How to check internet connectivity type in Universal Windows Platform

1. Check Internet Connection Availability

To check whether any network connection is established or not use GetIsNetworkAvailable method of NetworkInterface class.

bool isNetworkConnected = NetworkInterface.GetIsNetworkAvailable();

GetIsNetworkAvailable() -
Summary: Indicates whether any network connection is available.
Returns: true if a network connection is available; otherwise, false.

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2. Check Internet Connection Availability via WWLN (WiFi)

To check whether internet connected via WWAN use IsWlanConnectionProfile property of ConnectionProfile class

ConnectionProfile InternetConnectionProfile = NetworkInformation.GetInternetConnectionProfile();
bool isWLANConnection = (InternetConnectionProfile == null)?false:InternetConnectionProfile.IsWlanConnectionProfile;

IsWlanConnectionProfile
Summary: Gets a value that indicates if connection profile is a WLAN (WiFi) connection. This determines whether or not WlanConnectionProfileDetails is null.
Returns: Indicates if the connection profile represents a WLAN (WiFi) connection.

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3. Check Internet Connection Availability via WWAN (Mobile)

To check whether internet connected via WWAN use IsWwanConnectionProfile property ofConnectionProfile class

ConnectionProfile InternetConnectionProfile = NetworkInformation.GetInternetConnectionProfile();
bool isWLANConnection = (InternetConnectionProfile == null)?false:InternetConnectionProfile.IsWwanConnectionProfile;

IsWwanConnectionProfile
Summary: Gets a value that indicates if connection profile is a WWAN (mobile) connection. This determines whether or not WwanConnectionProfileDetails is null.
Returns: Indicates if the connection profile represents a WWAN (mobile) connection.

Reference
Hippiehunter Answer

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4. Check Metered network

To check whether Internet reachable via a metered connection or not, use GetConnectionCost method on NetworkInterface class.

var connectionCost = NetworkInformation.GetInternetConnectionProfile().GetConnectionCost();
if (connectionCost.NetworkCostType == NetworkCostType.Unknown 
        || connectionCost.NetworkCostType == NetworkCostType.Unrestricted)
{
    //Connection cost is unknown/unrestricted
}
else
{
   //Metered Network
}

Reference (More detailed answer here)
1. How to manage metered network cost constraints - MSDN
2. NetworkCostType Enum - MSDN

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5. Manage network availability changes

To sense the significant network availability changes, use eventNetworkStatusChanged of NetworkInformation class

// register for network status change notifications
 networkStatusCallback = new NetworkStatusChangedEventHandler(OnNetworkStatusChange);
 if (!registeredNetworkStatusNotif)
 {
     NetworkInformation.NetworkStatusChanged += networkStatusCallback;
     registeredNetworkStatusNotif = true;
 }

async void OnNetworkStatusChange(object sender)
{
    // get the ConnectionProfile that is currently used to connect to the Internet                
    ConnectionProfile InternetConnectionProfile = NetworkInformation.GetInternetConnectionProfile();

    if (InternetConnectionProfile == null)
    {
        await _cd.RunAsync(CoreDispatcherPriority.Normal, () =>
        {
            rootPage.NotifyUser("Not connected to Internet\n", NotifyType.StatusMessage);
        });
    }
    else
    {
        connectionProfileInfo = GetConnectionProfile(InternetConnectionProfile);
        await _cd.RunAsync(CoreDispatcherPriority.Normal, () =>
        {
            rootPage.NotifyUser(connectionProfileInfo, NotifyType.StatusMessage);
        });
    }
    internetProfileInfo = "";
}

References
Check Internet Connectivity - developerinsider.co

How to manage network connection events and changes in availability - MSDN

How to retrieve network connection information- MSDN

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Hope it helpful to someone.

I use NetworkInformation.GetInternetConnectionProfile().IsWlanConnectionProfile and IsWwanConnectionProfile. If neither is true, it should mean you're on Ethernet or something like that.

Keep in mind thatGetInternetConnectionProfile() can return null and can falsely return that there is an active internet connection when the connection is active but DHCP has failed.