The function of a network is generally related to the transport of items over the underlying graph. For example, in transportation networks, vehicles, trains, ships and aircraft deliver passengers and cargo to their destinations, while in telecommunication networks, IP packets are transmitted to enable data change between devices or users. To enhance the performance of item transmission in networks, such as reducing latency in telecommunication networks, we need to deepen the understanding of how a network supports efficient transport between two nodes and how transmission paths can be determined. Another important research topic concerns the design or construction of networks to meet users’ evolving demands, either by developing new networks from scratch or by modifying existing ones. In this dissertation, the analysis and design of networks for item transmission are considered in three distinct areas.

We first explore the item transmission in networks where the item travels along a single path, usually the shortest path in most practical scenarios. In Chapter 2, we focus on the geometric properties of shortest paths in soft random geometric graphs (SRGGs) in 2-dimensional Euclidean space. We demonstrate that shortest paths are aligned along geodesic curves connecting shortest path endpoints. To quantify the strength of the alignment, we introduce two metrics: the average distance to the geodesic from shortest path nodes and the average path stretch. Both metrics decrease in larger SRGGs with short-range connections. Additionally, we find that the alignment strength is nonmonotonic with respect to the average degree of the SRGG and identify network properties maximizing shortest path alignment. Based on these observations, we establish that the alignment of shortest paths may be sufficiently strong to allow the identification of shortest path nodes based on their proximity to geodesic curves. Extensive simulations confirm that our geometric-based approach predicts shortest path nodes even when node positions in the latent space are not precisely known.

The second part of this dissertation investigates inverse problems of the shortest paths. Given predetermined demands on the shortest path weight, such as upper bounds on delay between nodes in telecommunication networks, we focus on how to dimension the network, both topology and link weights, so that transport along the shortest path between any pair of nodes satisfies the given demands. Chapter 3 introduces the inverse all shortest path problem (IASPP), which asks for a weighted adjacency matrix of a graph such that all the elements in the corresponding shortest path weight matrix do not exceed those of the given demand matrix. We propose the Descending Order Recovery (DOR) algorithm, which exactly solves the IASPP. The network provided by DOR minimized the number of links and the sum of link weights across all networks with the same shortest path weight matrix. As a complement to DOR, a second algorithm, Omega based Link Removal (OLR), leverages effective resistance to solve the IASPP. Chapter 4 further extends IASPP to the modification of existing networks under changing transmission requirements. Inspired by practical scenarios, we study two extensions of IASPP. The first develops a new graph whose shortest path weight matrix equals a predetermined demand matrix while minimizing the total change in link weights from the original graph and an exact approach is proposed. The second variant considers a more constrained scenario in which the graph topology remains unchanged. We propose an algorithm, “distance basis sum” (DBS), for tree topologies commonly found in domains such as telecommunications and industrial networks. Compared with classical approaches, DBS can improve computational efficiency while constructing a graph close to the optimal solution.

In the third part of the dissertation, we investigate item transmission in networks where items propagate along multiple paths between two nodes. In Chapter 5, we study the expected number of nodes and links involved in transferring items between the source and destination in random graphs. Then, we investigate the power dissipation associated with transportation at link level. Further, we address how to construct a network given a predetermined end-to-end power dissipation, which reduces to the “inverse effective resistance problem” of finding a network whose effective resistance matrix matches a given demand. We propose a heuristic algorithm, “Resistor Gap Pruning” (RGP), which provides sparse networks closely approximating the demand effective resistance and performs consistently across different demand scenarios.

Community detection and graph partitioning have seamlessly integrated themselves into the fabric of network science by providing valuable insights into the structure, function, and dynamics of complex networks. In this thesis, a comprehensive performance comparison of the recently introduced Linear Clustering Process (LCP) is carried out against well-established clustering algorithms from literature. We evaluate its effectiveness using synthetic benchmarks commonly employed in the field, as well as real-world networks with both known and unknown community structures. Through our analysis, we reveal that the Linear Clustering Process consistently yields superior community partitions with optimized modularity when the clusters are well-defined compared to the majority of the assessed algorithms. Meanwhile, remarkably, this improved performance is achieved while maintaining computational complexity comparable to the simplest existing clustering algorithms. Furthermore, this thesis also provides an empirical approach for enhancing the performance of a variant of Linear Clustering Process on power-law networks.

