Asymptotic Results for an Epidemic Process on Random Graphs

1983 ◽  
pp. 301-306 ◽  
Author(s):  
Zvetan Ignatov Sofia
Keyword(s):  
Random Graphs ◽  
Asymptotic Results ◽  
Epidemic Process

scholarly journals Random Graphs and the Strong Convergence of Bootstrap Means

2000 ◽  
Vol 9 (4) ◽  
pp. 315-347 ◽  
Author(s):  
SÁNDOR CSÖRGŐ ◽  
WEI BIAO WU
Keyword(s):  
Strong Convergence ◽  
Sample Size ◽  
Random Graphs ◽  
Complete Convergence ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Moment Condition ◽  
Bootstrap Sample ◽  
Asymptotic Results

We consider graphs Gn generated by multisets [Iscr ]n with n random integers as elements, such that vertices of Gn are connected by edges if the elements of [Iscr ]n that the vertices represent are the same, and prove asymptotic results on the sparsity of edges connecting the different subgraphs Gn of the random graph generated by ∪∞n=1[Iscr ]n. These results are of independent interest and, for two models of the bootstrap, we also use them here to link almost sure and complete convergence of the corresponding bootstrap means and averages of related randomly chosen subsequences of a sequence of independent and identically distributed random variables with a finite mean. Complete convergence of these means and averages is then characterized in terms of a relationship between a moment condition on the bootstrapped sequence and the bootstrap sample size. While we also obtain new sufficient conditions for the almost sure convergence of bootstrap means, the approach taken here yields the first necessary conditions.

scholarly journals Exponential Random Graphs as Models of Overlay Networks

2009 ◽  
Vol 46 (1) ◽  
pp. 199-220 ◽  
Author(s):  
M. Draief ◽  
A. Ganesh ◽  
L. Massoulié
Keyword(s):  
Communication Networks ◽  
Random Graphs ◽  
Degree Distribution ◽  
Overlay Networks ◽  
Analytic Solution ◽  
Mean Value ◽  
Asymptotic Results ◽  
Graph Expansion ◽  
Exponential Random Graphs ◽  
Random Failures

In this paper we give an analytic solution for graphs withnnodes andE=cnlognedges for which the probability of obtaining a given graphGisµn(G) = exp (-β∑i=1ndi2), wherediis the degree of nodei. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.

scholarly journals Greedy Matching in Bipartite Random Graphs

2021 ◽  
Author(s):  
Nick Arnosti
Keyword(s):  
Random Graphs ◽  
Bipartite Graphs ◽  
Approximation Ratio ◽  
Prior Work ◽  
Asymptotic Results ◽  
Worst Case ◽  
Average Performance ◽  
Finite Graphs ◽  
Random Priority ◽  
Priority Order

This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].

scholarly journals Exponential Random Graphs as Models of Overlay Networks

2009 ◽  
Vol 46 (01) ◽  
pp. 199-220 ◽  
Author(s):  
M. Draief ◽  
A. Ganesh ◽  
L. Massoulié
Keyword(s):  
Communication Networks ◽  
Random Graphs ◽  
Degree Distribution ◽  
Overlay Networks ◽  
Analytic Solution ◽  
Mean Value ◽  
Asymptotic Results ◽  
Graph Expansion ◽  
Exponential Random Graphs ◽  
Random Failures

In this paper we give an analytic solution for graphs with n nodes and E = cn log n edges for which the probability of obtaining a given graph G is µn (G) = exp (- β ∑i=1 n d i 2), where d i is the degree of node i. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.

scholarly journals Revolutionaries and Spies on Random Graphs

2013 ◽  
Vol 22 (3) ◽  
pp. 417-432 ◽  
Author(s):  
DIETER MITSCHE ◽  
PAWEŁ PRAŁAT
Keyword(s):  
Network Security ◽  
Random Graphs ◽  
Large Range ◽  
Average Degree ◽  
Simplified Model ◽  
Asymptotic Results ◽  
Pursuit Evasion ◽  
Dense Graphs ◽  
Minimum Number

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of r revolutionaries tries to hold an unguarded meeting consisting of m revolutionaries. A team of s spies wants to prevent this forever. For given r and m, the minimum number of spies required to win on a graph G is the spy number σ(G,r,m). We present asymptotic results for the game played on random graphs G(n,p) for a large range of p = p(n), r=r(n), and m=m(n). The behaviour of the spy number is analysed completely for dense graphs (that is, graphs with average degree at least n1/2+ε for some ε > 0). For sparser graphs, some bounds are provided.

scholarly journals Bipartite Random Graphs and Cuckoo Hashing

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Reinhard Kutzelnigg
Keyword(s):  
Saddle Point ◽  
Bipartite Graph ◽  
Random Graphs ◽  
Hash Table ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Cuckoo Hashing ◽  
International Audience ◽  
Covering Tree ◽  
Number Of Cycles

International audience The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically $1-c(\varepsilon)/m+O(1/m^2)$ for some explicit $c(\varepsilon)$, where $m$ denotes the size of each of the two tables, $n=m(1- \varepsilon)$ is the number of keys and $\varepsilon \in (0,1)$. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.

Random Graphs

1998 ◽  
Author(s):  
V. F. Kolchin
Keyword(s):  
Random Graphs

Epidemiological Characteristics of Meningococcal Infection in Moscow

2020 ◽  
Vol 19 (2) ◽  
pp. 56-62
Author(s):  
M. I. Gritsay ◽  
M. A. Koroleva ◽  
N. N. Fomkina ◽  
I. S. Koroleva
Keyword(s):  
Young Adults ◽  
Mortality Rate ◽  
Incidence Rate ◽  
Meningococcal Disease ◽  
Invasive Meningococcal Disease ◽  
Meningococcal Infection ◽  
Epidemic Process ◽  
Epidemiological Features ◽  
Teenagers And Young Adults ◽  
Epidemiological Characteristics

Aims. The purpose of this study was to identify current epidemiological features of meningococcal infection in Moscow.Materials and methods. Cases of invasive meningococcal disease in Moscow from 2014 to 2018 and the biomaterial from patients with an invasive meningococcal disease were analyzed.Results. The features of the epidemic process of meningococcal disease in Moscow were revealed: increasing in the incidence rate involving teenagers and young adults into the epidemic process; meningococcal strains of serogroups W and A increased in the etiology of the invasive meningococcal disease; high mortality rate.Conclusions. It seems reasonable to recommend vaccination against meningococcal disease by including adolescents, young adults and persons over 65 years old.

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