The silsila's logic is borrowed directly from a much older Islamic discipline: the isnad, the chain of named transmitters that authenticates a report of the Prophet's words or actions, in which a hadith is only as trustworthy as the unbroken, named sequence of people who passed it down. Sufi orders adapted the same verification method to a different kind of transmission, using a named chain of masters, each authorized by the one before, to certify that a teaching or initiation genuinely descends from the order's founder and, through him, back toward the Prophet himself.
Most orders trace this chain through Ali, treating him as the channel through which the Prophet's esoteric teaching passed into the Sufi tradition, which is why so many silsilas converge on the same name a few links back from their own founder. The Naqshbandi order is the well known exception, tracing its chain instead through Abu Bakr, a distinction Naqshbandi writers have long treated as significant, since it ties their order's legitimacy to the same figure Sunni tradition already recognized as the Prophet's first political successor rather than to the line more associated with Shia claims. Written silsila charts, sometimes illuminated and hung in a lodge, served the same evidentiary function a certificate of ordination does elsewhere, and an initiate could be asked to recite the chain from memory as proof of standing within the order.