Properties and algorithms of the hyper-star graph and its related graphs

The hyper-star interconnection network was proposed in 2002 to overcome the drawbacks of the hypercube and its variations concerning the network cost, which is defined by the product of the degree and the diameter. Some properties of the graph such as connectivity, symmetry properties, embedding properties have been studied by other researchers, routing and broadcasting algorithms have also been designed. This thesis studies the hyper-star graph from both the topological and algorithmic point of view. For the topological properties, we try to establish relationships between hyper-star graphs with other known graphs. We also give a formal equation for the surface area of the graph. Another topological property we are interested in is the Hamiltonicity problem of this graph. For the algorithms, we design an all-port broadcasting algorithm and a single-port neighbourhood broadcasting algorithm for the regular form of the hyper-star graphs. These algorithms are both optimal time-wise. Furthermore, we prove that the folded hyper-star, a variation of the hyper-star, to be maixmally fault-tolerant.

Automatic Inference of Graph Models for Complex Networks with Genetic Programming

Complex networks can arise naturally and spontaneously from all things that act as a part of a larger system. From the patterns of socialization between people to the way biological systems organize themselves, complex networks are ubiquitous, but are currently poorly understood. A number of algorithms, designed by humans, have been proposed to describe the organizational behaviour of real-world networks. Consequently, breakthroughs in genetics, medicine, epidemiology, neuroscience, telecommunications and the social sciences have recently resulted. The algorithms, called graph models, represent significant human effort. Deriving accurate graph models is non-trivial, time-intensive, challenging and may only yield useful results for very specific phenomena. An automated approach can greatly reduce the human effort required and if effective, provide a valuable tool for understanding the large decentralized systems of interrelated things around us. To the best of the author's knowledge this thesis proposes the first method for the automatic inference of graph models for complex networks with varied properties, with and without community structure. Furthermore, to the best of the author's knowledge it is the first application of genetic programming for the automatic inference of graph models. The system and methodology was tested against benchmark data, and was shown to be capable of reproducing close approximations to well-known algorithms designed by humans. Furthermore, when used to infer a model for real biological data the resulting model was more representative than models currently used in the literature.

Network Similarity Measures and Automatic Construction of Graph Models using Genetic Programming

A complex network is an abstract representation of an intricate system of interrelated elements where the patterns of connection hold significant meaning. One particular complex network is a social network whereby the vertices represent people and edges denote their daily interactions. Understanding social network dynamics can be vital to the mitigation of disease spread as these networks model the interactions, and thus avenues of spread, between individuals. To better understand complex networks, algorithms which generate graphs exhibiting observed properties of real-world networks, known as graph models, are often constructed. While various efforts to aid with the construction of graph models have been proposed using statistical and probabilistic methods, genetic programming (GP) has only recently been considered. However, determining that a graph model of a complex network accurately describes the target network(s) is not a trivial task as the graph models are often stochastic in nature and the notion of similarity is dependent upon the expected behavior of the network. This thesis examines a number of well-known network properties to determine which measures best allowed networks generated by different graph models, and thus the models themselves, to be distinguished. A proposed meta-analysis procedure was used to demonstrate how these network measures interact when used together as classifiers to determine network, and thus model, (dis)similarity. The analytical results form the basis of the fitness evaluation for a GP system used to automatically construct graph models for complex networks. The GP-based automatic inference system was used to reproduce existing, well-known graph models as well as a real-world network. Results indicated that the automatically inferred models exemplified functional similarity when compared to their respective target networks. This approach also showed promise when used to infer a model for a mammalian brain network.

Object-Oriented Genetic Programming for the Automatic Inference of Graph Models for Complex Networks

Complex networks are systems of entities that are interconnected through meaningful relationships. The result of the relations between entities forms a structure that has a statistical complexity that is not formed by random chance. In the study of complex networks, many graph models have been proposed to model the behaviours observed. However, constructing graph models manually is tedious and problematic. Many of the models proposed in the literature have been cited as having inaccuracies with respect to the complex networks they represent. However, recently, an approach that automates the inference of graph models was proposed by Bailey [10] The proposed methodology employs genetic programming (GP) to produce graph models that approximate various properties of an exemplary graph of a targeted complex network. However, there is a great deal already known about complex networks, in general, and often specific knowledge is held about the network being modelled. The knowledge, albeit incomplete, is important in constructing a graph model. However it is difficult to incorporate such knowledge using existing GP techniques. Thus, this thesis proposes a novel GP system which can incorporate incomplete expert knowledge that assists in the evolution of a graph model. Inspired by existing graph models, an abstract graph model was developed to serve as an embryo for inferring graph models of some complex networks. The GP system and abstract model were used to reproduce well-known graph models. The results indicated that the system was able to evolve models that produced networks that had structural similarities to the networks generated by the respective target models.

