Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one

Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector that maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory, and statistical physics. The network is endowed with a probabilistic structure in which rewards are sampled from a distribution. Our aim is to identify sufficient conditions on the network structure and rewards distributions to guarantee average-case polynomiality of the underlying optimization problem. Additionally, we wish to characterize the efficiency of a decentralized solution generated on the basis of local information. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of graph models and reward function distributions. Specifically, for certain classes of models we prove that our algorithm is able to find a near-optimal solution with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a certain correlation decay (long-range independence) property, and we prove that this property is indeed exhibited by the models of interest. Our results have the following surprising implications in the area of average-case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three unless P D NP. Yet we show that the closely related Maximum Weight Independent Set problem for the same class of graphs admits a PTAS when the weights are independently and identically distributed with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one. Keywords: optimization; NP-hardness; long-range independenceNational Science Foundation (U.S.) (Grant CMMI-0726733

Consumers don't play dice, influence of social networks and advertisements

Empirical data of supermarket sales show stylised facts that are similar to stock markets, with a broad (truncated) Levy distribution of weekly sales differences in the baseline sales [R.D. Groot, Physica A 353 (2005) 501]. To investigate the cause of this, the influence of social interactions and advertisements are studied in an agent-based model of consumers in a social network. The influence of network topology was varied by using a small-world network, a random network and a Barabasi-Albert network. The degree to which consumers value the opinion of their peers was also varied. On a small-world and random network we find a phase-transition between an open market and a locked-in market that is similar to condensation in liquids. At the critical point, fluctuations become large and buying behaviour is strongly correlated. However, on the small world network the noise distribution at the critical point is Gaussian, and critical slowing down occurs which is not observed in supermarket sales. On a scale-free network, the model shows a transition between a gas-like phase and a glassy state, but at the transition point the noise amplitude is much larger than what is seen in supermarket sales. To explore the role of advertisements, a model is studied where imprints are placed on the minds of consumers that ripen when a decision for a product is made. The correct distribution of weekly sales returns follows naturally from this model, as well as the noise amplitude, the correlation time and cross-correlation of sales fluctuations. For particular parameter values, simulated sales correlation shows power law decay in time. The model predicts that social interaction helps to prevent aversion, and that products are viewed more positively when their consumption rate is higher.Comment: Accepted for publication in Physica

An Empirical Evaluation Of Social Influence Metrics

Predicting when an individual will adopt a new behavior is an important problem in application domains such as marketing and public health. This paper examines the perfor- mance of a wide variety of social network based measurements proposed in the literature - which have not been previously compared directly. We study the probability of an individual becoming influenced based on measurements derived from neigh- borhood (i.e. number of influencers, personal network exposure), structural diversity, locality, temporal measures, cascade mea- sures, and metadata. We also examine the ability to predict influence based on choice of classifier and how the ratio of positive to negative samples in both training and testing affect prediction results - further enabling practical use of these concepts for social influence applications.Comment: 8 pages, 5 figure

Structure learning of antiferromagnetic Ising models

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. samples. We first observe that the notoriously difficult problem of learning parities with noise can be captured as a special case of learning graphical models. This leads to an unconditional computational lower bound of $\Omega (p^{d/2})$ for learning general graphical models on $p$ nodes of maximum degree $d$, for the class of so-called statistical algorithms recently introduced by Feldman et al (2013). The lower bound suggests that the $O(p^d)$ runtime required to exhaustively search over neighborhoods cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., many recent papers on structure learning assume that the model has the correlation decay property. Indeed, focusing on ferromagnetic Ising models, Bento and Montanari (2009) showed that all known low-complexity algorithms fail to learn simple graphs when the interaction strength exceeds a number related to the correlation decay threshold. Our second set of results gives a class of repelling (antiferromagnetic) models that have the opposite behavior: very strong interaction allows efficient learning in time $O(p^2)$. We provide an algorithm whose performance interpolates between $O(p^2)$ and $O(p^{d+2})$ depending on the strength of the repulsion.Comment: 15 pages. NIPS 201