The basic reproductive number (R0) of an infectious disease is the expected number of secondary cases infected by the first case in an otherwise susceptible population. In this thesis, based on this fundamental concept, we propose a new method: the IIN-TN method that could calculate R0 of a network from the local structure of the network under the scenario of the SIS model. The analytical solution of this method is given in this thesis, and simulations are performed to verify the result. After the code implementation in MATLAB, the method is applied to a human contact network in reality. The results from the real-world network show that the R0 calculated by the IIN-TN method will be smaller than the value given by the traditional definition. Apart from that, we also find that the IIN-TN method is more effective for infectious diseases, which have a relatively larger infection probability of link and a higher effective infection rate.

Since the first introduction of 4G in the early 2010s, mobile data traffic has increased significantly, mainly because 4G networks can support more devices, services and applications, as well as greater cellular network coverage. With the advent of the 5G era, the mobile communication system will be further developed. Today, most countries and regions are already implementing their own 5G plans. Compared with China, South Korea and the United States, the three leading countries in the 5G market, the EU’s 5G plans are relatively slow.

The focus of this thesis is to explore various 5G deployment and migration options, as well as the current 5G status in different countries and regions. We start with current network technologies, compare and analyze various 5G deployment and migration options. Then we compare the 5G status of the EU with China, South Korea and the United States, and analyze the internal and external factors faced by the EU in the development of 5G. In these two analysis, we will consider several communication services that are currently used in the mobile network, including, but not limited to, voice/video calling and messaging, for both home usage and roaming usage. Finally, based on the above two analysis, suggestions for EU operators in 5G deployment and development are provided, along with recommendations for future research.

The inverse shortest path problem (ISPP) is a problem based on graph theory, that is to design link weights in a graph to satisfy that given paths are the shortest between the corresponding node pairs. It can be used in networks of complex systems to solve practical problems such as re-routing in transportation systems and reallocating resources in IP networks. This thesis proposes two new methods based on the simplex algorithm, the split path method (Sp) and the limit constraints method (Lc), to solve the inverse shortest path problem with one predefined path (ISPP-S). The different approaches used to find the constraints in these two methods affect the obtained optimal solution and running time. By analyzing the simulation results (running time, adjustment of link weight as well as the relationship between path length and running time) of these two algorithms in graphs with various structures, we can evaluate the efficiency of Sp and Lc and summarize their applicable networks. After modification, these two methods can be utilized to solve the extended inverse shortest path problem with multiple target paths (ISPP-M) and with target paths in a spanning tree (ISPP-T). In addition, the applicability of these two algorithms is also extended from directed graphs to undirected graphs. The application of Sp and Lc algorithms in a variety of empirical networks is also discussed. Through the analysis of the above problems and the experimental results, we discover that Sp is more suitable to solve ISPP with relatively few constraints (ISPP-S or ISPP in small-sized networks); Lc performs better when solving ISPP with relatively more constraints (ISPP-M or ISPP in large-sized networks).

With the proliferating networks, resource allocation based on Quality of Service (QoS) constraints mapping has been one of the difficulties faced by Internet Service Providers (ISP). The advent of new technologies such as virtualization and cloud computing have enabled the users to access content from anywhere around the world. This results in a need for an efficient and fast resource allocation method based on demand acting on the network. Many researchers have developed various algorithms to address this issue. However, they have considered only flow-based network properties, while the contemporary networks communicate mostly through path-based communication using the shortest paths. Inverse shortest path algorithm (ISPA) is a graph-theory based heuristic developed to solve this network resource allocation problem which considers both flow-based communication properties through Effective resistance matrix and path-based communication properties through shortest path matrix. Nevertheless, ISPA is so far implemented only for unweighted graphs. This study aims at assessing the feasibility of ISPA for weighted graphs by understanding the nature of Inverse shortest path problem (ISPP) bounds in weighted random graphs and implementing ISPA for weighted graphs. ISPP bounds behaviour is studied for weighted graphs by analyses of Q-norm distributions, probability of failure distributions and hopcount distributions. A feasibility condition verifying ISPP bounds is derived in relation with the input parameters of the weighted random graph. The solutions obtained through hop count distribution analysis are observed to be Poissonian in nature. Finally, ISPA is implemented for weighted graphs such that demand matrix can be resolved into a simple distance matrix to obtain solution which is a non-negative weighted Adjacency matrix and feasibility conditions are verified for the solutions obtained through ISPA.

