Complex Objects in the Polytopes of the Linear State-Space Process

A simple object (one point in $m$-dimensional space) is the resultant of the evolving matrix polynomial of walks in the irreducible aperiodic network structure of the first order DeGroot (weighted averaging) state-space process. This paper draws on a second order generalization the DeGroot model that allows complex object resultants, i.e, multiple points with distinct coordinates, in the convex hull of the initial state-space. It is shown that, holding network structure constant, a unique solution exists for the particular initial space that is a sufficient condition for the convergence of the process to a specified complex object. In addition, it is shown that, holding network structure constant, a solution exists for dampening values sufficient for the convergence of the process to a specified complex object. These dampening values, which modify the values of the walks in the network, control the system's outcomes, and any strongly connected typology is a sufficient condition of such control

Scale-free interpersonal influences on opinions in complex systems

An important side effect of the evolution of the human brain is an increased capacity to form opinions in a very large domain of issues, which become points of aggressive interpersonal disputes. Remarkably, such disputes are often no less vigorous on small differences of opinion than large differences. Opinion differences that may be measured on the real number line may not directly correspond to the subjective importance of an issue and extent of resistance to opinion change. This is a hard problem for field of opinion dynamics, a field that has become increasingly prominent as it has attracted more contributions to it from investigators in the natural and engineering sciences. The paper contributes a scale-free approach to assessing the extents to which individuals, with unknown heterogeneous resistances to influence, have been influenced by the opinions of others

Distributed Learning from Interactions in Social Networks

We consider a network scenario in which agents can evaluate each other according to a score graph that models some interactions. The goal is to design a distributed protocol, run by the agents, that allows them to learn their unknown state among a finite set of possible values. We propose a Bayesian framework in which scores and states are associated to probabilistic events with unknown parameters and hyperparameters, respectively. We show that each agent can learn its state by means of a local Bayesian classifier and a (centralized) Maximum-Likelihood (ML) estimator of parameter-hyperparameter that combines plain ML and Empirical Bayes approaches. By using tools from graphical models, which allow us to gain insight on conditional dependencies of scores and states, we provide a relaxed probabilistic model that ultimately leads to a parameter-hyperparameter estimator amenable to distributed computation. To highlight the appropriateness of the proposed relaxation, we demonstrate the distributed estimators on a social interaction set-up for user profiling.Comment: This submission is a shorter work (for conference publication) of a more comprehensive paper, already submitted as arXiv:1706.04081 (under review for journal publication). In this short submission only one social set-up is considered and only one of the relaxed estimators is proposed. Moreover, the exhaustive analysis, carried out in the longer manuscript, is completely missing in this versio

Dynamic Models of Appraisal Networks Explaining Collective Learning

This paper proposes models of learning process in teams of individuals who collectively execute a sequence of tasks and whose actions are determined by individual skill levels and networks of interpersonal appraisals and influence. The closely-related proposed models have increasing complexity, starting with a centralized manager-based assignment and learning model, and finishing with a social model of interpersonal appraisal, assignments, learning, and influences. We show how rational optimal behavior arises along the task sequence for each model, and discuss conditions of suboptimality. Our models are grounded in replicator dynamics from evolutionary games, influence networks from mathematical sociology, and transactive memory systems from organization science.Comment: A preliminary version has been accepted by the 53rd IEEE Conference on Decision and Control. The journal version has been submitted to IEEE Transactions on Automatic Contro

Novel Multidimensional Models of Opinion Dynamics in Social Networks

Unlike many complex networks studied in the literature, social networks rarely exhibit unanimous behavior, or consensus. This requires a development of mathematical models that are sufficiently simple to be examined and capture, at the same time, the complex behavior of real social groups, where opinions and actions related to them may form clusters of different size. One such model, proposed by Friedkin and Johnsen, extends the idea of conventional consensus algorithm (also referred to as the iterative opinion pooling) to take into account the actors' prejudices, caused by some exogenous factors and leading to disagreement in the final opinions. In this paper, we offer a novel multidimensional extension, describing the evolution of the agents' opinions on several topics. Unlike the existing models, these topics are interdependent, and hence the opinions being formed on these topics are also mutually dependent. We rigorous examine stability properties of the proposed model, in particular, convergence of the agents' opinions. Although our model assumes synchronous communication among the agents, we show that the same final opinions may be reached "on average" via asynchronous gossip-based protocols.Comment: Accepted by IEEE Transaction on Automatic Control (to be published in May 2017

Generalized Markovian Quantity Distribution Systems: Social Science Applications

We propose a model of Markovian quantity flows on connected networks that relaxes several properties of the standard compartmental Markov process. The motivation of our generalization are social science applications of the standard model that do not comport with its steady state predictions. The proposed generalization relaxes the predictions that every node belonging to the same nontrivial strong component of a network must acquire the same fraction of its members’ initial quantities and that the sink component(s) of the network must absorb all of the system’s available initial quantity. For example, when applied to refugee flows from a nation in chaos to other nations on a network with one or more sink nations, the standard model predicts that all the refugees will be eventually located in the sink(s) of the network and none that will permanently locate themselves in the nations along the paths to the sink(s). We illustrate this and several other social science applications to which our proposed model is applicable