Critical phenomena in networks: Report from the NetSci 2025 satellite symposium

This blog post explores how similar critical phenomena can be found in nature and society. At the first CPIN symposium during the International School and Conference on Network Science (NetSci 2025), researchers discussed models and empirical results regarding self-organization, phase transitions, and universal laws. Their contributions addressed neural, social, and transportation networks, among others, and their structural and dynamic behavior.

Dieser Blogbeitrag beschreibt, dass ähnliche kritische Phänomene sowohl in der Natur als auch in der Gesellschaft gefunden werden. Auf dem ersten CPIN-Symposium im Rahmen der International School and Conference on Network Science (NetSci 2025) diskutierten Forschende Modelle und empirische Ergebnisse zu Selbstorganisation, Phasenübergängen und universellen Gesetzmäßigkeiten. Ihre Beiträge behandelten u.a. neuronale, soziale und technische Netzwerke, deren Strukturen und Dynamiken.

DOI: 10.34879/gesisblog.2025.98

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According to an argument by Philip W. Anderson, chemical networks obey the laws of physics, biological networks obey the laws of chemistry, up to social networks obeying the laws of psychology. At each new level of complexity, new laws add to the lower-level laws. By virtue of this argument, we can expect similar phenomena in social, biological, and chemical networks.

Critical phenomena may be an example. They are associated with structures and dynamics that reside at or self-organize to a phase transition. A (second-order) phase transition is a state of its own between disorder and order. Critical phenomena include the divergence of particular parameters, power laws, fractals, and the possibility of similarly modeling different networks at the macro level. Critical phenomena in networks (CPIN) may be explained with established concepts of equilibrium statistical mechanics when networks are static. Non-equilibrium explanations may be required for temporal networks that constantly adapt to endogenous and exogenous stimuli.

On June 2nd, 2025, around 40 researchers participated in the first CPIN satellite symposium of the International School and Conference on Network Science (NetSci 2025) to advance the understanding of critical phenomena in networks through empirical analysis and modeling as well as interdisciplinary exchange.

In the first keynote lecture of the symposium, Stefan Bornholdt, professor of theoretical physics at the University of Bremen, described how networks can self-organize to a critical state by following two simple rules. If a node is in a disordered state (i.e., it constantly changes between activity and inactivity), it randomly loses one of its links; if a node is in an ordered state (i.e., it is constantly either active or inactive), it randomly gains a link. This model, designed to describe real neural networks, produces their key out-of-equilibrium dynamics. A critical state may be preferred because it allows the brain to easily balance inactivity and activity, Bornholdt argued.

In the second keynote lecture, Marija Mitrovic Dankulov, research professor at the University of Belgrade, introduced another mechanism by which networks can self-organize to a transition state. She has studied online social networks and reported the same kind of power-law dynamics as in neural networks. Using agent-based modeling, Dankulov showed that changes of social networks, the avalanches of activity that sweep the system, are driven by emotions or new ideas. Remarkably, her social networks as well as Bornholdt’s neural networks obey the same scaling laws as some physical systems. That means, vastly different networks can be modeled similarly – they belong to the same universality class, as physicists say.

System states at phase transitions typically exhibit fractal structures and dynamics. Rashid Bekhbudov from RWTH Aachen University addressed the question which complex networks are fractal. He showed that six networks from different domains of nature and society give rise to fractality and lose the small-world property when links that connect distant parts of a network (i.e., shortcuts) are removed. His initial results suggest that the small-world property emerges from the aggregation of interactions.

Minsuk Kim from Indiana University Bloomington presented the structural effects of removing shortest paths from random networks. When shortest paths of unlimited length are removed, networks transition into disconnected components much more abruptly than when paths of limited length are removed. When the network contains strongly-connected hubs, the phase transition is even preceded by a homogenization process in which hubs disappear. This suggests that shortest paths are strongly facilitated by the hubs, a meaningful finding in Kim’s context of transportation networks.

Game-theoretical models often exhibit phase transitions. Ren Manfredi from the IMT School for Advanced Studies Lucca presented a version of the Public Goods Game (PGG) where pro-social cooperation on scale-free networks emerges at a critical value of enhancement. The enhancement parameter that governs the phase transition is a positive return on pro-social behavior. If the PGG fairly represents a societal mechanism, its implications for social policy are profound: if pro-social societies are desired, they can be supported by increasing enhancement, and cooperation will emerge spontaneously. Manfredi’s contribution is that social influence is a kind of amplifier that reinforces behavioral tendencies in the model.

Two contributions specifically addressed critical transitions. These are processes in which networks exhibit abrupt topological transitions over time. Eva Rifà from Eurecat and colleagues have studied rumor spreading in social networks using the stochastic Maki-Thompson model. In this model, agents can be uninterested in a rumor, actively spread it, or no longer be interested in it. Rifà and colleagues explored whether an organic increase of rumor spreading can be distinguished from an externally stimulated increase (i.e., astroturfing). They found that only an organic process produces lag-based oscillations in autocorrelation. They applied their finding to the spreading of the rumor in 2012 that the Higgs boson was discovered. They found the oscillations and concluded that rumor spreading was due to the activation of previously existing scientists. Their method should be broadly applicable to diagnose critical transitions in time series.

Finally, Şiir Çınar Uysal from Bielefeld University proposed that one can predict if a system approaches a critical transition if it can be transformed into a weighted network. He showed how the time series of stock returns, inter-particle forces in jammed materials, and functional brain patterns can be transformed into such networks and how their signatures can be quantified using topological data analysis, a recent trend in applied mathematics. All three examples exhibit topological changes akin to critical transitions.

NetSci 2026 will take place in Boston, United States. Feel free to subscribe to our mailing list to be informed about the next CPIN symposium or to participate in planning it.