There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Social network analysis of animal behaviour : a promising tool for the study of sociality

What is discrete math? • The real numbers are continuous in the senses that: * between any two real numbers there is a real number • The integers do not share this property. In this sense the integers are lumpy, or " discrete " So discrete math is the study of mathematical objects that are discrete. " It's all the math that counts " Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a network connection there are no terminals. Propositional Logic Propositions are statements that are either true or false. Principles: Substituting an equivalent statement. Replacing a logic variable in a tautology. Defn algebraic proof

Discrete math = 離散数学

Recently, Ranjan proposed a novel public key encryption technique based on multiple chaotic systems [Phys Lett 2005;95]. Unfortunately, Wang soon gave a successful attack on its special case based on Parseval’s theorem [Wang K, Pei W, Zhou L, et al. Security of public key encryption technique based on multiple chaotic system. Phys Lett A, in press]. In this letter, we give an improved example which can avoid the attack and point out that Wang cannot find the essential drawback of the technique. However, further experimental result shows Ruanjan’s encryption technique is inefficient, and detailed theoretic analysis shows that the complexity to break the cryptosystem is overestimated.

Cryptanalysis of the public key encryption based on multiple chaotic systems

Cloud computing builds upon advances on virtualization and distributed computing to support cost-efficient usage of computing resources, emphasizing on resource scalability and on demand services. Moving away from traditional data-center oriented models, distributed clouds extend over a loosely coupled federated substrate, offering enhanced communication and computational services to target end-users with quality of service (QoS) requirements, as dictated by the future Internet vision. Toward facilitating the efficient realization of such networked computing environments, computing and networking resources need to be jointly treated and optimized. This requires delivery of user-driven sets of virtual resources, dynamically allocated to actual substrate resources within networked clouds, creating the need to revisit resource mapping algorithms and tailor them to a composite virtual resource mapping problem. In this paper, toward providing a unified resource allocation framework for networked clouds, we first formulate the optimal networked cloud mapping problem as a mixed integer programming (MIP) problem, indicating objectives related to cost efficiency of the resource mapping procedure, while abiding by user requests for QoS-aware virtual resources. We subsequently propose a method for the efficient mapping of resource requests onto a shared substrate interconnecting various islands of computing resources, and adopt a heuristic methodology to address the problem. The efficiency of the proposed approach is illustrated in a simulation/emulation environment, that allows for a flexible, structured, and comparative performance evaluation. We conclude by outlining a proof-of-concept realization of our proposed schema, mounted over the European future Internet test-bed FEDERICA, a resource virtualization platform augmented with network and computing facilities.

On the optimal allocation of virtual resources in cloud computing networks

We propose public-key encryption algorithms based on chaotic maps, which are generalization of well-known and commercially used algorithms: Rivest–Shamir–Adleman (RSA), ElGamal, and Rabin. For the case of generalized RSA algorithm we discuss in detail its software implementation and properties. We show that our algorithm is as secure as RSA algorithm.

Public-key encryption with chaos.

We study traffi c dynamics in growing scale-free networks. Both the scale-free structure of the network and the adaptive nature of the dynamics which controls traffi c in the network are considered in the model. The model is investigated with computer simulations and analytically for the case of a scale-free tree. For the scale-free tree, an exact formula and its power law approximation of the complementary cumulative distribution function of link load (edge betweenness) is presented. We examine whether the scaling properties of the network affect the performance of the transport mechanism and estimate the average number of competing transport mechanisms at bottlenecks. We fi that bottlenecks tend to appear on the periphery of the network as the performance increases. Various bandwidth allocation strategies are compared. We show that the best performance is achieved when capacity is distributed proportionally to the expected load of links. We demonstrate that it is necessary to study both the topology and the dynamics of the transport mechanism to understand the whole system.

/pdf/traffic-dynamics-in-scale-free-networks-4nqwjl7hc2.pdf

Traffic Dynamics in Scale-Free Networks

In this Rapid Communication we present an analytic study of sampled networks in the case of some important shortest-path sampling models. We present analytic formulas for the probability of edge discovery in the case of an evolving and a static network model. We also show that the number of discovered edges in a finite network scales much more slowly than predicted by earlier mean-field models. Finally, we calculate the degree distribution of sampled networks and we demonstrate that they are analogous to a destroyed network obtained by randomly removing edges from the original network.

Shortest path discovery of complex networks.

Node betweenness has been studied recently by a number of authors, but until now less attention has been paid to edge betweenness. In this paper, we present an exact analytic study of edge betweenness in evolving scale-free and non-scale-free trees. We aim at the probability distribution of edge betweenness under the condition that a local property, the in-degree of the "younger" node of a randomly selected edge, is known. En route to the conditional distribution of edge betweenness the exact joint distribution of cluster size and in-degree, and its one-dimensional marginal distributions have been presented in the paper as well. From the derived probability distributions the expectation values of different quantities have been calculated. Our results provide an exact solution not only for infinite, but for finite networks as well.

Distribution of edge load in scale-free trees.

ETOMIC is a network traffic measurement platform with high precision GPS-synchronized monitoring nodes. The infrastructure is publicly available to the network research community, supporting advanced experimental techniques by providing high precision hardware equipments and a Central Management System. Researchers can deploy their own active measurement codes to perform experiments on the public Internet. Recently, the functionalities of the original system were significantly extended and new generation measurement nodes were deployed. The system now also includes well structured data repositories to archive and share raw and evaluated data. These features make ETOMIC as one of the experimental facilities that support the design, development and validation of novel experimental techniques for the future Internet. In this paper we focus on the improved capabilities of the management system, the recent extensions of the node architecture and the accompanying database solutions.

/pdf/etomic-advanced-network-monitoring-system-for-future-8yzbpfvbb6.pdf

Attila Fekete

Papers

Public-key encryption with chaos.

Traffic Dynamics in Scale-Free Networks

Shortest path discovery of complex networks.

Distribution of edge load in scale-free trees.

ETOMIC advanced network monitoring system for future Internet experimentation