Maximizing the Size of the Giant

2012 ◽  
Vol 49 (04) ◽  
pp. 1156-1165
Author(s):  
Tom Britton ◽  
Pieter Trapman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Related Question ◽  
Configuration Model ◽  
Connected Component ◽  
The Social ◽  
Random Weight ◽  
The Mean ◽  
Positive Integers ◽  
Outbreak Size

Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X i with E(X i )=µ, and where there is an edge between i and j with probability 1-e-X i X j /(µ n), independently of other edges. The second model is the thinned configuration model in which the n vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = µ. The graph of interest is obtained by deleting edges independently with probability 1-p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1-q, where q depends on X or D and p. We investigate for which distributions X and D with given µ and p, 1-q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

scholarly journals Maximizing the Size of the Giant

2012 ◽  
Vol 49 (4) ◽  
pp. 1156-1165 ◽  
Author(s):  
Tom Britton ◽  
Pieter Trapman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Related Question ◽  
Configuration Model ◽  
Connected Component ◽  
The Social ◽  
Random Weight ◽  
The Mean ◽  
Positive Integers ◽  
Outbreak Size

Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight Xi with E(Xi)=µ, and where there is an edge between i and j with probability 1-e-XiXj/(µ n), independently of other edges. The second model is the thinned configuration model in which the n vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = µ. The graph of interest is obtained by deleting edges independently with probability 1-p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1-q, where q depends on X or D and p. We investigate for which distributions X and D with given µ and p, 1-q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

The Largest Component in Subcritical Inhomogeneous Random Graphs

2010 ◽  
Vol 20 (1) ◽  
pp. 131-154 ◽  
Author(s):  
TATYANA S. TUROVA
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Graph Model ◽  
Exact Formula ◽  
Connected Component ◽  
Subcritical Case ◽  
Random Graph Model ◽  
Asymptotic Size ◽  
Inhomogeneous Random Graphs ◽  
Inhomogeneous Random Graph

We study the ‘rank 1 case’ of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result complements the corresponding known result in the supercritical case. We provide some examples of applications of the derived formula.

Generating Random Networks and Graphs

2017 ◽  
Author(s):  
Ton Coolen ◽  
Alessia Annibale ◽  
Ekaterina Roberts
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Scientific Method ◽  
Preferential Attachment ◽  
Worked Examples ◽  
Configuration Model ◽  
Random Networks ◽  
Network Applications ◽  
Starting Point ◽  
Random Graph Generation

This book supports researchers who need to generate random networks, or who are interested in the theoretical study of random graphs. The coverage includes exponential random graphs (where the targeted probability of each network appearing in the ensemble is specified), growth algorithms (i.e. preferential attachment and the stub-joining configuration model), special constructions (e.g. geometric graphs and Watts Strogatz models) and graphs on structured spaces (e.g. multiplex networks). The presentation aims to be a complete starting point, including details of both theory and implementation, as well as discussions of the main strengths and weaknesses of each approach. It includes extensive references for readers wishing to go further. The material is carefully structured to be accessible to researchers from all disciplines while also containing rigorous mathematical analysis (largely based on the techniques of statistical mechanics) to support those wishing to further develop or implement the theory of random graph generation. This book is aimed at the graduate student or advanced undergraduate. It includes many worked examples, numerical simulations and exercises making it suitable for use in teaching. Explicit pseudocode algorithms are included to make the ideas easy to apply. Datasets are becoming increasingly large and network applications wider and more sophisticated. Testing hypotheses against properly specified control cases (null models) is at the heart of the ‘scientific method’. Knowledge on how to generate controlled and unbiased random graph ensembles is vital for anybody wishing to apply network science in their research.

scholarly journals On Resilience of Connectivity in the Evolution of Random Graphs

10.37236/7885 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Luc Haller ◽  
Miloš Trujić
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hitting Time ◽  
Connected Component ◽  
Empty Graph ◽  
Spanning Subgraph ◽  
Time Result ◽  
Graph Property

In this note we establish a resilience version of the classical hitting time result of Bollobás and Thomason regarding connectivity. A graph $G$ is said to be $\alpha$-resilient with respect to a monotone increasing graph property $\mathcal{P}$ if for every spanning subgraph $H \subseteq G$ satisfying $\deg_H(v) \leqslant \alpha \deg_G(v)$ for all $v \in V(G)$, the graph $G - H$ still possesses $\mathcal{P}$. Let $\{G_i\}$ be the random graph process, that is a process where, starting with an empty graph on $n$ vertices $G_0$, in each step $i \geqslant 1$ an edge $e$ is chosen uniformly at random among the missing ones and added to the graph $G_{i - 1}$. We show that the random graph process is almost surely such that starting from $m \geqslant (\tfrac{1}{6} + o(1)) n \log n$, the largest connected component of $G_m$ is $(\tfrac{1}{2} - o(1))$-resilient with respect to connectivity. The result is optimal in the sense that the constants $1/6$ in the number of edges and $1/2$ in the resilience cannot be improved upon. We obtain similar results for $k$-connectivity.

scholarly journals Client-Waiter Games on Complete and Random Graphs

10.37236/6039 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Oren Dean ◽  
Michael Krivelevich
Keyword(s):  
Positive Integer ◽  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Upper Bounds ◽  
Connected Component ◽  
Lower And Upper Bounds ◽  
Large Connected Component ◽  
Graph Property ◽  
Player Game

For a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

scholarly journals Encores on Cores

10.37236/1107 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Julie Cain ◽  
Nicholas Wormald
Keyword(s):  
Simple Model ◽  
Hybrid Model ◽  
Random Graph ◽  
Random Graphs ◽  
Degree Sequence ◽  
Configuration Model ◽  
Simple Derivation ◽  
Random Functions ◽  
Uniform Hypergraphs

We give a new derivation of the threshold of appearance of the $k$-core of a random graph. Our method uses a hybrid model obtained from a simple model of random graphs based on random functions, and the pairing or configuration model for random graphs with given degree sequence. Our approach also gives a simple derivation of properties of the degree sequence of the $k$-core of a random graph, in particular its relation to multinomial and hence independent Poisson variables. The method is also applied to $d$-uniform hypergraphs.

