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.

Some Balanced Colouring Algorithms For Examination Timetabling

1992 ◽  
pp. 57-63 ◽  
Author(s):  
Ghazali Sulong
Keyword(s):  
Random Graphs ◽  
Large Range ◽  
Graph Colouring ◽  
Examination Timetabling ◽  
Minimum Number

This paper describes some balanced colouring algorithms designed to construct examination schedules in such a way that :(1) all examination take place within a minimum number of days; (2) students are never scheduled to take two examinations at the same time; (3) the number of courses are scheduled into each period are approximately equal. These algorithms were tested on a large range of random graphs. Keywords: Balanced colouring,scheduling, random graphs,graph colouring

scholarly journals A VISIBILITY-BASED PURSUIT-EVASION PROBLEM

1999 ◽  
Vol 09 (04n05) ◽  
pp. 471-493 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JEAN-CLAUDE LATOMBE ◽  
STEVEN M. LAVALLE ◽  
DAVID LIN ◽  
RAJEEV MOTWANI
Keyword(s):  
Finite Graph ◽  
Initial Position ◽  
Moving Target ◽  
Multiply Connected ◽  
Pursuit Evasion ◽  
Minimum Number ◽  
Evasion Problem ◽  
Free Spaces ◽  
Simply Connected ◽  
Motion Strategy

This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simply-connected free spaces, it is shown that the minimum number of pursuers required is Θ( lg  n). For multiply-connected free spaces, the bound is [Formula: see text] pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a finite graph that is constructed on the basis of critical information changes. It has been implemented and computed examples are shown.

scholarly journals Optimal shattering of complex networks

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Nicole Balashov ◽  
Reuven Cohen ◽  
Avieli Haber ◽  
Michael Krivelevich ◽  
Simi Haber
Keyword(s):  
Complex Networks ◽  
Random Graphs ◽  
Polynomial Time ◽  
Optimal Performance ◽  
Regular Graphs ◽  
Scale Free ◽  
Scale Free Networks ◽  
Minimum Number

Abstract We consider optimal attacks or immunization schemes on different models of random graphs. We derive bounds for the minimum number of nodes needed to be removed from a network such that all remaining components are fragments of negligible size.We obtain bounds for different regimes of random regular graphs, Erdős-Rényi random graphs, and scale free networks, some of which are tight. We show that the performance of attacks by degree is bounded away from optimality.Finally we present a polynomial time attack algorithm and prove its optimal performance in certain cases.

scholarly journals Recurrent Neural Network Model Based on a New Regularization Technique for Real-Time Intrusion Detection in SDN Environments

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Marwan Ali Albahar
Keyword(s):  
Neural Network ◽  
Network Security ◽  
Intrusion Detection ◽  
Recurrent Neural Network ◽  
Network Performance ◽  
Integrated Approach ◽  
Regularization Technique ◽  
Model Based ◽  
Minimum Number ◽  
Sdr Model

Software-defined networking (SDN) is a promising approach to networking that provides an abstraction layer for the physical network. This technology has the potential to decrease the networking costs and complexity within huge data centers. Although SDN offers flexibility, it has design flaws with regard to network security. To support the ongoing use of SDN, these flaws must be fixed using an integrated approach to improve overall network security. Therefore, in this paper, we propose a recurrent neural network (RNN) model based on a new regularization technique (RNN-SDR). This technique supports intrusion detection within SDNs. The purpose of regularization is to generalize the machine learning model enough for it to be performed optimally. Experiments on the KDD Cup 1999, NSL-KDD, and UNSW-NB15 datasets achieved accuracies of 99.5%, 97.39%, and 99.9%, respectively. The proposed RNN-SDR employs a minimum number of features when compared with other models. In addition, the experiments also validated that the RNN-SDR model does not significantly affect network performance in comparison with other options. Based on the analysis of the results of our experiments, we conclude that the RNN-SDR model is a promising approach for intrusion detection in SDN environments.

scholarly journals Metric Dimension for Random Graphs

10.37236/2639 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Béla Bollobás ◽  
Dieter Mitsche ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Metric Dimension ◽  
Wide Range ◽  
Minimum Number ◽  
Vertex Set

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

scholarly journals Upper bounds on the non-3-colourability threshold of random graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Upper Bounds ◽  
Average Degree ◽  
Expected Number ◽  
International Audience ◽  
Full Analysis ◽  
A Minor ◽  
Minor Improvement

International audience We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

scholarly journals Blocks in Constrained Random Graphs with Fixed Average Degree

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Average Degree ◽  
Additional Restriction ◽  
Structural Constraints ◽  
Analytic Framework ◽  
Discrete Algorithms ◽  
International Audience ◽  
Sharp Concentration

International audience This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.

scholarly journals Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

10.37236/3198 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Wojciech Samotij
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Minimum Degree ◽  
Average Degree ◽  
Free Graph ◽  
Connected Graphs ◽  
Random Subgraph ◽  
Minimum Number ◽  
Random Subgraphs ◽  
Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

scholarly journals On-line List Colouring of Random Graphs

10.37236/5003 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alan Frieze ◽  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Constant Factor ◽  
Average Degree ◽  
Multiplicative Constant ◽  
Line List ◽  
Multiplicative Factor ◽  
On Line ◽  
Choice Number

In this paper, the on-line list colouring of binomial random graphs $\mathcal{G}(n,p)$ is studied. We show that the on-line choice number of $\mathcal{G}(n,p)$ is asymptotically almost surely asymptotic to the chromatic number of $\mathcal{G}(n,p)$, provided that the average degree $d=p(n-1)$ tends to infinity faster than $(\log \log n)^{1/3} (\log n)^2 n^{2/3}$. For sparser graphs, we are slightly less successful; we show that if $d \ge (\log n)^{2+\epsilon}$ for some $\epsilon>0$, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of $C$, where $C \in [2,4]$, depending on the range of $d$. Also, for $d=O(1)$, the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.

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