by Jess Behrens
© 2005-2018 Jess Behrens, All Rights Reserved
Because building a model like this is a process, I continually look for ways to improve it. The basis of the model is, of course, the vector and how it is converted into the win & loss networks (Chapter 3). Since my last post, I've discovered two additional large, structural queries that had a major impact on how the networks were 'wired'. While all of the queries are important and provide input into the interconnections of the model, the largest of them effectively serve as a backbone of connectivity.
Table 1. Structural Queries (40+ Teams), Network Type, & Weight
Table 1 shows all of the queries, 76 in total, that include 40 or more teams. Given that the vector covers 14 years and includes 934 teams, a query that includes 40 teams represents a little over 4% of the overall population, making it a significant contributor to the overall basic structure of connections within both the win & loss network.
As you can see, the largest query falls within the win network and includes 126 teams. Query ID is just that: the query identifier. It reads from left to right as:
[Index 1], [Range 1 start] - [Range 1 End], [Index 2], [Range 2 start] - [Range 2 End]
Weight is determined using the method described in Chapter 3 & is converted using Bayesian theory before calculating network centrality measures in Gephi. If you notice some overlap among the queries, that is intentional. Many times increasing the range of one index by one rank below and above a given range will produce a query that falls below the base sensitivity/specificity combination of 4. But, creating separate queries, one each for a rank above & below, will produce two separate viable queries. Since the goal is to create all possible connections, both queries are included.
The two queries I discovered after my last post both had 51 teams in them. As such, I re-ran the networks & the Hawk/Owl/Dove designation to see what impact these two additional structural queries had on the overall model. The answer is that there was not much of an effect on the species distribution within the overall population. However, it did alter the designation of some significant teams. Most salient to readers will be this past year where Virginia, who was originally designated as a Dove-Owl, was recast as a Hawk. What is interesting about this is the fact that UMBC remained an Owl.
This 'rewiring' makes sense given what we already know, and what I described, in Chapter 3: Hawks are at a statistical disadvantage in the first round, but have a strong statistical advantage in the Elite Eight. Only one Hawk out of 11 has beaten an Owl in the first round (Butler over South Alabama in 2008). Likewise, only 1 of 15 Owls has beaten a Hawk in the Elite Eight (Michigan State over Louisville in 2009). So, even in the switch, the general pattern remains.
I also wanted to test the effect of changing the Cost term as well as the number of simulation iterations on the evolutionary game theory output. As such, the next series of figures are based on a simulation ensemble with 10,000 iterations. Figure 1 is the cluster map showing the year to year
Figure 1. Cluster Map, Dove vs. Dove-Owl Fitness year to year regression slope, Cost = 1.5
population fitness regression slope of Doves & Dove-Owls when the cost term for Hawks is equal to 1.5 rounds. There are several significant things that come out of this cluster map.
First of all, the Hawk vs. Owl dominance shows up much starker than in previous cluster maps. There are 14 years in Figure 1, so the middle lies between 2007 & 2012 along the bottom. Six of the seven years on the left produced an Owl champion (p<0.05). Only Florida in 2007 was a Hawk. Likewise, 5 of the 7 teams on the right are Hawks (p<0.001). UConn was an Owl in both 2011 & 2014. But, 2011 & 2014 were both cluster 5 years as well. What is most significant about Figure 1 is it shows the relationship between the two weakest species in the system: Doves & Dove-Owls. In effect, Figure 1 shows that the number of weak teams has a dramatic impact on which teams are favored to win the tournament.
Figure 2. Doves vs. Dove-Owls, Hawk Dominant Years, Right Side of Figure 1
Figures 2 & 3 show the Dove vs. Dove-Owl best fit regression lines and associated Pearson correlation coefficients. While the Pearson for both is very slight (p=0.16 & p=0.17), they are both significant
Figure 3. Doves vs. Dove-Owls, Owl Dominant Years, Left Side of Figure 1
(p<0.0001). Thus, the switch from a competitive environment in Owl Dominant years to a cooperative environment in Hawk dominant years is a significant indicator of a change in the underlying processes.
The Lotka-Volterra (Figures 4 & 5) plots for Doves & Dove-Owls provide an even better picture of the effect of this switch from competition (Owl dominant) to mutualism (Hawk dominant). Figure 4 clearly
Figure 4. Dove vs. Dove-Owl, Lotka-Volterra Fitness Plots, Hawk Dominant Years, Cost = 1.5
shows how K1 (Hawk) & K2 (Owl) both grow with the number of Doves & Dove-Owls in the total population. Figure 5, on the other hand, shows a Dove-Owl dominant competition. In Figure 5,
Figure 5. Dove vs. Dove-Owl, Lotka-Volterra Fitness Plots, Owl Dominant Years, Cost = 1.5
Dove-Owls are dominant until they reach the X-Axis, which is expected given the nature of their strategies in the simulations. In Figure 4, this relationship is switched, and the Doves are dominant prior to x = 14, which is when Dove-Owl fitness first becomes positive.
Because the Hawks & Owls drive the tournament bus, so to speak, it is important to consider the nature of their relationship given the changes described in the first paragraph of this post. Several of the previous chapters (6, 7, 8, & 11) all deal explicitly with the nature of the competition between Hawks &
Figure 6. Hawk vs. Owl, Best Fit Linear Regression, Hawk Dominant Years, Cost = 1.5
Owls; Chapter 6 & 11 deal explicitly with the difference between linear and less linear years. While the strength of the relationship has altered (Figures 6 & 7), Hawk dominant years are still significantly less
Figure 7. Hawk vs. Owl, Best Fit Linear Regression, Owl Dominant Years, Cost = 1.5
linear (p=0.69) than Owl years (p=0.83), as expected given previous results. The difference has dropped 0.14 from 0.33, primarily due to the fact that as Figure 7 shows, the distribution is much more centered in the upper left (Owls do better than Hawks) in Owl dominant years than it is in Hawk dominant years (Figure 6).
As with the Doves & Dove-Owls, the relationships seen in Figures 6 & 7 are more clearly expressed using Lotka-Volterra plots. Figure 8 is the L-V plot for Hawk vs. Owl Inter-specific competition across all tournament years. It is a repeat of the Lotka-Volterra plot first posted and described in Chapter 13.
Figure 8. Hawk vs. Owl, Lotka-Volterra Fitness Plots, All Tournament Years, Cost = 1.5
As was described there, Figure 8 represents an unstable equilibrium between Hawks and Owls. Hawks are dominant in the space between K1 & K2 to the right of the intersection point. There are two tournament averages that fall in this area, 2015 & 2018. Thus, if you consider Figure 8 to be making a predicition, this region is 50/50 (2015 correct, 2018 incorrect). The opposite is true in the area between the two lines to the left of the intersection point. There are 5 tournament averages that fall in this area (2006, 2012, 2014, & 2016). Since 4 of those 5 are owls, this region goes 4 for 5. Areas above and below the two lines do not favor either strategy, but tend to be dominated by Hawk champions. There are 7 Tournament averages that fall in these areas (2017, 2013, 2009, 2007, 2005, 2010, & 2011). Four of those tournaments had Hawk Champions & 3 were Owls, which is pretty close to 50/50.
That's it for this post. Next, I will look at the impact of the system changes I've described here on major first round upsets & determining where the champion falls within the vector.
#ClusterMap #PredatorPrey #NCAATournament #SpeciesFitnessPlots #LotkaVolterra #MonteCarloSimulations #EvolutionaryGameTheory #basketball #NetworkAnalysis #Matplotlib #SpeciesCompetitionPlots #Seaborn #MensCollegeBasketball #NCAA