84,434 research outputs found
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Network-Configurations of Dynamic Friction Patterns
The complex configurations of dynamic friction patterns-regarding real time
contact areas- are transformed into appropriate networks. With this
transformation of a system to network space, many properties can be inferred
about the structure and dynamics of the system. Here, we analyze the dynamics
of static friction, i.e. nucleation processes, with respect to "friction
networks". We show that networks can successfully capture the crack-like shear
ruptures and possible corresponding acoustic features. We found that the
fraction of triangles remarkably scales with the detachment fronts. There is a
universal power law between nodes' degree and motifs frequency (for triangles,
it reads T(k)\proptok{\beta} ({\beta} \approx2\pm0.4)). We confirmed the
obtained universality in aperture-based friction networks. Based on the
achieved results, we extracted a possible friction law in terms of network
parameters and compared it with the rate and state friction laws. In
particular, the evolutions of loops are scaled with power law, indicating the
aggregation of cycles around hub nodes. Also, the transition to slow rupture is
scaled with the fast variation of local heterogeneity. Furthermore, the motif
distributions and modularity space of networks -in terms of withinmodule degree
and participation coefficient-show non-uniform general trends, indicating a
universal aspect of energy flow in shear ruptures
Superdiffusion in a class of networks with marginal long-range connections
A class of cubic networks composed of a regular one-dimensional lattice and a
set of long-range links is introduced. Networks parametrized by a positive
integer k are constructed by starting from a one-dimensional lattice and
iteratively connecting each site of degree 2 with a th neighboring site of
degree 2. Specifying the way pairs of sites to be connected are selected,
various random and regular networks are defined, all of which have a power-law
edge-length distribution of the form with the marginal
exponent s=1. In all these networks, lengths of shortest paths grow as a power
of the distance and random walk is super-diffusive. Applying a renormalization
group method, the corresponding shortest-path dimensions and random-walk
dimensions are calculated exactly for k=1 networks and for k=2 regular
networks; in other cases, they are estimated by numerical methods. Although,
s=1 holds for all representatives of this class, the above quantities are found
to depend on the details of the structure of networks controlled by k and other
parameters.Comment: 10 pages, 9 figure
Geometric spanners with small chromatic number
AbstractGiven an integer k⩾2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at most k. We prove that t(2)=3, t(3)=2, t(4)=2, and give upper and lower bounds on t(k) for k>4. We also show that for any ϵ>0, there exists a (1+ϵ)t(k)-spanner for P that has O(|P|) edges and chromatic number at most k. Finally, we consider an on-line variant of the problem where the points of P are given one after another, and the color of a point must be assigned at the moment the point is given. In this setting, we prove that t(2)=3, t(3)=1+3, t(4)=1+2, and give upper and lower bounds on t(k) for k>4
Structural transition in interdependent networks with regular interconnections
Networks are often made up of several layers that exhibit diverse degrees of
interdependencies. A multilayer interdependent network consists of a set of
graphs that are interconnected through a weighted interconnection matrix , where the weight of each inter-graph link is a non-negative real number . Various dynamical processes, such as synchronization, cascading failures
in power grids, and diffusion processes, are described by the Laplacian matrix
characterizing the whole system. For the case in which the multilayer
graph is a multiplex, where the number of nodes in each layer is the same and
the interconnection matrix , being the identity matrix, it has
been shown that there exists a structural transition at some critical coupling,
. This transition is such that dynamical processes are separated into
two regimes: if , the network acts as a whole; whereas when , the network operates as if the graphs encoding the layers were isolated. In
this paper, we extend and generalize the structural transition threshold to a regular interconnection matrix (constant row and column sum).
Specifically, we provide upper and lower bounds for the transition threshold in interdependent networks with a regular interconnection matrix
and derive the exact transition threshold for special scenarios using the
formalism of quotient graphs. Additionally, we discuss the physical meaning of
the transition threshold in terms of the minimum cut and show, through
a counter-example, that the structural transition does not always exist. Our
results are one step forward on the characterization of more realistic
multilayer networks and might be relevant for systems that deviate from the
topological constrains imposed by multiplex networks.Comment: 13 pages, APS format. Submitted for publicatio
Modularity of regular and treelike graphs
Clustering algorithms for large networks typically use modularity values to
test which partitions of the vertex set better represent structure in the data.
The modularity of a graph is the maximum modularity of a partition. We consider
the modularity of two kinds of graphs.
For -regular graphs with a given number of vertices, we investigate the
minimum possible modularity, the typical modularity, and the maximum possible
modularity. In particular, we see that for random cubic graphs the modularity
is usually in the interval , and for random -regular graphs
with large it usually is of order . These results help to
establish baselines for statistical tests on regular graphs.
The modularity of cycles and low degree trees is known to be close to 1: we
extend these results to `treelike' graphs, where the product of treewidth and
maximum degree is much less than the number of edges. This yields for example
the (deterministic) lower bound mentioned above on the modularity of
random cubic graphs.Comment: 25 page
Quantum transport on two-dimensional regular graphs
We study the quantum-mechanical transport on two-dimensional graphs by means
of continuous-time quantum walks and analyse the effect of different boundary
conditions (BCs). For periodic BCs in both directions, i.e., for tori, the
problem can be treated in a large measure analytically. Some of these results
carry over to graphs which obey open boundary conditions (OBCs), such as
cylinders or rectangles. Under OBCs the long time transition probabilities
(LPs) also display asymmetries for certain graphs, as a function of their
particular sizes. Interestingly, these effects do not show up in the marginal
distributions, obtained by summing the LPs along one direction.Comment: 22 pages, 11 figure, acceted for publication in J.Phys.
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