-It has recently been suggested [1] that in a two-level multiplex network, a gradual change in the value of the"interlayer" strength p can provoke an abrupt structural transition. The critical point p * at which this happens is system-dependent. In this article, we show in a similar way as in [2] that this is a consequence of the graph Laplacian formalism used in [1]. We calculate the evolution of p * as a function of system size for ER and RR networks. We investigate the behavior of structural measures and dynamical processes of a two-level system as a function of p, by Monte-Carlo simulations, for simple particle diffusion and for reaction-diffusion systems. We find that as p increases there is a smooth transition from two separate networks to a single one. We cannot find any abrupt change in static or dynamic behavior of the underlying system.Introduction. -In the past several years, single networks have been extensively studied [3-8] both regarding their structure and also regarding different interactions between their nodes. In real world systems, however, there may be more than one type of relationship for the same collection of objects constituting the network. Consider, for example, the communication and power networks in a given geographical region. In this case, a failure in some power station will affect not only the functionality of the power grid, but also the routing system for all computers that uses electrical power to sustain its functionality. Thus, more recently [9], effort has been applied to the so called "interdependent" or "interconnected" networks, meaning a system of two or more networks that are linked together. Several articles have been published investigating the properties of these systems [9,10] as well as the evolution of dynamical processes on them [11,12]. In their simplest form, they consist of two single networks having their nodes connected in a one-to-one configuration with each "interlink" having the same strength p (in the range of 0 < p < ∞). A first question of interest is how does the variation of the parameter p affect the global structural properties of the entire system. To this end, one can make use of the Laplacian matrix. According to graph theory, given a graph of N nodes, the adjacency matrix A is defined as:

The use of a nanoparticle (NP)-based antitumor drug carrier has been an emerging strategy for selectively delivering the drugs to the tumor area and, thus, reducing the side effects that are associated with a high systemic dose of antitumor drugs. Precise control of drug loading and release is critical so as to maximize the therapeutic index of the NPs. Here, we propose a simple method of synthesizing NPs with tunable drug release while maintaining their loading ability, by varying the polymer matrix density of amine- or carboxyl-functionalized hydrogel NPs. We find that the NPs with a loose matrix released more cisplatin, with up to a 33 times faster rate. Also, carboxyl-functionalized NPs loaded more cisplatin and released it at a faster rate than amine-functionalized NPs. We performed detailed Monte Carlo computer simulations that elucidate the relation between the matrix density and drug release kinetics. We found good agreement between the simulation model and the experimental results for drug release as a function of time. Also, we compared the cellular uptake between amine-functionalized NPs and carboxyl-functionalized NPs, as a higher cellular uptake of NPs leads to improved cisplatin delivery. The amine-functionalized NPs can deliver 3.5 times more cisplatin into cells than the carboxyl-functionalized NPs. The cytotoxic efficacy of both the amine-functionalized NPs and the carboxyl-functionalized NPs showed a strong correlation with the cisplatin release profile, and the latter showed a strong correlation with the NP matrix density.

We study the transient dynamics of an A+B→0 process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions q of cross couplings, the concentration of A (or B) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time t_{x}. By numerical and analytical arguments, we show that for symmetric and homogeneous structures t_{x}∝(〈k〉/q)log(〈k〉/q) where 〈k〉 is the mean degree of both networks. Being this behavior is in marked contrast with a purely diffusive process, where the mixing time would go simply like 〈k〉/q, we identify the logarithmic slowing down in t_{x} to be the result of a spontaneous mechanism of repulsion between the reactants A and B due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1μm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70–1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles.

We study the effect of the variation of reaction efficiency in binary reactions. We use the well-known A + B → 0 model, which has been extensively studied in the past. We perform simulations on this model where we vary the efficiency of reaction, i.e., when two particles meet they do not instantly react, as has been assumed in previous studies, but they react with a probability γ, where γ is in the range 0 < γ < 1. Our results show that at small γ values the system is reaction limited, but as γ increases it crosses over to a diffusion limited behavior. At early times, for small γ values, the particle density falls slower than for larger γ values. This fall-off goes over a crossover point, around the value of γ = 0.50 for high initial densities. Under a variety of conditions simulated, we find that the crossover point was dependent on the initial concentration but not on the lattice size. For intermediate and long times simulations, all γ values (in the depleted reciprocal density versus time plot) converge to the same behavior. These theoretical results are useful in models of epidemic reactions and epidemic spreading, where a contagion from one neighbor to the next is not always successful but proceeds with a certain probability, an analogous effect with the reaction probability examined in the current work.