Random Majority Opinion Diffusion: Stabilization Time, Absorbing States, and Influential Nodes

Consider a graph G with n nodes and m edges, which represents a social network, and assume that initially each node is blue or white. In each round, all nodes simultaneously update their color to the most frequent color in their neighborhood. This is called the Majority Model (MM) if a node keeps its color in case of a tie and the Random Majority Model (RMM) if it chooses blue with probability 1/2 and white otherwise. We prove that there are graphs for which RMM needs exponentially many rounds to reach a stable configuration in expectation, and such a configuration can have exponentially many states (i.e., colorings). This is in contrast to MM, which is known to always reach a stable configuration with one or two states in $O(m)$ rounds. For the special case of a cycle graph C_n, we prove the stronger and tight bounds of $\lceil n/2\rceil-1$ and $O(n^2)$ in MM and RMM, respectively. Furthermore, we show that the number of stable colorings in MM on C_n is equal to $\Theta(\Phi^n)$, where $\Phi = (1+\sqrt{5})/2$ is the golden ratio, while it is equal to 2 for RMM. We also study the minimum size of a winning set, which is a set of nodes whose agreement on a color in the initial coloring enforces the process to end in a coloring where all nodes share that color. We present tight bounds on the minimum size of a winning set for both MM and RMM. Furthermore, we analyze our models for a random initial coloring, where each node is colored blue independently with some probability $p$ and white otherwise. Using some martingale analysis and counting arguments, we prove that the expected final number of blue nodes is respectively equal to $(2p^2-p^3)n/(1-p+p^2)$ and pn in MM and RMM on a cycle graph C_n. Finally, we conduct some experiments which complement our theoretical findings and also lead to the proposal of some intriguing open problems and conjectures to be tackled in future work.Comment: Accepted in AAMAS 2023 (The 22nd International Conference on Autonomous Agents and Multiagent Systems

Two Phase Transitions in Two-Way Bootstrap Percolation

Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d-dimensional torus and identify the threshold values

Majority Opinion Diffusion in Social Networks: An Adversarial Approach

We introduce and study a novel majority-based opinion diffusion model. Consider a graph $G$, which represents a social network. Assume that initially a subset of nodes, called seed nodes or early adopters, are colored either black or white, which correspond to positive or negative opinion regarding a consumer product or a technological innovation. Then, in each round an uncolored node, which is adjacent to at least one colored node, chooses the most frequent color among its neighbors. Consider a marketing campaign which advertises a product of poor quality and its ultimate goal is that more than half of the population believe in the quality of the product at the end of the opinion diffusion process. We focus on three types of attackers which can select the seed nodes in a deterministic or random fashion and manipulate almost half of them to adopt a positive opinion toward the product (that is, to choose black color). We say that an attacker succeeds if a majority of nodes are black at the end of the process. Our main purpose is to characterize classes of graphs where an attacker cannot succeed. In particular, we prove that if the maximum degree of the underlying graph is not too large or if it has strong expansion properties, then it is fairly resilient to such attacks. Furthermore, we prove tight bounds on the stabilization time of the process (that is, the number of rounds it needs to end) in both settings of choosing the seed nodes deterministically and randomly. We also provide several hardness results for some optimization problems regarding stabilization time and choice of seed nodes.Comment: To appear in AAAI 202