In this paper, we introduce a classical algorithm for random sampling of permutations, drawing inspiration from the Steinhaus-Johnson-Trotter algorithm. Our approach takes a comprehensive view of permutation sampling by expressing them as products of adjacent transpositions. Building on this, we develop a quantum analogue of the classical algorithm using a quantum circuit model for random sampling of permutations. As an application, we present a quantum algorithm for the two-sample randomization test to assess the difference of means in classical data. Finally, we propose a nested corona product graph generative model for symmetric groups, which facilitates random sampling of permutations from specific sets of permutations through a quantum circuit model.

We propose modeling signed networks by considering two layers in a social network for generation of positive and negative links where both the layers comprise of identical set of nodes. The growth process is modeled based on preferential attachment, formation of links probabilistically asserting structural balance of local groups, and internal growth which happens without addition of new nodes. We prove that the degree distribution of a generated network follows a power-law whose exponent depends on the largest eigenvalue of a matrix which governs the dynamics of growth of degrees of nodes with respect to positive and negative links. A computable formula for average degree and lower-bounds for the number of balanced and unbalanced triads of modelled networks are also obtained. A method for structural reconstruction of real signed networks is formulated through estimation the values of the model parameters to generate the network that can inherit different structural properties of the corresponding real network. Experimental results show that our model which we term as 2L-SNM can replicate properties of several real world signed networks much more robustly than competitive state-of-the-art techniques.

We propose a parametric network generation model which we call network reconstruction model (NRM) for structural reconstruction of scale-free real networks with power-law exponent greater than 2 in the tail of its degree distribution. The reconstruction method for a real network is concerned with finding the optimal values of the model parameters by utilizing the powerlaw exponents of model network and the real network. The method is validated for certain real world networks. The usefulness of NRM in order to solve structural reconstruction problem is demonstrated by comparing its performance with some existing popular network generative models. We show that NRM can generate networks which follow edge-densification and densification power-law when the model parameters satisfy an inequality. Computable expressions of the expected number of triangles and expected diameter are obtained for model networks generated by NRM. Finally, we numerically establish that NRM can generate networks with shrinking diameter and modular structure when specific model parameters are chosen.