Thank you alex.veuthey!
While referencing the example you provided, I modified may Dataset class as following:
class MyOwnDataset(Dataset):
    def __init__(self, root, input_data_paths, 
                 transform=None, pre_transform=None):
        super(MyOwnDataset, self).__init__(root, transform, pre_transform)
        # Change to private variable
        #self.input_data_paths = input_data_paths     # original line
        self._input_data_paths = input_data_paths

        
    def _download(self):
        # Don't need to download
        pass
    
    @property
    def raw_file_names(self):
        # Change to private variable 
        # return self.input_data_paths    # original line
        return self.input_data_paths

    # My property setter here
    @raw_file_names.setter
    def raw_file_names(self, value):
        self.raw_file_names = value

    # no more changes...

But with no luck, similar error was returned:
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-6-d69608eea4a9> in <module>
      4 # Create dataset
      5 train_params = {'batch_size':1, 'shuffle': True, 'num_workers': 1}
----> 6 train_dataset = MyOwnDataset("/tmp/Data/train/", input_paths)

<ipython-input-5-c37d75e54a81> in __init__(self, root, input_data_paths, transform, pre_transform)
      2     def __init__(self, root, input_data_paths, 
      3                  transform=None, pre_transform=None):
----> 4         super(MyOwnDataset, self).__init__(root, transform, pre_transform)
      5         #self.input_data_paths = input_data_paths
      6         self._input_data_paths = input_data_paths

/usr/local/lib/python3.6/dist-packages/torch_geometric/data/dataset.py in __init__(self, root, transform, pre_transform, pre_filter)
     81 
     82         self._download()
---> 83         self._process()
     84 
     85     @property

/usr/local/lib/python3.6/dist-packages/torch_geometric/data/dataset.py in _process(self)
    119 
    120     def _process(self):
--> 121         if files_exist(self.processed_paths):  # pragma: no cover
    122             return
    123 

/usr/local/lib/python3.6/dist-packages/torch_geometric/data/dataset.py in processed_paths(self)
    108         r"""The filepaths to find in the :obj:`self.processed_dir`
    109         folder in order to skip the processing."""
--> 110         files = to_list(self.processed_file_names)
    111         return [osp.join(self.processed_dir, f) for f in files]
    112 

<ipython-input-5-c37d75e54a81> in processed_file_names(self)
     22     def processed_file_names(self):
     23         # Try to replicate names defined in process()
---> 24         return [ 'data_{}.pt'.format(i) for i in range(len(self.raw_paths)) ]
     25 
     26     def __len__(self):

/usr/local/lib/python3.6/dist-packages/torch_geometric/data/dataset.py in raw_paths(self)
    101     def raw_paths(self):
    102         r"""The filepaths to find in order to skip the download."""
--> 103         files = to_list(self.raw_file_names)
    104         return [osp.join(self.raw_dir, f) for f in files]
    105 

<ipython-input-5-c37d75e54a81> in raw_file_names(self)
     13     @property
     14     def raw_file_names(self):
---> 15         return self._input_data_paths
     16     @raw_file_names.setter
     17     def raw_file_names(self, value):

AttributeError: 'MyOwnDataset' object has no attribute '_input_data_paths'

I am still trying to figure out how properties are handled in torch_geometric.data.Dataset classes.