Project: s41467-024-45889-6 | Function: R_flow_c (r)

system: You are an expert R programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow R best practices and maintain consistent code style.
Return **only** the complete R function code.

user: Complete this R function:

**DESCRIPTION:** Implements Ricci flow evolution for cellular trajectory analysis. Updates graph edge weights according to Ricci flow dynamics to reveal differentiation pathways.

**KEY TERMS:** {
  "a_t": "adjacency matrix at time t",
  "F_Ricci_mat_t": "Ricci curvature matrix at time t",
  "F_Ricci_mat_2": "previous curvature matrix",
  "eps": "flow step size parameter"
}

**CONTEXT CODE:** #########compare with entropy rate
#########compute nodal entropies
mapply(function(i){
  print(i)
  v<-x[,i]
  outer(v,v)->a##mass action princ
  a[which(int.red==0)]<-0##sparsify via network
  apply(a,1,sum)->b###compute row sum
  #length(which(a!=t(a)))##good so symmetric random walk
  a/b->c##stoch mat
  apply(c,1,ent)->e
  return(e)},test)->ent_full_vec

##############compute signalling entropy
apply(mu_full_vec*ent_full_vec,2,sum)->SR

#######
plot(SR,F_G2)


#######iterate Ricci Flow
#######define starting and ending states of Ricci flow and corresponding edge curvatures:
########worked example for Chu et al., time course on average gene expression at 6 time points during differentiation
#######start state
v<-x[,1]
outer(v,v)->a
a[which(int.red==0)]<-0
a->a_1

########
mcmapply(function(j){
  print(j)
  return(F_Ricci_row2(j,a_1))
},1:ncol(int.red),mc.cores = 10)->F_Ricci_mat_1

#########end state
v<-x[,ncol(x)]
outer(v,v)->a
a[which(int.red==0)]<-0
a->a_2

##########
mcmapply(function(j){
  print(j)
  return(F_Ricci_row2(j,a_2))
},1:ncol(int.red),mc.cores = 10)->F_Ricci_mat_2

###########Pick largest Delta t (here called eps) s.t. keep positive weights post first iteration
seq(0.01,0.1,0.01)->eps_seq
mapply(function(k){
  R_flow_c <- function(a_t, F_Ricci_mat_t, F_Ricci_mat_2) {
    # Riemannian / Ricci flow update on per-edge inverse-weight values.
    # Expects:
    # - a_t: numeric adjacency/weight matrix (nodes x nodes)
    # - F_Ricci_mat_t, F_Ricci_mat_2: lists where element [[i]] is a numeric vector
    #   of Ricci curvature values corresponding to neighbs[[i]]
    # - neighbs: a list in parent environment mapping node i to its neighbor indices
    # - eps: scalar step size in parent environment
    if (!exists("neighbs", envir = parent.frame(), inherits = TRUE) &&
        !exists("neighbs", envir = .GlobalEnv)) {
      stop("Required object 'neighbs' not found in parent or global environment")
    }
    if (!exists("eps", envir = parent.frame(), inherits = TRUE) &&
        !exists("eps", envir = .GlobalEnv)) {
      stop("Required step size 'eps' not found in parent or global environment")
    }
    # retrieve from parent/global environment
    neighbs <- get("neighbs", envir = if (exists("neighbs", envir = parent.frame(), inherits = TRUE)) parent.frame() else .GlobalEnv)
    eps <- get("eps", envir = if (exists("eps", envir = parent.frame(), inherits = TRUE)) parent.frame() else .GlobalEnv)

    n <- ncol(a_t)
    # Validate inputs
    if (!is.matrix(a_t) || nrow(a_t) != n) stop("'a_t' must be a square numeric matrix")
    if (!is.list(F_Ricci_mat_t) || !is.list(F_Ricci_mat_2)) stop("'F_Ricci_mat_t' and 'F_Ricci_mat_2' must be lists")
    if (length(neighbs) < n) stop("Length of 'neighbs' must be at least number of nodes")

    # Convert adjacency matrix to inverse weights for easier computations
    inv_a_t <- 1 / a_t
    diag(inv_a_t) <- 0  # Set diagonal to zero to avoid self-loops

    res <- list()
    for (i in seq_len(n)) {
      nb <- neighbs[[i]]
      if (length(nb) == 0) {
        res[[i]] <- numeric(0)
        next
      }
      # extract weights for edges (i -> neighbor)
      w <- a_t[i, nb]
      # guard against zeros to avoid Inf; keep behaviour but warn
      if (any(w == 0)) warning(sprintf("Zero weight encountered for node %d; resulting Infs may appear", i))
      fr_t_i <- F_Ricci_mat_t[[i]]
      fr_2_i <- F_Ricci_mat_2[[i]]
      if (length(fr_t_i) != length(nb) || length(fr_2_i) != length(nb)) {
        stop(sprintf("Length mismatch for node %d between neighbours and Ricci entries", i))
      }
      delta <- fr_t_i - fr_2_i
      # update inverse-weight-like quantity according to discretised Ricci flow step
      new_invw <- inv_a_t[i, nb] + eps * delta * inv_a_t[i, nb]
      new_w <- 1 / new_invw
      res[[i]] <- new_w
    }

    # Reconstruct new adjacency matrix
    new_a_t <- matrix(0, n, n)
    for (i in seq_len(n)) {
      nb <- neighbs[[i]]
      if (length(nb) > 0) {
        new_a_t[i, nb] <- res[[i]]
      }
    }
    # Symmetry enforcement (undirected graph)
    new_a_t <- (new_a_t + t(new_a_t)) / 2

    return(new_a_t)
  }

  ##############itterate weights one step
  R_flow_c(a_1,F_Ricci_mat_1,F_Ricci_mat_2)->a_t
  return(length(which(unlist(a_t)<0))/length(unlist(a_t)))},1:length(eps_seq))->prop_zero
cbind(eps_seq,prop_zero) ##this will output the optimal eps, here 0.06

####################
eps<-0.06
####################

############define ricci flow
R_flow_c<-function(a_t,F_Ricci_mat_t,F_Ricci_mat_2){
  stop("Function implementation has been masked")
}

############
#define the 4 intermediate states
v<-x[,2]
outer(v,v)->a
a[which(int.red==0)]<-0
a->a_1_1

###
v<-x[,3]
outer(v,v)->a
a[which(int.red==0)]<-0
a->a_1_2

###
v<-x[,4]
outer(v,v)->a
a[which(int.red==0)]<-0
a->a_1_3



Complete the function implementation replacing "stop("Not implemented")".