Displacement of transport processes on networked topologies

Consider a particle whose position evolves along the edges of a network. One definition for the displacement of a particle is the length of the shortest path on the network between the current and initial positions of the particle. Such a definition fails to incorporate information of the actual path the particle traversed. In this work we consider another definition for the displacement of a particle on networked topologies. Using this definition, which we term the winding distance, we demonstrate that for Brownian particles, confinement to a network can induce a transition in the mean squared displacement from diffusive to ballistic behaviour, $\langle x^2(t) \rangle \propto t^2$ for long times. A multiple scales approach is used to derive a macroscopic evolution equation for the displacement of a particle and uncover a topological condition for whether this transition in the mean squared displacement will occur. Furthermore, for networks satisfying this topological condition, we identify a prediction of the timescale upon which the displacement transitions to long-time behaviour. Finally, we extend the investigation of displacement on networks to a class of anomalously diffusive transport processes, where we find that the mean squared displacement at long times is affected by both network topology and the character of the transport process.Comment: 22 pages, 8 figure

Teleoperation of passivity-based model reference robust control over the internet

This dissertation offers a survey of a known theoretical approach and novel experimental results in establishing a live communication medium through the internet to host a virtual communication environment for use in Passivity-Based Model Reference Robust Control systems with delays. The controller which is used as a carrier to support a robust communication between input-to-state stability is designed as a control strategy that passively compensates for position errors that arise during contact tasks and strives to achieve delay-independent stability for controlling of aircrafts or other mobile objects. Furthermore the controller is used for nonlinear systems, coordination of multiple agents, bilateral teleoperation, and collision avoidance thus maintaining a communication link with an upper bound of constant delay is crucial for robustness and stability of the overall system. For utilizing such framework an elucidation can be formulated by preparing site survey for analyzing not only the geographical distances separating the nodes in which the teleoperation will occur but also the communication parameters that define the virtual topography that the data will travel through. This survey will first define the feasibility of the overall operation since the teleoperation will be used to sustain a delay based controller over the internet thus obtaining a hypothetical upper bound for the delay via site survey is crucial not only for the communication system but also the delay is required for the design of the passivity-based model reference robust control. Following delay calculation and measurement via site survey, bandwidth tests for unidirectional and bidirectional communication is inspected to ensure that the speed is viable to maintain a real-time connection. Furthermore from obtaining the results it becomes crucial to measure the consistency of the delay throughout a sampled period to guarantee that the upper bound is not breached at any point within the communication to jeopardize the robustness of the controller. Following delay analysis a geographical and topological overview of the communication is also briefly examined via a trace-route to understand the underlying nodes and their contribution to the delay and round-trip consistency. To accommodate the communication channel for the controller the input and output data from both nodes need to be encapsulated within a transmission control protocol via a multithreaded design of a robust program within the C language. The program will construct a multithreaded client-server relationship in which the control data is transmitted. For added stability and higher level of security the channel is then encapsulated via an internet protocol security by utilizing a protocol suite for protecting the communication by authentication and encrypting each packet of the session using negotiation of cryptographic keys during each session

Topology-dependent density optima for efficient simultaneous network exploration

A random search process in a networked environment is governed by the time it takes to visit every node, termed the cover time. Often, a networked process does not proceed in isolation but competes with many instances of itself within the same environment. A key unanswered question is how to optimise this process: how many concurrent searchers can a topology support before the benefits of parallelism are outweighed by competition for space? Here, we introduce the searcher-averaged parallel cover time (APCT) to quantify these economies of scale. We show that the APCT of the networked symmetric exclusion process is optimised at a searcher density that is well predicted by the spectral gap. Furthermore, we find that non-equilibrium processes, realised through the addition of bias, can support significantly increased density optima. Our results suggest novel hybrid strategies of serial and parallel search for efficient information gathering in social interaction and biological transport networks.This work was supported by the EPSRC Systems Biology DTC Grant No. EP/G03706X/1 (D.B.W.), a Royal Society Wolfson Research Merit Award (R.E.B.), a Leverhulme Research Fellowship (R.E.B.), the BBSRC UK Multi-Scale Biology Network Grant No. BB/M025888/1 (R.E.B. and F.G.W.), and Trinity College, Cambridge (F.G.W.)

Wall mediated transport in confined spaces: Exact theory for low density

We present a theory for the transport of molecules adsorbed in slit and cylindrical nanopores at low density, considering the axial momentum gain of molecules oscillating between diffuse wall reflections. Good agreement with molecular dynamics simulations is obtained over a wide range of pore sizes, including the regime of single-file diffusion where fluid-fluid interactions are shown to have a negligible effect on the collective transport coefficient. We show that dispersive fluid-wall interactions considerably attenuate transport compared to classical hard sphere theory

Synchronization in networks with multiple interaction layers

The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multilayered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavor in mathematics and physics and has potential applications in several socially relevant topics, such as power grid engineering and neural dynamics. We propose a general framework to assess the stability of the synchronized state in networks with multiple interaction layers, deriving a necessary condition that generalizes the master stability function approach. We validate our method by applying it to a network of Rössler oscillators with a double layer of interactions and show that highly rich phenomenology emerges from this. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely arising from the true multilayer structure of the interactions among the units in the network

