Multifractal characterisation and analysis of complex networks

Complex networks have been studied extensively due to their relevance to many real-world systems such as the world-wide web, the internet, biological and social systems. During the past two decades, studies of such networks in different fields have produced many significant results concerning their structures, topological properties, and dynamics. Three well-known properties of complex networks are scale-free degree distribution, small-world effect and self-similarity. The search for additional meaningful properties and the relationships among these properties is an active area of current research. This thesis investigates a newer aspect of complex networks, namely their multifractality, which is an extension of the concept of selfsimilarity. The first part of the thesis aims to confirm that the study of properties of complex networks can be expanded to a wider field including more complex weighted networks. Those real networks that have been shown to possess the self-similarity property in the existing literature are all unweighted networks. We use the proteinprotein interaction (PPI) networks as a key example to show that their weighted networks inherit the self-similarity from the original unweighted networks. Firstly, we confirm that the random sequential box-covering algorithm is an effective tool to compute the fractal dimension of complex networks. This is demonstrated on the Homo sapiens and E. coli PPI networks as well as their skeletons. Our results verify that the fractal dimension of the skeleton is smaller than that of the original network due to the shortest distance between nodes is larger in the skeleton, hence for a fixed box-size more boxes will be needed to cover the skeleton. Then we adopt the iterative scoring method to generate weighted PPI networks of five species, namely Homo sapiens, E. coli, yeast, C. elegans and Arabidopsis Thaliana. By using the random sequential box-covering algorithm, we calculate the fractal dimensions for both the original unweighted PPI networks and the generated weighted networks. The results show that self-similarity is still present in generated weighted PPI networks. This implication will be useful for our treatment of the networks in the third part of the thesis. The second part of the thesis aims to explore the multifractal behavior of different complex networks. Fractals such as the Cantor set, the Koch curve and the Sierspinski gasket are homogeneous since these fractals consist of a geometrical figure which repeats on an ever-reduced scale. Fractal analysis is a useful method for their study. However, real-world fractals are not homogeneous; there is rarely an identical motif repeated on all scales. Their singularity may vary on different subsets; implying that these objects are multifractal. Multifractal analysis is a useful way to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns. However, the tools for multifractal analysis of objects in Euclidean space are not suitable for complex networks. In this thesis, we propose a new box covering algorithm for multifractal analysis of complex networks. This algorithm is demonstrated in the computation of the generalized fractal dimensions of some theoretical networks, namely scale-free networks, small-world networks, random networks, and a kind of real networks, namely PPI networks of different species. Our main finding is the existence of multifractality in scale-free networks and PPI networks, while the multifractal behaviour is not confirmed for small-world networks and random networks. As another application, we generate gene interactions networks for patients and healthy people using the correlation coefficients between microarrays of different genes. Our results confirm the existence of multifractality in gene interactions networks. This multifractal analysis then provides a potentially useful tool for gene clustering and identification. The third part of the thesis aims to investigate the topological properties of networks constructed from time series. Characterizing complicated dynamics from time series is a fundamental problem of continuing interest in a wide variety of fields. Recent works indicate that complex network theory can be a powerful tool to analyse time series. Many existing methods for transforming time series into complex networks share a common feature: they define the connectivity of a complex network by the mutual proximity of different parts (e.g., individual states, state vectors, or cycles) of a single trajectory. In this thesis, we propose a new method to construct networks of time series: we define nodes by vectors of a certain length in the time series, and weight of edges between any two nodes by the Euclidean distance between the corresponding two vectors. We apply this method to build networks for fractional Brownian motions, whose long-range dependence is characterised by their Hurst exponent. We verify the validity of this method by showing that time series with stronger correlation, hence larger Hurst exponent, tend to have smaller fractal dimension, hence smoother sample paths. We then construct networks via the technique of horizontal visibility graph (HVG), which has been widely used recently. We confirm a known linear relationship between the Hurst exponent of fractional Brownian motion and the fractal dimension of the corresponding HVG network. In the first application, we apply our newly developed box-covering algorithm to calculate the generalized fractal dimensions of the HVG networks of fractional Brownian motions as well as those for binomial cascades and five bacterial genomes. The results confirm the monoscaling of fractional Brownian motion and the multifractality of the rest. As an additional application, we discuss the resilience of networks constructed from time series via two different approaches: visibility graph and horizontal visibility graph. Our finding is that the degree distribution of VG networks of fractional Brownian motions is scale-free (i.e., having a power law) meaning that one needs to destroy a large percentage of nodes before the network collapses into isolated parts; while for HVG networks of fractional Brownian motions, the degree distribution has exponential tails, implying that HVG networks would not survive the same kind of attack.

