Abstract: The books should explain basic experimental reasons why the complex networks are relevant objects of modern society. Then, three Multi-Agent Systems placed on graphs and complex networks are studied by computer simulation. The author focused on Sznajd model of opinion formation, Minority Game and Scattering model of wealth in society. Finally, description of library Zarja, where all the simulations were prepared, is presented.
Abstract: We analyze a special class of $1$-D quantum walks (QWs) realized
using optical multi-ports. We assume non-perfect multi-ports showing
errors in the connectivity, i.e. with a small probability the multi-ports
can connect not to their nearest neighbor but to another multi-port
at a fixed distance -- we call this a jump.
We study two cases of QW with jumps where multiple displacements can emerge
at one timestep. The first case assumes time-correlated jumps (static
disorder). In the second case, we choose the positions of jumps randomly in time
(dynamic disorder).
The probability distributions of positions of the QW walker in both instances differ significantly: dynamic disorder leads to a Gaussian-like distribution, while for static disorder we find two distinct behaviors depending on the parity of jump size. In the case of even-sized jumps, the
distribution exhibits a three-peak profile around the position
of the initial excitation, whereas the probability distribution in
the odd case follows a Laplace-like discrete distribution modulated
by additional (exponential) peaks for long times. Finally, our numerical results indicate that by an appropriate mapping an universal functional behavior of the variance of the long-time probability distribution can be revealed with respect to the scaled average of jump size.
Abstract: We have simulated the model of Employment, Production and Consumption (EPC) using Monte Carlo. The EPC model is an agent based model that mimics very basic rules of industrial economy. From the perspective of physics, the nature of the interactions in the EPC model represents multi-agent interactions where the relations among agents follow the key laws for circulation of capital and money. Monte Carlo simulations of the stochastic model reveal phase transition in the model economy. The two phases are the phase with full unemployment and the phase with nearly full employment. The economy switches between these two states suddenly as a reaction to a slight variation in the exogenous parameter, thus the system exhibits strong non-linear behavior as a response to the change of the exogenous parameters.
Abstract: We examine the operation of a device for a public quantum key distribution network. The recipients attempt to determine whether or not their individual key copies, which are a sequence of coherent states, are identical. To quantify the success of the protocol we use a fidelity-based figure of merit and describe a method for increasing this in the presence of noise and imperfect detectors. We show that the fidelity may be written as the product of two factors: one that depends on the properties of the device setup and another that depends on the detectors used. We then demonstrate the effect various parameters have on the overall effective operation of the device.
Abstract: The Minority Game is adapted to study the "imitation dilemma", i.e. the tradeoff between local benefit and global harm coming from imitation. The agents are placed on a substrate network and are allowed to imitate more successful neighbours. Imitation domains, which are oriented trees, are formed. We investigate size distribution of the domains and in-degree distribution within the trees. We use four types of substrate: one-dimensional chain; Erdos-Renyi graph; Barabasi-Albert scale-free graph; Barabasi-Albert 'model A' graph. The behaviour of some features of the imitation network strongly depend on the information cost epsilon, which is the percentage of gain the imitators must pay to the imitated. Generally, the system tends to form a few domains of equal size. However, positive epsilon makes the system stay in a long-lasting metastable state with complex structure. The in-degree distribution is found to follow a power law in two cases of those studied: for Erdos-Renyi substrate for any epsilon and for Barabasi-Albert scale-free substrate for large enough epsilon. A brief comparison with empirical data is provided.
Abstract: The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochronibel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed 1/n dependence in the distribution of votes in Brazilian elections is provided.
