Macro- and mesoscale pattern interdependencies in complex networks

Abstract

<jats:p>Identifying and explaining the structure of complex networks at different scales has become an important problem across disciplines. At the mesoscale, modular architecture has attracted most of the attention. At the macroscale, other arrangements—e.g. nestedness or core–periphery—have been studied in parallel, but to a much lesser extent. However, empirical evidence increasingly suggests that characterizing a network with a unique pattern typology may be too simplistic, since a system can integrate properties from distinct organizations at different scales. Here, we explore the relationship between some of these different organizational patterns: two at the mesoscale (modularity and in-block nestedness); and one at the macroscale (nestedness). We show experimentally and analytically that nestedness imposes bounds to modularity, with exact analytical results in idealized scenarios. Specifically, we show that nestedness and modularity are interdependent. Furthermore, we analytically evidence that in-block nestedness provides a natural combination between nested and modular networks, taking structural properties of both. Far from a mere theoretical exercise, understanding the boundaries that discriminate each architecture is fundamental, to the extent that modularity and nestedness are known to place heavy dynamical effects on processes, such as species abundances and stability in ecology.</jats:p>

Mutualistic networks emerging from adaptive niche-based interactions

“…The correlation indicates that the two measures cannot be determined independently: one can find networks with very high nestedness (modularity) and very low modularity (nestedness), but not likely with both being high or low. In the case of both nontrivial nestedness and modularity (significantly higher than for the null model), modules may emerge with embedded nested structures 33,34 , as exemplified in Fig. 1c.…”

An ecological approach to structural flexibility in online communication systems

“…This behaviour can be explained by the mutual structural constraints that these two arrangements impose on each other, i.e., the upper bound for the co-existence of nested and modular structures. This bound is , which implies that extremely high values are incompatible with the high values of Q 47 . Note however that this does not impede many other regimes: it is perfectly possible (and actually frequent) that both Q and are extremely low (e.g., in Erdös-Rényi networks), or that both have intermediate values (see Supplementary Note 3 ).…”

“…The question remains, however, how a networked system can fluctuate so fast between two states which have often been considered incompatible 52 , 53 . The key to this puzzle is in-block nestedness, a hybrid modular-nested architecture that bridges the apparent antagonism between nestedness and modularity 47 .…”

Robustness to extinction and plasticity derived from mutualistic bipartite ecological networks

“…Our results are consistent with the trends reported in these metrics although the literature shows some conflicting outcomes. Modularity and nestedness are observed to be negatively correlated, as found in several bipartite networks 68 , at least for large connectance 43 and especially for pollination networks 69 . Nevertheless, note that this relation is not mathematically proven except for the case of some models as in 68 .…”

“…Modularity and nestedness are observed to be negatively correlated, as found in several bipartite networks 68 , at least for large connectance 43 and especially for pollination networks 69 . Nevertheless, note that this relation is not mathematically proven except for the case of some models as in 68 . Stability is negatively correlated with nestedness as reported in some studies 70,71 , but positively correlated with modularity, in agreement with previous studies 14,72,73 .…”