What happens to Greek society when Odysseus leaves his home in Ithaca to go on his epic journey?

The most noticeable change to Greek society in terms of
Homer's story The Odyssey is what occurs in Ithaca, specifically
with regard to Odysseus' home.

When Odysseus travels to
Troy to fight in the Trojan War, he is gone ten years. As he starts his voyage home, he
visits extraordinary places, is held captive for several years, and angers a god or two.
His journey home stretches out for ten more
years.

Meanwhile, things are happening at his home—stemming
from the href="http://dictionary.reference.com/browse/ignoble">ignoble behavior of
noblemen of Ithaca.

In Book Two, in front of an Assembly of
the people of Ithaca, Telemachus, son of Odysseus, describes two disasters that have
befallen his home. First, Odysseus has not returned from the war. Second, a group of
suitors—sons of the most prominent of Ithaca's citizens—have taken over his
home.

Like vultures sensing something dead, suitors for
Penelope's hand have descended upon her in the absence of her husband, and taken up
residence in Odysseus and Penelope's home. The men believe that Odysseus is
dead.

The suitors break the laws of hospitality not only by
coming uninvited, but also by staying—and refusing to leave.
Telemachus is really too young and too inexperienced to throw these unwanted guests out.
He was born just after Odysseus left, so he is not yet twenty. There are those in the
household who are loyal to Penelope, but others have welcomed these enemies of Odysseus
who want nothing other than to take possession not only of his home and his belongings,
but to take Penelope to wife as well.

So Telemachus goes to
the Assembly to ask their help. None of the men of the council come to his aid, but
several of the suitors verbally attack Telemachus. Antinous accuses Penelope of leading
all of them on with no intent of marrying any of
them.

Telemachus notes that to marry Penelope to another
man, he would have to send her home to her father, but he believes that this would be
unjust, and that the gods would punish him. He tells Antinous that if he is unhappy at
Odysseus' home, he should leave. If he does not, he will have to reckon with the
gods:

...If
you choose to take offense at this, leave the house and feast elsewhere at one another's
houses at your own cost turn and turn about. If, on the other hand, you elect to persist
in spunging upon one man, heaven help me, but Zeus shall reckon with you in full, and
when you fall in my father's house there shall be no man to avenge
you.

Two eagles from the
heavens fly into the room, fight and then depart. Halitherses, the prophet, sees the
eagles as an omen and delivers a prophecy. He warns the Assembly that Odysseus
will return; the suitors must leave Odysseus' home willingly; and,
that the suitors are not the only ones who need to listen to
him.

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...indeed [Odysseus] is close at hand to deal
out death and destruction, not on [the suitors] alone, but on many another of us who
live in Ithaca. Let us then be wise in time, and put a stop to this wickedness before he
comes.

Another of the
suitors, Eurymachus, threatens the prophet (that he should go home or his children will
"miss" him:

Go
home, old man, and prophesy to your own children, or it may be worse for
them.

Eurymachus tells him
that the eagles mean nothing—Odysseus is dead. In all, no one is
willing to help Telemachus. The noble society and its laws have declined in Odysseus'
absence, evident not only in how the suitors act, but in the way the men of the council
fail to stop the wicked behavior of the suitors.