Robust Control

The need to be tolerant to changes in the control systems or in the operational environment of systems subject to unknown disturbances has generated new control methods that are able to deal with the non-parametrized disturbances of systems, without adapting itself to the system uncertainty but rather providing stability in the presence of errors bound in a model. With this approach in mind and with the intention to exemplify robust control applications, this book includes selected chapters that describe models of H-infinity loop, robust stability and uncertainty, among others. Each robust control method and model discussed in this book is illustrated by a relevant example that serves as an overview of the theoretical and practical method in robust control

Mathematical analysis of k-path Laplacian operators on simple graphs

A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler.A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler

Limen, portal, network subjectivities

The ambiguity of the term 'network subjectivities' is enticing: is there a subjectivity proper to networks? Is that subjectivity itself distributed? The inference in each case is that there is or has been a subjectivity that is or was not networked in either sense. It follows then that there must be a passage between non- or pre-networked subjectivity and networked, a threshold. Anne Friedman's work would suggest that the key threshold is the human-computer interface, the physical display and the software interface that appears in it. Her analysis concerns the actual, but subjectivity also concerns the potential. To pursue that we need to look at imaginary (fictitious, experimental) accounts of thresholds between worlds. This paper looks at liminal spaces and portals in popular culture as evidence for the potential of subjectivities in transition between IRL or individualist subjectivity towards networks and networking. From Odysseus' visit to the gates of the underworld and Orpheus' rescue of Eurydice, all the way to Malevich's black icon, the liminal has been approached with ritual and trepidation. Transitions between worlds retain their magic in the Narnia books of C.S. Lewis and the tesseract of Interstellar, but lose their ritual and their power to demand awe and fear in the promise of easy, cost-free travel between the mundane world and any one of thousands of networked worlds. Such transitions, we know from our GPS trackers, do not involve a change of place; therefore they must involve a change of state, whose essence will be temporal rather than spatial, and thus also historical. The major historical change of the period covering the rise of network communications has been the rise of dataveillance, and its corollary, the real subsumption of consumption under capital. With the approaching exhaustion or higher risks associated with the extraction of natural resources, capital turns to the exploitation of human nature, notably through mapping behaviours as predictors of future activity, exploiting the creativity of interactors as unpaid sources of innovation, and personal debt. The derivatives market in debt producers the singular temporality of contemporary network capital and its subjectivity. This paper follows the histories that have produced this condition, including those of individuation, the construction of states of affairs as data, and the variety of terms frequently used as binary opposites of truth, each constitutive of a different kind of subject. This analysis prompts a definition of ideology as the intersection of the wishful and the paranoid, a position characterisable as subjunctive that should be taken as the typical form of network subjectivity as liminal. This in turn suggests two further hypotheses, that the category and the reality of the human has become environmental, that is treated as economic externality and as divorced from the core of the social; and that this alienated, subjunctive, mass subject is now in process of dissolving its links with the subjectivity as sovereignty, to replace it with the grounds for a new sociality. The challenge of the present conjuncture is therefore to construct a new 'we' capable of expressing the newly interdependent networks, not only of communications, but of ecological and technological imbrication of humans and non-humans, in a new politics