A major factor in the spreading of viruses is human-to-human transmission, and human mobility is clearly linked to the spreading process of epidemics. If we hope to understand the evolution of an epidemic, then we must also understand the underlying mobility process and the interaction between the two.

We propose the Markov modulated process (MMP) model as a tool for modeling mobility processes. We take a link-based approach to the MMP model, where modulating actions are a joint combination of adding and removing a number of links. We demonstrate that for such an approach, the states of the Markov process must encode not only a modulating action, but also the current number of links in the graph. We further show that in this case, the number of states will increase with O(N⁶) where N is the number of nodes. We refer to this as the "unaggregated" MMP model, and we introduce the concept of state aggregation to create the "aggregated" MMP model which only requires O(N²) states.

We demonstrate that both the aggregated and unaggregated MMP models are able to match the average number of links in the simulated mobility process, capturing the long-term dynamics of the mobility process with high accuracy. Both the aggregated and unaggregated MMP models are also able to match the average number of links added and removed between two time steps, capturing the short-term dynamics with high accuracy. Finally, we demonstrate that for certain mobility processes, the aggregated MMP model is able to produce results which are indistinguishable from the unaggregated MMP model while requiring just O(N²) states compared to O(N⁶).

This thesis consists of two parts which are connected by the central theme of epidemics. In the first part, a website is designed for forecasting the number of cases of COVID-19 in the Netherlands. The forecasting is performed using the Network-Inference-Based Prediction Algorithm (NIPA). The first part of the thesis documents the design process and some of the challenges faced along the way. An explanation of the NIPA algorithm is presented, and its implementation on the website is discussed.

The second part of this thesis shifts the focus to the modeling of mobility processes. Human-to-human transmission is often the dominant viral transmission vector, and therefore the spreading of the virus is inextricably linked to human mobility. The Markov-modulated process (MMP) model has recently been proposed as a novel method for modeling mobility processes. Yang has developed a so-called "aggregated MMP model" which is able to capture several key dynamics of mobility processes with high accuracy. This research builds on Yang's results and proposes the "quantized MMP model" as an extension of Yang's aggregated MMP model. Compared to the aggregated MMP model, the quantized MMP model reduces the number of required states by a factor equal to the quantization step size q, improving the efficiency of the model.

The behavior of the quantized MMP model for different step sizes q is compared with the aggregated MMP model. The results show that the quantization error has zero mean, allowing the quantized MMP model to achieve similarly high model accuracy. The SIR dynamics of the quantized MMP model for different step sizes are tested extensively and the disease spreading performance is compared with the mobility process. The suitability of the MMP model for forecasting the evolution of epidemics is evaluated and discussed.

The spreading process of diseases has been an important research topic for many years. It has profound effects on the development of human social behaviors. The underlying social network structure may change when individuals change their connection with other individuals in response to the epidemic. The classic susceptible-infected-susceptible (SIS) model is used to model the spread of an epidemic on a network, where all individuals are defined as nodes and the connections between the individuals are regarded as links. Besides the typical static network, the structure of the network can be related to the state of nodes (infected or susceptible) by link breaking and link creation processes. So the extended network, adaptive susceptible-infected-susceptible model (ASIS model) can be derived.

To study the spreading process on static networks and adaptive networks, we use stochas- tic simulations and mean field approximations. We assume that the spreading process over the network is a continuous-time discrete-state Markov process. But most recent works use the discrete-time simulator, which is actually an approximation of the process. In this report, we extend a existing continuous-time simulator towards adaptive networks. This existing simulator is based on the Gillespie algorithm. We perform the simulations using both discrete-time Markov chain and continuous-time Markov process. And based on the simulation results, we demonstrate that the continuous-time simulator has a better performance than the discrete-time simulator on modeling both static SIS network and ASIS network with high accuracy.

The second part of this work aims to study the characteristics of the ASIS network in the metastable state. We observe three possible states of the ASIS network: the endemic state, disease-free state and bistable state. The degree distribution of the graph follows a binomial distribution in some cases. By plenty of simulations with different parameters, we illustrate under what circumstances the degree distribution follows the binomial distribution.

