The ability of a mutant individual to overtake the whole of a population is one of the fundamental problems in evolutionary dynamics. Fixation probability and Average Fixation Time (AFT) are two important parameters to quantify this ability. In this paper we introduce an analytical approach for exact calculation of AFT. Using this method we obtain AFT for two types of evolutionary graphs: cycle graph, as a highly homogeneous graph and star graph, as a highly heterogeneous graph. We use symmetries of these graphs to calculate AFT. Analytical results are confirmed with simulation. We also examine the effect of adding some random edges to each of these structures.

We introduce the notion of a topological symmetry as a quantum mechanical symmetry involving a certain topological invariant. We obtain the underlying algebraic structure of the Z 2 -graded uniform topological symmetries of types (1, 1) and (2, 1). This leads to a novel derivation of the algebras of supersymmetry and p = 2 parasupersymmetry.

A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

In this work we investigate the stability of synchronized states for the Kuramoto model on scale-free and random networks in the presence of white noise forcing. We show that for a fixed coupling constant, the robustness of the globally synchronized state against the noise is dependent on the noise intensity on both kinds of networks. At low noise intensities the random networks are more robust against losing the coherency but upon increasing the noise, at a specific noise strength the synchronization among the population vanishes suddenly. In contrast, on scale-free networks the global synchronization disappears continuously at a much larger critical noise intensity respect to the random networks.

We give the definition and explore the algebraic structure of a class of quantum symmetries, called topological symmetries, which are generalizations of supersymmetry in the sense that they involve topological invariants similar to the Witten index. A topological symmetry (TS) is specified by an integer n > 1, which determines its grading properties, and an n-tuple of positive integers (m 1 , m 2 ,. .. , m n). We identify the algebras of supersymmetry, p = 2 parasupersymmetry, and fractional supersymmetry of order n with those of the Z 2-graded TS of type (1, 1), Z 2-graded TS of type (2, 1), and Z n-graded TS of type (1, 1,. .. , 1), respectively. We also comment on the mathematical interpretation of the topological invariants associated with the Z n-graded TS of type (1, 1,. .. , 1). For n = 2, the invariant is the Witten index which can be identified with the analytic index of a Fredholm operator. For n > 2, there are n independent integer-valued invariants. These can be related to differences of the dimension of the kernels of various products of n operators satisfying certain conditions.