Distance between very large discrete probability distributions I have 192 countries where each country has some value for 1 million attributes which sum up to 1 (a discrete probability distribution). For any one country most of the values for the attributes are 0. 

Now I am trying to find the distance/similarity between those countries using these attributes. I know we can use  Jensen Shannon Divergence between two discrete probability distributions to get a distance measure, but the caveat is that all the values have to be non-zero.

Given that there are zero valued attributes for the countries, is there any other suitable statistical distance measure that can help me to cluster these countries using these 1 million attributes?
Yes, plenty.

Get the book "encyclopedia of distances".

For example, you can use Histogram Intersection distance. Since your data is already normalized, that reduces to Manhattan distance, if I am not mistaken. Yes: this can be appropriate for distributions.