45,911 research outputs found
On Large-Scale Graph Generation with Validation of Diverse Triangle Statistics at Edges and Vertices
Researchers developing implementations of distributed graph analytic
algorithms require graph generators that yield graphs sharing the challenging
characteristics of real-world graphs (small-world, scale-free, heavy-tailed
degree distribution) with efficiently calculable ground-truth solutions to the
desired output. Reproducibility for current generators used in benchmarking are
somewhat lacking in this respect due to their randomness: the output of a
desired graph analytic can only be compared to expected values and not exact
ground truth. Nonstochastic Kronecker product graphs meet these design criteria
for several graph analytics. Here we show that many flavors of triangle
participation can be cheaply calculated while generating a Kronecker product
graph. Given two medium-sized scale-free graphs with adjacency matrices and
, their Kronecker product graph has adjacency matrix . Such
graphs are highly compressible: edges are represented in memory and can be built in a distributed setting from
small data structures, making them easy to share in compressed form. Many
interesting graph calculations have worst-case complexity bounds and often these are reduced to
for Kronecker product graphs, when a Kronecker formula can be derived yielding
the sought calculation on in terms of related calculations on and .
We focus on deriving formulas for triangle participation at vertices, , a vector storing the number of triangles that every vertex is involved
in, and triangle participation at edges, , a sparse matrix storing
the number of triangles at every edge.Comment: 10 pages, 7 figures, IEEE IPDPS Graph Algorithms Building Block
On The Multiparty Communication Complexity of Testing Triangle-Freeness
In this paper we initiate the study of property testing in simultaneous and
non-simultaneous multi-party communication complexity, focusing on testing
triangle-freeness in graphs. We consider the model,
where we have players receiving private inputs, and a coordinator who
receives no input; the coordinator can communicate with all the players, but
the players cannot communicate with each other. In this model, we ask: if an
input graph is divided between the players, with each player receiving some of
the edges, how many bits do the players and the coordinator need to exchange to
determine if the graph is triangle-free, or from triangle-free?
For general communication protocols, we show that
bits are sufficient to test triangle-freeness in
graphs of size with average degree (the degree need not be known in
advance). For protocols, where there is only one
communication round, we give a protocol that uses bits
when and when ; here, again, the average degree does not need to be
known in advance. We show that for average degree , our simultaneous
protocol is asymptotically optimal up to logarithmic factors. For higher
degrees, we are not able to give lower bounds on testing triangle-freeness, but
we give evidence that the problem is hard by showing that finding an edge that
participates in a triangle is hard, even when promised that at least a constant
fraction of the edges must be removed in order to make the graph triangle-free.Comment: To Appear in PODC 201
Sparse halves in dense triangle-free graphs
Erd\H{o}s conjectured that every triangle-free graph on vertices
contains a set of vertices that spans at most
edges. Krivelevich proved the conjecture for graphs with minimum degree at
least . Keevash and Sudakov improved this result to graphs with
average degree at least . We strengthen these results by showing
that the conjecture holds for graphs with minimum degree at least
and for graphs with average degree at least for some absolute . Moreover, we show that the
conjecture is true for graphs which are close to the Petersen graph in edit
distance.Comment: 23 page
The random k-matching-free process
Let be a graph property which is preserved by removal of edges,
and consider the random graph process that starts with the empty -vertex
graph and then adds edges one-by-one, each chosen uniformly at random subject
to the constraint that is not violated. These types of random
processes have been the subject of extensive research over the last 20 years,
having striking applications in extremal combinatorics, and leading to the
discovery of important probabilistic tools. In this paper we consider the
-matching-free process, where is the property of not
containing a matching of size . We are able to analyse the behaviour of this
process for a wide range of values of ; in particular we prove that if
or if then this process is likely to
terminate in a -matching-free graph with the maximum possible number of
edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds
on are essentially best possible, and we make a first step towards
understanding the behaviour of the process in the intermediate regime
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