The field of epidemiology encompasses a broad class of spreading phenomena, ranging from the seasonal influenza and the dissemination of fake news on online social media to the spread of neural activity over a synaptic network. The propagation of viruses, fake news and neural activity relies on the contact between individuals, social media accounts and brain regions, respectively. The contact patterns of the whole population result in a network. Due to the complexity of such contact networks, the understanding of epidemics is still unsatisfactory. In this dissertation, we advance the theory of epidemics and its applications, with a particular emphasis on the impact of the contact network. Our first contribution focusses on the analysis of the N-Intertwined Mean-Field Approximation (NIMFA) of the Susceptible-Infected-Susceptible (SIS) epidemic process on networks. We propose a geometric approach to clustering for epidemics on networks, which reduces the number of NIMFA differential equations from the network size N to the number m << N of clusters (Chapter 2). Specifically, we show that exact clustering is possible if and only if the contact network has an equitable partition, and we propose an approximate clustering method for arbitrary networks. Furthermore, for arbitrary contact networks, we derive the closed-form solution of the nonlinear NIMFA differential equations around the epidemic threshold (Chapter 3). Our solution reveals that the topology of the contact network is practically irrelevant for the epidemic outbreak around the epidemic threshold. Lastly, we study a discrete-time version of the NIMFA epidemic model (Chapter 4). We derive that the viral state is (almost always) monotonically increasing, the steady state is exponentially stable, and the viral dynamics is bounded by linear time-invariant systems. In the second part, we consider the reconstruction of the contact network and the prediction of epidemic outbreaks. We show that, for the stochastic SIS epidemic process on an individual level, the exact reconstruction of the contact network is impractical. Specifically, the maximum-likelihood SIS network reconstruction is NP-hard, and an accurate reconstruction requires a tremendous number of observations of the epidemic outbreak (Chapter 5). For epidemic models between groups of individuals, we argue that, in the presence of model errors, accurate long-term predictions of epidemic outbreaks are not possible, due to a severely ill-conditioned problem (Chapter 6). Nonetheless, short-term forecasts of epidemics are valuable, and we propose a prediction method which is applicable to a plethora of epidemic models on networks (Chapter 7). As an intermediate step, our prediction method infers the contact network from observations of the epidemic outbreak. Our key result is paradoxical: even though an accurate network reconstruction is impossible, the epidemic outbreak can be predicted accurately. Lastly, we apply our network-inference-based prediction method to the outbreak of COVID-19 (Chapter 8). The third part focusses on spreading phenomena in the human brain. We study the relation between two prominent methods for relating structure and function in the brain: the eigenmode approach and the series expansion approach (Chapter 9). More specifically, we derive closed-form expressions for the optimal coefficients of both approaches, and we demonstrate that the eigenmode approach is preferable to the series expansion approach. Furthermore, we study cross-frequency coupling in magnetoencephalography (MEG) brain networks (Chapter 10). By employing a multilayer network reconstruction method, we show that there are strong one-to-one interactions between the alpha and beta band, and the theta and gamma band. Furthermore, our results show that there are many cross-frequency connections between distant brain regions for theta-gamma coupling.

A synchromodal transport network is a transportation technique which aims to create more efficient and sustainable transportation plans, which utilizes different modes of transport, i.e., roadways, railways, and waterways synchronously. Synchromodal transport network is a complex network with interdependence. The aim of this thesis was to analyze the robustness and identify critical infrastructure in synchromodal transport network. In order to achieve the goal of the thesis. First, the synchromodal transport network is analyzed on the basis of its transportation characteristics and network topology. The analysis of transportation characteristics helped us to define a notion of node criticality. The node criticality quantifies the effect on the robustness of a transportation system due to the perturbation of an infrastructure. Then, a relationship is established between the node criticality and topological centrality metrics in order to quickly identify critical infrastructures in the synchromodal transport network. Lastly, a systematic framework is proposed and applied to the Dutch synchromodal transport network. The results from the case study are quite insightful. First, we observe that the distribution of node criticality exhibits a power-law distribution which implies that the Dutch synchromodal transport network tends to robustness against node perturbation. Second, we observe that the performance of the dutch synchromodal transport network is quite sensitive to the perturbation in the road network. Lastly, we identify that the weighted degree centrality, eigenvector centrality shows a high correlation with node criticality thus these centrality metrics can be used to identify critical infrastructure in the synchromodal transport network.

