🔗🎲 Random Graph Agent-Based Models (Erdős–Rényi): The Foundation of Stochastic Network Modeling

The Random Graph model, specifically of the Erdős–Rényi (ER) type, represents the foundational network architecture for studying infectious disease spread in stochastic settings. Within this Agent-Based Model (ABM) framework, the population is represented as a collection of nodes (agents), while contacts between individuals are represented as edges formed independently and uniformly at random with a fixed probability p.

This structural simplicity enables strong analytical insight into epidemic thresholds and phase transitions, while also providing a neutral baseline against which more realistic and heterogeneous network topologies can be systematically compared in mathematical epidemiology.

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🦠 1. Compartmental Structure and Flow

Agents in an Erdős–Rényi network ABM follow standard compartmental disease progressions, most commonly the Susceptible–Exposed–Infectious–Recovered (SEIR) structure, which explicitly incorporates a latent (non-infectious) period. Unlike deterministic compartmental models, all transitions occur as discrete stochastic events conditioned on the underlying random graph.

Agent State Flow:

• Susceptible (S): An agent transitions to the exposed state following effective contact with an infectious neighbor through an existing edge in the random graph.
• Exposed (E): The agent enters a latent phase during which it is infected but not yet infectious. The duration of this phase is governed by the latent rate σ.
• Infectious (I): The agent can transmit infection to susceptible neighbors connected by edges. The infectious period is governed by the recovery rate γ.
• Recovered (R): The agent is removed from the transmission process and is assumed to have acquired immunity.

A defining feature of the ER ABM is that transmission is strictly constrained to pre-existing random contacts defined at network generation. No spatial or social clustering beyond random chance is present.

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📐 2. Mathematical Formulation

As a stochastic Agent-Based Model, the ER network framework is governed by individual-level probabilities rather than global systems of Ordinary Differential Equations. Disease propagation depends jointly on the random topology induced by p and the biological parameters governing transmission and recovery.

The network consists of N agents, with each unordered pair connected independently with probability p. This construction yields a near-uniform (Poisson) degree distribution, with average degree
⟨k⟩ = p(N − 1).

Let 𝒩(a) denote the set of neighbors of agent a. The probability that a susceptible agent a transitions from S to E during a small time interval Δt depends on its infectious neighbors.

Infection Probability for Agent a:

P(S → E)ₐ = 1 − ∏ over b in 𝒩(a) of (1 − β_per · 𝟙(b infectious) · Δt)

Here, β_per denotes the per-link transmission probability per unit time, and the indicator function equals one if neighbor b is infectious and zero otherwise. This formulation captures the combined effect of multiple independent exposure risks from different neighbors.

Compartmental Rate Parameters:

The biological time scales governing disease progression are specified by constant rates and typically implemented as exponentially distributed waiting times:

Latency rate: σ = 1 / T_inc
Recovery rate: γ = 1 / T_inf

Parameter Definitions:

Parameter	Definition	Role in the Model
p	Probability of connection between two agents	Determines network topology and the average degree ⟨k⟩
β_per	Transmission probability per contact	Probability of infection given an infectious contact
σ	Latent period rate	Governs transition from Exposed to Infectious
γ	Recovery rate	Governs transition from Infectious to Recovered

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📊 3. Parameter Ranges (General Viral Disease)

Parameter values are chosen to reflect acute viral infections, particularly respiratory pathogens, and are calibrated to biologically plausible time scales and contact intensities.

Parameter	Typical Range	Unit	Interpretation
Recovery Rate (γ)	0.07 – 0.14	day⁻¹	Infectious period of approximately 7 to 14 days
Latency Rate (σ)	0.14 – 0.25	day⁻¹	Incubation period of approximately 4 to 7 days
Average Degree (⟨k⟩)	5 – 20	Contacts	Mean number of contacts per agent
Per-Link Transmission (β_per)	0.01 – 0.15	Dimensionless per contact	Calibrated to produce epidemic growth under ER mixing

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🚧 4. Applicability and Limitations

Applicability – When to Use Erdős–Rényi ABMs:

The ER random graph model is primarily used for theoretical analysis and benchmarking due to its mathematical tractability.

1. Epidemic Threshold Analysis: Provides a clean analytical setting for identifying critical transmission conditions required for epidemic emergence.
2. Baseline Comparisons: Serves as a reference model for evaluating how clustering, heterogeneity, or hubs in alternative network structures alter epidemic dynamics.
3. Highly Mixed Populations: Appropriate when contacts are effectively random and unstructured, such as transient interactions in large, anonymous crowds.

Key Assumptions and Weaknesses:

1. Homogeneous Degree Distribution: The Poisson-like degree distribution implies that most individuals have similar numbers of contacts, contradicting empirical evidence of strong heterogeneity in social networks.
2. Lack of Social Structure: The model does not capture households, workplaces, schools, or community clustering, which are fundamental drivers of real-world transmission.
3. Systematic Underestimation of Risk: The absence of highly connected individuals leads to underestimation of epidemic speed and final size compared with heterogeneous network models.

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📚 References

Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control.

Barabási, A.-L., & Albert, R. (1999). Emergence of Scaling in Random Networks.

Newman, M. E. J. (2002). The spread of epidemic disease on networks.

Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic spreading in scale-free networks.

Wallinga, J., Teunis, P., & Kretzschmar, M. (2006). Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents.