Sharp threshold functions via a coupling method

2013 ◽  
pp. 443-448
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Coupling Method ◽  
Sharp Threshold ◽  
Threshold Functions

scholarly journals Sharp Threshold Functions for Random Intersection Graphs via a Coupling Method

10.37236/523 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Perfect Matching ◽  
Hamilton Cycle ◽  
Coupling Method ◽  
New Method ◽  
Intersection Graphs ◽  
Sharp Threshold ◽  
Threshold Functions

We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for $k$–connectivity, perfect matching containment and Hamilton cycle containment.

scholarly journals The Coupling Method for Inhomogeneous Random Intersection Graphs.

10.37236/5186 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Coupling Method ◽  
Intersection Graphs ◽  
New Approach ◽  
Threshold Functions ◽  
Wide Family

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we improve and generalize  the coupling method introduced for random intersection graphs so that it may be used for a wider range of parameters. Using the new approach we are able to tighten the best known results concerning random intersection graphs and establish threshold functions for some monotone properties of inhomogeneous random intersection graphs. Considered properties are: $k$-connectivity, matching containment and hamiltonicity.

Random high-dimensional orders

1995 ◽  
Vol 27 (01) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order P k (n) on n points is defined by taking n points uniformly at random from [0,1] k . Previous work has concentrated on the case where k is constant: we consider the model where k increases with n. We pay particular attention to the height H k (n) of P k (n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in P k (n): if k – (t/log t!) log n tends to + ∞ then the probability that P k (n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of H k (n) for the entire range of k(n). We also consider the maximum degree of P k (n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k. We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of P k (n) for large k(n).

Random high-dimensional orders

1995 ◽  
Vol 27 (1) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order Pk(n) on n points is defined by taking n points uniformly at random from [0,1]k. Previous work has concentrated on the case where k is constant: we consider the model where k increases with n.We pay particular attention to the height Hk(n) of Pk(n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in Pk(n): if k – (t/log t!) log n tends to + ∞ then the probability that Pk(n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of Hk(n) for the entire range of k(n).We also consider the maximum degree of Pk(n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k.We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of Pk(n) for large k(n).

Threshold Functions for H-factors

1993 ◽  
Vol 2 (2) ◽  
pp. 137-144 ◽  
Author(s):  
Noga Alon ◽  
Raphael Yuster
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Threshold Function ◽  
Sharp Threshold ◽  
Threshold Functions ◽  
Spanning Subgraph ◽  
Degree Of A Vertex ◽  
Vertex Disjoint ◽  
Special Case

Let H be a graph on h vertices, and G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is , where the maximum is taken over all subgraphs (V′, E′) of H with |V′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H), then n−1/a(H) is a sharp threshold function for the property that the random graph G(n, p) contains an H-factor. That is, there are two positive constants c and C so that for p(n) = cn−1/a(H) almost surely G(n, p(n)) does not have an H-factor, whereas for p(n) = Cn−1/a(H), almost surely G(n, p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdős.

scholarly journals Hybrid Model for Time Series of Complex Structure with ARIMA Components

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1122
Author(s):  
Oksana Mandrikova ◽  
Nadezhda Fetisova ◽  
Yuriy Polozov
Keyword(s):  
Neural Network ◽  
Time Series ◽  
Hybrid Model ◽  
Complex Structure ◽  
Arima Model ◽  
Ionospheric Parameter ◽  
Arima Models ◽  
Complicated Structure ◽  
Threshold Functions ◽  
Anomalous Component

A hybrid model for the time series of complex structure (HMTS) was proposed. It is based on the combination of function expansions in a wavelet series with ARIMA models. HMTS has regular and anomalous components. The time series components, obtained after expansion, have a simpler structure that makes it possible to identify the ARIMA model if the components are stationary. This allows us to obtain a more accurate ARIMA model for a time series of complicated structure and to extend the area for application. To identify the HMTS anomalous component, threshold functions are applied. This paper describes a technique to identify HMTS and proposes operations to detect anomalies. With the example of an ionospheric parameter time series, we show the HMTS efficiency, describe the results and their application in detecting ionospheric anomalies. The HMTS was compared with the nonlinear autoregression neural network NARX, which confirmed HMTS efficiency.

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