Susceptible-Infected-Susceptible (SIS) model is commonly used to describe the spreading of virus on networks. However, a real-life epidemic process is not necessarily Markovian. The spreading of diseases, behaviors and information in real systems are sometimes dependent on the characteristics and current status of individuals. Thus it is far from enough to just consider Markovian processes. We need to consider a more general model with non-Markovian processes. Although some recent works focus on the SIS model with a non-Markovian infection process, systematic research on the non-Markovian curing process is still lacking. Therefore, this thesis project is to study the influence of the non-Markovian curing process on the performance of SIS viral spreading on networks. Through continuous-time SIS epidemics simulator, we find some dramatic effects of a non-exponential curing time (while still assuming an exponential infection time) on the prevalence and critical point of effective infection rate by considering Weibullean curing times with same mean, but different shape parameter α. For α ∈ [0.2, 10], the epidemic threshold satisfies τ_c = 1/λ₁, which is the same as the NIMFA conclusions of Markovian SIS process. Relatively, when α is too small, a large number of curing events synchronously happened at the beginning of the simulation, which will lead to collective deaths on finite network. The effect on initial condition of nodes further cause a decline on prevalence and an slow phase transition between healthy state and the meta-stable state. Furthermore, the heavy-tailed distribution of curing time leads to a small percent of nodes still surviving at the meta-stable state, even under a very low effective infection rate. The heavy-tailed distribution gives some nodes an extreme long curing time and thus can infect other nodes with a pretty small probability, thereby maintaining the virus' long-term spread in a small group of nodes. This spreading mode seems can explain some virus spreading phenomenon, like the spreading mode of hepatitis B virus (HBV). Additionally, when the shape parameter α of Weibull distribution is pretty large, the distribution of curing time is like a pulse or a Dirac delta function ( δ function), thus a huge amount of nodes can get synchronously recovered. We find when we control the successful curing probability p =1-1/e ≈0.632, the prevalence of pulse curing at the meta-stable state is equivalent to a Poisson curing process. Therefore, the pulse curing strategy can suppress the spreading of viruses and further save medical resources.

Spreading processes are ubiquitous in nature and societies, e.g. spreading of diseases and computer virus, propagation of messages, and activation of neurons. Computer viruses cause an enormous economic loss. Moreover, many illnesses/diseases still causing a serious threat to public health. For example, the outbreaks of circulating influenza strains cause millions of illness and deaths worldwide every year. Pronounced outbreaks of flu usually occur during winter. This recognized timing allows public health agencies to organize their flu-related mitigation and response activities to prepare for the winter flu season. Although the general wintertime peak of influenza incidence in temperate regions can be easily forecast, the specific intensity, duration, and time of individual local outbreaks are quite changeable. Even after an outbreak has begun, it is still difficult to predict the future characteristics of the epidemic curve. If the diseases/viruses outbreak characteristics could be reliably predicted, the public health response will be better coordinated.The goal is to develop a fast and accurate epidemic model to estimate, fit and forecast the spreading of an epidemic on a defined network. The aim is to conduct a study over viruses spreading phenomena both theoretically and numerically, then create a general model/algorithm that can be easily applied to different diseases and computer viruses. In this master thesis, we propose a new approach which can be used on real illness/viruses data (such as influenza) to estimate and forecast the epidemic more accurately. The approach is to use a model-inference system combining the network science, susceptible-infected-recovered-susceptible (SIRS) model, statistical filtering techniques and gradient descent. We are able to fit and estimate with a relatively low error compared to other algorithms. Moreover, we forecast the out-breaker with a high accuracy, four weeks before the true out-breaker on synthetic epidemic data. The model is evaluated on a regular graph, Erdös-Rényi graph, Watts-Strogatz small-world graph, & Barabási-Albert graph. Furthermore, the model is carried out on real-world epidemic data (influenza data) for four countries (the Netherlands, Germany, Belgium and the United Kingdom), from the years 2012 to 2017.

