934 research outputs found
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
Dismantling sparse random graphs
We consider the number of vertices that must be removed from a graph G in
order that the remaining subgraph has no component with more than k vertices.
Our principal observation is that, if G is a sparse random graph or a random
regular graph on n vertices with n tending to infinity, then the number in
question is essentially the same for all values of k such that k tends to
infinity but k=o(n).Comment: 7 page
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
The minimum bisection in the planted bisection model
In the planted bisection model a random graph with
vertices is created by partitioning the vertices randomly into two classes of
equal size (up to ). Any two vertices that belong to the same class are
linked by an edge with probability and any two that belong to different
classes with probability independently. The planted bisection model
has been used extensively to benchmark graph partitioning algorithms. If
for numbers that remain fixed as
, then w.h.p. the ``planted'' bisection (the one used to construct
the graph) will not be a minimum bisection. In this paper we derive an
asymptotic formula for the minimum bisection width under the assumption that
for a certain constant
Optimal Nested Test Plan for Combinatorial Quantitative Group Testing
We consider the quantitative group testing problem where the objective is to
identify defective items in a given population based on results of tests
performed on subsets of the population. Under the quantitative group testing
model, the result of each test reveals the number of defective items in the
tested group. The minimum number of tests achievable by nested test plans was
established by Aigner and Schughart in 1985 within a minimax framework. The
optimal nested test plan offering this performance, however, was not obtained.
In this work, we establish the optimal nested test plan in closed form. This
optimal nested test plan is also order optimal among all test plans as the
population size approaches infinity. Using heavy-hitter detection as a case
study, we show via simulation examples orders of magnitude improvement of the
group testing approach over two prevailing sampling-based approaches in
detection accuracy and counter consumption. Other applications include anomaly
detection and wideband spectrum sensing in cognitive radio systems
Finding community structure in networks using the eigenvectors of matrices
We consider the problem of detecting communities or modules in networks,
groups of vertices with a higher-than-average density of edges connecting them.
Previous work indicates that a robust approach to this problem is the
maximization of the benefit function known as "modularity" over possible
divisions of a network. Here we show that this maximization process can be
written in terms of the eigenspectrum of a matrix we call the modularity
matrix, which plays a role in community detection similar to that played by the
graph Laplacian in graph partitioning calculations. This result leads us to a
number of possible algorithms for detecting community structure, as well as
several other results, including a spectral measure of bipartite structure in
networks and a new centrality measure that identifies those vertices that
occupy central positions within the communities to which they belong. The
algorithms and measures proposed are illustrated with applications to a variety
of real-world complex networks.Comment: 22 pages, 8 figures, minor corrections in this versio
Asymptotic of geometrical navigation on a random set of points of the plane
A navigation on a set of points is a rule for choosing which point to
move to from the present point in order to progress toward a specified target.
We study some navigations in the plane where is a non uniform Poisson point
process (in a finite domain) with intensity going to . We show the
convergence of the traveller path lengths, the number of stages done, and the
geometry of the traveller trajectories, uniformly for all starting points and
targets, for several navigations of geometric nature. Other costs are also
considered. This leads to asymptotic results on the stretch factors of random
Yao-graphs and random -graphs
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