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If My.Computer.Network.IsAvailable = True Then
  Me.Text = "Сеть доступна"
Else
  Me.Text = "Сеть недоступна"
End If

If My.Computer.Network.Ping("**.**.***.2", 1000) Then

Private Declare Function InternetGetConnectedState Lib "wininet" (ByRef dwFlags As Integer, ByVal dwReserved As Long) As Integer
    Private Enum ConnectStates
        LAN = &H2
        Модем = &H1
        Прокси = &H4
        Отключен = &H20
        Конфигурировано = &H40
        RasInstalled = &H10
    End Enum
 
    Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
        Dim dwFlags As Integer
        Dim Connected As Boolean = (InternetGetConnectedState(dwFlags, 0&) <> 0)
        If Connected Then
            MsgBox("Этот компьютер подключен к сети Интернет")
            Dim ConnectionType As ConnectStates
            For Each ConnectionType In System.Enum.GetValues(GetType(ConnectStates))
                If (ConnectionType And dwFlags) = ConnectionType Then
                    MsgBox("Способ подключения: " & ConnectionType.ToString())
                End If
            Next
        End If
    End Sub