investigate the typical performance of the classical and
well-studied push model. Assume that initially one node in a given network
holds some piece of information. In each round, every one of the informed nodes
chooses independently a neighbor uniformly at random and transmits the message
to it.
In this paper we consider random networks where each vertex has degree d,
which is at least 3, i.e., the underlying graph is drawn uniformly at random
from the set of all d-regular graphs with n vertices. We show that with
probability 1 - o(1) the push model broadcasts the message to all nodes within
(1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In
particular, we can characterize precisely the effect of the node degree to the
typical broadcast time of the push model. Moreover, we consider pseudo-random
regular networks, where we assume that the degree of each node is very large.
There we show that the broadcast time is (1+o(1))C ln n with probability 1 -
o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.Comment: 18 page
On the Insertion Time of Cuckoo Hashing
Cuckoo hashing is an efficient technique for creating large hash tables with
high space utilization and guaranteed constant access times. There, each item
can be placed in a location given by any one out of k different hash functions.
In this paper we investigate further the random walk heuristic for inserting in
an online fashion new items into the hash table. Provided that k > 2 and that
the number of items in the table is below (but arbitrarily close) to the
theoretically achievable load threshold, we show a polylogarithmic bound for
the maximum insertion time that holds with high probability.Comment: 27 pages, final version accepted by the SIAM Journal on Computin
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