Q:

How is Abiather linked to the house of Eli?

In 1 Samuel 2, Eli is cursed for letting his sons defile the house of God:

27 Then a man of God came to Eli and said to him, “Thus says the LORD: ‘Did I not clearly reveal  Myself to the house of your father when they were in Egypt in Pharaoh’s house? 28 Did I not  choose him out of all the tribes of Israel to be My priest, to offer upon My altar, to burn incense,  and to wear an ephod before Me? And did I not give to the house of your father all the offerings of  the children of Israel made by fire? 29 Why do you kick at My sacrifice and My offering which I  have commanded in My dwelling place, and honor your sons more than Me, to make yourselves  fat with the best of all the offerings of Israel My people?’ 30 Therefore the LORD God of Israel  says: ‘I said indeed that your house and the house of your father would walk before Me forever.’  But now the LORD says: ‘Far be it from Me; for those who honor Me I will honor, and those who  despise Me shall be lightly esteemed. 31 Behold, the days are coming that I will cut off your arm  and the arm of your father’s house, so that there will not be an old man in your house.

Years later during the reign of king Solomon Abiather is removed from priesthood:

26 And to Abiathar the priest the king said, “Go to Anathoth, to your own fields, for you  are  deserving of death; but I will not put you to death at this time, because you carried the ark of the  Lord GOD before my father David, and because you were afflicted every time my father was  afflicted.” 27 So Solomon removed Abiathar from being priest to the LORD, that he might fulfill  the word of the LORD which He spoke concerning the house of Eli at Shiloh. (1 kings 2:26-28) 

It is said this was a fulfillment of the word the word spoken by the prophet to Eli. I have been trying to link the house of Eli to the house of Abiather but I was not successful!
How are they linked?

A:

Abiathar was evidently Eli's great-great-grandson(Abiathar the son of Ahimelech, the son of Ahitub, the son of Phinehas, the son of Eli - 1 Samuel 22:20, 14:3).

