128 research outputs found
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Maximum flow is approximable by deterministic constant-time algorithm in sparse networks
We show a deterministic constant-time parallel algorithm for finding an
almost maximum flow in multisource-multitarget networks with bounded degrees
and bounded edge capacities. As a consequence, we show that the value of the
maximum flow over the number of nodes is a testable parameter on these
networks.Comment: 8 page
Random local algorithms
Consider the problem when we want to construct some structure on a bounded
degree graph, e.g. an almost maximum matching, and we want to decide about each
edge depending only on its constant radius neighbourhood. We show that the
information about the local statistics of the graph does not help here. Namely,
if there exists a random local algorithm which can use any local statistics
about the graph, and produces an almost optimal structure, then the same can be
achieved by a random local algorithm using no statistics.Comment: 9 page
Hypotheses testing on infinite random graphs
Drawing on some recent results that provide the formalism necessary to
definite stationarity for infinite random graphs, this paper initiates the
study of statistical and learning questions pertaining to these objects.
Specifically, a criterion for the existence of a consistent test for complex
hypotheses is presented, generalizing the corresponding results on time series.
As an application, it is shown how one can test that a tree has the Markov
property, or, more generally, to estimate its memory
Faster and Simpler Distributed Algorithms for Testing and Correcting Graph Properties in the CONGEST-Model
In this paper we present distributed testing algorithms of graph properties
in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error
testing algorithms in the general graph model.
We first describe a general procedure for converting -testers with
a number of rounds , where denotes the diameter of the graph, to
rounds, where is the number of
processors of the network. We then apply this procedure to obtain an optimal
tester, in terms of , for testing bipartiteness, whose round complexity is
, which improves over the -round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for
cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects
the graph so that the corrected graph is acyclic. Note that, unlike a tester, a
corrector needs to mend the graph in many places in the case that the graph is
far from having the property.
In the second part of the paper we design algorithms for testing whether the
network is -free for any connected of size up to four with round
complexity of . This improves over the
-round algorithms for testing triangle freeness by
Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size by
Fraigniaud et al. (DISC 2016).
In the last part we generalize the global tester by Iwama and Yoshida (ITCS
2014) of testing -path freeness to testing the exclusion of any tree of
order . We then show how to simulate this algorithm in the CONGEST-model in
rounds
Every property is testable on a natural class of scale-free multigraphs
In this paper, we introduce a natural class of multigraphs called
hierarchical-scale-free (HSF) multigraphs, and consider constant-time
testability on the class. We show that a very wide subclass, specifically, that
in which the power-law exponent is greater than two, of HSF is hyperfinite.
Based on this result, an algorithm for a deterministic partitioning oracle can
be constructed. We conclude by showing that every property is constant-time
testable on the above subclass of HSF. This algorithm utilizes findings by
Newman and Sohler of STOC'11. However, their algorithm is based on the
bounded-degree model, while it is known that actual scale-free networks usually
include hubs, which have a very large degree. HSF is based on scale-free
properties and includes such hubs. This is the first universal result of
constant-time testability on the general graph model, and it has the potential
to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and
SF become more general. Typos were fixe
How to solve the cake-cutting problem in sublinear time
In this paper, we show algorithms for solving the cake-cutting problem in
sublinear-time. More specifically, we preassign (simple) fair portions to o(n)
players in o(n)-time, and minimize the damage to the rest of the players. All
currently known algorithms require Omega(n)-time, even when assigning a portion
to just one player, and it is nontrivial to revise these algorithms to run in
-time since many of the remaining players, who have not been asked any
queries, may not be satisfied with the remaining cake. To challenge this
problem, we begin by providing a framework for solving the cake-cutting problem
in sublinear-time. Generally speaking, solving a problem in sublinear-time
requires the use of approximations. However, in our framework, we introduce the
concept of "eps n-victims," which means that eps n players (victims) may not
get fair portions, where 0< eps =< 1 is an arbitrary constant. In our
framework, an algorithm consists of the following two parts: In the first
(Preassigning) part, it distributes fair portions to r < n players in
o(n)-time. In the second (Completion) part, it distributes fair portions to the
remaining n-r players except for the eps n victims in poly}(n)-time. There are
two variations on the r players in the first part. Specifically, whether they
can or cannot be designated. We will then present algorithms in this framework.
In particular, an O(r/eps)-time algorithm for r =< eps n/127 undesignated
players with eps n-victims, and an O~(r^2/eps)-time algorithm for r =< eps
e^{{sqrt{ln{n}}}/{7}} designated players and eps =< 1/e with eps n-victims are
presented.Comment: 15 pages, no figur
Interconnection network with a shared whiteboard: Impact of (a)synchronicity on computing power
In this work we study the computational power of graph-based models of
distributed computing in which each node additionally has access to a global
whiteboard. A node can read the contents of the whiteboard and, when activated,
can write one message of O(log n) bits on it. When the protocol terminates,
each node computes the output based on the final contents of the whiteboard. We
consider several scheduling schemes for nodes, providing a strict ordering of
their power in terms of the problems which can be solved with exactly one
activation per node. The problems used to separate the models are related to
Maximal Independent Set, detection of cycles of length 4, and BFS spanning tree
constructions
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