A classroom-based investigation of reciprocal teaching at the grade seven level

This study assessed the effectiveness of a reciprocal teaching program as a method of teaching reading comprehension, using narrative text material in a t.ypical grade seven classroom. In order to determine the effectiveness of the reciprocal teaching program, this method was compared to two other reading instruction approaches that, unlike rcciprocal teaching, did not include social interaction components. Two intact grade scven classes, and a grade seven teacher, participated in this study. Students were appropriately assigned to three treatment groups by reading achievement level as determined from a norm-referenced test. Training proceeded for a five week intervention period during regularly scheduled English periods. Throughout the program curriculum-based tests were administered. These tests were designed to assess comprehension in two distinct ways; namely, character analysis components as they relate to narrative text, and strategy use components as they contribute to student understanding of narrative and expository text. Pre, post, and maintenance tests were administered to measure overall training effects. Moreover, during intervention, training probes were administered in the last period of each week to evaluate treatment group performance. AU curriculum-based tests were coded and comparisons of pre, post, maintenance tests and training probes were presented in graph form. Results showed that the reciprocal group achieved some improvement in reading comprehension scores in the strategy use component of the tests. No improvements were observed for the character analysis components of the curriculum-based tests and the norm-referenced tests. At pre and post intervention, interviews requiring students to respond to questions that addressed metacomprehension awareness of study strategies were administered. The intelviews were coded and comparisons were made between the two intelVicws. No significant improvements were observed regarding student awareness of ten identified study strategies . This study indicated that reciprocal teaching is a viable approach that can be utilized to help students acquire more effective comprehension strategies. However, the maximum utility of the technique when administered to a population of grade seven students performing at average to above average levels of reading achievement has yet to be determined. In order to explore this issue, the refinement of training materials and curriculum-based measurements need to be explored. As well, this study revealed that reciprocal teaching placed heavier demands on the classroom teacher when compared to other reading instruction methods. This may suggest that innovative and intensive teacher training techniques are required before it is feasible to use this method in the classroom.

A school-based cognitive behavior modification intervention strategy for learned helpless mentally retarded students

This study assessed the usefulness of a cognitive behavior modification (CBM) intervention package with mentally retarded students in overcoming learned helplessness and improving learning strategies. It also examined the feasibility of instructing teachers in the use of such a training program for a classroom setting. A modified single subject design across individuals was employed using two groups of three subjects. Three students from each of two segregated schools for the mentally retarded were selected using a teacher questionnaire and pupil checklist of the most learned helpless students enrolled there. Three additional learned helplessness assessments were conducted on each subject before and after the intervention in order to evaluate the usefulness of the program in alleviating learned helplessness. A classroom environment was created with the three students from each school engaged in three twenty minute work sessions a week with the experimenter and a tutor experimenter (TE) as instructors. Baseline measurements were established on seven targeted behaviors for each subject: task-relevant speech, task-irrelevant speech, speech denoting a positive evaluation of performance, speech denoting a negative evaluation of performance, proportion of time on task, non-verbal positive evaluation of performance and non-verbal negative evaluation of performance. The intervention package combined a variety of CBM techniques such as Meichenbaum's (1977) Stop, Look and Listen approach, role rehearsal and feedback. During the intervention each subject met with his TE twice a week for an individual half-hour session and one joint twenty minute session with all three students, the experimentor and one TE. Five weeks after the end of this experiment one follow up probe was conducted. All baseline, post-intervention and probe sessions were videotaped. The seven targeted behaviors were coded and comparisons of baseline, post intervention, and probe testing were presented in graph form. Results showed a reduction in learned helplessness in all subjects. Improvement was noted in each of the seven targeted behaviors for each of the six subjects. This study indicated that mentally retarded children can be taught to reduce learned helplessness with the aid of a CBM intervention package. It also showed that CBM is a viable approach in helping mentally retarded students acquire more effective learning strategies. Because the TEs (Tutor experimenters) had no trouble learning and implementing this program, it was considered feasible for teachers to use similar methods in the classroom.

Properties and algorithms of the (n, k)-star graphs

The (n, k)-star interconnection network was proposed in 1995 as an attractive alternative to the n-star topology in parallel computation. The (n, k )-star has significant advantages over the n-star which itself was proposed as an attractive alternative to the popular hypercube. The major advantage of the (n, k )-star network is its scalability, which makes it more flexible than the n-star as an interconnection network. In this thesis, we will focus on finding graph theoretical properties of the (n, k )-star as well as developing parallel algorithms that run on this network. The basic topological properties of the (n, k )-star are first studied. These are useful since they can be used to develop efficient algorithms on this network. We then study the (n, k )-star network from algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms for basic communication, prefix computation, and sorting, etc. A literature review of the state-of-the-art in relation to the (n, k )-star network as well as some open problems in this area are also provided.

Properties and algorithms of the (n, k)-arrangement graphs

The (n, k)-arrangement interconnection topology was first introduced in 1992. The (n, k )-arrangement graph is a class of generalized star graphs. Compared with the well known n-star, the (n, k )-arrangement graph is more flexible in degree and diameter. However, there are few algorithms designed for the (n, k)-arrangement graph up to present. In this thesis, we will focus on finding graph theoretical properties of the (n, k)- arrangement graph and developing parallel algorithms that run on this network. The topological properties of the arrangement graph are first studied. They include the cyclic properties. We then study the problems of communication: broadcasting and routing. Embedding problems are also studied later on. These are very useful to develop efficient algorithms on this network. We then study the (n, k )-arrangement network from the algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms such as prefix sums computation, sorting, merging and basic geometry computation: finding convex hull on the (n, k )-arrangement graph. A literature review of the state-of-the-art in relation to the (n, k)-arrangement network is also provided, as well as some open problems in this area.

Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

Rings, Group Rings, and Their Graphs

We associate some graphs to a ring R and we investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of the graphs associated to R. Let Z(R) be the set of zero-divisors of R. We define an undirected graph ᴦ(R) with nonzero zero-divisors as vertices and distinct vertices x and y are adjacent if xy=0 or yx=0. We investigate the Isomorphism Problem for zero-divisor graphs of group rings RG. Let Sk denote the sphere with k handles, where k is a non-negative integer, that is, Sk is an oriented surface of genus k. The genus of a graph is the minimal integer n such that the graph can be embedded in Sn. The annihilating-ideal graph of R is defined as the graph AG(R) with the set of ideals with nonzero annihilators as vertex such that two distinct vertices I and J are adjacent if IJ=(0). We characterize Artinian rings whose annihilating-ideal graphs have finite genus. Finally, we extend the definition of the annihilating-ideal graph to non-commutative rings.