# SVT - Singular Vector of Tensor Centrality

#### Definition

Singular Vector of Tensor $(SVT)$ centrality is used to quantitatively evaluate the importance of nodes connected by different types of links in multilayer networks. It applys a novel iterative method to obtain four alternative metrics that can quantify the hub and authority scores of nodes and layers in multilayer networked systems. $SVT$ centrality measure obtains by integrating these four metrics.

$$U_i={1\over L}{\underset{\alpha=1}{\overset{L}{\sum}}} U_i^{\alpha}(1\leq i \leq N)$$

were $U_i^{\alpha}=U^{(1)}(i). U^{(1)}(\alpha)+ U^{(3)}(i). U^{(4)}(\alpha) (1\leq i \leq N; (1\leq {\alpha} \leq L))$ represents the $SVT$ centrality of node $i$ in layer $\alpha$ when accounting for the whole interconnected structure.Here, $U^{(i)}=(i=1, 2, 3, 4)$ are obtained by ranking all components in descending order of $u^{(i)}=(i=1, 2, 3, 4)$, respectively.

A lower $SVT$ centrality indicates a relatively greater importance for each node within the whole network.

In multilayer networks, this method considers four measures for two roles of each node and two roles each layer, namely, the authority and hub of nodes and the authority and hub of layers. Using a fourth-order tensor to represent the multilayer networks, motivated by the iterative idea of the $HITS$ method.

$$U_i={1\over L}{\underset{\alpha=1}{\overset{L}{\sum}}} U_i^{\alpha}(1\leq i \leq N)$$

were $U_i^{\alpha}=U^{(1)}(i). U^{(1)}(\alpha)+ U^{(3)}(i). U^{(4)}(\alpha) (1\leq i \leq N; (1\leq {\alpha} \leq L))$ represents the $SVT$ centrality of node $i$ in layer $\alpha$ when accounting for the whole interconnected structure.Here, $U^{(i)}=(i=1, 2, 3, 4)$ are obtained by ranking all components in descending order of $u^{(i)}=(i=1, 2, 3, 4)$, respectively.

A lower $SVT$ centrality indicates a relatively greater importance for each node within the whole network.

In multilayer networks, this method considers four measures for two roles of each node and two roles each layer, namely, the authority and hub of nodes and the authority and hub of layers. Using a fourth-order tensor to represent the multilayer networks, motivated by the iterative idea of the $HITS$ method.

#### References

- Wang D., Zou X., 2018. A new centrality measure of nodes in multilayer networks under the framework of tensor computation. Applied Mathematical Modelling, 54, pp.46-63. DOI: 10.1016/j.apm.2017.07.012