Store all the id's of the target elements in an array and then you can loop the above code for your result:
var start = 'logo';
var end = ['link1','link2','link3',....];
for(var i=0;i<end.length;i++){
jsPlumb.connect({
source:start,
target:end[i],
connector: [ "Flowchart", {cornerRadius:1} ],
paintStyle:{
lineWidth:5, strokeStyle:'#3E2522' },
anchors: [[1.02, 0.5, 0, 1], [-0.02, 0.5, 0, 0]],
endpointStyle: { radius:0.5 }
})
}