Hsun-Yi Hsieh
A network is composed of nodes and edges. With information of nodes and edges, we can compute various properties of a network.
However, there are times that we do not know whether there is an edge between two nodes. One possible thing that we can do is to generate as many networks as we can and test which one makes the most sense to a special occasion.
This sounds like a daunting task... It is likely to do with a computer and some simple math, however.
I create this Shiny APP for users to generate as many networks as they would like to based on the 'distance' between two nodes.
Hopefully this APP will be accompanied with R package ulnet, standing for 'unlimited networks'. Or, at least, I can add the functions into some existing R package.
The Shiny APP uses the spatial data of an ant species as an example. Each node on the plot is an ant nest. It has its coordinate on the plot. Each node is seperate from another with a specific distance.
Link to the old version of the Shiny APP
The user can use the slider on the left to set up the 'critical distance', measured in meter, to generate a network. Two nests belong to the same compartment if their distance is less than this 'critical distance'. They should be signified with the same color on the plot.
(1) User's data-input on the Shiny APP. (2) User's set-up of distance range, interval and threshold (3) The exclusion of nodes located within the threashold distance to plot margins (4) A spatial network with node information (5) Zooming function (in and out) accompanied with node information
(1) Source code will be available on GitHub (2) Hopefully I will make a R package to accommodate functions
The application of the fundamental matrix of the Markov chain.
\[ S = I + T + T^2 + ...\] \[S-I = T + T^2 + ... = TS\] \[(I-T)S = I\] \[ S = (I-T)^{-1}\]
More detailed math will be provided.
More trafficed nodes by natural enemies would be less stable. The ant-scale mutualism would be more likely to be driven to extinction at highly trafficed nodes.
A less trafficed node, in contrast, would more likely to become an ant-scale mutualism patch.
Likewise, a patch close to other nodes would be more likely to go extinct. A node that is strongly conntected to other nodes would also be more likely to go extinct.
All of the above contribute to the observations of detectable ant-scale mutualism patches in the field.
(However, think about this - An ant nest that is weakly connected to other ant nests would have weaker 'collective' protection. So it would be a hump-shaped curve? Should we predice to see a greater probability of observing mutualism pathes in 'intermediately' connected nodes?)
## [1] 229 6
## [1] 114 6
Degree centrality The number of ties that a node has.
Betweenness centrality Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes.
Closeness centrality
In connected graphs there is a natural distance metric between all pairs of nodes, defined by the length of their shortest paths. The farness of a node x is defined as the sum of its distances from all other nodes, and its closeness was defined by Bavelas as the reciprocal of the farness.
In summer 2006, there were 343 trees with ant nests. Among them, 114 had ant-scale mutualism.
Randomly sample 114 of the 229 records of 'no mutualism', combining with the 114 records of "with mutualism".
Model selections based on three variables: betweenness centrality, closeness centrality and degree centrality.
Obtain the prediction error of the best model. Simulate for 1000 times.
The same procedure is repeated against networks generated by a series of 'threshold distance' to obtain the spatial scale that explains the most of the appearance of ant-scale mutualism.
D5 --> Very bad, none of the predictors does the work. D10 --> prediction error = 41.64% D12 --> prediction error = 38.98% D13 --> prediction error = 37.13% D14 --> prediction error = 40.33% D15 --> prediction error = 40.6% D17 --> prediction error = 42%
Note: D stands for 'critical distance', the number after it is the distance in meter (ex. 5m, 10m, 12m, etc.)
##
## Call:
## glm(formula = predator_incidence ~ c, family = binomial)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.0852 -0.7188 -0.6829 -0.6656 1.7978
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.115e+01 7.053e+00 -1.581 0.114
## c 1.145e+06 8.070e+05 1.418 0.156
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 124.81 on 113 degrees of freedom
## Residual deviance: 122.88 on 112 degrees of freedom
## AIC: 126.88
##
## Number of Fisher Scoring iterations: 4
##
## Call:
## glm(formula = predator_incidence ~ d, family = binomial)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.8711 -0.7467 -0.7177 -0.6895 1.7625
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.31553 0.33496 -3.927 8.59e-05 ***
## d 0.04517 0.07624 0.592 0.554
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 124.81 on 113 degrees of freedom
## Residual deviance: 124.47 on 112 degrees of freedom
## AIC: 128.47
##
## Number of Fisher Scoring iterations: 4
Ulanowicz, R. E. & Puccia, C. J. Mixed trophic impacts in ecosystems, Coenoses 5(1): 7-16, 1990.
Samantha Tyner & Heike Hofmann (2015). geomnet: Network Visualization in the 'ggplot2' Framework. R package version 0.0.1. http://CRAN.R-project.org/package=geomnet