Emergence of simple and complex contagion dynamics from weighted belief networks

TL;DR

The paper presents a belief-network based model in which each agent's internal coherence drives contagion alongside social influence, enabling both simple and complex contagion to emerge from cognitive dynamics. Using a Markov-state analytical framework and extensive network experiments (including Watts-Strogatz and stochastic block models), it shows that simple contagion tends to spread fastest in random-like networks, while complex contagion benefits from clustering and modularity, even revealing an optimal modularity regime in certain settings. A key insight is that resistance to conflicting beliefs, arising from coherence-seeking dynamics, acts as a mechanism for complex contagion in the presence of social reinforcement. This framework thus links cognitive processes to macro-diffusion patterns and offers a principled basis for understanding misinformation spread and behavior change in real-world social networks.

Abstract

Social contagion is a ubiquitous and fundamental process that drives individual and social changes. Although social contagion arises as a result of cognitive processes and biases, the integration of cognitive mechanisms with the theory of social contagion remains an open challenge. In particular, studies on social phenomena usually assume contagion dynamics to be either simple or complex, rather than allowing it to emerge from cognitive mechanisms, despite empirical evidence indicating that a social system can exhibit a spectrum of contagion dynamics -- from simple to complex -- simultaneously. Here, we propose a model of interacting beliefs, from which both simple and complex contagion dynamics can organically arise. Our model also elucidates how a fundamental mechanism of complex contagion -- resistance -- can come about from cognitive mechanisms.

Figures (12)

• Figure 1: (a) An illustration of the simple and complex contagion mechanisms, as reflected in the adoption probability’s dependence on the number of adopted neighbors. Simple contagion (orange curve) is characterized by a concave shape and exhibits "diminishing returns" due to the independence of the exposure’s impact. On the other hand, the complex contagion curve (blue curve) shows a sharp increase due to the reinforcement. (b) An individual’s belief system is described as a network of interacting beliefs, where nodes represent concepts and weighted edges between them represent beliefs. Beliefs describe the (personal) degree of coherence that agents attribute between pairs of concepts. The color of the edge indicates the belief polarity and the thickness indicates its strength. (c) The stability of each triad in an individual’s belief network is modeled using the social balance theory. (d) Each belief network has a bias toward internal coherence (stable triads).
• Figure 2: (a,b,c) Both simple and complex contagion dynamics emerge from the weighted belief network model. (a) An unstable hub surrounded by a fixed fraction of stable neighbors. (b) A stable hub surrounded by a fixed fraction of stable neighbors of a different kind. (c) The probability that the hub changes to the new belief system exhibits the characteristic behaviors of simple (orange) and complex (blue) contagion ($N=40, M=39,\sigma=0.2,\alpha=1.5,\beta=1$). The probability is calculated by running the simulation 50 times and calculating the proportion of times the hub “flipped.” Error bars, indicating SD, are then obtained by repeating this process 10 times. Dotted lines are results from numerical simulations and solid lines are results from the analytical calculations. Inset: The numerical simulations carried out with the belief systems of neighbors of the hub which are initially similar to it are allowed to vary during the simulation. The robustness of results across different $\alpha$ and $\beta$ is shown in fig. S1 and across different $\sigma$ is shown in fig. S3. (d,e) Intuition for the effect of social influence on an individual’s belief system. (d) An unstable belief system can easily be collapsed into a stable state with social influence. (e) On the contrary, a stable individual’s belief system resists social influence that destabilizes it. Repeated exposures are necessary to push the individual into an unstable state.
• Figure 3: The state machine for Scenario 1 in the star graph setup with $\alpha=1.5$ and $\beta=1$. $u (v)$ is proportional to the probability that the hub receives a belief from neighbors dissimilar (similar) to its initial state
• Figure 4: The impact of network structure on the belief cascade. We run the two scenarios (“simple” versus “complex”) on two networks (clustered large world versus random small world). As predicted by the theory of simple and complex contagion, we observe that a new stabilizing belief (simple contagion) spreads better in a random small world than in a clustered large world, whereas a new stable belief system that competes with an existing stable belief system (complex contagion) spreads better in a clustered large world rather than a random small world. Yellow nodes represent the seeds whose belief systems do not change during the course of the simulation (zealots), and purple represents the rest of the population. (a) Watts-Strogatz model with rewiring probability $P = 0$ results in a clustered large world, whereas $P = 1$ results in a random small world network. (b) Scenario 1: Simple contagion spreads faster in a random network ($P = 1$) than in a clustered network ($P = 0)$. (c) Scenario 2: Complex contagion spreads faster on a clustered network ($P = 0$) than on a random network ($P = 1$). $\alpha=2,\beta=1,N=100,\bar{k}=10,\sigma=0.2,\rho_0=0.08$. Curves are obtained from averaging 10 ensembles. See the section S4 for varying $\rho_0$.
• Figure 5: The case of complex contagion (two competing stable belief systems) also exhibits the optimal modularity behavior. (a,b) An illustration of the initial condition of the simulation. The population (purple nodes) has a stable belief system and the zealots (yellow nodes), constrained to a single community, have a stable belief system of a different kind. (c) The mixing parameter $\Omega$ determines the strength of community structure in the network. (d) The phase diagram exhibits optimal modularity with three phases: no diffusion (dark blue), local diffusion in the seed community (green), and global diffusion (yellow). (e) Cross sections of the phase diagram. Error bars indicate standard error. The following parameters are used: $N=100, M=1500,\sigma=0.2,\alpha=2,\beta=1,40$ ensembles.