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Diffusion and Contagion in Networks with Heterogeneous Agents and Homophily
Matthew O. Jackson, Dunia Lopez-Pintado
TL;DR
The paper asks how homophily and heterogeneous adoption or infection proclivities affect whether behaviors spread from small seeds. It develops a general diffusion framework and a heterogeneous SIS illustration, showing that homophily can facilitate diffusion by giving a self-sustaining type a foothold that reaches the wider population.
Problem
How homophily and heterogeneous adoption or infection proclivities affect whether behaviors spread from small seeds.
Method
The paper develops a general diffusion model with heterogeneous types, adoption proclivities, and biased interactions, including heterogeneous SIS contagion as a special case.
Results
Homophily can facilitate diffusion from a small seed by allowing a type that adopts independently to generate a foothold for wider spread.
Takeaways & Limitations
Diffusion depends jointly on homophily, types’ adoption or infection proclivities, and within-type interaction patterns.
Takeaways & Limitations
The illustrative analysis restricts some cases to symmetric within-type meeting probabilities and allows types to differ in spreading rates.
Abstract
from arXiv · showhide
We study how a behavior (an idea, buying a product, having a disease, adopting a cultural fad or a technology) spreads among agents in an a social network that exhibits segregation or homophily (the tendency of agents to associate with others similar to themselves). Individuals are distinguished by their types (e.g., race, gender, age, wealth, religion, profession, etc.) which, together with biased interaction patterns, induce heterogeneous rates of adoption. We identify the conditions under which a behavior diffuses and becomes persistent in the population. These conditions relate to the level of homophily in a society, the underlying proclivities of various types for adoption or infection, as well as how each type interacts with its own type. In particular, we show that homophily can facilitate diffusion from a small initial seed of adopters.
1 Introduction
The paper studies diffusion in heterogeneous societies where homophily shapes interactions among types with different adoption proclivities. It shows that homophily can facilitate diffusion from small seeds, including by helping an initially favorable group generate a critical mass.
- Model and questions: The model incorporates heterogeneous adoption proclivities and biased interactions across types while encompassing SIS contagion and strategic-complementarity models.It analyzes how preferences or proclivities for adoption interact with cross-type interaction biases.
- Small-seed diffusion: The analysis asks whether a behavior spreads from a very small introduction in a heterogeneous, homophilous society.The focal two-type case considers one group that would foster diffusion in isolation and another that would not.
- Small-seed diffusion: Homophily can facilitate diffusion by allowing the group that would foster it independently to build the critical mass needed for wider spread.Greater within-group interaction lets diffusion get started in the favorable group before reaching the broader society.
- Small-seed diffusion: In societies with less homophily, diffusion from small initial seeds can fail.The result contrasts with the facilitation effect of sufficiently strong within-type interaction.
- Many types: For many types, diffusion is characterized by the largest eigenvalue of an interaction matrix tracking types’ initial adoption rates.A sufficient condition is that a sufficiently homophilous type or group of types would adopt on its own.
2 An Illustrative Example: The Heterogeneous SIS model with Two Types
The heterogeneous two-type SIS model studies whether infection from a small seed becomes endemic when groups differ in vulnerability, interaction patterns, and degree distributions. Its conditions show that homophily can enable diffusion when one type sustains it in isolation but the other does not.
- Model setup: The two-type SIS example models infection spreading and recovery in a population divided into groups such as the young and old.Susceptible agents can become infected through interactions, while infected agents recover and become susceptible again.
- Model setup: The model allows types to differ in spreading rates, degree distributions, and interaction patterns.The spreading rate is λ = ν/δ, while degree describes the number of meetings per period.
- Definition and homophily: Diffusion from a small seed means that arbitrarily few infected individuals lead to a nontrivial steady-state infection rate.Homophily is represented by π, the probability that an agent meets someone of the same type.
- Diffusion condition: Diffusion occurs in the two-type SIS model if λ1λ2 > 1/(e d1 e d2), or if λ1λ2 < 1/(e d1 e d2) and π > π0.These are the two alternatives stated by Theorem 1; the displayed notation preserves the source’s formatting.
