I am sorry to tell that it does not work:
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-25-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-24-f615466d757b> 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 

/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-24-f615466d757b> in processed_file_names(self)
     21     def processed_file_names(self):
     22         # Try to replicate names defined in process()
---> 23         return [ 'data_{}.pt'.format(i) for i in range(len(self.raw_paths)) ]
     24 
     25     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-24-f615466d757b> in raw_file_names(self)
     12     @property
     13     def raw_file_names(self):
---> 14         return self.__input_data_paths
     15     @raw_file_names.setter
     16     def raw_file_names(self, value):

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

I am not very clear about what you mean by “same name as property” in init() function. Do you mean I should change my variable name “input_data_paths” to “raw_file_names”, like this?
class MyOwnDataset(Dataset):
    def __init__(self, root, input_data_paths, 
                 transform=None, pre_transform=None):
        super(MyOwnDataset, self).__init__(root, transform, pre_transform)
        self.raw_file_names = input_data_paths

    @property
    def raw_file_names(self):
        return self.__raw_file_names

    @raw_file_names.setter
    def raw_file_names(self, value):
        self._raw_file_names = value

Still, the same error.
AttributeError: 'MyOwnDataset' object has no attribute '_MyOwnDataset__raw_file_names'