# Multilayer Measures

Multilayer measures can have different measure shapes (nodal, global or binodal) and measure scopes (unilayer, superglobal, or bilayer). A superglobal measure is a measure that provides a unique result for the whole multilayer network, whereas a bilayer provides a result for each pair of layers. An unilayer measure is that one that provides a result for each layer, in that case, you can see the full list of available measures in Graph measures.

Furthermore, we will indicate to which kind of graph a given measure belongs by using (= weighted graphs) or B (= binary graphs), and D (= directed graphs) or U (= undirected graphs). If no letter is indicated it means that the measure applies to both cases.

Core Structure

Multirichness (nodal): The multirichness of a node is the sum of the edges that connect nodes of degree k or higher in all layers. The relevance of each layer is

controlled by the coefficients c that are between 0 and 1; the default coefficients are (1/number of layers).

Multiplex Core-Periphery (nodal): The multiplex core-periphery of a node is the value of the rank corresponding to the maximum multirichness nodes. It returns 1 for a node belonging to the core and zero otherwise. The relevance of each layer is controlled by the multirichness coefficients that are between 0 and 1, and add up to one; the default coefficients are (1/number of layers).

Multi RichClub Degree (nodal): The multi rich-club degree of a node at level k is the sum of the edges that connect nodes of degree k or higher in all layers. The relevance of each layer is controlled by the coefficients c that are between 0 and 1; the default coefficients are (1/number of layers).

Multi RichClub Strength (nodal): The multi rich-club strength of a node at level s is the sum of the weighted edges that connect nodes of strength s or higher in all layers. The relevance of each layer is controlled by the coefficients c that are between 0 and 1; the default coefficients are (1/number of layers).

Clustering and Community Structure

Multiplex Triangles (nodal): The multiplex triangles are calculated as the total number of a node’s neighbor pairs that are connected to each other between each pair of layers. In weighted graphs, the multiplex triangles are calculated as the geometric mean of the weights of the edges forming the multiplex triangle.

Multiplex Clustering Coefficient (nodal): The two-multiplex clustering coefficient of a node i is the fraction of two-multiplex triangles (triangles that use edges from two different layers) with a vertex in node i and the number of one-triads centered in i.

Battiston F, Nicosia V, Latora V. Structural measures for multiplex networks. Physical Review E. 2014 Mar 12;89(3):032804.

Average Multiplex Clustering Coefficient (global): The average two-multiplex clustering coefficient is the average of the two-multiplex clustering coefficient of all nodes.

Multilayer Community Structure (nodal): The multilayer community structure of a multilayer graph is a subdivision of the network into non-overlapping groups of nodes which maximizes the number of within-group edges, and minimizes the number of between-group edges. It is calculated using the code of the Generalized Louvain community detection algorithm implemented in the genlouvain package.

Lucas G. S. Jeub, Marya Bazzi, Inderjit S. Jutla, and Peter J. Mucha,
“A generalized Louvain method for community detection implemented in MATLAB,” https://github.com/GenLouvain/GenLouvain (2011-2019).

Multilayer Modularity (nodal): The multilayer modularity of a multilayer graph is the quality of the resulting partition of the multilayer network.

Lucas G. S. Jeub, Marya Bazzi, Inderjit S. Jutla, and Peter J. Mucha,
“A generalized Louvain method for community detection implemented in MATLAB,” https://github.com/GenLouvain/GenLouvain (2011-2019).

Dynamic Community Structure

Flexibility (nodal): The flexibility of each node is calculated as the number of times that it changes community assignment, normalized by the total possible number of changes. In ordered multilayer networks (e.g. temporal, changes are possible only between adjacent layers, whereas in categorical multilayer networks, community assignment changes are possible between any pairs of layers.

Average Flexibility (global): The average flexibility is the average of the flexibility of all nodes in a multilayer network.

Persistence (Global): The persistence of a multilayer network is calculated as the normalized sum of the number of nodes that do not change community assignments. It varies between 0 and 1. In categorical multilayer networks, it is the sum over all pairs of layers of the number of nodes that do not change community assignments, whereas in ordinal multilayer networks (e.g. temporal), it is the number of nodes that do not change community assignments between consecutive layers.

