112 research outputs found
Complex Objects in the Polytopes of the Linear State-Space Process
A simple object (one point in -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
Two Steps to Obfuscation
This note addresses the historical antecedents of the 1998 PageRank measure
of centrality. An identity relation links it to 1990-1991 models of Friedkin
and Johnsen
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
Dynamic Social Balance and Convergent Appraisals via Homophily and Influence Mechanisms
Social balance theory describes allowable and forbidden configurations of the
topologies of signed directed social appraisal networks. In this paper, we
propose two discrete-time dynamical systems that explain how an appraisal
network \textcolor{blue}{converges to} social balance from an initially
unbalanced configuration. These two models are based on two different
socio-psychological mechanisms respectively: the homophily mechanism and the
influence mechanism. Our main theoretical contribution is a comprehensive
analysis for both models in three steps. First, we establish the well-posedness
and bounded evolution of the interpersonal appraisals. Second, we fully
characterize the set of equilibrium points; for both models, each equilibrium
network is composed by an arbitrary number of complete subgraphs satisfying
structural balance. Third, we establish the equivalence among three distinct
properties: non-vanishing appraisals, convergence to all-to-all appraisal
networks, and finite-time achievement of social balance. In addition to
theoretical analysis, Monte Carlo validations illustrates how the non-vanishing
appraisal condition holds for generic initial conditions in both models.
Moreover, numerical comparison between the two models indicate that the
homophily-based model might be a more universal explanation for the formation
of social balance. Finally, adopting the homophily-based model, we present
numerical results on the mediation and globalization of local conflicts, the
competition for allies, and the asymptotic formation of a single versus two
factions
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Structural balance emerges and explains performance in risky decision-making.
Polarization affects many forms of social organization. A key issue focuses on which affective relationships are prone to change and how their change relates to performance. In this study, we analyze a financial institutional over a two-year period that employed 66 day traders, focusing on links between changes in affective relations and trading performance. Traders' affective relations were inferred from their IMs (>2 million messages) and trading performance was measured from profit and loss statements (>1 million trades). Here, we find that triads of relationships, the building blocks of larger social structures, have a propensity towards affective balance, but one unbalanced configuration resists change. Further, balance is positively related to performance. Traders with balanced networks have the "hot hand", showing streaks of high performance. Research implications focus on how changes in polarization relate to performance and polarized states can depolarize
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Mathematical Structures in Group Decision-Making on Resource Allocation Distributions.
Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group's array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group's initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group's implicit initial decision space
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