Complete Graph Network Research Articles

Less is different: Why sparse networks with inhibition differ from complete graphs.

In neuronal systems, inhibition contributes to stabilizing dynamics and regulating pattern formation. Through developing mean-field theories of neuronal models, using complete graph networks, inhibition is commonly viewed as one "control parameter" of the system, promoting an absorbing phase transition. Here, we show that, for low connectivity sparse networks, inhibition weight is not a control parameter of the absorbing transition. We present analytical and simulation results using generic stochastic integrate-and-fire neurons that, under specific restrictions, become other simpler stochastic neuron models common in literature, which allows us to show that our results are valid for those models as well. We also give a simple explanation about why the inhibition role depends on topology, even when the topology has a dimensionality greater than the critical one. The absorbing transition independence of the inhibitory weight may be an important feature of a sparse network, as it will allow the network to maintain a near-critical regime, self-tuning average excitation, but at the same time have the freedom to adjust inhibitory weights for computation, learning, and memory, exploiting the benefits of criticality.

Minimum-Energy Distributed Consensus Control of Multiagent Systems: A Network Approximation Approach

This paper investigates the problem of minimizing the global energy cost for multiagent systems to achieve consensus over undirected network topologies. Conventional optimization problems so defined require all-to-all network topologies for interagent communications, which precludes the use of distributed control or otherwise demands that the network be complete. To circumvent this difficulty, we introduce a network approximation (NA) scheme in the optimization criterion, which tends to render the optimization problem to one seemingly over a complete graph network, thus removing the all-to-all communication requirement. With the NA cost, we show that a distributed optimal consensus algorithm always exists for any given connected network topology, which can be determined by solving a single-agent-level parametric algebraic Ricatti equation (PARE). We also investigate the performance of the optimal consensus algorithm, focusing on the minimal energy cost required to achieve consensus optimally, and the speed at which consensus is achieved. Furthermore, for certain more special yet worthy cases, such as single-integrator, double-integrator, and first-order unstable agents, we derive explicit expressions for the energy cost and the consensus speed. It can be seen from these results that the energy cost can be made arbitrarily small for single-integrator and double-integrator systems under the optimal distributed control. On the other hand, for the first-order unstable agents, the energy cost increases and consensus speed decreases monotonically with the value of the agent's real unstable pole.

Linking Consumer Opinions with Reservation Prices in an Agent-Based Model of Innovation Diffusion

We extend a recently developed agent-based model of innovation diffusion by linking the opinions of potential consumers with their market behavior via the concept of reservation prices. Through a dynamic mechanism that takes into account social influence, the agents in our model can both increase or decrease their product appraisal. Considering complete graph network structures and using mean-field treatment we find that the model can exhibit a plethora of scenarios, observed empirically but not attainable within the classical Bass model. We also show the existence of a critical market price above which the innovation cannot diffuse.

AN IMPROVED COURNOT COMPETITION MODEL: CONSIDERATION OF MARKET SHARE OBJECTIVE

Cournot competition model calculates profit by company's product quantity, but it does not take market share objective into consideration. It has obstacles of considering nonlinear aspects because the partial derivative equations are very hard to solve. Our algorithm both improves Cournot's model with consideration of market share ratio, and avoids the difficulty of solving several partial derivative equations directly. In our algorithm, we give a parameter σ to describe a company's consideration of market share objective, formulate the competition between companies, and calculate new market share ratio using parameter σ. We consider two network structures in our model: complete graph network and star graph network. For complete graph network, our calculation and numerical analysis is based on the case of 3 companies, but the result for general number of companies, n, is also discussed. We give a series of numerical results of new market share ratio after given the initial market share ratio and the parameter σ, and show that the initial market share ratio and new market share ratio are highly linear. Another important property in this network is the convergence of market share ratio when n goes to infinity. For star graph network, we give calculation for general cases and show that n = 3, 4 are the only numbers of companies that have positive increase of the market share ratio of the company.

Distributed Lossy Averaging

In this paper, an information theoretic formulation of the distributed averaging problem previously studied in computer science and control is presented. We assume a network with m nodes each observing a white Gaussian noise (WGN) source. The nodes communicate and perform local processing with the goal of computing the average of the sources to within a prescribed mean squared error distortion. The network rate distortion function R* (D) for a two-node network with correlated Gaussian sources is established. A general cutset lower bound on R*(D) is established and shown to be achievable to within a factor of 2 via a centralized protocol over a star network. A lower bound on the network rate distortion function for distributed weighted-sum protocols, which is larger in order than the cutset bound by a factor of log m, is established. An upper bound on the network rate distortion function for gossip-base weighted-sum protocols, which is only log log m larger in order than the lower bound for a complete graph network, is established. The results suggest that using distributed protocols results in a factor of log m increase in order relative to centralized protocols.

Identity-based conference key distribution systems in broadcast networks

Identity-based conference key distribution systems in a complete graph network or a star network are proposed. Messages among two or more users are authenticated using each user's identification information. Security and transmission efficiency for the proposed schemes are also shown.