12,981 research outputs found
Synchronization in Random Geometric Graphs
In this paper we study the synchronization properties of random geometric
graphs. We show that the onset of synchronization takes place roughly at the
same value of the order parameter that a random graph with the same size and
average connectivity. However, the dependence of the order parameter with the
coupling strength indicates that the fully synchronized state is more easily
attained in random graphs. We next focus on the complete synchronized state and
show that this state is less stable for random geometric graphs than for other
kinds of complex networks. Finally, a rewiring mechanism is proposed as a way
to improve the stability of the fully synchronized state as well as to lower
the value of the coupling strength at which it is achieved. Our work has
important implications for the synchronization of wireless networks, and should
provide valuable insights for the development and deployment of more efficient
and robust distributed synchronization protocols for these systems.Comment: 5 pages, 4 figure
Guarantees for Spontaneous Synchronization on Random Geometric Graphs
The Kuramoto model is a classical mathematical model in the field of
non-linear dynamical systems that describes the evolution of coupled
oscillators in a network that may reach a synchronous state. The relationship
between the network's topology and whether the oscillators synchronize is a
central question in the field of synchronization, and random graphs are often
employed as a proxy for complex networks. On the other hand, the random graphs
on which the Kuramoto model is rigorously analyzed in the literature are
homogeneous models and fail to capture the underlying geometric structure that
appears in several examples.
In this work, we leverage tools from random matrix theory, random graphs, and
mathematical statistics to prove that the Kuramoto model on a random geometric
graph on the sphere synchronizes with probability tending to one as the number
of nodes tends to infinity. To the best of our knowledge, this is the first
rigorous result for the Kuramoto model on random geometric graphs
Synchronizability of random rectangular graphs
Random rectangular graphs (RRGs) represent a generalization of the random geometric graphs in which the nodes are embedded into hyperrectangles instead of on hypercubes. The synchronizability of RRG model is studied. Both upper and lower bounds of the eigenratio of the network Laplacian matrix are determined analytically. It is proven that as the rectangular network is more elongated, the network becomes harder to synchronize. The synchronization processing behavior of a RRG network of chaotic Lorenz system nodes is numerically investigated, showing complete consistence with the theoretical results
Shaping bursting by electrical coupling and noise
Gap-junctional coupling is an important way of communication between neurons
and other excitable cells. Strong electrical coupling synchronizes activity
across cell ensembles. Surprisingly, in the presence of noise synchronous
oscillations generated by an electrically coupled network may differ
qualitatively from the oscillations produced by uncoupled individual cells
forming the network. A prominent example of such behavior is the synchronized
bursting in islets of Langerhans formed by pancreatic \beta-cells, which in
isolation are known to exhibit irregular spiking. At the heart of this
intriguing phenomenon lies denoising, a remarkable ability of electrical
coupling to diminish the effects of noise acting on individual cells.
In this paper, we derive quantitative estimates characterizing denoising in
electrically coupled networks of conductance-based models of square wave
bursting cells. Our analysis reveals the interplay of the intrinsic properties
of the individual cells and network topology and their respective contributions
to this important effect. In particular, we show that networks on graphs with
large algebraic connectivity or small total effective resistance are better
equipped for implementing denoising. As a by-product of the analysis of
denoising, we analytically estimate the rate with which trajectories converge
to the synchronization subspace and the stability of the latter to random
perturbations. These estimates reveal the role of the network topology in
synchronization. The analysis is complemented by numerical simulations of
electrically coupled conductance-based networks. Taken together, these results
explain the mechanisms underlying synchronization and denoising in an important
class of biological models
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
Fundamentals of Large Sensor Networks: Connectivity, Capacity, Clocks and Computation
Sensor networks potentially feature large numbers of nodes that can sense
their environment over time, communicate with each other over a wireless
network, and process information. They differ from data networks in that the
network as a whole may be designed for a specific application. We study the
theoretical foundations of such large scale sensor networks, addressing four
fundamental issues- connectivity, capacity, clocks and function computation.
