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

Mitigating Misinformation Spreading in Social Networks Via Edge Blocking

The wide adoption of social media platforms has brought about numerous benefits for communication and information sharing. However, it has also led to the rapid spread of misinformation, causing significant harm to individuals, communities, and society at large. Consequently, there has been a growing interest in devising efficient and effective strategies to contain the spread of misinformation. One popular countermeasure is blocking edges in the underlying network. We model the spread of misinformation using the classical Independent Cascade model and study the problem of minimizing the spread by blocking a given number of edges. We prove that this problem is computationally hard, but we propose an intuitive community-based algorithm, which aims to detect well-connected communities in the network and disconnect the inter-community edges. Our experiments on various real-world social networks demonstrate that the proposed algorithm significantly outperforms the prior methods, which mostly rely on centrality measures