Shares of Sony rose on Monday, a day after the firm said it would resume some services on its PlayStation Network this week and offer customer incentives following the theft of personal information from 78 million user accounts.

Many PlayStation users around the world were frustrated that the first warning of one of the largest Internet security break-ins ever came a week after Sony detected a problem with the network on April 19.

Analysts said it was too early to say whether the measures the consumer electronics giant unveiled on Sunday would be enough to stop disgruntled gamers leaving the network and warned restoring faith in its security system would take time.

Sony has touted online services as a way of leveraging the synergies between its unique combination of hardware and content, including films and music as well as games.

Damage has been done to Sony whatever the scale of the content giveaway at this point, and Sony is facing a prolonged effort to regain customer trust, said Jay Defibaugh, director of equities research at MF Global in Tokyo.

Anything that undermines consumer willingness to divulge credit card details to Sony is a problem for the network strategy, he added.

Sony said on Sunday it would offer some free content, including 30 days of free membership to a premium service to existing users and in some regions pay credit card-renewal fees, but added compensation would only be paid if users suffered damage.

The news sparked thousands of comments on the official PlayStation fan page on Facebook, some of them from users who said they would switch to Microsoft's Xbox Live games network.

Shares in Sony were up 2 percent to 2,305 yen, after falling 4.5 percent on Thursday, ahead of a holiday on Friday. But analysts said concerns about the leak would weigh on investor sentiment.

At minimum, having to suspend the service, fix its problems and deal with the aftermath, looks set to cost (Sony) tens of billions of yen, said analyst Nobuo Kurahashi of Mizuho Investors Securities.

I don't think anyone knows where they will be able to absorb this loss, nor how much it will be, and that'll weigh on share prices going ahead, he added.

The incident has sparked legal action and investigations by authorities in North America and Europe, home to almost 90 percent of the users of the network, which enables gamers to download software and compete with other members.

Sony is the latest Japanese company to come under fire for how it has disclosed bad news.

Tokyo Electric Power Co was criticized for how it handled the nuclear crisis after the March 11 earthquake. Last year, Toyota Motor Corp was slammed for being less than forthright about problems over a massive vehicle recall.

(Reporting by James Topham and Isabel Reynolds; Editing by Edwina Gibbs and Joseph Radford)