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This dissertation establishes mathematical foundations for the properties exhibited by generalizations of cliques, as well as algorithms to find such objects in a network. Cliques are a model of an ideal group with roots in social network analysis. They have since found applications as a part of grouping mechanisms in computer vision, coding theory, experimental design, genomics, economics, and telecommunications among other fields. Because only groups with ideal properties form a clique, they are often too restrictive for identifying groups in many real-world networks. This motivated the introduction of clique relaxations that preserve some of the various defining properties of cliques in relaxed form. There are six clique relaxations that are the focus of this dissertation: s-clique, s-club, s-plex, k-core, quasi-clique, and k-connected subgraphs. Since cliques have found applications in so many fields, research into these clique relaxations has the potential to steer the course of much future research. The focus of this dissertation is on bringing organization and rigorous methodology to the formation and application of clique relaxations. We provide the first taxonomy focused on how the various clique relaxations relate on key structural properties demonstrated by groups. We also give a framework for how clique relaxations can be formed. This equips researchers with the ability to choose the appropriate clique relaxation for an application based on its structural properties, or, if an appropriate clique relaxation does not exist, form a new one. In addition to identifying the structural properties of the various clique relaxations, we identify properties and prove propositions that are important computationally. These assist in creating algorithms to find a clique relaxation quickly as it is immersed in a network. We give the first ever analysis of the computational complexity of finding the maximum quasi-clique in a graph. Such analysis identifies for researchers the appropriate set of computational tools to solve the maximum quasiclique problem. We further create a polynomial time algorithm for identifying large 2-cliques within unit disk graphs, a special class of graphs often arising in communication networks. We prove the algorithm to have a guaranteed 1=2-approximation ratio and finish with computational results.

Many chaotic and complicated systems cannot be analysed by traditional methods. In 1987 P.Bak, C.Tang, and K.A.Wiesenfeld developed a new concept called Self-Organised Criticality (SOC) to explain the behaviour of composite systems containing a large number of elements that interact over a short range. In general this theory applies to complex systems that naturally evolve to a critical state in which a minor event starts a chain reaction that can affect any number of elements in the system. It was later shown that many complex phenomena such as flux pinning in superconductors, dynamics of granular systems, earthquakes, droplet formation and biological evolution show signs of SOC. The dynamics of complex systems in nature often occurs in terms of punctuation, or avalanches rather than following a smooth, gradual path. Extremal dynamics is used to model the temporal evolution of many different complex systems. Specifically the Bak-Sneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models. This thesis considers extremal dynamics at constant flux where M>1 smallest barriers are simultaneously updated as opposed to models in the limit of zero flux where only the smallest barrier is updated. For concreteness, we study the Bak-Sneppen (BS) evolution model [Phys. Rev. Lett. 71, 4083 (1993)]. M=1 corresponds to the original BS model. The aim of the present work is to understand analytically through mean field theory the random neighbour version of the generalised BS model and verify the results against the computer simulations. This is done in order to scrutinise the trustworthiness of our numerical simulations. The computer simulations are found to be identical with results obtained from the analytical approach. Due to this agreement, we know that our simulations will produce reliable results for the nearest neighbour version of the generalised BS model. Since the nearest neighbour version of the generalised BS model cannot be solved analytically, we have to rely on simulations. We investigate the critical behaviour of both versions of the model using the scaling theory. We look at various distributions and their scaling properties, and also measure the critical exponents accurately verifying whether the scaling relations holds. The effect of increasing from M=1 to M>1 is surprising with dramatic decrease in size of the scaling regime.

In this thesis I claim that dominant realist, interpretive and postmodern research methodologies, taken together, provide necessary but not sufficient tools for use within educational research. Understandings of material, social and linguistic worlds do not, in themselves, cater for teachers' pragmatic needs to consider (a) the social consequences of educational practices, both their own and those of the institutions within which they work, and (b) the complexity of teaching in a postmodern world. I draw on ideas from pragmatism, post-structuralism, critical pedagogy, complexity theory, reflective practice, and personal experience in order to invite the emergence (or social construction) of new phenomena: these I hope, may enable teachers and other educationalists to take a vibrant part in ongoing debates and actions concerning educational policy and practice. I argue that the assumption that educational theory can be applied in practice is flawed and needs to be replaced by theory which recognises the dynamic nature of theory-in-practice: all theory is data within practice. This is a late-career thesis written by a practitioner with an unusually broad experience of the New Zealand educational system. I argue that the purpose of theory is to guide practice, that practice must drive theory, and that theory and practice need to join together to focus on the consequences of planned actions. This is neo-pragmatism, but, as stated thus far, it is not enough for my purposes because it does not include a commitment to social justice. Praxis is a term which ties emancipatory political goals to theory-and-practice. I invite the construction of the understandings of praxitioner activities where collective praxis and individual praxis might co-emerge in the interests of social justice. I promote the expansion of fresh discourses through research into collective praxis within teaching and teacher education.

