where Scatt - means average scattering for clusters and Dens_bw - inter-cluster density.
Lower value -> better clustering.

where distance - distances between cluster centers, k - weighting coefficient equal to distance(Cmax).
Lower value -> better clustering.

pip install --upgrade s-dbw

from s_dbw import S_Dbw
score = S_Dbw(X, labels, centers_id=None, method='Tong', alg_noise='bind',
centr='mean', nearest_centr=True, metric='euclidean')

from s_dbw import SD
score = SD(X, labels, k=1.0, centers_id=None,  alg_noise='bind',centr='mean', nearest_centr=True, metric='euclidean')

• X : array-like, shape (n_samples, n_features)
  List of n_features-dimensional data points. Each row corresponds to a single data point.
• labels : array-like, shape (n_samples,)
  Predicted labels for each sample (-1 - for noise).
• centers_id : array-like, shape (n_samples,)
  The center_id of each cluster's center. If None - cluster's center calculate automatically.
• alg_noise : str,
  Algorithm for recording noise points.
  'comb' - combining all noise points into one cluster (default)
  'sep' - definition of each noise point as a separate cluster
  'bind' - binding of each noise point to the cluster nearest from it
  'filter' - filtering noise points
• centr : str,
  cluster center calculation method (mean (default) or median)
• nearest_centr : bool,
  The centroid corresponds to the cluster point closest to the geometric center (default: True).
• metric : str,
  The distance metric, can be ‘braycurtis’, ‘canberra’, ‘chebyshev’, ‘cityblock’, ‘correlation’,
  ‘cosine’, ‘dice’, ‘euclidean’, ‘hamming’, ‘jaccard’, ‘kulsinski’, ‘mahalanobis’, ‘matching’, ‘minkowski’,
  ‘rogerstanimoto’, ‘russellrao’, ‘seuclidean’, ‘sokalmichener’, ‘sokalsneath’, ‘sqeuclidean’, ‘wminkowski’,‘yule’.
  Default is ‘euclidean’.

• method : str,
  S_Dbw calc method:
  'Halkidi' - original paper [1]
  'Kim' - see [2]
  'Tong' - see [3]

• k: float, The weighting coefficient equal to distance(Cmax). It is necessary for evaluating solutions with vary number of clusters because distance(C) depends on number of clusters [4].

score : float
The resulting S_Dbw or SD score.

1. M. Halkidi and M. Vazirgiannis, “Clustering validity assessment: Finding the optimal partitioning of a data set,” in ICDM, Washington, DC, USA, 2001, pp. 187–194.
2. Youngok Kim and Soowon Lee. A clustering validity assessment Index. PAKDD’2003, Seoul, Korea, April 30–May 2, 2003, LNAI 2637, 602–608
3. Tong, J. & Tan, H. J. Electron.(China) (2009) 26: 258. https://doi.org/10.1007/s11767-007-0151-8
4. Halkidi, Maria & Vazirgiannis, Michalis & Batistakis, Yannis. (2000). Quality Scheme Assessment in the Clustering Process. LNCS (LNAI). 1910. 265-276. 10.1007/3-540-45372-5_26.