Statistical modelling of wind effects on signal propagation for wireless sensor networks

A wireless sensor network system must have the ability to tolerate harsh environmental conditions and reduce communication failures. In a typical outdoor situation, the presence of wind can introduce movement in the foliage. This motion of vegetation structures causes large and rapid signal fading in the communication link and must be accounted for when deploying a wireless sensor network system in such conditions. This thesis examines the fading characteristics experienced by wireless sensor nodes due to the effect of varying wind speed in a foliage obstructed transmission path. It presents extensive measurement campaigns at two locations with the approach of a typical wireless sensor networks configuration. The significance of this research lies in the varied approaches of its different experiments, involving a variety of vegetation types, scenarios and the use of different polarisations (vertical and horizontal). Non–line of sight (NLoS) scenario conditions investigate the wind effect based on different vegetation densities including that of the Acacia tree, Dogbane tree and tall grass. Whereas the line of sight (LoS) scenario investigates the effect of wind when the grass is swaying and affecting the ground-reflected component of the signal. Vegetation type and scenarios are envisaged to simulate real life working conditions of wireless sensor network systems in outdoor foliated environments. The results from the measurements are presented in statistical models involving first and second order statistics. We found that in most of the cases, the fading amplitude could be approximated by both Lognormal and Nakagami distribution, whose m parameter was found to depend on received power fluctuations. Lognormal distribution is known as the result of slow fading characteristics due to shadowing. This study concludes that fading caused by variations in received power due to wind in wireless sensor networks systems are found to be insignificant. There is no notable difference in Nakagami m values for low, calm, and windy wind speed categories. It is also shown in the second order analysis, the duration of the deep fades are very short, 0.1 second for 10 dB attenuation below RMS level for vertical polarization and 0.01 second for 10 dB attenuation below RMS level for horizontal polarization. Another key finding is that the received signal strength for horizontal polarisation demonstrates more than 3 dB better performances than the vertical polarisation for LoS and near LoS (thin vegetation) conditions and up to 10 dB better for denser vegetation conditions.

Context-aware multicast protocol for emergency message dissemination in vehicular networks

Road traffic accidents can be reduced by providing early warning to drivers through wireless ad hoc networks. When a vehicle detects an event that may lead to an imminent accident, the vehicle disseminates emergency messages to alert other vehicles that may be endangered by the accident. In many existing broadcast-based dissemination schemes, emergency messages may be sent to a large number of vehicles in the area and can be propagated to only one direction. This paper presents a more efficient context aware multicast protocol that disseminates messages only to endangered vehicles that may be affected by the emergency event. The endangered vehicles can be identified by calculating the interaction among vehicles based on their motion properties. To ensure fast delivery, the dissemination follows a routing path obtained by computing a minimum delay tree. The multicast protocol uses a generalized approach that can support any arbitrary road topology. The performance of the multicast protocol is compared with existing broadcast protocols by simulating chain collision accidents on a typical highway. Simulation results show that the multicast protocol outperforms the other protocols in terms of reliability, efficiency, and latency.