This thesis explores how the discourse of complexity thinking can be used to foster emergence in curriculum and practice-based research. The curriculum-related exploration focused specifically on games of chase as one facet of early childhood curriculum. It investigated using complexity thinking firstly, to occasion emergence (that is, create a new phenomenon) in children’s games of chase at an early childhood centre and secondly, to describe this emergence. The research-related exploration focused on creating an emergent methodology which is underpinned by complexity thinking. In this thesis report, I present a series of emergent curriculum-related phenomena that arose during the explorations, that is, an emergent game, a local curriculum theory for games of chase, the concepts of local curriculum theory, curriculum design and curriculum dynamics, and a curriculum vision. I also present an understanding of emergent methodology and two methodological innovations in the form of the Research Data Management System and the Visual Summary. This research involved taking the role of a volunteer teacher-researcher-curriculum designer at an early childhood centre to play games of chase with children. This role was informed by and contributed to a curriculum design that focused on designing the teaching and learning environment to occasion emergence in learning and curriculum. The games of chase curriculum contributed to children’s learning, my own learning and the general rhythm of life at the centre. The children learnt to distinguish between children who were playing and those who were not. They also learnt different ways to tag people in a game. In addition, the children and I developed a game playing routine before playing each game. This routine involved putting on tag belts, discussing what game we were playing and how we were going to play it. We played three different games of chase, starting with tag, followed by What is the time Mr(s) Wolf?, and finally the emergent game Big A, Little A. The stories of emergence are described in visual, descriptive and narrative texts organised into curriculum stories, teaching stories and children’s learning stories. Curriculum stories describe the activities that unfolded. Teaching stories present stories of teaching while learning stories are stories of children’s learning. These stories represent views of the enacted curriculum as activity, teaching and learning respectively. Taken together, the stories present a description of the curriculum dynamics that unfolded at the centre in relation to games of chase. This thesis shows that a local curriculum theory for games of chase at the centre emerged from the complex interactions of curriculum design and curriculum dynamics that unfolded at the centre. It also articulates the emergent concepts of local curriculum theory, curriculum design and curriculum dynamics using the language of complexity. This thesis also presents the local curriculum theory as a curriculum vision. This vision involves a shift in thinking about curriculum as either a set “course to be run” or the “path created in the running” (currere) to embracing curriculum as both “the space for running” and currere. It is a vision that values both children’s and teachers’ interests, focuses on teachers and children exploring depth and breadth of a curriculum domain together, enables teachers to follow, generate and sustain children’s interest in the explorations, and is generative, flexible and future-focused. This thesis conceptualises an emergent methodology as a methodology for emergence which (1) involves the researcher actively striving to foster emergence in research, (2) is brought forth in the interactions between the designed and enacted facets of methodology, (3) is local to a particular research project, and (4) emerges from the interactions of several related strategies. This thesis can be seen as an attempt to change the language game of curriculum by using the language of complexity throughout the thesis. In so doing, it not only enables the reader to talk about the discourse of complexity thinking, it also enables the reader to experience the discourse and the emergence of the curriculum-related phenomena and the methodological innovations that are the focus of this thesis. Finally, this thesis argues that using the discourse of complexity thinking in teaching and research can be enabling. It can enable the teacher and/or researcher to be creative, flexible and ethical within the constraints of his/her professional and personal life.

This dissertation investigates interactions between gender and number and gender and evaluative morphology in a sample of 100 African languages, and provides a method for assessing the role that these interactions play in the grammatical complexity of gender systems. The dissertation is organised around three research foci. First, the dissertation surveys patterns of interaction between gender and number along the following dimensions: exponence, syncretism, indexation, correlations in type of marking, and gender assignment. The study provides evidence for the possibility that nominal features are organised in a relevance hierarchy. In addition, the study shows that animacy and lexical plurality play a crucial role in the distribution of special patterns of plural indexation. The study also shows that pervasive indexation systems in the language sample always involve both gender and number. Finally, the study shows how gender assignment can be used as a means for encoding variation in the countability properties of nouns and noun phrases. Second, the dissertation surveys patterns of interaction between gender and evaluative morphology in the languages of the sample. Two types of interactions are found. The study shows that the distribution of the two types depends on three factors: the type of gender system, the number of gender distinctions and the possibility of assigning a noun to more than one gender. Third, the dissertation investigates the role that interactions of gender and number and gender and evaluative morphology play in the absolute complexity of gender. The study proposes a metric for gender complexity and uses this metric to compute complexity scores for the languages of the sample. The results suggest that the gender systems of the language sample lean toward high complexity, that genealogically related languages have the same or similar complexity scores, and that the distribution of the outliers can often be understood as the result of language contact.

