Bubble Shuffle - Weighted Shuffle

It's possible to conceive of a modification to bubble sort where the "swap" occurs randomly with probability p, rather than by performing a comparison. The result could be called a "bubble shuffle". Elements near the front would be likely to remain there, but have a chance to be shifted to the back of the list.

Modifying a bubble sort stolen from the internet you could come up with the following:

import random

def bubble_shuffle(arr, p):
    arr = copy.copy(arr)
    n = len(arr) 
  
    # Traverse through all array elements 
    for i in range(n-1): 
    # range(n) also work but outer loop will repeat one time more than needed. 
  
        # Last i elements are already in place 
        for j in range(0, n-i-1): 
  
            # traverse the array from 0 to n-i-1 
            # Swap if random number [0, 1] is less than p
            if random.random() < p:
                arr[j], arr[j+1] = arr[j+1], arr[j]
    return arr

This algorithm is order n-squared... but the probability of an element ending up in any particular spot should be computable ahead of time, so it doesn't "need" to be n-squared. Is there a more efficient approach which could be taken?

I'd considered sampling from a geometric distribution and add this to the original index (plus (len(n)-i-1)/len(n) to break ties), but this doesn't give the same dynamic.

from Bubble Shuffle - Weighted Shuffle