178 research outputs found
Critical dynamics of the k-core pruning process
We present the theory of the k-core pruning process (progressive removal of
nodes with degree less than k) in uncorrelated random networks. We derive exact
equations describing this process and the evolution of the network structure,
and solve them numerically and, in the critical regime of the process,
analytically. We show that the pruning process exhibits three different
behaviors depending on whether the mean degree of the initial network is
above, equal to, or below the threshold _c corresponding to the emergence of
the giant k-core. We find that above the threshold the network relaxes
exponentially to the k-core. The system manifests the phenomenon known as
"critical slowing down", as the relaxation time diverges when tends to
_c. At the threshold, the dynamics become critical characterized by a
power-law relaxation (1/t^2). Below the threshold, a long-lasting transient
process (a "plateau" stage) occurs. This transient process ends with a collapse
in which the entire network disappears completely. The duration of the process
diverges when tends to _c. We show that the critical dynamics of the
pruning are determined by branching processes of spreading damage. Clusters of
nodes of degree exactly k are the evolving substrate for these branching
processes. Our theory completely describes this branching cascade of damage in
uncorrelated networks by providing the time dependent distribution function of
branching. These theoretical results are supported by our simulations of the
-core pruning in Erdos-Renyi graphs.Comment: 12 pages, 10 figure
Avalanche Collapse of Interdependent Network
We reveal the nature of the avalanche collapse of the giant viable component
in multiplex networks under perturbations such as random damage. Specifically,
we identify latent critical clusters associated with the avalanches of random
damage. Divergence of their mean size signals the approach to the hybrid phase
transition from one side, while there are no critical precursors on the other
side. We find that this discontinuous transition occurs in scale-free multiplex
networks whenever the mean degree of at least one of the interdependent
networks does not diverge.Comment: 4 pages, 5 figure
Bootstrap Percolation on Complex Networks
We consider bootstrap percolation on uncorrelated complex networks. We obtain
the phase diagram for this process with respect to two parameters: , the
fraction of vertices initially activated, and , the fraction of undamaged
vertices in the graph. We observe two transitions: the giant active component
appears continuously at a first threshold. There may also be a second,
discontinuous, hybrid transition at a higher threshold. Avalanches of
activations increase in size as this second critical point is approached,
finally diverging at this threshold. We describe the existence of a special
critical point at which this second transition first appears. In networks with
degree distributions whose second moment diverges (but whose first moment does
not), we find a qualitatively different behavior. In this case the giant active
component appears for any and , and the discontinuous transition is
absent. This means that the giant active component is robust to damage, and
also is very easily activated. We also formulate a generalized bootstrap
process in which each vertex can have an arbitrary threshold.Comment: 9 pages, 3 figure
Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks
We introduce the heterogeneous--core, which generalizes the -core, and
contrast it with bootstrap percolation. Vertices have a threshold which
may be different at each vertex. If a vertex has less than neighbors it
is pruned from the network. The heterogeneous--core is the sub-graph
remaining after no further vertices can be pruned. If the thresholds are
with probability or with probability , the process
forms one branch of an activation-pruning process which demonstrates
hysteresis. The other branch is formed by ordinary bootstrap percolation. We
show that there are two types of transitions in this heterogeneous--core
process: the giant heterogeneous--core may appear with a continuous
transition and there may be a second, discontinuous, hybrid transition. We
compare critical phenomena, critical clusters and avalanches at the
heterogeneous--core and bootstrap percolation transitions. We also show that
network structure has a crucial effect on these processes, with the giant
heterogeneous--core appearing immediately at a finite value for any
when the degree distribution tends to a power law with
.Comment: 10 pages, 4 figure
Localization and Spreading of Diseases in Complex Networks
Using the SIS model on unweighted and weighted networks, we consider the disease localizationphenomenon. In contrast to the well-recognized point of view that diseases infect a finite fractionof vertices right above the epidemic threshold, we show that diseases can be localized on a finitenumber of vertices, where hubs and edges with large weights are centers of localization. Our resultsfollow from the analysis of standard models of networks and empirical data for real-world networks
Correlations in interacting systems with a network topology
We study pair correlations in cooperative systems placed on complex networks.
We show that usually in these systems, the correlations between two interacting
objects (e.g., spins), separated by a distance , decay, on average,
faster than . Here is the mean number of the
-th nearest neighbors of a vertex in a network. This behavior, in
particular, leads to a dramatic weakening of correlations between second and
more distant neighbors on networks with fat-tailed degree distributions, which
have a divergent number in the infinite network limit. In this case, only
the pair correlations between the nearest neighbors are observable. We obtain
the pair correlation function of the Ising model on a complex network and also
derive our results in the framework of a phenomenological approach.Comment: 5 page
Analytical and experimental studies of pneumatic vibration exciter in inertia vibroabrasive machining of parts based on beryllium oxide
In the article the results of studying a vibration exciter for vibration machining of the surface of the parts with a pneumatic vibratory drive, which showed its high efficiency when used in highly explosive and associated with fire risk productions, as well as for various technological processes of inertia machining parts based on beryllium oxide, are presented. As a result of the carried out analytical studies of an installation with a pneumatic vibratory drive, a mathematical model of vibration machining process dynamics was determined.The adequacy of the mathematical model of the compressed air flow effect on the roller in the pneumatic grinding set, as well as the roller mass effect on the oscillation amplitude was proved experimentally.The oscillation amplitude dependencies on pressure and the feeding nozzle diameter, when vibration process technology (vibration grinding, vibration polishing and others) of machining of parts is realized, are presented
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