A recent growth in textual analysis research in the accounting and finance literature relies heavily on context to draw conclusions about the readability or sentiment of the text under study. Yet the complexity of the text used in the financial disclosure is also relevant in evaluating readability and sentiment. Experimental results in this dissertation thesis show that a change in annual report complexity is associated with a change in the probability that a subject will comprehend the information being communicated in the disclosure. Specifically, increasing the complexity of an annual report disclosure dampens the probability that a subject will understand good news disclosures and accentuates the probability that a subject will understand bad news disclosures. Experimental results in this dissertation thesis also demonstrate that a change in annual report complexity is associated with a change in the probability that a subject will be optimistic about the nature of the news being communicated in the disclosure. Specifically, an increase in the complexity of an annual report disclosure reduces the probability that a subject will be optimistic about neutral news disclosures, decreases the probability that a subject will be optimistic about good news disclosures, and increases the probability that a subject will be optimistic about bad news disclosures. Further, experimental results show that subjects utilize the Financial Statements, Management's Discussion and Analysis, and Business Data sections of the annual report more frequently than the Notes to Financial Statements section of the annual report. These results should be of interest to regulators, public corporations, and readers of annual report disclosures.

South African schools face many challenges as they are inundated with dysfunctional behaviour. The research on South African schools indicates that behavioural challenges such as disobedience, swearing, truancy violence and bullying are evident in many school contexts. From a reductionist, individualist approach, the focus when addressing these challenges is often on causal factors and dysfunctional individuals rather than on ways in which people relate and interact in schools. It is however evident from a social ecological perspective, that in order to facilitate social change, we need to understand people’s experiences of social interaction in schools as an important context for the enhancement of wellbeing. The first phase of the PhD project is a base‐line exploration of the learners’, educators’ and parents’ experiences of relating and interacting in school communities. A qualitative phenomenological investigation was applied in combination with a cross‐sectional descriptive survey design. 1170 learners, ages ranging from 11 to 18 years, 150 parents and 85 educators, from 12 South African schools, participated in the research. The participants completed written assignments that were analysed through the application of global analysis followed by thematic analysis. The findings indicated that enabling ways of relating and interacting were patterned by active engagement and acknowledgement of people. Disenabling social interaction was patterned by disengagement and disregard for people. The findings indicated that both enabling and disenabling ways of relating and interacting, play a crucial role in the enhancement of mental wellbeing in schools, and suggest that schools need to focus more seriously on the ways in which people in schools relate and interact on the everyday micro‐levels of social interaction, as suggested by complexity theory. The second phase of the study comprised a more in‐depth investigation into nurturing and restraining relationships between parents, learners and educators in a school community. A single instrumental case study design was applied to gain an indepth understanding of the complex dynamic interactions between the members of the school community. All the learners and educators in the school were involved during the work sessions. Nominal group technique was applied to obtain information about their perceptions of relationships in the school community. The work sessions were followed by focus group interviews with 18 educators, 40 learners, the management team, six members of the administrative and terrain staff and two parents. A thematic analysis of the data indicated that nurturing relationships could be understood with reference to connectedness: respect, care and transparent communication; whilst restrained relationships could be understood with reference to limited connectedness between people: abuse of power, shifting of responsibility and disrespect for one another. The findings indicated the need for a sensitive, empathic and non‐patronising approach to people in school communities that acknowledge that restrained relationships are inevitably part of the human interaction and understand schools in terms of inter‐subjective recursive processes that pattern the relationships between the members of the school community. In the third phase, the findings of the first two stages of the study were integrated with theoretical perspectives and critical reflections on the findings to conceptualise a relationship‐focused approach to the co‐construction of an enabling school community. The approach encompasses the facilitation of continuous conversations using identified facets of interrelatedness as focal points for the understanding of being together in school communities on a meta‐level. It is recommended that the implementation of a relationship‐focused approach conceptualised in this study, should be considered as an alternative approach for dealing with the challenges associated with human behaviour that currently prevail in schools. Further research on the implementation of the approach in schools is recommended. / Thesis (Ph.D. (Psychology))--North-West University, Potchefstroom Campus, 2011

With an abundance of social network data being released, the need to protect sensitive information within these networks has become an important concern of data publishers. In this thesis we focus on the popular notion of k-anonymization as applied to social network graphs. Given such a network N, the problem we study is to transform N to N', such that some property P of each node in N' is attained by at least k-1 other nodes in N'. We study edge-labeled, vertex-labeled and unlabeled graphs, since instances of each occur in real-world social networks. Our main contributions are as follows 1. When looking at edge additions, we show that k-label sequence anonymity of arbitrary edge-labeled graphs is NP-complete, and use this fact to prove hardness results for many other recently introduced notions of anonymity. We also present interesting hardness results and algorithms for labeled and unlabeled bipartite graphs. 2. When looking at node additions, we show that on vertex-labeled graphs, the problem is NP-complete. For unlabeled graphs, we give an efficient (near-linear) algorithm and show that it gives solutions that are optimal modulo k, a guarantee that is novel in the literature. We examine anonymization both from its theoretical foundations and empirically, showing that our proposed algorithms for anonymization maintain structural properties shown to be necessary for graph analysis. / Graduate