Migration of networks in multi-cloud environment

Tese de mestrado, Engenharia Informática (Arquitetura, Sistemas e Redes de Computadores) Universidade de Lisboa, Faculdade de Ciências, 2018A forma como os centros de dados e os recursos computacionais são geridos tem vindo a mudar. O uso exclusivo de servidores físicos e os complexos processos para provisionamento de software são já passado, sendo agora possível e simples usar recursos de uma terceira parte a pedido, na nuvem (cloud). A técnica central que permitiu esta evolução foi a virtualização, uma abstração dos recursos computacionais que torna o software mais independente do hardware em que é executado. Os avanços tecnológicos nesta área permitiram a migração de máquinas virtuais, agilizando ainda mais os processos de gestão e manutenção de recursos. A possibilidade de migrar máquinas virtuais libertou o software da infraestrutura física, facilitando uma série de tarefas como manutenção, balanceamento de carga, tratamento de faltas, entre outras. Hoje em dia a migração de máquinas virtuais é uma ferramenta essencial para gerir clouds, tanto públicas como privadas. Os sistemas informáticos de grande escala existentes na cloud são complexos, compostos por múltiplas partes que trabalham em conjunto para atingir os seus objectivos. O facto de os sistemas estarem intimamente ligados coloca pressão nos sistemas de comunicação e nas redes que os suportam. Esta dependência do sistema na infraestrutura de comunicação vem limitar a flexibilidade da migração de máquinas virtuais. Isto porque actualmente a gestão de uma rede é pouco flexível, limitando por exemplo a migração de VMs a uma subrede ou obrigando a um processo de reconfiguração de rede para a migração, um processo difícil, tipicamente manual e sujeito a falhas. Idealmente, a infraestrutura de que as máquinas virtuais necessitam para comunicar seria também virtual, permitindo migrar tanto as máquinas virtuais como a rede virtual. Abstrair os recursos de comunicação permitiria que todo o sistema tivesse a flexibilidade de ser transferido para outro local. Neste sentido foi recentemente proposta a migração de redes usando redes definidas por software (SDN), um novo paradigma que separa a infraestrutura de encaminhamento (plano de dados) do plano de controlo. Numa SDN a responsabilidade de tomar as decisões de controlo fica delegada num elemento logicamente centralizado, o controlador, que tem uma visão global da rede e do seu estado. Esta separação do plano de controlo do processo de encaminhamento veio facilitar a virtualização de redes. No entanto, as recentes propostas de virtualização de redes usando SDN apresentam limitações. Nomeadamente, estas soluções estão limitadas a um único centro de dados ou provedor de serviços. Esta dependência é um problema. Em primeiro lugar, confiar num único provedor ou cloud limita a disponibilidade, tornando efectivamente o provedor num ponto de falha único. Em segundo lugar, certos serviços ficam severamente limitados por recorrerem apenas a uma cloud, devido a requisitos especiais (de privacidade, por exemplo) ou mesmo legais (que podem obrigar a que, por exemplo, dados de utilizadores fiquem guardados no próprio país). Idealmente, seria possível ter a possibilidade de tirar partido de múltiplas clouds e poder, de forma transparente, aproveitar as vantagens de cada uma delas (por exemplo, umas por apresentarem custos mais reduzidos, outras pela sua localização). Tal possibilidade garantiria uma maior disponibilidade, visto que a falha de uma cloud não comprometeria todo o sistema. Além disso, poderia permitir baixar os custos porque seria possível aproveitar a variação dos preços existente entre clouds ao longo do tempo. Neste contexto multi-cloud um dos grandes desafios é conseguir migrar recursos entre clouds de forma a aproveitar os recursos existentes. Num ambiente SDN, em particular, a migração de redes é problemática porque é necessario que o controlador comunique com os elementos físicos da rede para implementar novas políticas e para que estes possam informar o controlador de novos eventos. Se a capacidade de comunicação entre o controlador e os elementos de rede for afectada (por exemplo, devido a latências elevadas de comunicação) o funcionamento da rede é também afectado. O trabalho que aqui propomos tem como objectivo desenvolver algoritmos de orquestração para migração de redes virtuais, com o objectivo de minimizar as latências na comunicação controlador-switches, em ambientes multi-cloud. Para esse efeito foi desenvolvida uma solução óptima, usando programação linear, e várias heurísticas. A solução de programação linear, sendo óptima, resulta na menor disrupção possível da ligação ao controlador. No entanto, a complexidade computacional desta solução limita a sua usabilidade, levando a tempos de execução elevados. Por esta razão são prospostas heurísticas que visam resolver o problema em tempo útil e de forma satisfatória. Os resultados das nossas experiências mostram que nas várias topologias testadas algumas heurísticas conseguem resultados próximos da solução óptima. O objectivo é atingido com tempos de execução consideravelmente inferiores.The way datacenters and computer resources are managed has been changing, from bare metal servers and complex deployment processes to on-demand cloud resources and applications. The main technology behind this evolution was virtualization. By abstracting the hardware, virtualization decoupled software from the hardware it runs on. Virtual machine (VM) migration further increased the flexibility of management and maintenance procedures. Tasks like maintenance, load balancing and fault handling were made easier. Today, the migration of virtual machines is a fundamental tool in public and private clouds. However as VMs rarely act alone, when the VMs migrate, the virtual networks should migrate too. Solutions to this problem using traditional networks have several limitations: they are integrated with the devices and are hard to manage. For these reasons the logical centralisation offered by Software-Defined Networking (SDN) architectures has been shown recently as an enabler for transparent migration of networks. In an SDN a controller remotely controls the network switches by installing flow rules that implement the policies defined by the network operator. Recent proposals are a good step forward but have problems. Namely, they are limited to a single data center or provider. The user’s dependency on a single cloud provider is a fundamental limitation. A large number of incidents involving accidental and malicious faults in cloud infrastructures show that relying on a single provider can lead to the creation of internet-scale single points of failures for cloud-based services. Furthermore, giving clients the power to choose how to use their cloud resources and the flexibility to easily change cloud providers is of great value, enabling clients to lower costs, tolerate cloud-wide outages and enhance security. The objective of this dissertation is therefore to design, implement and evaluate solutions for network migration in an environment of multiple clouds. The main goal is to schedule the migration of a network in such a way that the migration process has the least possible impact on the SDN controller’s ability to manage the network. This is achieved by creating a migration plan that aims to minimize the experienced control plane latency (i.e., the latency between the controller and the switches). We have developed an optimal solution based on a linear program, and several heuristics. Our results show that it is possible to achieve results close to the optimal solution, within reasonable time frames