shortest.js

function shortest(graph)
{
    var paths = [];
    var counts = graph.map(_ => 0);
    
    for (var startNode of graph.keys())
    {
        paths[startNode] = [];
        var visited = [startNode];
        
        function visit(node, path)
        {
            if (path.length)
            {
                visited.push(node);
                if (!paths[node] || !paths[node][startNode]) // record this path unless it's the reverse of an existing one 
                {
                    paths[startNode][node] = path;
                    for (var n of path.slice(0, path.length-1))
                    {
                        counts[n]++;
                    }
                }
            }
            
            for (var nextNode of graph[node])
            {
                if (visited.indexOf(nextNode) == -1)
                {
                    visit(nextNode, path.concat([nextNode]));
                }
            }
        }
        
        visit(startNode, []);
    }
    
    var sortedCounts = counts.slice().sort((a, b) => b - a);   
    return "node " + counts.indexOf(sortedCounts[0]) + " is on " + sortedCounts[0] + " paths"; 
}

var test = [[1,4], [2,3], [1], [1], [0,5], [4]];
var result = shortest(test);
console.log(result);