The Effects of Randomness on the Stability of Node Embeddings

Below we provide an overview of all our experiments and how they can be executed.

Experiments

All Python code (when not explicitly marked other) must be run with python 3.6, since the used tensorflow version 1 is not available in pypi.org for newer python versions. All used external libraries are with version numbers in requirements.txt.

We recommend to run the code within a virtual enviroment and install requirements with in it:

    python3.6 -m venv venv
    source venv/bin/activate
    pip install -r requirements.txt

Embedding Generation

Scripts for all our used embedding methods (GraphSAGE, LINE, HOPE, node2vec, SDNE) are located in the train/ directory. If you do not want to specify a directory in which the embeddings are saved, there needs to be subdirectory results/. They can be called as follows:

• GraphSAGE [Source]:

  python3 -m train.graphsage {LOCATION OF GRAPH} -num-embeddings=30  --seed {SEED}

  The reference implementation linked above is not compatible with newer versions of python and networkx. We ported the source code accordingly, but made no further changes to the code. The code is located at lib/graphsage_embedding/.

• HOPE [Source]:

  python3 -m train.hope_generate_embeddings {LOCATION OF GRAPH} --num_embeddings 30 --seed {SEED}

  An installation of GEM is required. Please see their documentation on how to install GEM.

• LINE [Source]:

  python3 -m train.line_generate_embeddings {LOCATION OF GRAPH} --num_embeddings 30 --seed {SEED}

  A compiled version of LINE in lib/line_embedding/COMPILED_LINE/ is needed (line, normalize and reconstruct). We changed line.cpp to add an argument to seed the initialization of the embedding. Please see their documentation on how to install and compile LINE, but use our modified version. The file can be found in lib/line_embedding/line.cpp.

• node2vec [Source]:

  python3 -m train.node2vec {LOCATION OF GRAPH} -num-embeddings=30 -no-alias --seed {SEED}

  We made changes to the reference implementation, in particular to the random walk procedure. The changed code can be found in lib/node2vec_embedding/randomwalk.py.

• SDNE [Source]:

  The SDNE reference implementation requires python version 2.7 for execution. For detailed package versions see below.

  python2 -m train.sdne_generate_embeddings -c {LOCATION OF CONFIG FILE}

  The following package versions are required to properly execute the SDNE implementation:

  Library	Version
  python	==2.7
  tensorflow[-gpu]	==1.12.0
  scikit-learn	==0.20.4
  configparser	*

  We slighlty adjusted the config file structure given by the reference implementation as well as the model to fix random seeding. Full details for configuration parameters can be found in the reference implementation. The following changes were made to the configuration structure:

  • [Output]:
    • removed embedding_filename since the filename for a single embedding is fixed
    • added num_embeddings (expecting an int) to set the numbers of embeddings that will be created (usually set to 30)
  • [Model_Setup]:
    • seed (expecting an int) added to fix a random seed (useful for reproducibility)

  Exemplary Pipfile (specifying the required package versions) and .ini config files are included at /lib/sdne_embeddings/.

Embeddings will be stored in plain text in train/results/ as {EMBEDDING ALGORITHM}_{GRAPH NAME}_{ITERATION}.emb.

In our case, all graphs were stored inside a graphs/ directory, such that {LOCATION OF GRAPH} was one of the following:

• barabasi_albert_graphs/ba_N{NUM}_d01.edgelist with {NUM} in 1000, 2000, 4000, 8000, 16000, 32000, 64000, 128000.
• barabasi_albert_graphs/ba_N8000_d{DEG}.edgelist with {DEG} in 00025, 0005, 001, 002, 005, 02, 05, 1.
• BlogCatalog/
• Facebook
• Protein/
• Cora/
• watts_strogatz_graphs/ba_N{NUM}_d01.edgelist with {NUM} in 1000, 2000, 4000, 8000, 16000, 32000, 64000, 128000.
• watts_strogatz_graphs/ba_N8000_d{DEG}.edgelist with {DEG} in 00025, 0005, 001, 002, 005, 02, 05, 1.
• Wikipedia/

For reproducibility, an optional seed parameter (integer) can be part of the arguments. This will set the random number generators from numpy, tensorflow etc. Keep in mind, that for LINE, the seed will only fix the random initialization of the embeddings due to the usage of ASGD (only restricting ourselves to a single thread would eliminate this randomness).

