Nothing

```
#' two node solution
#'
#' Uses a newton-Raphson to solve the two node solution. Seldom used outside of a 'setse' function
#'
#' @details a helper function inside the main setse suite of functions. However it can be used to solve two node graphs
#'
#' @param g the graph must be two nodes connected by a single edge
#' @param Prep The output of setse_data_prep. provides the inputs needed to do the two node solution
#' @param auto_setse_mode outputs an additional list element "memory_df" to make it compatible with the setse_auto function
#'
#' @return
#' A list of two elements node_embeddings and network_dynamics. if `auto_setse_mode==TRUE`
#' then a third element is returned the memory_df dataframe
#'
#' @noRd
#'
two_node_solution <- function(g, Prep = Prep, auto_setse_mode = FALSE){
start_time <- Sys.time()
#If the the force of the two nodes is effectively identical then the solution angle is zero.
#This also covers the case of the forces being close to 0
#It should be noted this means there should be a minimum force in the system to account for the nodes.
#Very large networks may have small force values if normalised to one.
if(isTRUE(all.equal(Prep$node_embeddings$force[1], Prep$node_embeddings$force[2]))){
solution_angle <-0
} else {
#uses the non-linear optimiser from minpack.lm to find the solution to the two node special case, this is much faster
solution_angle <- minpack.lm::nlsLM(Force ~ ForceV_from_angle(target_angle, k = k, d = d),
start = c(target_angle = pi/4),
data = list(Force = abs(Prep$node_embeddings$force[1]), k = Prep$Link$k, d = Prep$Link$distance),
upper = pi/2) %>% stats::coefficients()
}
stop_time <- Sys.time()
temp <- Prep$node_embeddings #%>%
# mutate(elevation = ifelse(force>0,
# tan(solution_angle)/2, #height above mid point
# -tan(solution_angle)/2 ), #height below mid-point
# net_force = 0,
# acceleration = 0,
# # k * the extension of the edge due to stretching *
# net_tension = ifelse(force>0,
# -abs(Prep$node_embeddings$force[1]),
# abs(Prep$node_embeddings$force[1]) )
# )
temp$elevation <-ifelse(temp$force>0,
tan(solution_angle)/2, #height above mid point
-tan(solution_angle)/2 )
temp$net_force <- 0
temp$acceleration <- 0
temp$net_tension <- ifelse(temp$force>0,
-abs(Prep$node_embeddings$force[1]),
abs(Prep$node_embeddings$force[1])
)
#The static force needs to be set to 0 otherwise it can appear as if the network has not converged
temp$static_force <-0
Out <- list(network_dynamics = tibble::tibble(t = 0,
Iter = 0,
static_force = 0,
kinetic_force = 0),
node_embeddings = temp,
time_taken = tibble::tibble(time_diff = stop_time - start_time, nodes = 2, edges = 1)
)
if(auto_setse_mode){
#the memory_df data frame
#everything is basically NA but it allows easier post processing
Out$memory_df<-tibble::tibble(iteration = 1,
error = NA,
perc_change = NA,
log_ratio = NA,
common_drag_iter = NA,
tstep = NA,
direction = 1,
target_area = NA,
res_stat = NA,
upper = NA,
lower = NA,
best_log_ratio =NA,
stable = NA)
}
return(Out)
}
```

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