Some Considerations on Six Degrees of Separation from A Theoretical Point of View

In this article we discuss six degrees of separation, which has been proposed by Milgram, from a theoretical point of view. Simply if one has $k$ friends, the number $N$ of indirect friends goes up to $\sim k^d$ in $d$ degrees of separation. So it would easily come up to population of whole world. That, however, is unacceptable. Mainly because of nonzero clustering coefficient $C$, $N$ does not become $\sim k^d$. In this article, we first discuss relations between six degrees of separation and the clustering coefficient in the small world network proposed by Watt and Strogatz\cite{Watt1},\cite{Watt2}. Especially, conditions that $(N)>$ (population of U.S.A or of the whole world) arises in the WS model is explored by theoretical and numerical points of view. Secondly we introduce an index that represents velocity of propagation to the number of friends and obtain an analytical formula for it as a function of $C$, $K$, which is an average degree over all nodes, and some parameter $P$ concerned with network topology. Finally the index is calculated numerically to study the relation between $C$, $K$ and $P$ and $N$.Comment: 10 pages, 5 figure

Cooperation in Evolutionary Public Goods Game on Complex Networks with Topology Change

The evolution of cooperation among unrelated individuals in human and animal societies remains a challenging issue across disciplines. It is an important subject also in the evolutionary game theory to research how cooperation arises. The subject has been extensively studied especially in Prisonars' dilemma game(PD) and the emergence of cooperation is important subject also in public goods game(PGG). In this article, we consider evolutionary PGG on complex networks where the topology of the networks varies under the infulence of game dynamics. Then we study what effects on the evolution of player's strategies, defection and cooretation and the average payoff does the interaction between the game dynamics and the network topology bring. By investigating them by making computer simulations, we intend to clear in what situations cooperation strategy is promoted or preserved. We also intend to investigate the infuluence of the interaction on the average payoff over all players. Furthermore how initial networks are transformed to final networks by the evolution through the influences of PGG dynamics is invistigated.Comment: 10 pages, 24 figure

Does Parrondo Paradox occur in Scale Free Networks? -A simple Consideration-

Parrondo's paradox occurs in sequences of games in which a winning expectation may be obtained by playing the games in a random order, even though each game in the sequence may be lost when played individually. Several variations of Parrondo's games apparently with paradoxical property have been introduced; history dependence, one dimensional line, two dimensional lattice and so on. In this article, we examine whether Parrondo's paradox occurs or not in scale free networks. This is interesting as an empirical study, since scale free networks are ubiquitous in our real world. First some simulation results are given and after that theoretical studies are made. As a result, we mostly confirm that Parrondo's paradox can not occur in the naive case, where the game has the same number of parameters as the original Parrondo's game.Comment: 11 pages, 11 figure

Differential Geometry and Integrability of the Hamiltonian System of a Closed Vortex Filament

The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequense of constants of motion in involution. An algebraic framework is given for the aim of describing differential geometry of this system. A geometrical structure related to the integrability of this system is revealed. It is not a bi-Hamiltonian structure but similar one. As a related topic, a remark on the inspection of J.Langer and R.Perline, J.Nonlinear Sci.1, 71 (1991), is given.Comment: Title is changed ('a' is added). A mistake (absence of stating F-linearity condition of a skew-adjoint op., Sect.4) is corrected. One reference is added. The other changes are minor. 12 pages, LaTe

Braess like Paradox in a Small World Network

Braess \cite{1} has been studied about a traffic flow on a diamond type network and found that introducing new edges to the networks always does not achieve the efficiency. Some researchers studied the Braess' paradox in similar type networks by introducing various types of cost functions. But whether such paradox occurs or not is not scarcely studied in complex networks. In this article, I analytically and numerically study whether Braess like paradox occurs or not on Dorogovtsev-Mendes network\cite{2}, which is a sort of small world networks. The cost function needed to go along an edge is postulated to be equally identified with the length between two nodes, independently of an amount of traffic on the edge. It is also assumed the it takes a certain cost $c$ to pass through the center node in Dorogovtsev-Mendes network. If $c$ is small, then bypasses have the function to provide short cuts. As result of numerical and theoretical analyses, while I find that any Braess' like paradox will not occur when the network size becomes infinite, I can show that a paradoxical phenomenon appears at finite size of network.Comment: 5 pages,5 figure

Comments on Six Degrees of Separation based on the le Pool and Kochen Models

In this article we discuss six degrees of separation, which has been suggested by Milgram's famous experiment\cite{Milg},\cite{Milg2}, from a theoretical point of view again. Though Milgram's experiment was partly inspired to Pool and Kochen's study \cite{Pool} that was made from a theoretical point of view. At the time numerically detailed study could not be made because computers and important concepts, such as the clustering coefficient, needed for a network analysis nowadays, have not yet developed. In this article we devote deep study to the six degrees of separation based on some models proposed by Pool and Kochen by using a computer, numerically. Moreover we estimate the clustering coefficient along the method developed by us \cite{Toyota1} and extend our analysis of the subject through marrying Pool and Kochen's models to our method.Comment: 7 pages, 8 figure

GeV Photon Beams for Nuclear/Particle Physics

Production of a GeV photon beam by laser backward-Compton scattering has been playing an important role as a tool for nuclear and particle physics experiments. Its production techniques are now established at electron storage rings, which are increasing worldwide. A typical photon intensity has reached $\sim$ 10 $^6$ sec$^{-1}$. In the present article, the LEPS beamline facility at SPring-8 is mainly described with an overview of experimental applications, for the purpose to summarize the GeV photon beam production. Finally, possible future upgrades are discussed with new developments of laser injection.Comment: 30 pages, 13 figure

The Evolution of Cellar Automaton based on Dilemmma Games with Selfish Strategy

We have proposed two new evolutionary rules on spatio-iterated games that is not mimic evolution of strategies, and mainly discussed the Prisoner's Dilemma game \cite{toyota2} by the two evoutionary rules \cite{toyota3}. In this paper we focus the first rule, that is, the selfish evolutionary rule for various dilemma games. In contrast to the Prisoner's Dilemma, there are gererally rich pase structures in the dilemma games. First we analytically clear the structure to present phase diagrams in the various dilemma games. Forthermore we simulate the time evolution of the soatio-games in the some representatives of the parameters according to the phase diagrams. Including some mutations, detail investigations are made by a computer simulation for five kinds of initial configurations. As results we find some dualities and game invariant properties. They show a sort of bifurcation as a mutation parameter are varied. In the path from one period to two one some common features are observed in most of games and some chaotic behaviors appear in the middle of the transition. Lastly we estimate the total hamiltonian, which is defined by the sum of the total payoff of all agents in the system, and show that the chaotic period is best from the perspective of the payoff. We also made some primitive discussions on them.Comment: 15 pages, 16 figure

Parrondo Paradox in Scale Free Networks

Parrondo's paradox occurs in sequences of games in which a winning expectation may be obtained by playing the games in a random order, even though each game in the sequence may be lost when played individually. Several variations of Parrondo's games with paradoxical property have been introduced. In this paper, I examine whether Parrondo's paradox occurs or not in scale free networks. Two models are discussed by some theoretical analyses and computer simulations. As a result, I prove that Parrondo's paradox occurs only in the second model.Comment: 4 pages, 4 figures, appear in Proceedings of ITC-CSCC201

A Class of FRT Quantum Groups and Funq_q(G2_2) as a Special Case

Citations are updated; referred papers are increased. An error right after the eq.~(27) is corrected, and several chages (not serious) are made.Comment: 15 page