Graph Sources

We downloaded the empirical graphs for our study at these locations:

• BlogCatalog: http://socialcomputing.asu.edu/datasets/BlogCatalog3
• Cora: http://konect.uni-koblenz.de/networks/subelj_cora
• Facebook: https://snap.stanford.edu/data/gemsec-Facebook.html (we only used the government_edges.csv graph from this dataset)
• Protein: https://snap.stanford.edu/node2vec
• Wikipedia: https://snap.stanford.edu/node2vec/

Similarity Tests

All scripts related to our similarity experiments are located in the corresponding directory similarity_tests/.

For some experiments, graph-tool is required, which is not automatically installed when installing the dependencies from our requirements.txt file. Consult their official documentation on how to install graph-tool on your system. We used version 2.29 in our experiments.

For all experiments, a set of embeddings - corresponding to the graph(s) in the experiment - is required. We conduct experiments to measure the similarity of two different embeddings on the same graph by the same algorithm. Our measures (with abbreviations to be used in the similarity tests script) are

• Aligned cosine similarity cossim

• Naive k-nearest-neighbor overlap knn

• Jaccard similarity of k-nearest-neighbor sets jaccard

• Ranking Invariance of k-nearest-neighbor sets ranks

• Ranked Neighborhood Stability of k-nearest-neighbor sets rns

• k-NN Angle Divergence angdiv

• Second order cosine similarity 2ndcos

• Variants of aligned cosine similarity cosine similarity with respect to different kind of neighbors

To run all experiments and see the results (for all graphs and algorithms), do the following:

1. Use lib/tools/get_node_info.py to generate files containing information for nodes of a graph. The graphs have to be available in graphml format. Also, a functional installation of graph-tool is required.
2. Run the similarity test script (running for all graphs). For a detailed description of the command, see the corresponding section.
3. Use the IPython notebooks to visualize the results:
   • general_stability_plots.ipynb: Distribution plots of similarity values.
   • neighbor_variance_plots_agg.ipynb: Average mean absolute deviation in degrees for different kind of neighbors.
   • node_centrality_plots.ipynb: Similarity measure over node property (e.g. mean aligned cosine similarity over coreness).
   • synth_plots.ipynb: Similarity measures over a graph property such as density or size.

Additionally, tables.ipynb can be used to generate tables of different similarity measures. The columns are graphs; rows are algorithms. The cell values are aggregated in one of the following ways:

• Mean of means
• Standard deviation of means
• Mean of standard deviations

Similarity Test Script

python3 similarity_tests.py -a {ALGORITHMS} -d {GRAPHS} -t {TESTS} --knn-size {KNN_SIZE} --num-nodes {NUM_NODES} --emb-dir {PATH_TO_EMBEDDINGS} --nodeinfo-dir {PATH_TO_NODEINFO_DIRECTORY} --results-dir {PATH_TO_SAVE_DIRECTORY}

where {ALGORITHMS} is one or more of line, hope, sdne, graphsage, node2vec. If the option -a is omitted, all algorithms will be used in the experiment. {GRAPHS} are identifiers of graphs, e.g., ba_N8000_d1 or cora. If the option -d is omitted, the experiment will run with default values for the names, that might not match your file names. The -t option specifies which experiments will be conducted. {TESTS} is one or more of cossim, knn, jaccard, ranks, rns, angdiv, 2ndcos, cos, orthtf, tfcos, lintf, lincos. If omitted, all experiments will be run.

The size of the neighborhoods for the nearest-neighbor-based measures can be specified with --knn-size.

With --num-nodes, the number of nodes of a graph that will be used in the experiment can be specified. Providing a negative value will select all nodes and corresponds to default behavior. {PATH_TO_EMBEDDINGS}, {PATH_TO_NODEINFO_DIRECTORY}, {PATH_TO_SAVE_DIRECTORY} are required to run the script. Note that --results-dir must have a subdirectory called procrustes_matrices.

Additional convenience parameters are documented within the argparse of the script.

