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Fig. 6. Classification rates in percent, averaged on 100 runs, obtained on partially labelled graphs, for an increasing labelling rate of 20, 35, 50, 65, 80 ) and 95%. Results are reported for the SoP {euin-over-peths) kernel, the NCT (normalized commute-time) kernel, the RSP (randomized shortest-path) kernel, the CT (commute-time) kernel, the DM (diffusion map) kernel, the RL (regularized laplacian) kernel, the Netkit (NetKit) framework and the KNN algorithm. The graphs show the results obtained on the washington, wisconsin, texas and cornell WebKB data sets, on the two industries datasets and on the IMDb and Cora data sets [83]. Notice that the results of the RSP kernel are not reported on the largest datasets (industries and CoRA) because of prohibitive computation time.

Figure 6 Classification rates in percent, averaged on 100 runs, obtained on partially labelled graphs, for an increasing labelling rate of 20, 35, 50, 65, 80 ) and 95%. Results are reported for the SoP {euin-over-peths) kernel, the NCT (normalized commute-time) kernel, the RSP (randomized shortest-path) kernel, the CT (commute-time) kernel, the DM (diffusion map) kernel, the RL (regularized laplacian) kernel, the Netkit (NetKit) framework and the KNN algorithm. The graphs show the results obtained on the washington, wisconsin, texas and cornell WebKB data sets, on the two industries datasets and on the IMDb and Cora data sets [83]. Notice that the results of the RSP kernel are not reported on the largest datasets (industries and CoRA) because of prohibitive computation time.

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where the Lagrange parameter » plays the role of a tempera- ture (A = T) and 6 = 1/4 is the inverse temperature. Thus, as expected, short paths g (having small E(g)) are favoured in that they have a large probability of being followed. Indeed, from Equation (5), we clearly observe that when 6 — 0, the paths probabilities reduce to the probabilities generated by the natural random walk on the graph (characterized by the transition probabilities pj, as defined in Equation {i}). In this case, Jo — 0 as well. On the other hand, when 6 is large, the first term in the exponential dominates the second one, so that the probability distribution defined by Equation is biased towards short paths (the most likely paths are the shortest ones). Notice that, in the sequel, it will be assumed that the user provides the value of the parameter 6 instead of Jo, with 9 > 0. Notice that the model could be derived thanks to a maximum entropy principle instead [70], [71]. where P™() represents the probability of following the path when walking according to the natural random walk, i.e. using transition probabilities pif, (see Equation {i}). Here, Jo > 0 is provided a priori by the user, according to the desired degree of randomness he is willing to concede. By minimizing the Lagrange function over the set of path probabilities P(g)
where the Lagrange parameter » plays the role of a tempera- ture (A = T) and 6 = 1/4 is the inverse temperature. Thus, as expected, short paths g (having small E(g)) are favoured in that they have a large probability of being followed. Indeed, from Equation (5), we clearly observe that when 6 — 0, the paths probabilities reduce to the probabilities generated by the natural random walk on the graph (characterized by the transition probabilities pj, as defined in Equation {i}). In this case, Jo — 0 as well. On the other hand, when 6 is large, the first term in the exponential dominates the second one, so that the probability distribution defined by Equation is biased towards short paths (the most likely paths are the shortest ones). Notice that, in the sequel, it will be assumed that the user provides the value of the parameter 6 instead of Jo, with 9 > 0. Notice that the model could be derived thanks to a maximum entropy principle instead [70], [71]. where P™() represents the probability of following the path when walking according to the natural random walk, i.e. using transition probabilities pif, (see Equation {i}). Here, Jo > 0 is provided a priori by the user, according to the desired degree of randomness he is willing to concede. By minimizing the Lagrange function over the set of path probabilities P(g)
Fig. 1. A graph with one shortest-path between i and j. The edge weight: are assumed to be 1.
Fig. 1. A graph with one shortest-path between i and j. The edge weight: are assumed to be 1.