Langevin dynamics modeling of the water diffusion tensor in partially aligned collagen networks

In this work, a Langevin dynamics model of the diffusion of water in articular cartilage was developed. Numerical simulations of the translational dynamics of water molecules and their interaction with collagen fibers were used to study the quantitative relationship between the organization of the collagen fiber network and the diffusion tensor of water in model cartilage. Langevin dynamics was used to simulate water diffusion in both ordered and partially disordered cartilage models. In addition, an analytical approach was developed to estimate the diffusion tensor for a network comprising a given distribution of fiber orientations. The key findings are that (1) an approximately linear relationship was observed between collagen volume fraction and the fractional anisotropy of the diffusion tensor in fiber networks of a given degree of alignment, (2) for any given fiber volume fraction, fractional anisotropy follows a fiber alignment dependency similar to the square of the second Legendre polynomial of cos(θ), with the minimum anisotropy occurring at approximately the magic angle (θMA), and (3) a decrease in the principal eigenvalue and an increase in the transverse eigenvalues is observed as the fiber orientation angle θ progresses from 0◦ to 90◦. The corresponding diffusion ellipsoids are prolate for θ < θMA, spherical for θ ≈ θMA, and oblate for θ > θMA. Expansion of the model to include discrimination between the combined effects of alignment disorder and collagen fiber volume fraction on the diffusion tensor is discussed.

Establishing initial trust in autonomous delay tolerant networks without centralised PKI

A Delay Tolerant Network (DTN) is one where nodes can be highly mobile, with long message delay times forming dynamic and fragmented networks. Traditional centralised network security is difficult to implement in such a network, therefore distributed security solutions are more desirable in DTN implementations. Establishing effective trust in distributed systems with no centralised Public Key Infrastructure (PKI) such as the Pretty Good Privacy (PGP) scheme usually requires human intervention. Our aim is to build and compare different de- centralised trust systems for implementation in autonomous DTN systems. In this paper, we utilise a key distribution model based on the Web of Trust principle, and employ a simple leverage of common friends trust system to establish initial trust in autonomous DTN’s. We compare this system with two other methods of autonomously establishing initial trust by introducing a malicious node and measuring the distribution of malicious and fake keys. Our results show that the new trust system not only mitigates the distribution of fake malicious keys by 40% at the end of the simulation, but it also improved key distribution between nodes. This paper contributes a comparison of three de-centralised trust systems that can be employed in autonomous DTN systems.

Key Management in Wireless Sensor Networks

This chapter presents a comparative survey of recent key management (key distribution, discovery, establishment and update) solutions for wireless sensor networks. We consider both distributed and hierarchical sensor network architectures where unicast, multicast and broadcast types of communication take place. Probabilistic, deterministic and hybrid key management solutions are presented, and we determine a set of metrics to quantify their security properties and resource usage such as processing, storage and communication overheads. We provide a taxonomy of solutions, and identify trade-offs in these schemes to conclude that there is no one-size-fits-all solution.

Complexity of increasing the secure connectivity in wireless ad hoc networks

We consider the problem of maximizing the secure connectivity in wireless ad hoc networks, and analyze complexity of the post-deployment key establishment process constrained by physical layer properties such as connectivity, energy consumption and interference. Two approaches, based on graph augmentation problems with nonlinear edge costs, are formulated. The first one is based on establishing a secret key using only the links that are already secured by shared keys. This problem is in NP-hard and does not accept polynomial time approximation scheme PTAS since minimum cutsets to be augmented do not admit constant costs. The second one extends the first problem by increasing the power level between a pair of nodes that has a secret key to enable them physically connect. This problem can be formulated as the optimal key establishment problem with interference constraints with bi-objectives: (i) maximizing the concurrent key establishment flow, (ii) minimizing the cost. We prove that both problems are NP-hard and MAX-SNP with a reduction to MAX3SAT problem.

Improving power quality in low-voltage networks containing distributed energy resources

Severe power quality problems can arise when a large number of single-phase distributed energy resources (DERs) are connected to a low-voltage power distribution system. Due to the random location and size of DERs, it may so happen that a particular phase generates excess power than its load demand. In such an event, the excess power will be fed back to the distribution substation and will eventually find its way to the transmission network, causing undesirable voltage-current unbalance. As a solution to this problem, the article proposes the use of a distribution static compensator (DSTATCOM), which regulates voltage at the point of common coupling (PCC), thereby ensuring balanced current flow from and to the distribution substation. Additionally, this device can also support the distribution network in the absence of the utility connection, making the distribution system work as a microgrid. The proposals are validated through extensive digital computer simulation studies using PSCADTM