- Corollaries: If diffusion occurs within each type in isolation, it also occurs with interaction; if neither type diffuses alone, interaction does not create diffusion.These are the first two consequences of the theorem.
- Corollaries: When one type diffuses alone and the other does not, sufficiently high homophily can produce diffusion across the whole population.In that case, higher introspection lets the favorable group start diffusion and spread it to the wider society.
3 The General Model
The general model represents diffusion among heterogeneous agents whose types shape interaction biases, adoption proclivities, and degree distributions. It studies when small initial adoption grows, using continuous-time dynamics and a local stability condition.
- Model primitives: Agents are characterized by type, degree, and active or passive state, with type-specific meeting patterns and adoption proclivities.Types determine cross-type meeting probabilities, while degree measures meetings per period and may vary within types.
- Model primitives: The interaction matrix Π records the probability that type i meets type j, and its primitiveness permits infection to reach every group.The model allows directional meetings and degree-biased sampling, including uniform and reciprocal-meeting cases.
- Diffusion process: Adoption and reversion rates are governed by functions f_i(d,a) and g_i(d,a), which vary with degree and the number of active agents met.The model requires no adoption without exposure, non-decreasing adoption with exposure, positive adoption for some degree, and possible reversion without exposure.
- Diffusion process: The framework includes SIS, relative-threshold, aggregate-threshold, and imitation processes as special cases.These examples differ in how exposure affects activation and deactivation rates.
- Dynamics and analysis: The continuous-time system tracks active shares by type and degree and is used as an analytically tractable alternative to a stochastic discrete system.Transition rates depend on the probability that an agent samples an active agent, and steady states arise from a fixed-point calculation.
- Small-seed diffusion: Diffusion from a small seed occurs when small positive adoption vectors grow, and the condition is independent of the initial seed’s distribution across types.Near the all-passive state, the authors use local linearization and identify growth when the system moves to a vector at least as large as the starting vector.
4 Analysis
The analysis characterizes diffusion from a small seed using interaction patterns and type-specific adoption rates. Homophily can enable population-wide diffusion when a sufficiently self-reinforcing type or group provides an initial foothold.
- Two types: If one type diffuses in isolation while the other does not, sufficiently high homophily makes diffusion reach the entire population.
- Many types: Theorem 3 states that diffusion occurs if and only if the largest eigenvalue of the interaction matrix A exceeds 1.
- Many types: If every type diffuses when isolated, interaction among types also produces diffusion; if no type diffuses alone, interaction cannot produce it.
- Many types: If some type satisfies π_iixi > 1, diffusion occurs from a small seed.
- Many types: A self-interacting subset that can sustain diffusion can establish a toehold from which diffusion spreads through the population.
- Many types: When all types share the same near-zero adoption or infection rate x, diffusion occurs if and only if x > 1, regardless of interaction details.
5 Concluding Remarks
The paper focuses on whether a new behavior spreads from a small initial seed in networks with homophily. Its central conclusion is that homophily can facilitate infection or contagion, while its effect on eventual prevalence remains unresolved.
- Scope and contribution: The paper studies diffusion of a new behavior beginning from a small initial seed, emphasizing homophily rather than degree distribution.
- Main insight: Homophily can facilitate infection or contagion from a small seed.
- Open question: The eventual fraction of adopters may depend on homophily in complicated ways, and increasing homophily could decrease overall infection despite facilitating initial infection.
Appendix
The appendix supplies proofs connecting small-seed diffusion to eigenvalue conditions and derives the two-type cases underlying the main theorems. It also uses Figure 1 to organize the relevant parameter regions.
- General proof: For primitive A, the maximum eigenvalue is positive and has a positive eigenvector, enabling the instability argument.
- General proof: The proof links diffusion from a small seed to instability of the zero-activity state and to the largest eigenvalue of A exceeding 1.
- Two-type proof: The appendix derives the two-type diffusion condition by computing A's eigenvalues and substituting a11 = πx1, a22 = πx2, a12 = (1 −π)x2, and a21 = (1 −π)x1.
- Two-type proof: Figure 1 relates the key expressions used in the proof of Theorem 2 and partitions the relevant parameter regions.