Furthermore, the expected number of times the link k — k’ and the link | — lI’ are traversed together along a path is and this quantity is a measure of covariance between link k — k’ and link 1 > I’. Finally, the covariance measure between node k’ and node l’ is simply defined from Equation as
Furthermore, the expected number of times the link k — k’ and the link | — lI’ are traversed together along a path is and this quantity is a measure of covariance between link k — k’ and link 1 > I’. Finally, the covariance measure between node k’ and node l’ is simply defined from Equation as
Fig. 2. Linear correlation between the SoP betweenness and Freeman’s & Newman’s betweenness measures, obtained on Zachary’s karate club (upper) and the dolphin network (lower) for different values of 0 < 0 < 10.
Fig. 2. Linear correlation between the SoP betweenness and Freeman’s & Newman’s betweenness measures, obtained on Zachary’s karate club (upper) and the dolphin network (lower) for different values of 0 < 0 < 10.
_ The SoP measure is based on the same diffusing hypothesis. However, in the present case, to assign a label to a node, we take into account all the other nodes of the graph. The way a node is correlated to the others is tuned through the 0 parameter. A high 6 value means that we consider only shortest paths. Considering only shortest paths gives small chance to two nodes to appear together on the same path, which leads to a constantly low correlation. This can be observed on Figure 4(c)) where we display the SoP matrix obtained for the graph of Figure |3| The correlation between nodes is low and quite the same between all pair of nodes so that discriminating between nodes is difficult. However, a lower 0 behaves better for discriminating between nodes: in this case, longer paths are taken into account. This behavior can be observed on Figure A(a)| The left-hand side nodes and the right-hand side nodes are well clustered. We may still perform better using a 6 of, for example 0.1, which will give more weight to close nodes and improve the diffusion of unlabelled nodes. The Figure|4(b) shows that for a 6 of 0.1 the left hand-side nodes are less correlated to right-hand side nodes than for a near-zero 6 value (Figure while still clustering nodes. This is confirmed by the experiments shown in Figure |5}| where we present the influence of the @ parameter on the classification rate on the WebKB-washington data set for increasing labelling rates. On this data set, the best 6 parameter is located around 0.2 and 0.8 according to the labelling rate considered. Fig. 3. One simple example graph, for illustration.
_ The SoP measure is based on the same diffusing hypothesis. However, in the present case, to assign a label to a node, we take into account all the other nodes of the graph. The way a node is correlated to the others is tuned through the 0 parameter. A high 6 value means that we consider only shortest paths. Considering only shortest paths gives small chance to two nodes to appear together on the same path, which leads to a constantly low correlation. This can be observed on Figure 4(c)) where we display the SoP matrix obtained for the graph of Figure |3| The correlation between nodes is low and quite the same between all pair of nodes so that discriminating between nodes is difficult. However, a lower 0 behaves better for discriminating between nodes: in this case, longer paths are taken into account. This behavior can be observed on Figure A(a)| The left-hand side nodes and the right-hand side nodes are well clustered. We may still perform better using a 6 of, for example 0.1, which will give more weight to close nodes and improve the diffusion of unlabelled nodes. The Figure|4(b) shows that for a 6 of 0.1 the left hand-side nodes are less correlated to right-hand side nodes than for a near-zero 6 value (Figure while still clustering nodes. This is confirmed by the experiments shown in Figure |5}| where we present the influence of the @ parameter on the classification rate on the WebKB-washington data set for increasing labelling rates. On this data set, the best 6 parameter is located around 0.2 and 0.8 according to the labelling rate considered. Fig. 3. One simple example graph, for illustration.
Fig. 5. Curves of the classification rates obtained for different values of the parameter 0 (x-axis) and different labelling rates. Fig. 4. SoP correlation matrix computed on the graph of Figure [3] for respectively 0 — 0 (a), 0 = 0.1 (b), and 0 — co (c). Dark red colors indicate high correlations while dark blue colors indicate low correlations.
Fig. 5. Curves of the classification rates obtained for different values of the parameter 0 (x-axis) and different labelling rates. Fig. 4. SoP correlation matrix computed on the graph of Figure [3] for respectively 0 — 0 (a), 0 = 0.1 (b), and 0 — co (c). Dark red colors indicate high correlations while dark blue colors indicate low correlations.
The expected number of passages through node k’, that is, the betweenness measure (Equation (15), is
The expected number of passages through node k’, that is, the betweenness measure (Equation (15), is
Related topics:
Information SystemsProbability Distribution & ApplicationsSupervised ClassificationBetweenness CentralityElectrical and Electronic EngineeringBoltzmann distribution
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