scholarly journals The Diameter of Sparse Random Graphs

2010 ◽  
Vol 19 (5-6) ◽  
pp. 835-926 ◽  
Author(s):  
OLIVER RIORDAN ◽  
NICHOLAS WORMALD
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Branching Process ◽  
Correction Term ◽  
Separate Analysis ◽  
Wide Range ◽  
Range Of Functions ◽  
Finite Distance ◽  
Additive Correction

In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ > 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (4) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

scholarly journals On the Number of Solutions in Random Graphk-Colouring

2018 ◽  
Vol 28 (1) ◽  
pp. 130-158 ◽  
Author(s):  
FELICIA RASSMANN
Keyword(s):  
Phase Transition ◽  
Asymptotic Distribution ◽  
Random Graph ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Wide Range

Letk⩾ 3 be a fixed integer. We exactly determine the asymptotic distribution of lnZk(G(n, m)), whereZk(G(n, m)) is the number ofk-colourings of the random graphG(n, m). A crucial observation to this end is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles inG(n, m). Our result holds for a wide range of average degrees, and forkexceeding a certain constantk0it covers all average degrees up to the so-calledcondensation phase transition.

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (04) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

scholarly journals АНАЛІЗ СУЧАСНОГО СТАНУ РОЗВИТКУ ВИСОТНИХ БЕЗПІЛОТНИХ ЛІТАЛЬНИХ АПАРАТІВ

2019 ◽  
pp. 28-41
Author(s):  
Тамара Павловна Цепляева ◽  
Антон Юрьевич Мигунов
Keyword(s):  
High Altitude ◽  
Unmanned Aerial Vehicles ◽  
Statistical Data ◽  
Unmanned Vehicles ◽  
Future Research ◽  
Aerial Vehicles ◽  
Wide Range ◽  
Range Of Functions

The development of unmanned aerial vehicles, at this time, is of great interest, both of the largest aircraft building companies and design enthusiasts, and among the total amount of development, the volume of high-altitude unmanned vehicles occupies one of the leading positions. In this regard, a very topical issue is the analysis of existing developments and the determination of the vector of future research in this direction. High-altitude unmanned vehicles have a wide range of functions, both in the field of military and civilian use. The work collected and analyzed statistical data of high-altitude unmanned aerial vehicles to determine advances in the design of high-altitude unmanned aerial vehicles (UAVs). The current classification of UAVs was considered, as a result of the analysis of statistical data, options for its expansion were proposed. The flight characteristics of high-altitude UAVs are described. There are charts and tables showing the place of highaltitude UAVs in their total number. Also, flight performance, aerodynamic schemes and engine type, which are the most rational for high-altitude unmanned aerial vehicles according to their purpose and class, are defined.

scholarly journals Complex roots of polynomials and their computation with the help of Scientific Calculators

BIBECHANA ◽  
2012 ◽  
Vol 9 ◽  
pp. 18-27
Author(s):  
Mohd Yusuf Yasin
Keyword(s):  
Number System ◽  
Complex Numbers ◽  
Real Numbers ◽  
Engineering Technology ◽  
The Real ◽  
Complex Roots ◽  
Wide Range ◽  
Range Of Functions ◽  
Imaginary Number

Real numbers are something which are associated with the practical life. This number system is one dimensional. Situations arise when the real numbers fail to provide a solution. Perhaps the Italian mathematician Gerolamo Cardano is the first known mathematician who pointed out the necessity of imaginary and complex numbers. Complex numbers are now a vital part of sciences and are used in various branches of engineering, technology, electromagnetism, quantum theory, chaos theory etc. A complex number constitutes a real number along with an imaginary number that lies on the quadrature axis and gives an additional dimension to the number system. Therefore any computation based on complex numbers, is usually complex because both the real and imaginary parts of the number are to be simultaneously dealt with. Modern scientific calculators are capable of performing on a wide range of functions on complex numbers in their COMP and CMPLX modes with an equal ease as with the real numbers. In this work, the use of scientific calculators (Casio brand) for efficient determination of complex roots of various types of equations is discussed. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7148 BIBECHANA 9 (2013) 18-27

scholarly journals Planting Colourings Silently

2016 ◽  
Vol 26 (3) ◽  
pp. 338-366 ◽  
Author(s):  
VICTOR BAPST ◽  
AMIN COJA-OGHLAN ◽  
CHARILAOS EFTHYMIOU
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Random Variable ◽  
Average Degree ◽  
Fixed Integer ◽  
Wide Range ◽  
Concentration Result

