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Networks that have power-law connectivity, commonly referred to as the scale-free networks, are an important class of complex networks. A heterogeneous mean-field approximation has been previously proposed for the Ising model of the Barab��si-Albert model of scale-free networks with classical spins on the nodes wherein it was shown that the critical temperature for such a system scales logarithmically with network size. For finite sizes, there is no criticality for such a system and hence no true phase transition in terms of singular behavior. Further, in the thermodynamic limit, the mean-field prediction of an infinite critical temperature for the system may exclude any true phase transition even then. Nevertheless, with an eye on potential applications of the model on biological systems that are generally finite, one may still try to find approximations that describe the relevant observables quantitatively. Here we present an alternative, approximate formulation for the description of the Ising model of a Barab��si-Albert Network. Using the classical definition of magnetization, we show that Ising models on a network can be well-approximated by a long-range interacting homogeneous Ising model wherein each node of the network couples to all other spins with a strength determined by the mean degree of the Barab��si-Albert Network. In such an effective long-range Ising model of a Barab��si-Albert Network, the critical temperature is directly proportional to the number of preferentially attached links added to grow the network. The proposed model describes the magnetization of the majority of the sites with average or smaller than average degree better compared to the heterogeneous mean-field approximation. The long-range Ising model is the only homogeneous description of Barab��si-Albert networks that we know of., 30 pages, 6 figures, 2 tables, preliminary results of this work has been presented in the form of a talk at the Conference on Complex Systems 2019

The central question of systems biology is to understand how individual components of a biological system such as genes or proteins cooperate in emerging phenotypes resulting in the evolution of diseases. As living cells are open systems in quasi-steady state type equilibrium in continuous exchange with their environment, computational techniques that have been successfully applied in statistical thermodynamics to describe phase transitions may provide new insights to emerging behavior of biological systems. Here we will systematically evaluate the translation of computational techniques from solid-state physics to network models that closely resemble biological networks and develop specific translational rules to tackle problems unique to living systems. Hence we will focus on logic models exhibiting only two states in each network node. Motivated by the apparent asymmetry between biological states where an entity exhibits boolean states i.e. is active or inactive, we present an adaptation of symmetric Ising model towards an asymmetric one fitting to living systems here referred to as the modified Ising model with gene-type spins. We analyze phase transitions by Monte Carlo simulations and propose mean-field solution of modified Ising model of a network type that closely resembles real-world network, the Barab��si-Albert model of scale-free networks. We show that asymmetric Ising models show similarities to symmetric Ising models with external field and undergoes a discontinuous phase transition of the first-order and exhibits hysteresis. The simulation setup presented here can be directly used for any biological network connectivity dataset and is also applicable for other networks that exhibit similar states of activity. This is a general statistical method to deal with non-linear large scale models arising in the context of biological systems and is scalable to any network size., 30 pages, 8 figures, presented in poster form in SBHD'18, LA, USA; as a talk in SIAM CSE'19

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatio-temporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrate that memory changes the properties of the spatio-temporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatio-temporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory., 10 pages,2 figures

Geometric phase in the wave function is important with regard to quantum non-locality and adiabatic evolution. We study the confinement of a particle by three-dimensional isotropically moving walls, of relevance to experimental trapping techniques, via a proposed approach that explains the physical origin of the geometric Dirac phase induced in the wave function. This phase depends only on the relative rate of change of the spatial scale factor. The approach also yields the class of external potentials that replicate adiabatic evolution in finite time. As illustrative examples, we consider uniform and accelerating walls, and the case where the Dirac phase is due to cosmic expansion.