Diffusion tensor of water in partially aligned fibre networks

Monte Carlo simulations were used to investigate the relationship between the morphological characteristics and the diffusion tensor (DT) of partially aligned networks of cylindrical fibres. The orientation distributions of the fibres in each network were approximately uniform within a cone of a given semi-angle (θ0). This semi-angle was used to control the degree of alignment of the fibres. The networks studied ranged from perfectly aligned (θ0 = 0) to completely disordered (θ0 = 90°). Our results are qualitatively consistent with previous numerical models in the overall behaviour of the DT. However, we report a non-linear relationship between the fractional anisotropy (FA) of the DT and collagen volume fraction, which is different to the findings from previous work. We discuss our results in the context of diffusion tensor imaging of articular cartilage. We also demonstrate how appropriate diffusion models have the potential to enable quantitative interpretation of the experimentally measured diffusion-tensor FA in terms of collagen fibre alignment distributions.

Sampling designs on stream networks using the pseudo-Bayesian approach

Monitoring stream networks through time provides important ecological information. The sampling design problem is to choose locations where measurements are taken so as to maximise information gathered about physicochemical and biological variables on the stream network. This paper uses a pseudo-Bayesian approach, averaging a utility function over a prior distribution, in finding a design which maximizes the average utility. We use models for correlations of observations on the stream network that are based on stream network distances and described by moving average error models. Utility functions used reflect the needs of the experimenter, such as prediction of location values or estimation of parameters. We propose an algorithmic approach to design with the mean utility of a design estimated using Monte Carlo techniques and an exchange algorithm to search for optimal sampling designs. In particular we focus on the problem of finding an optimal design from a set of fixed designs and finding an optimal subset of a given set of sampling locations. As there are many different variables to measure, such as chemical, physical and biological measurements at each location, designs are derived from models based on different types of response variables: continuous, counts and proportions. We apply the methodology to a synthetic example and the Lake Eacham stream network on the Atherton Tablelands in Queensland, Australia. We show that the optimal designs depend very much on the choice of utility function, varying from space filling to clustered designs and mixtures of these, but given the utility function, designs are relatively robust to the type of response variable.

Modelling for the electricity distribution network

Global awareness for cleaner and renewable energy is transforming the electricity sector at many levels. New technologies are being increasingly integrated into the electricity grid at high, medium and low voltage levels, new taxes on carbon emissions are being introduced and individuals can now produce electricity, mainly through rooftop photovoltaic (PV) systems. While leading to improvements, these changes also introduce challenges, and a question that often rises is ‘how can we manage this constantly evolving grid?’ The Queensland Government and Ergon Energy, one of the two Queensland distribution companies, have partnered with some Australian and German universities on a project to answer this question in a holistic manner. The project investigates the impact the integration of renewables and other new technologies has on the physical structure of the grid, and how this evolving system can be managed in a sustainable and economical manner. To aid understanding of what the future might bring, a software platform has been developed that integrates two modelling techniques: agent-based modelling (ABM) to capture the characteristics of the different system units accurately and dynamically, and particle swarm optimization (PSO) to find the most economical mix of network extension and integration of distributed generation over long periods of time. Using data from Ergon Energy, two types of networks (3 phase, and Single Wired Earth Return or SWER) have been modelled; three-phase networks are usually used in dense networks such as urban areas, while SWER networks are widely used in rural Queensland. Simulations can be performed on these networks to identify the required upgrades, following a three-step process: a) what is already in place and how it performs under current and future loads, b) what can be done to manage it and plan the future grid and c) how these upgrades/new installations will perform over time. The number of small-scale distributed generators, e.g. PV and battery, is now sufficient (and expected to increase) to impact the operation of the grid, which in turn needs to be considered by the distribution network manager when planning for upgrades and/or installations to stay within regulatory limits. Different scenarios can be simulated, with different levels of distributed generation, in-place as well as expected, so that a large number of options can be assessed (Step a). Once the location, sizing and timing of assets upgrade and/or installation are found using optimisation techniques (Step b), it is possible to assess the adequacy of their daily performance using agent-based modelling (Step c). One distinguishing feature of this software is that it is possible to analyse a whole area at once, while still having a tailored solution for each of the sub-areas. To illustrate this, using the impact of battery and PV can have on the two types of networks mentioned above, three design conditions can be identified (amongst others): · Urban conditions o Feeders that have a low take-up of solar generators, may benefit from adding solar panels o Feeders that need voltage support at specific times, may be assisted by installing batteries · Rural conditions - SWER network o Feeders that need voltage support as well as peak lopping may benefit from both battery and solar panel installations. This small example demonstrates that no single solution can be applied across all three areas, and there is a need to be selective in which one is applied to each branch of the network. This is currently the function of the engineer who can define various scenarios against a configuration, test them and iterate towards an appropriate solution. Future work will focus on increasing the level of automation in identifying areas where particular solutions are applicable.

