Cascades towards noise-induced transitions on networks revealed using information flows

TL;DR

The paper tackles how endogenously generated metastable transitions arise in complex networks governed by Boltzmann-Gibbs dynamics, without external forcing. It introduces two node roles—initiator nodes that propagate short-term fluctuations and stabilizer nodes that encode long-term memory—measured via two information-theoretic quantities: integrated mutual information $\mu(s_i)$ and asymptotic information $\omega(s_i)$. Through exact computations on small networks and controlled interventions, the authors reveal a domino-like cascade where initiators trigger tipping points and stabilizers guide the system into a new attractor, with the role score $r_i$ delineating initiators from stabilizers. The information-centric framework enables data-driven estimation of information flows and suggests targeted interventions to promote or inhibit systemic transitions, offering insights with potential impact across neuroscience, social dynamics, and beyond.

Abstract

Complex networks, from neuronal assemblies to social systems, can exhibit abrupt, system-wide transitions without external forcing. These endogenously generated ``noise-induced transitions'' emerge from the intricate interplay between network structure and local dynamics, yet their underlying mechanisms remain elusive. Our study unveils two critical roles that nodes play in catalyzing these transitions within dynamical networks governed by the Boltzmann-Gibbs distribution. We introduce the concept of ``initiator nodes'', which absorb and propagate short-lived fluctuations, temporarily destabilizing their neighbors. This process initiates a domino effect, where the stability of a node inversely correlates with the number of destabilized neighbors required to tip it. As the system approaches a tipping point, we identify ``stabilizer nodes'' that encode the system's long-term memory, ultimately reversing the domino effect and settling the network into a new stable attractor. Through targeted interventions, we demonstrate how these roles can be manipulated to either promote or inhibit systemic transitions. Our findings provide a novel framework for understanding and potentially controlling endogenously generated metastable behavior in complex networks. This approach opens new avenues for predicting and managing critical transitions in diverse fields, from neuroscience to social dynamics and beyond.

Figures (11)

• Figure 1: A dynamical network governed by kinetic Ising dynamics produces multistable behavior. (a) A typical trajectory is shown for a kite network for which each node is governed by the Ising dynamics with $\beta \approx 0.534$. The panels show system configurations $S_i \in S$ as the system approaches the tipping point (orange to purple to red). For the system to transition between attractor states, it has to cross an energy barrier (c). (b) The dynamics of the system can be represented as a graph. Each node represents a system configuration $S_i \in S$ such as depicted in (a). The probability for a particular system configuration $p(S)$ is indicated with a color; some states are more likely than others. The trajectory from (a) is visualized. Dynamics that move towards the tipping point (midline) destabilize the system, whereas moving away from the tipping point are stabilizing dynamics. (c) The stationary distribution of the system is bistable. Crossing the tipping point requires crossing a high energy states (dashed line). Transitions between the attractor states are infrequent and rare. For more information on the numerical simulations see \ref{['method:appendix']}.
• Figure 2: (a-e) Information flows as distance to tipping point. Far away from the tipping point most information processing occurs in low degree nodes (f,g). As the system moves towards the tipping point, the information flows increase and the information flows move towards higher degrees. (f) Integrated mutual information as function of distance to tipping point. The graphical inset plots show how noise is introduced far away from the tipping point in the tail of the kite graph. As the system approaches the tipping point, the local information dynamics move from the tail to the core of the kite. (g) A rise in asymptotic information indicates the system is close to a tipping point. At the tipping point, the decay maximizes as trajectories stabilize into one of the two attractor states.
• Figure 3: The tipping point is initiated from the bottom up. Each node is colored according to state 0 (black) and state 1 (yellow) Shown is a trajectory towards the tipping point that maximizes $\sum_{{t=1}}^{{5}} \log p(S^{{t+1}} | S^t, S^0 =\{0\}, \langle S^5 \rangle ) = 0.5)$. As the system approaches the tipping point, low degree nodes flip first, and recruit "higher" degree nodes to further destabilize the system and push it towards a tipping point. In total 30240 trajectories that reach the tipping point in 5 steps, and there are 10 trajectories that have the same maximized values as the trajectory shown in this figure.
• Figure 4: (a) Shown are the conditional probabilities at time $t=10$ relative to the tipping point. The shared information between the hub node 3 and the tail node 8 is similar but importantly caused through different sources. The hub (node 3) has high certainty on that the system macrostate will be the same sign as its state. In contrast, node 8 has high certainty that the system macrostate will be opposite to its state at the tipping point. This is caused by the interaction between the network structure and the system dynamics whereby the most likely trajectories to the tipping point from the stable regime is mediated by the noise-induced dynamics from the tail to the core in the kite graph (see main text).(b) Successful metastable transitions are affected by network structure. Successful metastable transitions are those for which the sign of the macrostate is not the same prior and after the tipping point, e.g. the system going from the 0 macrostate side to the +1 macrostate side or vice versa. Shown here are the number of successful metastable transitions for \ref{['fig:interventions']} under control and pinning interventions on the nodes in the kite graph.
• Figure 5: For a system to cross a tipping point, two distinct types of nodes are essential: stabilizers, which contain information about the system's next attractor state and facilitate transitions between states; and initiators, which propagate noise through the system. (a) The effect of causal pinning interventions on node 0 states in Erdos-Renyi graphs ($N = 100$, 10 nodes each, $p = 0.2$, 6 seeds) is shown. Normalized system fluctuations (second moment) and time spent below the tipping point relative to the control are presented per network to indicate the effect of the pinning interventions. Pinning initiators increases tipping points, while pinning stabilizers prevents tipping and increases noise above the tipping point. For more details on role approximation, see \ref{['sec:roles']}. (b) To exemplify the effect of the causal interventions in (a) typical system trajectories under pinning interventions on a node for the kite graph are shown. Colors reflect intervention on corresponding nodes in the inset kite graph. Initiator-based interventions remove fluctuations below the tipping point ($<0.5$) and increase fluctuations above, whereas stabilizer-based interventions stabilize tipping points while increasing noise.