Note: Ranked neighborhood stability rnscan only be computed if Jaccard similarity jaccard and ranking invariance ranks for the same value of --knn-size have been computed beforehand.

Plots

• Plotting of similarity test results is partly integrated with conducting the experiments. All plots are generated in IPython notebooks.
  • general_stability_plots.ipynb: Distribution plots of similarity values.
  • neighbor_variance_plots_agg.ipynb: Average mean absolute deviation in degrees for different kind of neighbors.
  • node_centrality_plots.ipynb: Similarity measure over node property (e.g. mean aligned cosine similarity over coreness).
• synth_plots.ipynb: Similarity measures over a controlled graph property such as density or size.

Downstream Classification

All scripts related to our experiments for downstream classification are located in the corresponding directory downstream_classification/.

We tested the performance of AdaBoost, Decision Trees, Neural Networks and Random Forests on all considered graphs and all generated embeddings.

We made experiments conserning the change in accuracy over different embeddings for the same graph using the same embedding algorithm and how much predictions overlap (i) for multiple runs of the classifier and (ii) for runs on different algorithms. These experiments can be run as follows:

Node Classification

• Accuracy:

  python3 -m downstream_classification.classify {LOCATION OF GRAPH} {LOCATION OF ALL EMBEDDINGS} -model=one_hidden

• Overlap between multiple runs of the classifier on the same embedding:

  python3 -m downstream_classification.overlap_classifier {LOCATION OF GRAPH} {LOCATION OF EMBEDDING} adaboost decision_tree neural_network random_forest

  We repeated this experiment for five different embeddings for each graph and embedding method. We then averaged the results over these five different embeddings.

• Overlap between different embeddings:

  python3 -m downstream_classification.overlap_embeddings {LOCATION OF GRAPH} {EMBEDDINGS} {CLASSIFIER}

  where {EMBEDDINGS} is the list of all embeddings of an embedding algorithm for the specified graph. {CLASSIFIER} can be one of adaboost, decision_tree, neuronal_network and random_forest.

Results for all downstream tasks are placed in downstream_classification/results/. For accuracy results, the results are stored in subfolders with the respective classifier name as dirname. The file contains as first line information about the task, and then for each run (cross-validation and repetitions) a line with training and test accuracy.

Results for the other tasks are stored inside the compare_embedding_errors subfolder, for the second task prefixed with self_. The file contains a upper triangle matrix with the pairwise number of nodes with different predictions.

Link Prediction

The experiments are analogue to those on node classification

For Link Prediction, 10% of the edges of graph must be deleted before computing the embeddings. These deleted edges are later used for traning, together with sampled non-edges. The Graphs can be reduced via

    python -m downstream_classification.remove_edges_from_graph {GRAPH_NAME} [--relabel-nodes]

The remove takes the larges connected component and only deletes edges such that the resulting graph keeps connected. The option --relabel-nodes is important when the graph is initinally not connect. When set, the nodes will be relabeled with ascending integer numbers, since it is needed for comnputing the stable core.

The corresponding class are:

• Accuracy (AUC)

    python -m downstream_classification.link_prediction {PATH_TO_REDUCED_GRAPHS} {PATH_TO_EMBEDDING} -classifiers { CLASSIFIER }

All individual predictions are saved, so this scipt also generates all needed information to plot stable core

Plots

Results for the downstream classification tasks can be visualized in two plots:

• Accuracy results are visualized in a letter plot, code can be found in the plot_scripts/plot_accuracies.py script.
• Results of both overlap experiments are plotted in one graph, which can be generated by executing downstream_classification/plot_avg.overlap.py. The plots will be saved inside the plots/ directory. Execute the script as follows:
• Link Prediction plots can be created with plot_scripts/plot_link_prediction.py and the corresponding table core with downstream_classification/plot_link_stable_core

All scripts should be called as modules from the base directory, e.g,

python3 -m downstream_classification.plot_avg_overlap

node2vec case study

The experiment for node2vec can be found within the folder lib/node2vec_pytorch_zerlegt

Additional Comment

If you do not want to run the files from the top-level directory, it may be necessary to add the directory to your $PYTHONPATH.