It really depends on your data. From the website you showed, the clusters are don't have the same structure as the ring.

From [review paper on DBSCAN][1]:

1. The model introduced by DBSCAN uses a simple minimum density level estimation, based on a threshold for the number of neighbors, minPts, within the radius ε (with an arbitrary distance measure).

2. Objects with more than minPts neighbors within this radius (including the query point) are considered to be a core point. The intuition of DBSCAN is to find those areas, which satisfy this minimum density, and which are separated by areas of lower density.

3. All neighbors within the ε radius of a core point are considered to be part of the same cluster as the core point (called direct density reachable). If any of these neighbors is again a core point, their neighborhoods are transitively included (density reachable).

So with a lower eps, it has no problem finding the inner ring. If you want to capture the outer ring, it's a matter of finding the distance that allows the core points of in the outer to be connected:

import seaborn as sns
fig, axs = plt.subplots(2, 3,figsize = (12,6))
axs = axs.reshape(-1)
EPS = [0.08,0.1,0.12,0.14,0.16,0.18]
for i in range(len(EPS)):
    db = DBSCAN(eps=EPS[i],min_samples=5).fit(X)
    labels = db.labels_
    sns.scatterplot(x=X[:,0],y=X[:,1],ax=axs[i],hue=pd.Categorical(labels))
    axs[i].set_title("eps = "+str(EPS[i]))
    axs[i].legend(loc='upper right', ncol=3,prop={'size': 6})

[![enter image description here][2]][2]

So we can see that as we increase eps, the members of the outer rings are connected. In this example, most likely something like 0.14 would work (you have some unassigned). [1]: http://www.ccs.neu.edu/home/vip/teach/DMcourse/2_cluster_EM_mixt/notes_slides/revisitofrevisitDBSCAN.pdf [2]: https://i.sstatic.net/a3Whc.png

It really depends on your data. From the website you showed, the clusters are don't have the same structure as the ring.

From review paper on DBSCAN:

1. The model introduced by DBSCAN uses a simple minimum density level estimation, based on a threshold for the number of neighbors, minPts, within the radius ε (with an arbitrary distance measure).

2. Objects with more than minPts neighbors within this radius (including the query point) are considered to be a core point. The intuition of DBSCAN is to find those areas, which satisfy this minimum density, and which are separated by areas of lower density.

3. All neighbors within the ε radius of a core point are considered to be part of the same cluster as the core point (called direct density reachable). If any of these neighbors is again a core point, their neighborhoods are transitively included (density reachable).

So with a lower eps, it has no problem finding the inner ring. If you want to capture the outer ring, it's a matter of finding the distance that allows the core points of in the outer to be connected:

import seaborn as sns
fig, axs = plt.subplots(2, 3,figsize = (12,6))
axs = axs.reshape(-1)
EPS = [0.08,0.1,0.12,0.14,0.16,0.18]
for i in range(len(EPS)):
    db = DBSCAN(eps=EPS[i],min_samples=5).fit(X)
    labels = db.labels_
    sns.scatterplot(x=X[:,0],y=X[:,1],ax=axs[i],hue=pd.Categorical(labels))
    axs[i].set_title("eps = "+str(EPS[i]))
    axs[i].legend(loc='upper right', ncol=3,prop={'size': 6})

So we can see that as we increase eps, the members of the outer rings are connected. In this example, most likely something like 0.14 would work (you have some unassigned).