Firefighter is a discrete-time dynamic process that models the spread of a virus or rumour through a network. The name “Firefighter” arises from the initial analogy being the spread of fire among the vertices of a graph. Given a graph G, the process begins at time t = 0 when one or more vertices of G spontaneously “catch fire”. At each subsequent time step, a collection of b ≥ 1 “firefighters” defend a set of vertices which are not burning, and then the fire spreads from each burning vertex to all of its undefended neighbours. There are many possible objectives one could have, for example minimizing the expected number of vertices burned when the fire breaks out at a random location or locations, finding the maximum number of vertices that can be saved from burning if the fire breaks out at known locations, minimizing the length of the process, or bounding the proportion of vertices that can be saved from burning. It is also possible to consider multiple objectives that may be in conflict. There are a great number of papers in the literature which address these, and other, issues in terms of computational complexity, algorithms, approximation, asymptotics, heuristics, and more. The main purpose of this thesis is to survey developments on Firefighter and its variants which have appeared in the literature subsequent to a previous survey that appeared in 2009 [S. Finbow and G. MacGillivray. The firefighter problem: A survey of results, directions and questions. Australas. J. Comb., 43, 2009]. The thesis concludes with a list of open problems and future directions from the previous survey, annotated with references for papers that have made progress on those topics since then. / Graduate

The slow render times of global illumination algorithms make them impractical in most commercial and academic settings. We propose a novel framework for calculating the computational complexity of global illumination algorithms and show that no other recent improvements have reduced this complexity. We further show that many algorithms use a tree as their rendering paradigm. We propose a new rendering algorithm, pipe casting, which calculates light paths using a graph instead of a tree. Pipe casting significantly reduces both computational complexity and actual render time of rendering. Using an L2 pixel-wise error comparison, on average our algorithm can render a variety of scenes at the same error as traditional algorithms but in about 50% of the time.

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The human brain is considered to be the most complex and powerful information-processing device in the known universe. The fundamental concepts behind the physics of complex systems motivate scientists to investigate the human brain as a collective property emerging from the interaction of thousand agents. In this dissertation, I investigate the emergence of cooperation-induced properties in a system of interacting units. I demonstrate that the neural network of my research generates a series of properties such as avalanche distribution in size and duration coinciding with the experimental results on neural networks both in vivo and in vitro. Focusing attention on temporal complexity and fractal index of the system, I discuss how to define an order parameter and phase transition. Criticality is assumed to correspond to the emergence of temporal complexity, interpreted as a manifestation of non-Poisson renewal dynamics. In addition, I study the transmission of information between two networks to confirm the criticality and discuss how the network topology changes over time in the light of Hebbian learning.

Counselor cognitive complexity is a counselor’s ability to recognize and organize multiple characteristics that might affect client needs. I examined whether various admissions criteria–Graduate Record Examination (GRE) Verbal, Quantitative, and Analytical Writing scores; previous coursework grade point averages; and faculty co-leaders’ admissions group interview ratings–for 182 applicants to a southwestern U.S. CACREP-accredited master’s counseling program predicted cognitive complexity scores on a modified Counselor Cognitions Questionnaire (CCQ). Participants were predominantly ages 20 to 30 years (91.8%), female (91.8%), and White (81.3%). Multiple regression analyses showed statistical significance with small effect sizes: the admissions criteria together significantly predicted cognitive complexity differentiation (p = .033), accounting for 6.6% of variance, and cognitive complexity integration (p = .003), accounting for 9.8% of variance. The small effect sizes and low variance percentages support the idea that cognitive complexity measured by the modified CCQ is a substantially different phenomenon from commonly-assessed academic aptitude and personality characteristics. If future researchers confirm these findings with additional samples, subsequent researchers could determine whether one or both domains of cognitive complexity, either alone or in combination with one or more of the commonly used admissions criteria, could help counselor educators better predict which applicants will be successful in master’s programs and the counseling field.

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