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Phase transitions (PTs) are generally classified into second-order and first-order transitions, each exhibiting different intrinsic properties. For instance, a first-order transition exhibits latent heat and hysteresis when a control parameter is increased and then decreased across a transition point, whereas a second-order transition does not. Recently, hybrid percolation transitions (HPTs) are issued in diverse complex systems, in which the features of first-order and second-order PTs occur at the same transition point. Thus, the question whether hysteresis appears in an HPT arises. Herein, we investigate this fundamental question with a so-called restricted Erd\H{o}s--R\'enyi random network model, in which a cluster fragmentation process is additionally proposed. The hysteresis curve of the order parameter was obtained. Depending on when the reverse process is initiated, the shapes of hysteresis curves change, and the critical behavior of the HPT is conserved throughout the forward and reverse processes.

The Kuramoto model exhibits different types of synchronization transitions depending on the type of natural frequency distribution. To obtain these results, the Kuramoto self-consistency equation (SCE) approach has been used successfully. However, this approach affords only limited understanding of more detailed properties such as the stability and finite size effect. Here, we extend the SCE approach by introducing an effective potential, that is, an integral version of the SCE. We examine the landscape of this effective potential for second-order, first-order, and hybrid synchronization transitions in the thermodynamic limit. In particular, for the hybrid transition, we find that the minimum of effective potential displays a plateau across the region in which the order parameter jumps. This result suggests that the effective free energy can be used to determine a type of synchronization transition. For finite systems, the effective potential contains local minima at which the system can be trapped. Using numerical simulations, we determine the stability of the system as a function of system size and simulation time., 12 pages, 13 figures

The features of animal population dynamics, for instance, flocking and migration, are often synchronized for survival under large-scale climate change or perceived threats. These coherent phenomena have been explained using synchronization models. However, such models do not take into account asynchronous and adaptive updating of an individual's status at each time. Here, we modify the Kuramoto model slightly by classifying oscillators as leaders or followers, according to their angular velocity at each time, where individuals interact asymmetrically according to their leader/follower status. As the angular velocities of the oscillators are updated, the leader and follower status may also be reassigned. Owing to this adaptive dynamics, oscillators may cooperate by taking turns acting as a leader or follower. This may result in intriguing patterns of synchronization transitions, including hybrid phase transitions, and produce the leader-follower switching pattern observed in bird migration patterns.

The synchronization pattern of a fully connected competing Kuramoto model with a uniform intrinsic frequency distribution $g(\omega)$ was recently considered. This competing Kuramoto model assigns two coupling constants with opposite signs, $K_1 < 0$ and $K_2 > 0$, to the $1-p$ and $p$ fractions of nodes, respectively. This model has a rich phase diagram that includes incoherent, $\pi$, and traveling wave (TW) phases and a hybrid phase transition with abnormal properties that occurs through an intermediate metastable $\pi$ state. Here, we consider the competing Kuramoto model on Erd\H{o}s--R\'enyi (ER) random networks. Numerical simulations and the mean-field solution based on the annealed network approximation reveal that in this case, when the mean degree of the random networks is large, the features of the phase diagram and transition types are consistent overall with those on completely connected networks. However, when the mean degree is small, the mean-field solution is not consistent with the numerical simulation results; specifically, the TW state does not occur, and thus the phase diagram is changed, owing to the strong heterogeneity of the local environment. By contrast, for the original Kuramoto oscillators, the annealed mean-field solution is consistent with the numerical simulation result for ER networks.

The Kuramoto model with mixed signs of couplings is known to produce a traveling-wave synchronized state. Here, we consider an abrupt synchronization transition from the incoherent state to the traveling-wave state through a long-lasting metastable state with large fluctuations. Our explanation of the metastability is that the dynamic flow remains within a limited region of phase space and circulates through a few active states bounded by saddle and stable fixed points. This complex flow generates a long-lasting critical behavior, a signature of a hybrid phase transition. We show that the long-lasting period can be controlled by varying the density of inhibitory/excitatory interactions. We discuss a potential application of this transition behavior to the recovery process of human consciousness.

Understanding of a hybrid percolation transitions (HPTs) induced by cluster coalescence, exhibiting a jump in the giant cluster size and a critical behavior of finite clusters, is fundamental and intriguing. Here, we uncover the underlying mechanism using the so-called restricted-random network model, in which clusters are ranked by size and partitioned into small- and large-cluster sets. As clusters are merged and their rankings are updated, they may move back and forth across the set boundary. The intervals of these crossings exhibit a self-organized critical (SOC) behavior with two power-law exponents. During this process, a bump is formed and eliminated in the cluster size distribution, characterizing the criticality of the HPT. This SOC behavior is in contrast to the critical branching process, which governs the avalanche dynamics of the HPT in the pruning process. Finally, we find that a burst of such crossing events occurs and signals the upcoming abrupt transition.

The Hopfield model is a pioneering neural network model with associative memory retrieval. The analytical solution of the model in mean field limit revealed that memories can be retrieved without any error up to a finite storage capacity of $O(N)$, where $N$ is the system size. Beyond the threshold, they are completely lost. Since the introduction of the Hopfield model, the theory of neural networks has been further developed toward realistic neural networks using analog neurons, spiking neurons, etc. Nevertheless, those advances are based on fully connected networks, which are inconsistent with recent experimental discovery that the number of connections of each neuron seems to be heterogeneous, following a heavy-tailed distribution. Motivated by this observation, we consider the Hopfield model on scale-free networks and obtain a different pattern of associative memory retrieval from that obtained on the fully connected network: the storage capacity becomes tremendously enhanced but with some error in the memory retrieval, which appears as the heterogeneity of the connections is increased. Moreover, the error rates are also obtained on several real neural networks and are indeed similar to that on scale-free model networks.