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.

scholarly journals Buyer Inspired Meta-Heuristic Optimization Algorithm

2020 ◽  
Vol 10 (1) ◽  
pp. 194-219 ◽  
Author(s):  
Sanjoy Debnath ◽  
Wasim Arif ◽  
Srimanta Baishya
Keyword(s):  
Optimization Algorithm ◽  
Optimization Technique ◽  
Statistical Test ◽  
Heuristic Optimization ◽  
Product Choice ◽  
Exploration And Exploitation ◽  
The Social ◽  
Engineering Problems ◽  
The Mean

AbstractNature inspired swarm based meta-heuristic optimization technique is getting considerable attention and established to be very competitive with evolution based and physical based algorithms. This paper proposes a novel Buyer Inspired Meta-heuristic optimization Algorithm (BIMA) inspired form the social behaviour of human being in searching and bargaining for products. In BIMA, exploration and exploitation are achieved through shop to shop hoping and bargaining for products to be purchased based on cost, quality of the product, choice and distance to the shop. Comprehensive simulations are performed on 23 standard mathematical and CEC2017 benchmark functions and 3 engineering problems. An exhaustive comparative analysis with other algorithms is done by performing 30 independent runs and comparing the mean, standard deviation as well as by performing statistical test. The results showed significant improvement in terms of optimum value, convergence speed, and is also statistically more significant in comparison to most of the reported popular algorithms.

The Subjective Present and Its Modulation in Clinical Contexts

2013 ◽  
Vol 1 (2) ◽  
pp. 239-259 ◽  
Author(s):  
Wolfgang Tschacher ◽  
Fabian Ramseyer ◽  
Claudia Bergomi
Keyword(s):  
Perceptual Grouping ◽  
Statistical Significance ◽  
Temporal Distribution ◽  
Future Research ◽  
Implicit Processes ◽  
The Social ◽  
Gestalt Perception ◽  
Psychotherapeutic Techniques ◽  
The Mean ◽  
The Individual

Time is a basic dimension in psychology, underlying behavior and experience. Timing and time perception constitute implicit processes that are often inaccessible to the individual person. Research in this field has shown that timing is involved in many areas of clinical significance. In the projects presented here, we combine timing with seemingly different fields of research, such as psychopathology, perceptual grouping, and embodied cognition. Focusing on the time scale of the subjective present, we report findings from three different clinical studies: (1) We studied perceived causality in schizophrenia patients, finding that perceptual grouping (‘binding’, ‘Gestalt formation’), which leads to visual causality perceptions, did not distinguish between patients and healthy controls. Patients however did integrate context (provided by the temporal distribution of auditory context stimuli) less into perceptions, in significant contrast to controls. This is consistent with reports of higher inaccuracy in schizophrenia patients’ temporal processing. (2) In a project on auditory Gestalt perception we investigated auditory perceptual grouping in schizophrenia patients. The mean dwell time was positively related to how much patients were prone to auditory hallucinations. Dwell times of auditory Gestalts may be regarded as operationalizations of the subjective present; findings thus suggested that patients with hallucinations had a shorter present. (3) The movement correlations of interacting individuals were used to study the non-verbal synchrony between therapist and patient in psychotherapy sessions. We operationalized the duration of an embodied ‘social present’ by the statistical significance of such associations, finding a window of roughly 5.7 seconds in conversing dyads. We discuss that temporal scales of nowness may be modifiable, e.g., by mindfulness. This yields promising goals for future research on timing in the clinical context: psychotherapeutic techniques may alter binding processes, hence the subjective present of individuals, and may affect the social present in therapeutic interactions.

scholarly journals An Improved Teaching-Learning-Based Optimization with the Social Character of PSO for Global Optimization

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Feng Zou ◽  
Debao Chen ◽  
Jiangtao Wang
Keyword(s):  
Optimization Problems ◽  
Global Solutions ◽  
Function Optimization ◽  
Improved Method ◽  
Social Character ◽  
The Social ◽  
Teaching Learning Based Optimization ◽  
The Mean ◽  
The Individual ◽  
Teaching Learning

An improved teaching-learning-based optimization with combining of the social character of PSO (TLBO-PSO), which is considering the teacher’s behavior influence on the students and the mean grade of the class, is proposed in the paper to find the global solutions of function optimization problems. In this method, the teacher phase of TLBO is modified; the new position of the individual is determined by the old position, the mean position, and the best position of current generation. The method overcomes disadvantage that the evolution of the original TLBO might stop when the mean position of students equals the position of the teacher. To decrease the computation cost of the algorithm, the process of removing the duplicate individual in original TLBO is not adopted in the improved algorithm. Moreover, the probability of local convergence of the improved method is decreased by the mutation operator. The effectiveness of the proposed method is tested on some benchmark functions, and the results are competitive with respect to some other methods.

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