Complexity of Eliminating (Majority) Illusion in Directed Networks

Majority illusion is an undesirable phenomenon in social networks in which agents incorrectly perceive a minority opinion as dominant. This can severely distort collective behavior and decision-making. We study the fundamental question of detecting whether a social network allows for a majority illusion. Formally, in the $q$-Majority Illusion problem, we ask whether there exists a binary labeling of agents in which at least a $q$-fraction of agents have the majority of neighbors with the minority label. We investigate how various structural properties of the underlying social network influence the tractability of this question, and provide a detailed map of its computational complexity.

In this paper, we study the detection of an echo chamber in a social network, i.e., the identification of a set of nodes that agree on a topic, while disagreeing with the rest of nodes. We argue that this problem is different from other social network analysis problems such as community detection, and from other graph problems such as maximum graph cut and maximum clique. To the best of our knowledge, we are the first to formalize the objective function of echo chamber detection, by using the theory of Fourier transforms of set functions (Stobbe and Krause, 2012). We propose scalable semidefinite relaxation, solved via an interior point method and sparse linear algebra. Experimentally, our algorithm recovers the ground truth echo chamber better than competing methods on small synthetic experiments. Our algorithm produces echo chambers with better network properties than competing methods on large real-world datasets. To independently validate our proposed objective function, we show that our algorithm finds echo chambers with more agreements with suspended users than competing methods on a small real-world dataset.

We address the problem of inferring a directed network from nodal measurements generated by linear diffusion dynamics on the sought graph. Observations are modeled as the outputs of a graph convolutional filter, i.e., a polynomial (with unknown coefficients) of a local diffusion graph-shift operator encoding the latent graph topology, excited with an ensemble of independent graph signals with arbitrarily-correlated nodal components. Unlike prior efforts that considered undirected graphs and white signal excitations, here the graph-shift operator and the observations' covariance matrix are not simultaneously diagonalizable. In this challenging context, we first rely on measurements of the output signals along with prior statistical information on the inputs to identify the diffusion filter. Such system identification problem involves solving a system of quadratic matrix equations, which we show is identifiable under spectral-diversity assumptions on the input covariances. For algorithmic purposes we recast it as a smooth quadratic minimization subject to Stiefel manifold constraints. Subsequent identification of the network topology given the graph filter estimate boils down to finding a sparse and structurally admissible shift that commutes with the given filter, thus, forcing the latter to be a polynomial in the sought graph-shift operator. A joint graph filter and topology identification algorithm is also proposed, which alternates between the aforementioned steps in a mutually reinforcing fashion to offer improved sample complexity. Numerical tests corroborate the effectiveness of the proposed algorithms in recovering synthetic digraphs and real-data case studies, and illustrate their potential utility on urban mobility analyses as well as portfolio optimization.