The reason why there us an absence of the description of the roman bishops authority in the canons of nicaea and various other ecumenical councils is because (as many fathers testify) it was not given by a council. It is a divine charism arising from his primacy that the pope is the highest authority in the church.
The Church historian Socrates Scholasticus relates the following in the 4th century :
Church History 2:8:
Maximus, however, bishop of Jerusalem; who had succeeded Macarius, did not attend, recollecting that he had been deceived and induced to subscribe the deposition of Athanasius. Neither was Julius, bishop of the great Rome, there, nor had he sent a substitute, Although an ecclesiastical canon commands that the churches shall not make any ordinances against the opinion of the bishop of Rome.
Church History 2:15:
Athanasius, meanwhile, after a lengthened journey, at last reached Italy. The western division of the empire was then under the sole power of Constans, the youngest of Constantine’s sons, his brother Constantine having been slain by the soldiers, as was before stated. At the same time also Paul, bishop of Constantinople, Asclepas of Gaza, Marcellus of Ancyra, a city of the Lesser Galatia, and Lucius of Adrianople, having been accused on various charges, and expelled from their several churches arrived at the imperial city. There each laid his case before Julius, bishop of Rome. He on his part, by virtue of the Church of Rome’s peculiar privilege, sent them back again into the East, fortifying them with commendatory letters and at the same time restored to each his own place, and sharply rebuked those by whom they had been deposed. Relying on the signature of the bishop Julius, the bishops departed from Rome, and again took possession of their own churches, forwarding the letters to the parties to whom they were addressed.
Church History 2:17:
On the receipt of these contradictory communications, Julius first replied to the bishops who had written to him from Antioch, complaining of the acrimonious feeling they had evinced in their letter, and charging them with a violation of the canons, because they had not requested his attendance at the council, seeing that the ecclesiastical law required that the churches should pass no decisions contrary to the views of the bishop of Rome: he then censured them with great severity for clandestinely attempting to pervert the faith; in addition, that their former proceedings at Tyre were fraudulent, because the investigation of what had taken place at Mareotes was on one side of the question only; not only this, but that the charge respecting Arsenius had plainly been proved a false charge. Such and similar sentiments did Julius write in his answer to the bishops convened at Antioch…
Also Bishop St. Optatus of Milevis (June 4) says in** 367** [On the Schism of the Donatists 2:2 in PL 11:946A-947A]:
You cannot then deny that you do know that upon Peter first in the City of Rome was bestowed the Episcopal Chair [Cathedra], on which sat Peter, the Head of all the Apostles (for which reason he was called Cephas), that, in this one Chair, unity should be preserved by all, lest the other Apostles might claim–each for himself–separate Chairs, so that he who should set up a second Chair against the unique Chair would already be a schismatic and a sinner. Well then, on the one Chair, which is the first of the Endowments, Peter was the first to sit.