Lucas G. S. Jeub, Marya Bazzi, Inderjit S. Jutla, and Peter J. Mucha,
“A generalized Louvain method for community detection implemented in MATLAB,” https://github.com/GenLouvain/GenLouvain (2011-2019).

Degree and Strength

Overlapping Degree (nodal): The overlapping degree of a graph is the sum of the degrees of a node in all layers.

Average Overlapping Degree (global): The average overlapping degree of a graph is the average of the sum of the degrees of a node in all layers.

Overlapping In-Degree (nodal): The overlapping in-degree of a graph is the sum of the in-degrees of a node in all layers.

Average Overlapping In-Degree (global): The average overlapping in-degree of a graph is the average of the sum of the in-degrees of a node in all layers.

Overlapping Out-Degree (nodal): The overlapping out-degree of a graph is the sum of the out-degrees of a node in all layers.

Average Overlapping Out-Degree (global): The average overlapping out-degree of a graph is the average of the sum of the out-degrees of a node in all layers.

Overlapping Strength (nodal): The overlapping strength of a graph is the sum of the strengths of a node in all layers.

Average Overlapping Strength (global): The average overlapping strength of a graph is the average of the sum of the strengths of a node in all layers.

Overlapping In-Strength (nodal): The overlapping in-strength of a graph is the sum of the in-strengths of a node in all layers.

Average Overlapping In-Strength (global): The average overlapping in-strength of a graph is the average of the sum of the in-strengths of a node in all layers.

Overlapping Out-Strength (nodal): The overlapping out-strength of a graph is the sum of the out-strengths of a node in all layers.

Average Overlapping Out-Strength (global): The average overlapping out-strength of a graph is the average of the sum of the out-strengths of a node in all layers.

Multiplex Integration

Multiplex Participation Coefficient (nodal): The multiplex participation is the nodal homogeneity of the number of neighbours of a node across the layers. It is calculated as $P_{i}= \frac{L}{L-1}\left [ 1 - \sum_{\alpha=1}^{L}\left (\frac{k_{i}^{[\alpha]}}{o_{i}} \right )^{2}\right ]$ where L is the number of layers, $k_{i}^{[\alpha]}$ is the degree in the $\alpha$-th layer and $latex o_{i}$ is the overlapping degree of the node. $latex P_{i}$ = 1 when the degree is the same in all layers and $latex P_{i}$ = 0 when a node has non-zero degree in only one layer.

Battiston F, Nicosia V, Latora V. Structural measures for multiplex networks. Physical Review E. 2014 Mar 12;89(3):032804.

Multiplex In-Participation Coefficient (nodal): The multiplex in-participation is the homogeneity of the number of inward neighbours of a node across the layers.

Multiplex Out-Participation Coefficient (nodal): The multiplex out-participation is the homogeneity of the number of outward neighbours of a node across the layers.

Average Multiplex Participation Coefficient (global): The average multiplex participation of a graph is the average homogeneity of its number of neighbours across the layers.

Weighted Multiplex Participation Coefficient (nodal): The weighted multiplex participation of a graph is the nodal homogeneity of its number of neighbours across the layers.

Weighted Multiplex In-Participation Coefficient (nodal): The weighted multiplex in-participation of a graph is the nodal homogeneity of its number of neighbours across the layers.

Weighted Multiplex Out-Participation Coefficient (nodal): The weighted multiplex out-participation of a graph is the nodal homogeneity of its number of neighbours across the layers.

Weighted Average Multiplex Participation Coefficient (global): The weighted multiplex participation of a graph is the average homogeneity of its number of neighbours across the layers.

Overlap

Degree Overlap (nodal): The degree overlap is the number of edges connected to a node in all layers. Connection weights are ignored in calculations.

Average Degree overlap (global): The average degree overlap is the average of the number of edges connected to a node in all layers. Connection weights are ignored in calculations.

Edge overlap (binodal): The edge overlap of a graph is the fraction of layers in which an edge between a pair of nodes exists. Connection weights are ignored in calculations.

Weighted Edge overlap (binodal): The weighted edge overlap of a graph is the average weight of an edge across all layers.