The following variables are available in profile stats block (it's in template/scripts/BxBaseServiceProfiles.php file):

        $aVars = array(
            'profile_id' => $oProfile->id(),
            'profile_url' => $oProfile->getUrl(),
            'profile_edit_url' => $oProfile->getEditUrl(),
            'profile_title' => $oProfile->getDisplayName(),
            'profile_title_attr' => bx_html_attribute($oProfile->getDisplayName()),
            'profile_ava_url' => $oProfile->getAvatar(),
            'profile_unit' => $oProfile->getUnit(0, array('template' => array(
                'name' => 'unit_wo_info',
                'size' => 'ava'
            ))),
            'profile_acl_title' => _t($aAclInfo['name']),
            'profile_acl_icon' => $aAclInfo['icon'],
            'menu' => BxDolMenu::getObjectInstance('sys_profile_stats')->getCode(),
        );

However there is menu which consist of menu items which can be configured via Studio menu builder.