Detecting Dynamic States of Temporal Networks Using Connection Series Tensors

Figure 4

A schematic illustration of our proposed similarity measure for connection series. Given two connection series, say and , we keep the sequential order of each series unchanged and find the best matching between them. To do this, we roll them into rings and fix one of them, say , as the inner ring, and the other one, say , as the outer ring. Then we rotate the outer ring clockwise element by element until we make a full rotation and obtain multiple cases regarding one-to-one correspondence between elements in the respective rings. In this example, we can obtain 5 matching cases. Note that the number of matching cases is equal to the length of connection series (i.e., length of time window). Then we count the number of matched elements in each case and choose the one that has the most matched elements as the best matching between these two given connection series. Here, case 3 provides the best matching between and , in which the number of matched elements is 4. Finally, we divide the number of matched elements in the best matching case (case 3) by the length connection series (length of the window), where in the example shown here. The final similarity between and is 0.8. The bottom left panel in a light-yellow frame gives a special case when two connection series are different in length, say and . In this case, we fix the shorter one as the inner ring and the longer one as the outer ring, respectively. Then we still keep the one-to-one correspondence of elements in the respective rings and leave the missing digits in the inner ring empty (the red square in the bottom left panel). The rest of the process of finding the best matching is the same as previously. For the final step, we divide the max number of matched elements in the best matching case by the length of the longer connection series and then obtain the similarity between the two given connection series of different lengths.