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The entire procedure for calculating the role descriptors is sketched in Figure 11.2 and can be summarized as outlined in Algorithm 9. The input for the method is the entire graph dataset G, a list of node labels L, a list of teleportation parametersαs, and the approximation threshold . For each of the nodes v in G, the algorithm stacks the entropy-based representations of the corresponding APPR vectors, denoted as pαv, to generate the role descriptor ofv, i.e., embv. The algorithm finally fits a classification model on the collection of role descriptors and retrieves the resulting model for node classification.

in terms of accuracy as well as to demonstrate the efficiency in terms of computation time.

Structural Node Embedding

We first consider the structural node embeddings and compare the embed-dings retrieved by our approach to the representations retrieved by RolX, GraphWave and struc2vec. For all competitors, we used the implementations and recommended configurations as published by the authors. Theα param-eter which is required to be set for our approach is ranged from 0.1 to 0.9 with a step size of0.1. For each of the parameters, we get a one dimensional descriptor for every node, referred to as APPRrolesα=i. We also construct higher dimensional representations by combining the node descriptors for all values ofαper node. We refer to this configuration as APPRrolesstacked. The approximation threshold for the computations of the APPR vectors is set to= 1E−4.

Barbell Graph

In Figure 11.3 we provide a visual analysis of how well the APPR based node representations are able to embed the structural properties of the node neighborhoods. Figure 11.3(a) depicts the barbell graph while the remaining plots show 1-dimensional structural node embeddings. For visualization pur-poses, we project higher dimensional embeddings into 1-dimensional spaces by using PCA to be able to discuss the outcomes in comparison to our 1-dimensional representations. Therefore, we use the node identifiers on the y-axes in our plots to spread the depicted 1-dimensional embeddings along the y-axes such that they do not cover each other. Figures 11.3(b)-11.3(d) show the structural node embeddings for the competitors when projecting them onto the first principal component and normalizing the values (cf. x-axes). The first three images in the lower row depict the results for our approach when using different values for α. Precisely, we show the normal-ized results forα ∈ {0.9,0.5,0.1}, which means that we range the exploration of the node neighborhoods from local (α = 0.9) to spacious (α = 0.1). In Figure 11.3(h), we visualize the node embeddings when stacking the descrip-tors for all values of α and projecting these embeddings into 1-dimensional space, again using PCA.

The first thing we want to emphasize is that since we do not use any external evaluation measures in this experiment due to lack of ground truth and it is thus difficult to say which of the methods work best on this dataset.

This also strongly depends on how to define the roles in this graph. What

we can clearly see is that RolX and GraphWave reveal only very few roles, i.e., most nodes of the chain are considered to have the same role. This also applies to struc2vec as we can identify three clusters in the 1-dimensional projection. However, differently to the other methods, struc2vec identifies the node in the center of the chain as an own role. Given our representations, we can see that our approach is much more flexible in terms of role identification.

When using a rather large value for α, i.e., defining roles only based on very local neighborhoods, our method also considers the chain elements to have the same role. However, when decreasing the value ofα, i.e., enlarging the neighborhoods based on which to define the roles, we can observe that the roles of the nodes in the chain are considered to differ more and more.

Nevertheless, the chain nodes that have the same hop distance from the center of the chain are always considered to have the same role. Given that roles are not always defined precisely, this is a very desirable property of our approach that RolX or struc2vec cannot fulfill. GraphWave might be able to have this flexibility but due to operating on the spectra of the graphs rather than in the vertex domain, it might be difficult to set the corresponding parameter appropriately.

Summarizing the insights revealed by this experiment, we can state that the choice of the value forα might have a significant impact on the outcome.

In other words, using a smallαvalue leads to an accurate distinction between node roles, while rather large values of α do not distinguish as accurately between different roles.

Mirrored Karate Network

Next, we consider the mirrored Karate network for which we measure the performance of the embedding methods by doing 1-NN-range queries. Recall that this network consists of two copies of the Zachary’s Karate network and an additional edge which links a randomly chosen node with its copy. Given a set of structural node embeddingsE that contains one embedding for each node of the mirrored karate network, the goal of this experiment is to identify the copy of the query node among its 1-nearest-neighbors. Note that we call this query 1-NN-range query to emphasize that the set of 1-nearest-neighbors might be of size greater than 1. Precisely, we compute the nearest neighbor ofor each of the query pointsq∈ E, and subsequently perform a range query aroundq with distance dist(o, q).

The results given in Table 11.1 show the accuracy and average size of the queries’ candidate sets, i.e., ∅|C|. Except for struc2vec, all methods achieve an accuracy of 100%. However, when considering the precision that we measure by the size of the candidate set of the 1-NN-range query, we can

see that our approach gives the best result on average.

Airport Traffic Networks

The results for the two airport traffic networks can be reviewed in Table 11.2. As proposed in [217] we employ one-vs-rest classifications with 90-10 train-test splits and report the mean accuracy and standard deviation over 10 runs. Additionally, the table contains the runtimes in milliseconds for each method including the preprocessing steps for struc2vec and our approach.

RolX and GraphWave do not have preprocessing steps if executing their standard configurations1.

For both networks, we can notice that although our proposed method out-performs RolX and GraphWave in terms of accuracy, the scores achieved with struc2vec are slightly better. Note that the results for the European airports network are comparable across all methods. However, regarding the runtime our method clearly outperforms the competitors. Compared to struc2vec our approach is more than 2’300 times faster, even when considering the APPRrolesstacked configuration, on the USA airports network2. It should be noted that the accuracy of our approach might be further improved by spending more time for preprocessing the local neighborhoods. As we use an approximate version of the Personalized PageRank for engineering the local neighborhoods, it is possible to decrease the approximation threshold to get more accurate node descriptors. However, this comes at the cost of increased computation time.

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