The Geometry of Cyclical Social Trends

Abstract

We investigate the emergence of periodic behavior in opinion dynamics and its underlying geometry. For this, we use a bounded-confidence model with contrarian agents in a convolution social network. This means that agents adapt their opinions by interacting with their neighbors in a time-varying social network. Being contrarian, the agents are kept from reaching consensus. This is the key feature that allows the emergence of cyclical trends. We show that the systems either converge to nonconsensual equilibrium or are attracted to periodic or quasi-periodic orbits. We bound the dimension of the attractors and the period of cyclical trends. We exhibit instances where each orbit is dense and uniformly distributed within its attractor. We also investigate the case of randomly changing social networks.

Figures (11)

• Figure 1: The evolution of 20,000 random points in an HK system.
• Figure 2: Typical attractors.
• Figure 3: Two orbits of a single agent around its attractor.
• Figure 4: A triangle identity.
• Figure 5: Why self-confidence slows down the dynamics: proof by contradiction.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4
• Theorem 5
• Lemma 6
• Theorem 7
• Lemma 8
• Theorem 9
• Theorem 10