Observability analysis of electric networks considering branch impedance

This paper presents an analytical method to analyze the effect of X to R ratio as well as impedance value of branches on observability of a network based on un-decoupled formulation of state estimation (SE) and null space of matrices. The results showed that the X to R ratio of branches had no effect on the observability of networks. In addition, it was shown that observability of some networks was affected by impedance values while some others were not affected. In addition, for branch observability analysis of radial network, a simple and quick method is developed. Illustrative examples of the network under transmission and distribution voltages demonstrate the effectiveness of the proposed methods.

Primary control level of parallel distributed energy resources converters in system of multiple interconnected autonomous microgrids within self-healing networks

To minimise the number of load sheddings in a microgrid (MG) during autonomous operation, islanded neighbour MGs can be interconnected if they are on a self-healing network and an extra generation capacity is available in the distributed energy resources (DER) of one of the MGs. In this way, the total load in the system of interconnected MGs can be shared by all the DERs within those MGs. However, for this purpose, carefully designed self-healing and supply restoration control algorithm, protection systems and communication infrastructure are required at the network and MG levels. In this study, first, a hierarchical control structure is discussed for interconnecting the neighbour autonomous MGs where the introduced primary control level is the main focus of this study. Through the developed primary control level, this study demonstrates how the parallel DERs in the system of multiple interconnected autonomous MGs can properly share the load of the system. This controller is designed such that the converter-interfaced DERs operate in a voltage-controlled mode following a decentralised power sharing algorithm based on droop control. DER converters are controlled based on a per-phase technique instead of a conventional direct-quadratic transformation technique. In addition, linear quadratic regulator-based state feedback controllers, which are more stable than conventional proportional integrator controllers, are utilised to prevent instability and weak dynamic performances of the DERs when autonomous MGs are interconnected. The efficacy of the primary control level of the DERs in the system of multiple interconnected autonomous MGs is validated through the PSCAD/EMTDC simulations considering detailed dynamic models of DERs and converters.

Effectiveness of reactive power capability of photo voltaic inverters to maintain voltage profile in a residential distribution feeder

Large penetration of rooftop PVs has resulted in unacceptable voltage profile in many residential distribution feeders. Limiting real power injection from PVs to alleviate over voltage problem is not feasible due to loss of green power and hence corresponding revenue loss. Reactive capability of the PV inverter can be a solution to address over voltage and voltage dip problems to some extent. This paper proposes an algorithm to utilize reactive capability of PV inverters and investigate their effectiveness for voltage improvement based on R/X ratio of the feeder. The length and loading level of the feeder for a particular R/X ratio to have acceptable voltage profile is also investigated. This can be useful for suburban design and residential distribution planning. Furthermore, coordination among different PVs using residential smart meters via a substation based controller is also proposed.

Securing multicast groups in ad hoc networks

We propose a reliable and ubiquitous group key distribution scheme that is suitable for ad hoc networks. The scheme has self-initialisation and self-securing features. The former feature allows a cooperation of an arbitrary number of nodes to initialise the system, and it also allows node admission to be performed in a decentralised fashion. The latter feature allows a group member to determine the group key remotely while maintaining the system security. We also consider a decentralised solution of establishing secure point-to-point communication. The solution allows a new node to establish a secure channel with every existing node if it has pre-existing secure channels with a threshold number of the existing nodes.