To begin with, a sensor network must be connected so that information can
indeed be exchanged between nodes. The connectivity graph of an ad-hoc network
is modeled as a random graph and the critical range for asymptotic connectivity
is determined, as well as the critical number of neighbors that a node needs to
connect to. Next, given connectivity, we address the issue of how much data can
be transported over the sensor network. We present fundamental bounds on
capacity under several models, as well as architectural implications for how
wireless communication should be organized.
Temporal information is important both for the applications of sensor
networks as well as their operation.We present fundamental bounds on the
synchronizability of clocks in networks, and also present and analyze
algorithms for clock synchronization. Finally we turn to the issue of gathering
relevant information, that sensor networks are designed to do. One needs to
study optimal strategies for in-network aggregation of data, in order to
reliably compute a composite function of sensor measurements, as well as the
complexity of doing so. We address the issue of how such computation can be
performed efficiently in a sensor network and the algorithms for doing so, for
some classes of functions.Comment: 10 pages, 3 figures, Submitted to the Proceedings of the IEE
A Gossip Algorithm based Clock Synchronization Scheme for Smart Grid Applications
The uprising interest in multi-agent based networked system, and the numerous
number of applications in the distributed control of the smart grid leads us to
address the problem of time synchronization in the smart grid. Utility
companies look for new packet based time synchronization solutions with Global
Positioning System (GPS) level accuracies beyond traditional packet methods
such as Network Time Proto- col (NTP). However GPS based solutions have poor
reception in indoor environments and dense urban canyons as well as GPS antenna
installation might be costly. Some smart grid nodes such as Phasor Measurement
Units (PMUs), fault detection, Wide Area Measurement Systems (WAMS) etc.,
requires synchronous accuracy as low as 1 ms. On the other hand, 1 sec accuracy
is acceptable in management information domain. Acknowledging this, in this
study, we introduce gossip algorithm based clock synchronization method among
network entities from the decision control and communication point of view. Our
method synchronizes clock within dense network with a bandwidth limited
environment. Our technique has been tested in different kinds of network
topologies- complete, star and random geometric network and demonstrated
satisfactory performance
The geometry of spontaneous spiking in neuronal networks
The mathematical theory of pattern formation in electrically coupled networks
of excitable neurons forced by small noise is presented in this work. Using the
Freidlin-Wentzell large deviation theory for randomly perturbed dynamical
systems and the elements of the algebraic graph theory, we identify and analyze
the main regimes in the network dynamics in terms of the key control
parameters: excitability, coupling strength, and network topology. The analysis
reveals the geometry of spontaneous dynamics in electrically coupled network.
Specifically, we show that the location of the minima of a certain continuous
function on the surface of the unit n-cube encodes the most likely activity
patterns generated by the network. By studying how the minima of this function
evolve under the variation of the coupling strength, we describe the principal
transformations in the network dynamics. The minimization problem is also used
for the quantitative description of the main dynamical regimes and transitions
between them. In particular, for the weak and strong coupling regimes, we
present asymptotic formulas for the network activity rate as a function of the
coupling strength and the degree of the network. The variational analysis is
complemented by the stability analysis of the synchronous state in the strong
coupling regime. The stability estimates reveal the contribution of the network
connectivity and the properties of the cycle subspace associated with the graph
of the network to its synchronization properties. This work is motivated by the
experimental and modeling studies of the ensemble of neurons in the Locus
Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive
performance and behavior
Reconstructing directed and weighted topologies of phase-locked oscillator networks
The formalism of complex networks is extensively employed to describe the
dynamics of interacting agents in several applications. The features of the
connections among the nodes in a network are not always provided beforehand,
hence the problem of appropriately inferring them often arises. Here, we
present a method to reconstruct directed and weighted topologies (REDRAW) of
networks of heterogeneous phase-locked nonlinear oscillators. We ultimately
plan on using REDRAW to infer the interaction structure in human ensembles
engaged in coordination tasks, and give insights into the overall behavior
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