Link-usage asymmetry and collective patterns emerging from rich-club organization of complex networks

Abstract

In models of excitable dynamics on graphs, excitations can travel in both directions of an undirected link. However, as a striking interplay of dynamics and network topology, excitations often establish a directional preference. Some of these cases of “link-usage asymmetry” are local in nature and can be mechanistically understood, for instance, from the degree gradient of a link (i.e., the difference in node degrees at both ends of the link). Other contributions to the link-usage asymmetry are instead, as we show, self-organized in nature, and strictly nonlocal. This is the case for excitation waves, where the preferential propagation of excitations along a link depends on its orientation with respect to a hub acting as a source, even if the link in question is several steps away from the hub itself. Here, we identify and quantify the contribution of such self-organized patterns to link-usage asymmetry and show that they extend to ranges significantly longer than those ascribed to local patterns. We introduce a topological characterization, the hub-set-orientation prevalence of a link, which indicates its average orientation with respect to the hubs of a graph. Our numerical results show that the hub-set-orientation prevalence of a link strongly correlates with the preferential usage of the link in the direction of propagation away from the hub core of the graph. Our methodology is embedding-agnostic and allows for the measurement of wave signals and the sizes of the cores from which they originate.

Two classes of functional connectivity in dynamical processes in networks

“…Given our hypothesis that high is associated with large-scale patterns (e.g. excitation waves [ 20 ]), the distinction between chaotic and complex behaviour may have a strong effect on SC/FC correlations. This aspect requires further investigation and coupled electronic chaotic oscillators may be an interesting test case for this, as they provide a high level of realism, in particular for brain dynamics, together with a detailed understanding of their collective behaviours [ 265 , 266 ].…”

“…In these cases, a good fit between simulated and empirical FC is obtained only when the model is tuned to operate close to a critical point of dynamic operation [ 212 , 213 ], indicating that FC manifests a peculiar dynamical regime, certainly shaped but not fully constrained by structure. For instance, waves [ 19 , 20 ] or ‘connectome harmonics’ [ 214 ] shape large-scale coordinated activity and, hence, FC. The importance of being tuned into specific dynamical regimes to account for the qualitative features of large-scale coordinated activity has also been confirmed by minimal models, with reduced realism but enhanced possibility to rigorously understand mechanisms [ 27 , 28 , 215 ].…”

“…In the case of the cortical area network, the excitable dynamics as well as the oscillatory dynamics can be seen as stylized but plausible dynamical probes and, in fact, those have been previously employed to explore such network royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210486 structures [20,[35][36][37]. But also chaotic dynamics, as in the third row of figure 1, have been used to study neuronal connectivity patterns [38,39].…”

“…a loss of SC/FC correlation may indicate pathological brain activity patterns [ 18 ]) and highly informative starting points for a mechanistic understanding of the system (e.g. revealing highly connected elements—hubs—as centres of self-organized excitation waves in scale-free graphs [ 19 , 20 ]).…”

“…Key results include that the topological overlap [ 30 ] is highly associated with functional connectivity based on simultaneous activity, , and that via this mechanism—a clustering of high topological overlap values within modules—modular graphs display high SC/FC correlations, while scale-free graphs tend to display low, or even systematically negative SC/FC correlations with this definition of FC [ 28 , 30 ]. Furthermore, a large asymmetry of the sequential activation matrix (which is the foundation of functional connectivity based on sequential activation, ) can be associated with self-organized waves around hubs [ 20 ]. Additionally the role of cycles for organizing SC/FC correlations has been investigated [ 31 ] and in the deterministic limit of the model, a theoretical framework for predicting SC/FC correlations has been established [ 30 ].…”