Critical transitions in degree mixed networks: A discovery of forbidden tipping regions in networked spin systems

Abstract

Critical transitions can be conceptualized as abrupt shifts in the state of a system typically induced by changes in the system’s critical parameter. They have been observed in a variety of systems across many scientific disciplines including physics, ecology, and social science. Because critical transitions are important to such a diverse set of systems it is crucial to understand what parts of a system drive and shape the transition. The underlying network structure plays an important role in this regard. In this paper, we investigate how changes in a network’s degree sequence impact the resilience of a networked system. We find that critical transitions in degree mixed networks occur in general sooner than in their degree homogeneous counterparts of equal average degree. This relationship can be expressed with parabolic curves that describe how the tipping point changes when the nodes of an initially homogeneous degree network composed only of nodes with degree k1 are replaced by nodes of a different degree k2. These curves mark clear tipping boundaries for a given degree mixed network and thus allow the identification of possible tipping intersections and forbidden tipping regions when comparing networks with different degree sequences.

Modular tipping points: How local network structure impacts critical transitions in networked spin systems

“…configured two-degree network (Fig 4, r � 0, black solid line) tips on average earlier than a randomly configured one-degree network of equal average degree, i.e. random k-regular network (black dashed line), confirming the results presented in a previous study which showed that randomly placed low-degree nodes disproportionally destabilize a networked system [15]. We also observe that a highly assortative two-degree network (Fig 4, r � 1, cyan solid line), transitions in cascades, confirming the results presented in [16,17].…”

“…For example, some system components may be embedded in highly resilient structures allowing them to transition quite late in response to changes in a control parameter, while others are embedded more weakly causing them to respond earlier to changes in a control parameter. In this context, it has been observed that changes in the network structure, such as modifying connections or altering node degrees, can significantly alter the onset of a system's critical transition [15], and that peripheral components in the networked system appear to be more sensitive to changes in an external or control parameter [16]. In particular, previous results show that:…”

“…A randomly networked system composed of multiple node degrees transitions as a unified whole and it transitions earlier in a response to changes in a control parameter than a comparable one-degree network of equal average degree, i.e. a random k-regular graph [15].…”

“…In this case, the clear and the turbid states correspond to the ball resting in either one of the potential wells and the change of the lake's nutrient load correspond to changes in the stability landscape of the system. Another example from the field of physics is the ferromagnetic transition where a ferrous material shifts from a magnetic to a non-magnetic state in response to changes in the material's temperature or between a negative and a positive magnetic state in response to changes in an external magnetic field [8,12,15,16]. A more recent example from sociology describes belief traps as a system of opinions, where the resilience of a belief system changes as the objective evidence for or against the belief is increased, creating a critical transition between two stable yet opposing opinions [10].…”

Longitudinal Network Changes and Phenoconversion Risk in Isolated REM Sleep Behavior Disorder

“…It is also noteworthy that assortativity, the link-averaged correlation coe cient of degree centrality for pairs of connected nodes in a network [16][17][18] , is abnormally increased in iRBD beginning at the second timepoint for PDRP and the third timepoint for PDCP. This metric captures changes in connectional diversity that render networks more vulnerable to random attacks, fragmentation, and critical transitions [19][20][21] . We have previously described progressive increases in PDRP assortativity in multiple groups of PD patients 22 .…”