Epidemic models are applied to describe epidemic processes such as the spreading of infectious viruses, opinions and fake news on real-life or online
social networks, and to analyse the epidemic processes mathmatically. The viral state evolution is closely related to the underlying network topology. Therefore, the network topology is of vital importance to describing the viral state of each individual in a network. This master thesis focuses on the network reconstruction problem of the NIMFA approximation of the Susceptible-Infected-Susceptible (SIS) epidemic process. Given the viral state series generated by the NIMFA epidemic process, we aim to estimate the adjacency matrix A of the underlying network given that the spreading parameters are known. In this thesis, we estimate the adjacency matrix of the network from the viral states by a constrained linear least-squares formulation. Our algorithm gives an accurate estimate of the adjacency matrix provided that suciently many epidemic outbreaks are observed.

Freight transport is an essential component of the economy, as among other things it ensures the availability of finished goods to consumers. Synchromodal transport is a new transport method that aims to use real-time and flexible switching among different modes of transport according to the latest logistics information, to utilise the different modes of transport efficiently. But what are the effects of disruptions of the infrastructure used by synchromodal transport on freight transport? As some recent disruptions in the Netherlands have shown, the negative effects can be considerable. The effects of disruptions on the functioning of multimodal transport networks have scarcely been researched. This research gap is addressed in this thesis, by analysing the robustness of the Dutch synchromodal freight network, comprising of the inland waterway, road, railway and container terminal infrastructure.

First an overview is given of the elements of the synchromodal transport infrastructure. Apart from the infrastructure elements of each mode of transport, two important characteristics are identified: the interconnection and the interdependence. Container terminals are the interconnection between different modes of transport, as they facilitate the transshipment of containers. Infrastructure elements whose functioning influences multiple modes of transport (e.g. bridges) are the interdependence between different modes of transport. Subsequently, this overview of infrastructure elements is used to make a macroscopic graph model in which all relevant infrastructure elements are represented. Using the random removal of graph elements, the robustness of the synchromodal network is analysed. Three case studies are used to study the Dutch synchromodal network: the corridor between Rotterdam and Antwerp, the corridor Rotterdam and Duisburg, and the domestic freight transport in the Netherlands.

The robustness analysis of the three case studies lead to the following conclusions. Of the three modes of transport, the modality road is the most robust. The inland waterway modality is less robust and the rail modality is the least robust. It should be noted that the rail modality provides a relatively cheap alternative and offers the capabilities to transport containers further into Europe than the inland waterways. The ability to transship containers between different modes of transport (i.e. interconnection) has a significant positive effect on the robustness. The interdependence has a smaller negative effect on the robustness. Generally, the new alternative paths offered by the addition of a mode of transport (even without interconnection) outweigh the negative effects of the interdependence.

These findings make it clear that the use of the synchromodal transport method, increases the robustness of the freight transport network. During this research, many possibilities for future research were identified. As the research gap for analysing the robustness of a synchromodal transport network was very large at the start of this research, many things can still be researched after this research. The importance of freight transport warrants additional research on this topic. According to this research, due to mode-free booking and the real-time and flexible switching among different modalities, synchromodal transport offers a robust freight transport method.

This master thesis focuses on the particular problem about the cut-size property of different networks. The scope of networks is from trivial network models (e.g., random graph) to real networks (e.g., Power grid network and Facebook network). The Susceptible-Infected-Susceptible (SIS) epidemic model can describe the spreading processes of information or diseases on networks. Within this thesis, we explore the cut-size property of networks and seek the relations between the cut-size property and the spreading behaviors. Our result deduces the cut-size property and the relevant physical meanings of the real networks. In such a way, a deeper understanding of the cut-size would help researchers to obtain insights of real networks. Our results may also contribute to the study of control of dynamic processes on networks.

As new technologies continue to find their way into everyday life, the world becomes more and more connected. Airplanes and other means of transportation provide global connections in the physical world, while the omnipresence of the Internet means that information is shared around the globe, easier than ever before. But not only these man-made systems are
distinctly connected, other complex systems like the human brain, or metabolic networks are successfully being studied from the perspective of their constituting connections. The combining concept in all these examples is the structure of the problem at hand: each system consists of interacting elementary components at the lowest level, from which a network structure emerges at the global level. The study of such networked systems, their observed features and the wide range of related analysis tools is commonly referred to as Network
Science.

