Dinámica del modelo financiero

As an example application we consider the problem of classifying networks with different num- bers of nodes according to their generative models. To that end we consider: (i) Weighted Erd˝os- R´enyi random networks [144] with connection probabilityp=0.5 and edge weights random and uniformly chosen from the unit interval [0, 1]. (ii) Random geometric networks where nodes are placed at random in the unit circle and edge weights are of the form exp(−d(i,j)2/2σ2), where

d(i,j)is the distance between vertices iand jin the unit circle andσ2is a kernel width parame- ter. We set σ2 = 0.5. (iii) Random feature networks where edge weights are determined by the Pearson correlation coefficientρij between a pair of corresponding featuresui,uj ∈Rd. Features

are randomly chosen standard white Gaussian vectors in a space of dimension d = 5 and edge weights are chosen asrX(i,j) = (1+ρij)/2. Observe that in all three cases edge weights measure

the relationship between pairs of vertices and take values in the unit interval.

We start with networks of equal size|X| =25 and construct 20 random networks for each afore- mentioned type. We then use the MDS method in Figure 46 [50, 51] to approximate the embedding network distancedE defined in Definition 27. To evaluate the effectiveness of considering interi-

ors of networks described in Section 6.2, we add midpoints for all edges in a given network; it is apparent that any pair of networks with interiors defined in this way would form a regular sample pair. Approximations of the embedding network distancedEbetween these networks with

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(a) with interiors, 25 nodes

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(b) without interiors, 25 nodes

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(c) with interiors, 20 to 25 nodes

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(d) without interiors, 20 to 25 nodes

Figure 48: Two dimensional Euclidean embeddings of the networks constructed from three dif- ferent models with different number of nodes with respect to the approximation to the network embedding distance. In the embeddings, red circles denote networks constructed from the Erd˝os- R´enyi model, blue diamonds represent networks constructed from the unit circle model, and black squares the networks from the correlation model.

beddings [146] of the network metric approximations with and without interiors, respectively1. In both figures we see the emergence of clusters corresponding to each generative model, but the clusters are more clear when interiors are added – random networks are denoted by red circles, geometric networks with blue diamonds, and feature networks with black squares. For a formal performance evaluation we conduct unsupervised hierarchical clustering with Ward’s linkage [156] method upon the approximated embedding network distancedE; results are drawn

using linear boundaries on the corresponding figures. There are 4 misclassifications when no in- teriors are added but only one misclassification after addition of interiors. This is an error rate of 1/60≈1.67%. Similar results are obtained with the use of other unsupervised learning methods. We further consider mixes in which the number of nodes ranges in the integer set {20 . . . , 25}. Four networks are randomly generated for each type and each number of nodes, resulting in 60 1_{The visualization embeddings minimize the sum of squares of the inter-point distances. Visualizations look similar} for other common choices of embedding distortion measures.

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(a) graphs with 2 clusters

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(b) graphs with 3 clusters

Figure 49: Two dimensional Euclidean embeddings of unweighted graphs generated with stochas- tic block models, with interiors added to the weighted networks. (a) Graphs with 150 nodes,S=2 clusters, in-diagonal connecting probability p = 0.2, and three different off-diagonal connecting probability q = 0.05 (red circles), q = 0.1 (blue diamonds),q = 0.2 (black squares). (b) Graphs with 150 nodes, S = 3 clusters, in-diagonal connecting probability p = 0.5, and three different off-diagonal connecting probabilityq =0.1 (red circles),q= 0.2 (blue diamonds),q=0.4 (black squares). In each figure, ten graphs are generated for each type of models.

networks in total. Interiors are examined similarly as before by adding midpoints for all edges in a given network. Figure 48 (c) and (d) illustrate the two dimensional Euclidean embeddings of the network metric approximations with and without interiors respectively. Despite the fact that networks with the same model have different number of nodes, dissimilarities between network distance approximations are smaller when their underlying networks are from the same process. As in comparison in Figure 47, considering interiors result in a more distinctive clustering pattern. An unsupervised classification would yield 2 out of 60 errors (3.33%) for networks with interiors added and 6 errors (10%) without interiors.

These results illustrate that: (i) Comparing networks by using embedding distances succeeds in identifying networks with different generative models. (ii) Adding interiors to networks to form regular sample pairs as in Section 6.2 would yield better approximations to the actual network distances. We must observe that other methods would exhibit similar success in this classification task. Weighted motifs [16] yields 1 misclassification for networks with constant size and 4 mis- classifications for networks of varying sizes. Comparison with persistent homologies [76] yields no errors for networks with constant sizes and 3 errors for networks with varying sizes. This means alternative methods are comparable although they seem to be worse when networks of different sizes are considered. We will see in Section 6.3.4 that this is indeed the case. This is as expected because the strength of embedding distances is precisely on the possibility of embedding a network into another network of a larger size.