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1. Triadic percolation on multilayer networks

   https://doi.org/10.1103/yvtg-wnn4  

   Sun, Hanlin; Radicchi, Filippo; Bianconi, Ginestra (January 2026, Physical Review E)

   Triadic interactions are special types of higher-order interactions that occur when regulator nodes modulate the interactions between other two or more nodes. In the presence of triadic interactions, a percolation process occurring on a single-layer network becomes a full fledged dynamical system, characterized by period doubling and a route to chaos. Here we generalize the model to multilayer networks and name it as the multilayer triadic percolation (MTP) model. We find a much richer dynamical behavior of the MTP model than its single-layer counterpart. MTP displays a Neimark-Sacker bifurcation, leading to oscillations of arbitrarily large period or pseudoperiodic oscillations. Moreover, MTP admits period-two oscillations without negative regulatory interactions, whereas single-layer systems only display discontinuous hybrid transitions. This comprehensive model offers new insights on the importance of regulatory interactions in real-world systems such as brain networks, climate, and ecological systems. 

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2. A generalized Flory-Stockmayer kinetic theory of connectivity percolation and rigidity percolation of cytoskeletal networks

   https://doi.org/10.1371/journal.pcbi.1010105  

   Bueno, Carlos; Liman, James; Schafer, Nicholas P.; Cheung, Margaret S.; Wolynes, Peter G. (May 2022, PLOS Computational Biology)

   Gov, Nir (Ed.)

   Actin networks are essential for living cells to move, reproduce, and sense their environments. The dynamic and rheological behavior of actin networks is modulated by actin-binding proteins such as α-actinin, Arp2/3, and myosin. There is experimental evidence that actin-binding proteins modulate the cooperation of myosin motors by connecting the actin network. In this work, we present an analytical mean field model, using the Flory-Stockmayer theory of gelation, to understand how different actin-binding proteins change the connectivity of the actin filaments as the networks are formed. We follow the kinetics of the networks and estimate the concentrations of actin-binding proteins that are needed to reach connectivity percolation as well as to reach rigidity percolation. We find that Arp2/3 increases the actomyosin connectivity in the network in a non-monotonic way. We also describe how changing the connectivity of actomyosin networks modulates the ability of motors to exert forces, leading to three possible phases of the networks with distinctive dynamical characteristics: a sol phase, a gel phase, and an active phase. Thus, changes in the concentration and activity of actin-binding proteins in cells lead to a phase transition of the actin network, allowing the cells to perform active contraction and change their rheological properties. 

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3. Adaptive Shrinkage Estimation for Streaming Graphs

   Ahmed, Nesreen; Duffield, Nick (October 2020, Advances in neural information processing systems)

   null (Ed.)

   Networks are a natural representation of complex systems across the sciences, and higher-order dependencies are central to the understanding and modeling of these systems. However, in many practical applications such as online social networks, networks are massive, dynamic, and naturally streaming, where pairwise interactions among vertices become available one at a time in some arbitrary order. The massive size and streaming nature of these networks allow only partial observation, since it is infeasible to analyze the entire network. Under such scenarios, it is challenging to study the higher-order structural and connectivity patterns of streaming networks. In this work, we consider the fundamental problem of estimating the higher-order dependencies using adaptive sampling. We propose a novel adaptive, single-pass sampling framework and unbiased estimators for higher-order network analysis of large streaming networks. Our algorithms exploit adaptive techniques to identify edges that are highly informative for efficiently estimating the higher-order structure of streaming networks from small sample data. We also introduce a novel James-Stein shrinkage estimator to reduce the estimation error. Our approach is fully analytic, computationally efficient, and can be incrementally updated in a streaming setting. Numerical experiments on large networks show that our approach is superior to baseline methods. 

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4. Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes

   https://doi.org/10.1038/s42005-022-01062-3  

   Kilic, Bengier Ülgen; Taylor, Dane (December 2022, Communications Physics)

   Abstract Cascades over networks (e.g., neuronal avalanches, social contagions, and system failures) often involve higher-order dependencies, yet theory development has largely focused on pairwise-interaction models. Here, we develop a ‘simplicial threshold model’ (STM) for cascades over simplicial complexes that encode dyadic, triadic and higher-order interactions. Focusing on small-world models containing both short- and long-range k -simplices, we explore spatio-temporal patterns that manifest as a frustration between local and nonlocal propagations. We show that higher-order interactions and nonlinear thresholding coordinate to robustly guide cascades along a k -dimensional generalization of paths that we call ‘geometrical channels’. We also find this coordination to enhance the diversity and efficiency of cascades over a simplicial-complex model for a neuronal network, or ‘neuronal complex’. We support these findings with bifurcation theory and data-driven approaches based on latent geometry. Our findings provide fruitful directions for uncovering the multiscale, multidimensional mechanisms that orchestrate the spatio-temporal patterns of nonlinear cascades. 

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5. Geometrically frustrated assembly at finite temperature: Phase transitions from self-limiting to bulk states

   https://doi.org/10.1103/bj18-bphb  

   Hackney, Nicholas W; Grason, Gregory (December 2025, Physical Review E)

   Geometric frustration is recognized to generate complex morphologies in self-assembling particulate and molecular systems. In bulk states, frustration drives structured arrays of topological defects. In the dilute limit, these systems have been shown to form a novel state of self-limiting assembly, in which the equilibrium size of multiparticle domains are finite and well defined. In this article we employ Monte Carlo simulations of a recently developed 2D lattice model of geometrically frustrated assembly [Hackney et al., Phys. Rev. X 13, 041010 (2023)] to study the phase transitions between the self-limiting and defect bulk phase driven by two distinct mechanisms: (1) increasing concentration and (2) decreasing temperature or frustration. The first transition is mediated by a concentration-driven percolation transition of self-limiting, wormlike domains into an intermediate heterogeneous network mesophase, which gradually fills in at high concentration to form a quasiuniform defect bulk state. We find that the percolation threshold is weakly dependent on frustration and shifts to higher concentration as frustration is increased, but depends strongly on the ratio of cohesion to elastic stiffness in the model. The second transition takes place between self-limiting assembly at high-temperature or frustration and phase separation into a condensed bulk state at low temperature or frustration. We consider the competing influences that translational and conformational entropy have on the critical temperature or frustration and show that the self-limiting phase is stabilized at higher frustrations and temperatures than previously expected. Taken together, this understanding of the transition pathways from self-limiting to bulk defect phases of frustrated assembly allows us to map the phase behavior of this 2D minimal model over the full range of concentration. 

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