arXiv subjects

Arnab Basu

Publications and source records attributed to Arnab Basu.

4 recordsLinked to original sources

arXiv2022

On the Rise and Fall of Online Social Networks

The rise and fall of online social networks recently generated an enormous amount of interest among people, both inside and outside of academia. Gillette [Businessweek magazine, 2011] did a detailed analysis of MySpace, which started losing its popularity since 2008. Cannarella and Spechler [ArXiv, 2014] used a model of disease spread to explain the rise and fall of MySpace. In this paper, we present a graph theoretical model that may be able to provide an alternative explanation for the rise and fall of online social networks. Our model is motivated by the well-known Barabasi-Albert model of generating random scale-free networks using preferential attachment or `rich-gets-richer' phenomenon. As shown by our empirical analysis, we conjecture that such an online social network growth model is inherently flawed as it fails to maintain the stability of such networks while ensuring their growth. In the process, we also conjecture that our model of preferential attachment also exhibits scale-free phenomenon.

cs.SI↗

arXiv2016

Generic Behaviour of Strongly Reinforced Polya Urns : Convergence and Stability

We consider, as proposed and studied in Hofstad et.\ al.\ \cite{HHKR}, a class of graph-based "interacting urn"-type Polya urn model inspired by neuronal processing in the brain where a signal enters the brain at some (randomly) chosen neuron and is transmitted to a (random) single neighbouring neuron with a probability depending on the relative `efficiency' of the synapses connecting the neurons, and in doing so the efficiency of the utilized synapse is improved/reinforced. We study the structures (or architectures) and relative efficiency of the neuronal networks that can arise from repeating this process a very large number of times in a "strong reinforcement regime". Under the most general conditions, we prove in the affirmative a part of the main open conjecture in \cite{HHKR} i.e. the zero probability of convergence of the corresponding "urn process" to any 'unstable' equilibrium. Under very generic conditions, i.e., for an open and dense subset of parameters with full measure, we also prove the full open conjecture by showing the finiteness of the equilibrium set and hence the unit probability of convergence of the "urn process" to some 'stable' equilibrium.

math.PR↗

arXiv2016

Nonzero-sum Risk-sensitive Stochastic Games on a Countable State Space

The infinite horizon risk-sensitive discounted-cost and ergodic-cost nonzero-sum stochastic games for controlled Markov chains with countably many states are analyzed. For the discounted-cost game, we prove the existence of Nash equilibrium strategies in the class of Markov strategies under fairly general conditions. Under an additional geometric ergodicity condition and a small cost criterion, the existence of Nash equilibrium strategies in the class of stationary Markov strategies is proved for the ergodic-cost game.

math.OC↗

arXiv2014

Nash Equilibria via Duality and Homological Selection

Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$, we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincaré Duality in addition to the Dold-Thom Theorem.

math.AT↗