Letk⩾ 3 be a fixed integer and letZk(G) be the number ofk-colourings of the graphG. For certain values of the average degree, the random variableZk(G(n,m)) is known to be concentrated in the sense that$\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability (Achlioptas and Coja-Oghlan,Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees,$\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability foranydiverging function$\omega=\omega(n)\ra\infty$. Forkexceeding a certain constantk0this result covers all average degrees up to the so-calledcondensation phase transitiondk,con, and this is best possible. As an application, we show that the experiment of choosing ak-colouring of the random graphG(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

scholarly journals The Critical Phase for Random Graphs with a Given Degree Sequence

2008 ◽  
Vol 17 (1) ◽  
pp. 67-86 ◽  
Author(s):  
M. KANG ◽  
T. G. SEIERSTAD
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Branching Process ◽  
Degree Sequence ◽  
Singularity Analysis ◽  
Fixed Degree ◽  
Critical Phase ◽  
Before And After

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

scholarly journals A Simple Branching Process Approach to the Phase Transition in $G_{n,p}$

10.37236/2588 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Branching Process ◽  
Process Approach ◽  
Giant Component ◽  
Asymptotic Size ◽  
The Way

It is well known that the branching process approach to the study of the random graph $G_{n,p}$ gives a very simple way of understanding the size of the giant component when it is fairly large (of order $\Theta(n)$). Here we show that a variant of this approach works all the way down to the phase transition: we use branching process arguments to give a simple new derivation of the asymptotic size of the largest component whenever $(np-1)^3n\to\infty$.

DINAMIKA POLITIK PERSETUJUAN BERSAMA TENTANG PENETAPAN DESA ADAT DI KABUPATEN ROKAN HULU TAHUN 2014

2017 ◽  
Vol 15 (1) ◽  
pp. 21
Author(s):  
Haryo Suganda ◽  
Raja Muhammad Amin
Keyword(s):  
Regulatory Region ◽  
Field Research ◽  
Analysis Techniques ◽  
Political Dynamics ◽  
Political Interests ◽  
Wide Range ◽  
Information Method ◽  
Qualitative Descriptive ◽  
The Village

This study is motivated the identification of policies issued by the regional Governmentof Rokan Hulu in the form of Regulatory region number 1 by 2015 on the determination of thevillage and Indigenous Village. Political dynamics based on various interests against themanufacture of, and decision-making in the process of formation of the corresponding localregulations determination of Indigenous Villages in the Rokan Hulu is impacted to a verysignificantamount of changes from the initial draft of the number i.e. 21 (twenty one) the villagebecame Customary 89 (eighty-nine) the Indigenous Villages who have passed. Type of thisresearch is a qualitative descriptive data analysis techniques. The research aims to describe theState of the real situation in a systematic and accurate fact analysis unit or related research, aswell as observations of the field based on the data (information). Method of data collectionwas done with interviews, documentation, and observations through fieldwork (field research).The results of the research on the process of discussion of the draft local regulations andmutual agreement about Designation of Indigenous Villages in the Rokan Hulu is, showed thatthe political dynamics that occur due to the presence of various political interests, rejectionorally by Villagers who were judged to have met the requirements of Draft Regulations to beformulated and the area for the set to be Indigenous Villages, and also there is a desire fromsome villages in the yet to Draft local regulations in order to set the Indigenous village , there isa wide range of interests of these aspects influenced the agreement to assign the entire localVillage which is in the Rokan Hulu become Indigenous village, and the village of Transmigrationinto administrative Villages where the initiator of the changes in the number of IndigenousVillages in the Rokan Hulu it is the desire of the local Government of its own.

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