ON THE EXISTENCE OF A GIANT COMPONENT IN SCHEMES OF ALLOCATING PARTICLES

2002 ◽  
pp. 99-104
Keyword(s):  
Giant Component

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

The Cognitive Social Network in Dreams: Transitivity, Assortativity, and Giant Component Proportion Are Monotonic

2015 ◽  
Vol 40 (3) ◽  
pp. 671-696 ◽  
Author(s):  
Hye Joo Han ◽  
Richard Schweickert ◽  
Zhuangzhuang Xi ◽  
Charles Viau-Quesnel
Keyword(s):  
Social Network ◽  
Giant Component

The cover time of the giant component of a random graph

2008 ◽  
Vol 32 (4) ◽  
pp. 401-439 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Giant Component ◽  
Cover Time

scholarly journals Low degree connectivity of ad-hoc networks via percolation

2010 ◽  
Vol 42 (02) ◽  
pp. 559-576
Author(s):  
Emilio De Santis ◽  
Fabrizio Grandoni ◽  
Alessandro Panconesi
Keyword(s):  
Ad Hoc Networks ◽  
Maximum Degree ◽  
High Probability ◽  
Ad Hoc ◽  
Classical Problem ◽  
Universal Constant ◽  
Giant Component ◽  
Second Phase ◽  
Connected Network ◽  
Hoc Networks

Consider the following classical problem in ad-hoc networks. Suppose that n devices are distributed uniformly at random in a given region. Each device is allowed to choose its own transmission radius, and two devices can communicate if and only if they are within the transmission radius of each other. The aim is to (quickly) establish a connected network of low average and maximum degree. In this paper we present the first efficient distributed protocols that, in poly-logarithmically many rounds and with high probability, set up a connected network with O(1) average degree and O(log n) maximum degree. Our algorithms are based on the following result, which is a nontrivial consequence of classical percolation theory. Suppose that each device sets up its transmission radius in order to reach the K closest devices. There exists a universal constant K (independent of n) such that, with high probability, there will be a unique giant component (i.e. a connected component of size Θ(n)). Furthermore, all remaining components will be of size O(log2 n). This leads to an efficient distributed probabilistic test for membership in the giant component, which can be used in a second phase to achieve full connectivity.

scholarly journals Component Evolution in Random Intersection Graphs

10.37236/935 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael Behrisch
Keyword(s):  
Real World ◽  
Linear Order ◽  
Graph Model ◽  
Vertex Degree ◽  
Intersection Graph ◽  
Giant Component ◽  
Logarithmic Order ◽  
Intersection Graphs ◽  
Similarity Network ◽  
Random Intersection Graph

We study the evolution of the order of the largest component in the random intersection graph model which reflects some clustering properties of real–world networks. We show that for appropriate choice of the parameters random intersection graphs differ from $G_{n,p}$ in that neither the so-called giant component, appearing when the expected vertex degree gets larger than one, has linear order nor is the second largest of logarithmic order. We also describe a test of our result on a protein similarity network.

scholarly journals Network Analysis for Economics and Finance: An Application to Firm Ownership

2021 ◽  
pp. 331-355
Author(s):  
Janina Engel ◽  
Michela Nardo ◽  
Michela Rancan
Keyword(s):  
Network Analysis ◽  
Network Structure ◽  
Giant Component ◽  
Model Data ◽  
Firm Ownership ◽  
Network Metrics ◽  
Different Levels ◽  
Significant Concentration

AbstractIn this chapter, we introduce network analysis as an approach to model data in economics and finance. First, we review the most recent empirical applications using network analysis in economics and finance. Second, we introduce the main network metrics that are useful to describe the overall network structure and characterize the position of a specific node in the network. Third, we model information on firm ownership as a network: firms are the nodes while ownership relationships are the linkages. Data are retrieved from Orbis including information of millions of firms and their shareholders at worldwide level. We describe the necessary steps to construct the highly complex international ownership network. We then analyze its structure and compute the main metrics. We find that it forms a giant component with a significant number of nodes connected to each other. Network statistics show that a limited number of shareholders control many firms, revealing a significant concentration of power. Finally, we show how these measures computed at different levels of granularity (i.e., sector of activity) can provide useful policy insights.

scholarly journals Typical Distances in Ultrasmall Random Networks

2012 ◽  
Vol 44 (2) ◽  
pp. 583-601 ◽  
Author(s):  
Steffen Dereich ◽  
Christian Mönch ◽  
Peter Mörters
Keyword(s):  
Power Law ◽  
Preferential Attachment ◽  
Shortest Paths ◽  
Giant Component ◽  
Configuration Model ◽  
Random Networks ◽  
Power Law Exponent ◽  
Extra Factor ◽  
Typical Distances

We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1))log log N / (-log(τ − 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

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