Inequality and Network Formation Games

This paper addresses the matter of inequality in network formation games. We employ a quantity that we are calling the Nash Inequality Ratio (NIR), defined as the maximal ratio between the highest and lowest costs incurred to individual agents in a Nash equilibrium strategy, to characterize the extent to which inequality is possible in equilibrium. We give tight upper bounds on the NIR for the network formation games of Fabrikant et al. (PODC '03) and Ehsani et al. (SPAA '11). With respect to the relationship between equality and social efficiency, we show that, contrary to common expectations, efficiency does not necessarily come at the expense of increased inequality.Comment: 27 pages. 4 figures. Accepted to Internet Mathematics (2014

Progressive modularization: Reframing our understanding of typical and atypical language development

The ability to acquire language is a critical part of human development. Yet there is no consensus on how the skill emerges in early development. Does it constitute an innately-specified, language-processing module or is it acquired progressively? One of Annette Karmiloff-Smith’s (1938–2016) key contributions to developmental science addresses this very question. Karmiloff-Smith persistently maintained that the process of development itself constitutes a crucial factor in phenotypic outcomes. She proposed that cognitive modules gradually emerge through a developmental process – ‘progressive modularization’. This concept helped to advance the field beyond the stale nature–nurture controversy. It enabled language researchers to develop more nuanced transactional frameworks that take seriously the integration of genes and environment. In homage to Karmiloff-Smith, the current article describes the importance of her work to the field of developmental psychology and language research. It examines how the concept of progressive modularization could be applied to language development as well as how it has greatly advanced our understanding of language difficulties in children with neurodevelopmental disorders. Finally, it discusses how Karmiloff-Smith’s approach is inspiring current and future research

Degree Distribution of Competition-Induced Preferential Attachment Graphs

We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cutoffs. This is the first explanation of how preferential attachment can arise from a more basic underlying mechanism of local competition. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a ``fertility'' parameter. We prove that the general model has a power-law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters. Limiting cases of the general model include the standard preferential attachment model without cutoff and the uniform attachment model.Comment: 24 pages, one figure. To appear in the journal: Combinatorics, Probability and Computing. Note, this is a long version, with complete proofs, of the paper "Competition-Induced Preferential Attachment" (cond-mat/0402268