Homer's epics, The Iliad and The Odyssey, were important to Tolstoy in his youth and again in his middle age, when he read them in their original Greek. These narrative poems about the Trojan War and its aftermath, composed around 700 B.C.E., may seem far removed from the world of Tolstoy's novels. Yet, in them, as in everything he read, Tolstoy found kindred features and made them his own.
Tolstoy admired, among other things, the way that Homer gave "a picture of manner and customs based on historical event." This is what he himself had done in War and Peace. The epic proportions of War and Peace, its depiction of the chaos and suffering of warfare, its concern with collective and individual identity, its extended similes, along with the way it blurs the boundaries between war and peace, between the masculine and the feminine, between enemy and friend—all these are features of War and Peace that make it a worthy successor to Homer's epics.
The plot of Anna Karenina, like The Iliad, is set in motion by adultery. According to legend, the Trojan War was waged because Helen of Troy, the wife of Menelaus, falls in love with Paris and follows him home to Troy, leaving her husband and child behind. Menelaus and the other Greeks wage war to get her back. At various points they ask whether it was all worth it. Anna Karenina contains a number of references to the adultery of Helen of Troy. The cuckolded Karenin is tempted to behave heroically, but he's not sure what this entails. What measures should he take? Challenging Vronsky to a duel is one possibility. But forgiveness is another.
Whereas adultery sets The Iliad in motion, the goal of The Odyssey is marriage and family happiness. This epic describes Odysseus's tortuous homecoming after the Trojan War ends. His faithful wife and dutiful son are waiting for him back in Ithaca and this is what draws him home, despite the temptations along the way. Levin's plotline is more along the lines of The Odyssey and it arrives at the same destination.