scholarly journals Clique Listing Algorithms and Characteristics of Cliques in Random Graphics

2021 ◽  
Author(s):  
Sonal Patel
Keyword(s):  
Graph Theory ◽  
Random Graphs ◽  
Input Graph ◽  
Exponential Time ◽  
Practical Limit

In this thesis we address three main problems in clique detection in the area of Graph Theory. i) Most of current methods for clique detection are time consuming (can take exponential time to the size of input graph), so there is a practical limit on size of input graph. In this thesis we propose three different methods for estimating the number of cliques. We examine these methods for various graphs and conclude that they efficiently find the number of cliques within 5% error typically. ii) We compare various versions of the Bron-Kerbosch (BK) clique listing algorithm to discover a method of combining the best features of different versions. We test our new versions of BK for various inputs. iii) We study the characteristics of cliques in random graphs as a function of size and density.

scholarly journals Clique Listing Algorithms and Characteristics of Cliques in Random Graphics

2021 ◽  
Author(s):  
Sonal Patel
Keyword(s):  
Graph Theory ◽  
Random Graphs ◽  
Input Graph ◽  
Exponential Time ◽  
Practical Limit

In this thesis we address three main problems in clique detection in the area of Graph Theory. i) Most of current methods for clique detection are time consuming (can take exponential time to the size of input graph), so there is a practical limit on size of input graph. In this thesis we propose three different methods for estimating the number of cliques. We examine these methods for various graphs and conclude that they efficiently find the number of cliques within 5% error typically. ii) We compare various versions of the Bron-Kerbosch (BK) clique listing algorithm to discover a method of combining the best features of different versions. We test our new versions of BK for various inputs. iii) We study the characteristics of cliques in random graphs as a function of size and density.

scholarly journals Erdos – Renyi Random Graph and Machine Learning Based Botnet Detection

2020 ◽  
Vol 9 (5) ◽  
pp. 1037-1040
Keyword(s):  
Machine Learning ◽  
Probability Theory ◽  
Graph Theory ◽  
Complex Network ◽  
Random Graphs ◽  
Connected Graph ◽  
Free Graph ◽  
Botnet Detection ◽  
Scale Free ◽  
Made In

Basically large networks are prone to attacks by bots and lead to complexity. When the complexity occurs then it is difficult to overcome the vulnerability in the network connections. In such a case, the complex network could be dealt with the help of probability theory and graph theory concepts like Erdos – Renyi random graphs, Scale free graph, highly connected graph sequences and so on. In this paper, Botnet detection using Erdos – Renyi random graphs whose patterns are recognized as the number of connections that the vertices and edges made in the network is proposed. This paper also presents the botnet detection concepts based on machine learning.

scholarly journals Structural Parameterizations of Clique Coloring

Algorithmica ◽  
2021 ◽  
Author(s):  
Lars Jaffke ◽  
Paloma T. Lima ◽  
Geevarghese Philip
Keyword(s):  
Lower Bound ◽  
Maximal Clique ◽  
Time Algorithm ◽  
Input Graph ◽  
Exponential Time ◽  
Exponential Time Hypothesis ◽  
Tree Decompositions

AbstractA clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed $$q \ge 2$$ q ≥ 2 , we give an $$\mathscr {O}^{\star }(q^{{\mathsf {tw}}})$$ O ⋆ ( q tw ) -time algorithm when the input graph is given together with one of its tree decompositions of width $${\mathsf {tw}} $$ tw . We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is $$\mathsf {XP}$$ XP parameterized by clique-width.

scholarly journals Triangle-Free Subgraphs of Random Graphs

2017 ◽  
Vol 27 (2) ◽  
pp. 141-161
Author(s):  
PETER ALLEN ◽  
JULIA BÖTTCHER ◽  
YOSHIHARU KOHAYAKAWA ◽  
BARNABY ROBERTS
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Random Graphs ◽  
Minimum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Spanning Subgraph

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p−1n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p−1n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

scholarly journals Toppling numbers of complete and random graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Anthony Bonato ◽  
William B. Kinnersley ◽  
Pawel Pralat
Keyword(s):  
Graph Theory ◽  
Differential Equations ◽  
Random Graphs ◽  
Complete Graph ◽  
Complete Graphs ◽  
Asymptotic Bounds ◽  
Large N ◽  
Chip Firing ◽  
International Audience ◽  
Person Game

Graph Theory International audience We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

Sparse random matrices: spectral edge and statistics of rooted trees

2001 ◽  
Vol 33 (1) ◽  
pp. 124-140 ◽  
Author(s):  
A. Khorunzhy
Keyword(s):  
Graph Theory ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Random Matrices ◽  
Random Graphs ◽  
Rooted Trees ◽  
Theory Method ◽  
Sparse Random Matrices ◽  
Spectral Edge ◽  
Spectral Statistics

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices AN,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥AN,p∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥AN,p/√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of AN,p. They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.

Poisson convergence and random graphs

1982 ◽  
Vol 92 (2) ◽  
pp. 349-359 ◽  
Author(s):  
A. D. Barbour
Keyword(s):  
Graph Theory ◽  
Poisson Distribution ◽  
Random Graphs ◽  
Factorial Moment ◽  
Expected Number ◽  
Factorial Moments ◽  
Random Graph Theory ◽  
The Mean ◽  
Fundamental Paper ◽  
Combinatorial Methods

Approximation by the Poisson distribution arises naturally in the theory of random graphs, as in many other fields, when counting the number of occurrences of individually rare and unrelated events within a large ensemble. For example, one may be concerned with the number of times that a particular small configuration is repeated in a large graph, such questions being considered, amongst others, in the fundamental paper of Erdös and Rényi (4). The technique normally used to obtain such approximations in random graph theory is based on showing that the factorial moments of the quantity concerned converge to those of a Poisson distribution as the size of the graph tends to infinity. Since the rth factorial moment is just the expected number of ordered r-tuples of events occurring, it is particularly well suited to evaluation by combinatorial methods. Unfortunately, such a technique becomes very difficult to manage if the mean of the approximating Poisson distribution is itself increasing with the size of the graph, and this limits the scope of the results obtainable.

A PROCESSOR EFFICIENT CONNECTIVITY ALGORITHM ON RANDOM GRAPHS

1994 ◽  
Vol 04 (01n02) ◽  
pp. 29-36
Author(s):  
S.B. YANG ◽  
S.K. DHALL ◽  
S. LAKSHMIVARAHAN
Keyword(s):  
Parallel Algorithm ◽  
Random Graphs ◽  
Connected Components ◽  
Input Graph ◽  
Random Input ◽  
Expected Time ◽  
Erew Pram ◽  
Pram Model

In this paper we present a randomized parallel algorithm for finding the connected components of a random input graph with n vertices in which the edges are chosen with probability p such that [Formula: see text]. The algorithm has O(log2 n) expected time using only O(n) processors on the EREW PRAM model. The probability that the output of our algorithm is correct is at least 1–0.6k, where k is a constant.

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