arXiv subjects

Somya Singh

Publications and source records attributed to Somya Singh.

3 recordsLinked to original sources

arXiv2026

Opinion Dynamics with Memory Loss and Communication Delays

We propose a novel framework for modeling binary opinions (0 or 1) of individuals connected through a weighted directed network, where edge weights quantify interpersonal influence. Unlike classical models that assume complete access to previously expressed opinions, our framework allows individuals to update their biases using structured memory sets that capture limited and delayed information exchange. To analyze these opinion differences, we introduce a mathematically tractable notion of relative bias between pairs of individuals. The relative biases evolve according to a linear update rule involving past expressed opinions specified by the memory sets. We define the belief of an individual as the probability of expressing opinion 1 and derive a time-delayed dynamical system governing the evolution of network beliefs. We establish its asymptotic behavior and characterize its properties. The framework is further extended to networks containing bots, which maintain fixed biases while influencing neighboring individuals. We quantify the effect of bots by comparing the fixed points of the dynamics in their presence and absence. Finally, simulations illustrate the influence of memory, network structure, and bot interactions on the resulting opinion dynamics.

math.PR↗

arXiv2024

Generating Preferential Attachment Graphs via a Pólya Urn with Expanding Colors

We introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number (potentially time-varying and non-integer) of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.

math.PR↗

arXiv2021

A Finite Memory Interacting Pólya Contagion Network and its Approximating Dynamical Systems

We introduce a new model for contagion spread using a network of interacting finite memory two-color Pólya urns, which we refer to as the finite memory interacting Pólya contagion network. The urns interact in the sense that the probability of drawing a red ball (which represents an infection state) for a given urn, not only depends on the ratio of red balls in that urn but also on the ratio of red balls in the other urns in the network, hence accounting for the effect of spatial contagion. The resulting network-wide contagion process is a discrete-time finite-memory ($M$th order) Markov process, whose transition probability matrix is determined. The stochastic properties of the network contagion Markov process are analytically examined, and for homogeneous system parameters, we characterize the limiting state of infection in each urn. For the non-homogeneous case, given the complexity of the stochastic process, and in the same spirit as the well-studied SIS models, we use a mean-field type approximation to obtain a discrete-time dynamical system for the finite memory interacting Pólya contagion network. Interestingly, for $M=1$, we obtain a linear dynamical system which exactly represents the corresponding Markov process. For $M>1$, we use mean-field approximation to obtain a nonlinear dynamical system. Furthermore, noting that the latter dynamical system admits a linear variant (realized by retaining its leading linear terms), we study the asymptotic behavior of the linear systems for both memory modes and characterize their equilibrium. Finally, we present simulation studies to assess the quality of the approximation purveyed by the linear and non-linear dynamical systems.

math.DS↗