114,106 research outputs found
Jigsaw percolation: What social networks can collaboratively solve a puzzle?
We introduce a new kind of percolation on finite graphs called jigsaw
percolation. This model attempts to capture networks of people who innovate by
merging ideas and who solve problems by piecing together solutions. Each person
in a social network has a unique piece of a jigsaw puzzle. Acquainted people
with compatible puzzle pieces merge their puzzle pieces. More generally, groups
of people with merged puzzle pieces merge if the groups know one another and
have a pair of compatible puzzle pieces. The social network solves the puzzle
if it eventually merges all the puzzle pieces. For an Erd\H{o}s-R\'{e}nyi
social network with vertices and edge probability , we define the
critical value for a connected puzzle graph to be the for which
the chance of solving the puzzle equals . We prove that for the -cycle
(ring) puzzle, , and for an arbitrary connected puzzle
graph with bounded maximum degree, and for
any . Surprisingly, with probability tending to 1 as the network size
increases to infinity, social networks with a power-law degree distribution
cannot solve any bounded-degree puzzle. This model suggests a mechanism for
recent empirical claims that innovation increases with social density, and it
might begin to show what social networks stifle creativity and what networks
collectively innovate.Comment: Published at http://dx.doi.org/10.1214/14-AAP1041 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
A structural analysis of the A5/1 state transition graph
We describe efficient algorithms to analyze the cycle structure of the graph
induced by the state transition function of the A5/1 stream cipher used in GSM
mobile phones and report on the results of the implementation. The analysis is
performed in five steps utilizing HPC clusters, GPGPU and external memory
computation. A great reduction of this huge state transition graph of 2^64
nodes is achieved by focusing on special nodes in the first step and removing
leaf nodes that can be detected with limited effort in the second step. This
step does not break the overall structure of the graph and keeps at least one
node on every cycle. In the third step the nodes of the reduced graph are
connected by weighted edges. Since the number of nodes is still huge an
efficient bitslice approach is presented that is implemented with NVIDIA's CUDA
framework and executed on several GPUs concurrently. An external memory
algorithm based on the STXXL library and its parallel pipelining feature
further reduces the graph in the fourth step. The result is a graph containing
only cycles that can be further analyzed in internal memory to count the number
and size of the cycles. This full analysis which previously would take months
can now be completed within a few days and allows to present structural results
for the full graph for the first time. The structure of the A5/1 graph deviates
notably from the theoretical results for random mappings.Comment: In Proceedings GRAPHITE 2012, arXiv:1210.611
A More Reliable Greedy Heuristic for Maximum Matchings in Sparse Random Graphs
We propose a new greedy algorithm for the maximum cardinality matching
problem. We give experimental evidence that this algorithm is likely to find a
maximum matching in random graphs with constant expected degree c>0,
independent of the value of c. This is contrary to the behavior of commonly
used greedy matching heuristics which are known to have some range of c where
they probably fail to compute a maximum matching
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