Complex Systems

Complex systems comprise large ensembles of interacting components whose collective behaviour cannot be deduced from any single part. Whether they are neural circuits, ecological webs, social networks or supply-chain infrastructures, such systems exhibit non-linear dynamics, self-organisation and emergent patterns that persist outside simple equilibrium. Small perturbations may dissipate harmlessly or cascade into systemic transitions, depending on feedback architecture and internal resilience. Key features include adaptive flows of information, matter or energy across evolving structures; modular organisation that enables rapid reconfiguration; and phase-transition thresholds that distinguish orderly regimes from disordered or critical states. A complex-systems perspective recognises that structure and function are inseparable: network motifs and feedback loops both constrain and enable the repertoire of possible behaviours. This outlook has proven essential for understanding phenomena as diverse as cardiac arrhythmias, market crashes, viral rumours, and climate tipping points, and for designing interventions that leverage or temper intrinsic systemic tendencies.

Research from Nature Portfolio

Recent studies have refined methods for building synthetic temporal networks that faithfully reproduce the bursty activity found in empirical data by prescribing target inter-event time distributions for nodes and links while accommodating arbitrary underlying topologies. Complementary work has introduced higher-order contagion models in simplicial complexes with non-Markovian memory, revealing how group interactions and history-dependent recovery jointly shift epidemic thresholds and resilience in social spreading. Further investigations have mapped the temporal-topological correlations of multi-node events across real systems, demonstrating that interactions of differing orders cluster both in time and across network neighbourhoods and that individuals maintain consistent activity levels across group sizes. These advances bridge theoretical frameworks with data-driven characterisations of dynamic processes in richly structured settings.

Research from all publishers

An analysis of information flow in a mobile drone swarm applied temporal-network techniques to quantify message delivery delays as a function of mobility patterns and communication ranges, extending classic graph algorithms to rapidly varying links and informing design principles for autonomous swarming platforms. In the realm of online interaction, researchers have uncovered that heavy-tailed distributions of clicks and replies in digital forums are interdependent, suggesting that micro-level engagement skews co-generate collective burstiness in user activity. Studies of high-intensity group messaging have identified shifts between bimodal, double-power-law and single-power-law inter-event distributions, attributing these transitions to the interplay of circadian rhythms and intense collective engagement. Together, these works illustrate the broad applicability of temporal-network concepts—from engineered robotic systems to social media—and highlight how endogenous rhythms and network topology shape dynamic patterns.

Complex Systems publication trend

The graph below shows the total number of articles in complex systems across all publications each year (not limited to Nature Index journals).

Technical terms

Complex system: A network of many components whose interactions produce collective dynamics not evident from individual parts.

Emergence: The appearance of new large-scale patterns or properties that cannot be inferred by examining individual components alone.

Self-organisation: The spontaneous formation of ordered structures or behaviours through internal feedback without external control.

Temporal network: A graph in which nodes and edges activate and deactivate over time, encoding the timing and sequence of interactions.

Burstiness: A pattern of activity characterised by rapid clusters of events separated by long inactive periods, yielding heavy-tailed inter-event times.

Higher-order interaction: A simultaneous relationship among three or more nodes, modelled by structures such as simplicial complexes or hyperedges.

Non-Markovian process: A stochastic process in which future transitions depend on the history of past events, not solely on the current state.

References

1. Temporal-topological properties of higher-order evolving networks. Scientific Reports (2023).
2. Information Transmission in a Drone Swarm: A Temporal Network Analysis. Drones (2024).
3. Constructing temporal networks with bursty activity patterns. Nature Communications (2023).
4. Higher-order non-Markovian social contagions in simplicial complexes. Communications Physics (2024).
5. Universal features of correlated bursty behaviour. Scientific Reports (2012).
6. Communication activity in a social network: relation between long-term correlations and inter-event clustering. Scientific Reports (2012).
7. Complex Systems and Complexity Thinking.