Relevance of the Minimum Degree to Dynamic Fluctuation in Strongly Heterogeneous Networks

Abstract

The fluctuation of dynamic variables in complex networks is known to depend on the dimension and the heterogeneity of the substrate networks. Previous studies, however, have reported inconsistent results for the scaling behavior of fluctuation in strongly heterogeneous networks. To understand the origin of this conflict, we study the dynamic fluctuation on scale-free networks with a common small degree exponent but different mean degrees and minimum degrees constructed by using the configuration model and the static model. It turns out that the global fluctuation of dynamic variables diverges algebraically and logarithmically with the system size when the minimum degree is one and two, respectively. Such different global fluctuations are traced back to different, linear and sub-linear, growth of local fluctuation at individual nodes with their degrees, implying a crucial role of degree-one nodes in controlling correlation between distinct hubs.

Giant component in a configuration-model power-law graph with a variable number of links

“…Given such a crucial role of the degree exponent in the critical phenomena on PL graphs, a natural question arises: Is the large-k behavior the only property of the degree distribution D(k) relevant to critical phenomena? There are various relevant factors beyond the degree distribution, such as degreedegree correlation [21] or the spectral dimensions [22][23][24], but we are here focused on whether different degree distributions sharing the same degree exponent could lead to different criti- * deoksun.lee@inha.ac.kr cal phenomena from described above and how much different if they do. The answers are not immediately clear, comparing critical phenomena on the PL graphs with degree distributions in different form but sharing the same degree exponent.…”