We study a class of nonlinear dynamical systems to develop efficient algorithms. As an efficient algorithm, interior point method based on Newton's method is well-known for solving convex programming problems which include linear, quadratic, semidefinite and lp-programming problems. On the other hand, the geodesic of information geometry is represented by a continuous Newton's method for minimizing a convex function called divergence. Thus, we discuss a relation between information geometry and convex programming in a related family of continuous Newton's method. In particular, we consider the α-projection problem from a given data onto an information geometric submanifold spanned with power-functions. In general, an information geometric structure can be induced from a standard convex programming problem. In contrast, the correspondence from information geometry to convex programming is slightly complicated. We first present there exists a same structure between the α-projection and semidefinite programming problems. The structure is based on the linearities or autoparallelisms in the function space and the space of matrices, respectively. However, the α-projection problem is not a form of convex programming. Thus, we reformulate it to a lp-programming and the related ones. For the reformulated problems, we derive self-concordant barrier functions according to the values of α. The existence of a polynomial time algorithm is theoretically confirmed for the problem. Furthermore, we present the coincidence with the gradient vectors for the divergence and a modified barrier function. These results connect a part of nonlinear and algorithm theories by the discreteness of variables.

The iterative random subdivision of rectangles is used as a generation model of networks in physics, computer science, and urban planning. However, these researches were independent. We consider some relations in them, and derive fundamental properties for the average lifetime depending on birth-time and the balanced distribution of rectangle faces.

We numerically investigate that optimal robust onion-like networks can emerge even with the constraint of surface growth in supposing a spatially embedded transportation or communication system. To be onion-like, moderately long links are necessary in the attachment through intermediations inspired from a social organization theory.

Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the fractal dimension by using the distribution without huge computations. This method can be applied to self-similar tilings based on a stochastic process.

From an information geometric viewpoint, we investigate a characteristic of the submanifold of a mixture or exponential family in the manifold of finite discrete distributions. Using the characteristic, we derive a direct calculation method for an em-geodesic in the submanifold. In this method, the value of the primal parameter on the geodesic can be obtained without iterations for a gradient system which represents the geodesic. We also derive the similar algorithms for both problems of parameter estimation and functional extension of the submanifold for a data in the ambient manifold. These theoretical approaches from geometric analysis will contribute to the development of an efficient algorithm in computational complexity.

For realistic scale-free networks, we investigate the traffic properties of stochastic routing inspired by a zero-range process known in statistical physics. By parameters α and δ, this model controls degree-dependent hopping of packets and forwarding of packets with higher performance at more busy nodes. Through a theoretical analysis and numerical simulations, we derive the condition for the concentration of packets at a few hubs. In particular, we show that the optimal α and δ are involved in the trade-off between a detour path for α < 0 and long wait at hubs for α > 0; In the low-performance regime at a small δ, the wandering path for α < 0 better reduces the mean travel time of a packet with high reachability. Although, in the high-performance regime at a large δ, the difference between α > 0 and α < 0 is small, neither the wandering long path with short wait trapped at nodes (α = -1), nor the short hopping path with long wait trapped at hubs (α = 1) is advisable. A uniformly random walk (α = 0) yields slightly better performance. We also discuss the congestion phenomena in a more complicated situation with packet generation at each time step.