In this thesis, the specific problem of how diseases spread over networks is addressed. Better understanding this spreading behavior has significant practical importance, i.e. for the prediction and control of disease prevalence, and poses many interesting theoretical challenges. In the context of modeling epidemics on networks, we formulate the Universal Mean-Field Framework. This new and theoretically well-founded framework unifies and generalizes a number of existing approximate models, and brings forth new approaches to bound the approximations. Apart from the work on epidemics, some new insights are explored in the context of the connections between electrical circuits, networks and simplices (higher-dimensional triangles). These deep theoretical equivalences allow the tools and intuitions from electrical circuits and geometry to be used in the study of networks. A comprehensive introduction and discussion of the equivalent representations and their connections is given. Additionally, we derive a new formula for the volume of a hyperacute simplex and propose to use this volume as a network-robustness measure.

Over the last two decades the field of network science has been evolving fast. Many useful applications in a wide variety of disciplines have been found. The application of network science to the brain initiated the interdisciplinary field of complex brain networks. On a macroscopic level, brain regions are taken as nodes in a network. The analysis of pairwise connections between the brain regions as links has provided a new perspective on many problems. The application of network science to neuroscience data helped, for example, to identify the disruptions due to different neurological disorders when comparing healthy and abnormal brain networks. In this dissertation, we focus on the macroscopic level of brain regions and analyze their pairwise connections from a network science perspective. We address different general research questions from network science and exploit their application possibilities towards brain networks. Due to different measurement techniques, one can construct many different representations of brain networks. We thereby distinguish between the structural and functional brain network. Structural brain networks map the anatomical connections between the regions, which we could interpret as the ’streets’ of the brain. On top of these streets, we can measure the traffic with techniques like e.g. magnetoencephalography (MEG) or functional Magnetic Resonance Imaging (fMRI) resulting in so-called functional brain networks. However, the relation between the structural and the functional brain networks is still insufficiently understood. The first main research question of this dissertation focuses on the functional network layer and tries to identify the most important links and motifs of these networks. For this purpose, we propose the union of shortest path trees (USPT) as a new sampling method extracting all the shortest paths of a network (Chapter 2 and 3). After constructing the USPT, we compare the individual functional brain networks of multiple sclerosis patients and healthy controls (Chapter 2). Furthermore, we generalize this sampling method and present a new ranking of all the links based on the USPT (Chapter 3). Regarding the higher-order building blocks of the functional brain networks, we analyze the so-called information flow motifs based on MEG data from different frequency bands (Chapter 4). After researching the local properties of the functional brain networks, we analyze the influence of the underlying structural connections on the emerging information flow. Thus, the second main research question concerns the relationship between the functional and the underlying structural connectivity. Specifically, we analyze which topological properties of the structural networks drive the functional interactions. First, this question is approached in a mathematical and straightforward manner by assuming that an analytic function between the two networks exists (Chapter 5). We investigate this mapping function and its reverse by evaluating empirical individual and group-averaged multimodal data sets. A second approach towards the structure-function relationship employs a simple model of activity spread. The epidemic spreading model is applied on the human connectome to investigate the global patterns of directional information flow in brain networks (Chapter 6). The main focus here lies on the pairwise measure of transfer entropy to investigate the influence of one brain region on another. We present the results for the local and global outcomes of the dynamic spreading process aiming to identify the driving structural properties behind the observed global patterns.

The Internet consists of many network elements that direct packets on the correct path leading towards the destination. This process of finding and following a path to the destination is called routing. Routing is not infallible and packets may get lost: the current Internet cannot give any quality guarantees regarding the packets it transports. However, many new multi-media applications (e.g., VoIP) cannot properly operate without such guarantees. Finding paths that can meet such demands is called Quality of Service (QoS) routing. This thesis identifies several algorithmic concepts of QoS routing, which are all incorporated into our exact SAMCRA algorithm. The first large-scale performance evaluation of QoS algorithms indicates that the SAMCRA algorithm performs best. Besides SAMCRA, also algorithms for multicast QoS routing and link-disjoint QoS routing are proposed in this thesis. QoS routing is NP-complete, which means that to find the exact solution, algorithms require, in the worst case, a running time that cannot be bounded by a polynomial function. This thesis also analyzes the complexity of QoS routing and argues that it is feasible in practice. Hence, exact algorithms like SAMCRA should be used instead of heuristics. Finally, the dynamics of QoS routing are discussed and some preliminary work in this area is provided. Here, too, SAMCRA outperformed the other implemented algorithms.