Concentric Characterization of Complex Networks
Measurements
This session presents a summary of the main concentric measurements calculated over rings and balls. Further details can be found in references.
Concentric number of nodes\(n_d(i)\)  Number of nodes that belongs to the ring \(R_d(i)\). 
Concentric number of edges\(e_d(i)\)  Number of edges connecting nodes inside ring \(R_d(i)\). 
Concentric node degree\(k_d(i)\)  Number of edges extending from ring \(R_d(i)\) to \(R_{d+1}(i)\). 
Concentric clustering coefficient\(cc_d(i)\)  Is the division of the number of edges existent in ring Rd(i) by the total number of possible edges in this ring, or \(cc_d(i) = 2 {e_d(i)\over n_d(i) (n_d(i)1)}\) . 
Convergence ratio\(C_d(i)\)  The ratio between the concentric node degree \(k_d(i)\) and the number of nodes at the next concentric level \(n_{d+1}(i)\), i.e. \(C_d(i)={k_d(i) \over n_{d+1}(i)} \) . 
Intraring node degree\(A_d(i)\)  The average of degree of the nodes at the ring \(R_d(i)\) considering only the edges lying in the ring \(R_d(i)\). Note that the sum of the degree of the nodes equals \(2e_d(i)\), then \(A_d(i) = {2 e_d(i)\over n_d(i)}\) . 
Interring node degree\(E_d(i)\)  The ratio between the concentric node degree \(k_d(i)\) and the number of nodes at the same ring, \(n_{d}(i)\), i.e. \(E_d(i)={k_d(i) \over n_{d}(i)} \) . 
Concentric common degree\(H_d(i)\)  The average of the degree considering all the connections of nodes at a specific ring, which yields the expression: \(H_d(i) = {k_{d1}(i) + k_{d}(i)+ 2 e_d(i)\over n_d(i)}\) . 
\(rs\) clustering coefficient\(cc_{rs}(i)\)  The concentric clustering coefficient spanning from ring \(R_r(i)\) up to \(R_s(i)\),including intermediate rings. It can be defined in terms of:
as \(cc_{rs}(i)=2 {e_{rs}(i) \over n_{rs}(i)(n_{rs}(i)1) }\) .
