Contagion Networks: Models, Mechanisms, Inference

• Contagion networks are structured representations of spreading processes that differentiate between simple and complex contagion using pairwise, multiplex, and higher-order interactions.
• They integrate temporal dynamics and spatial embeddings to analyze propagation phenomena like wavefront propagation, synchronized activation, and reentrant phase transitions.
• Recent extensions apply contagion frameworks to multi-agent AI, enabling source detection, mechanism inference, and control strategies through cascade data analysis.

Contagion networks are networked representations of spreading processes in which contagion, broadly construed, refers to anything that can spread infectiously from peer to peer, including communicable diseases, rumors, misinformation, ideas, innovations, bank failures, electrical blackouts, and, in recent work, evaluator bias in multi-agent LLM systems (Juul et al., 2019, Liu, 18 Jun 2026). The topic spans pairwise epidemic-like diffusion, threshold and higher-order reinforcement, multiplex and temporal contact systems, pathway-based mesoscopic descriptions, and inverse problems such as source detection, mechanism inference, and network reconstruction from cascade data (Cozzo et al., 2013, Tan et al., 2023).

1. Conceptual foundations

A central distinction in the literature is between simple contagion and complex contagion. In simple contagion, one infected neighbor can be enough for transmission; in complex contagion, multiple reinforcing exposures are needed, either through threshold mechanisms or through explicit group interactions (Cencetti et al., 2023). This distinction is not merely terminological. It determines which local structures facilitate spread, whether hubs are privileged early adopters, and whether infection patterns remain stable under parameter changes.

Several works emphasize that node-level states and final epidemic size are insufficient summaries of a contagion network. In the SI setting studied by Juul and Strogatz, each infected node has a single parent, so the realized spreading process defines a directed epidemic tree rooted at the initial seed; a mutation at a random infected node then affects its downstream descendants, making the epidemic tree the natural object for quantifying the “cone of influence” of local change (Juul et al., 2019). This proposes a mesoscopic theory of contagion intermediate between microscopic node states and macroscopic outbreak size.

The same shift away from one-shot state change appears in work on repeated activation. There, the relevant question is not only whether actors adopt, but whether they repeatedly reactivate and become temporally aligned, producing bursts of collective visibility. In that formulation, activation is cyclical rather than absorbing, and the core phenomenon is coordination through recurrent pulses rather than irreversible switching (Piedrahita et al., 2017). A plausible implication is that contagion networks are often better understood as dynamical infrastructures for pathway formation and synchronization than as static channels for one-time adoption.

2. Structural substrates and network representations

The field uses several nonequivalent structural representations. A single-layer network encodes one interaction type; an aggregated network projects multiple layers into one graph; a multiplex network retains separate layers and identity coupling between replicas of the same actor (Cozzo et al., 2013). In the contact-based multiplex SIS framework, the relevant operator is the supra-contact probability matrix $\bar R$, and the critical point is

$$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$

where $\bar{\Lambda}_{\max}$ is the largest eigenvalue of $\bar R$ (Cozzo et al., 2013). When one layer has the largest layerwise contact eigenvalue, that dominant layer controls onset. The same work argues that replacing the multiplex by its aggregated graph is flawed: it lowers the estimated threshold and can overestimate prevalence, in some numerical examples by nearly a factor of two (Cozzo et al., 2013).

Higher-order representations generalize graphs by allowing interactions beyond dyads. In a hypergraph, hyperedges can contain any number of nodes; in a simplicial complex, higher-order interactions satisfy downward inclusion (Arruda et al., 2024). The review literature shows that many higher-order SIS/SIR-like models can be written as

$$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$

where the structural choice enters through the hypergraph $\mathcal H$, the size-weighting $\lambda^\ast(|e_j|)$, and the nonlinear interaction kernel $f_j^i(\{Y\})$ (Arruda et al., 2024). This unifies pairwise SIS, simplicial contagion, power-law kernels, and critical-mass threshold models within one formalism.

Temporal networks form a third substrate class. In the activity-driven model, each node $i$ has activity $a_i$, becomes active with probability $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$0, creates $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$1 links, and all edges are deleted at the next step (Liu et al., 2013). There the uncontrolled SIS threshold depends on activity moments rather than static degree moments: $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$2 This makes activity, not time-aggregated degree alone, the central organizing variable for contagion control on time-varying networks (Liu et al., 2013).

Spatially embedded contagion networks introduce a further split between geometric edges, which respect latent geometry, and non-geometric edges, which act as shortcuts. In noisy geometric networks and torus-embedded networks, this supports two qualitatively distinct spreading mechanisms: wavefront propagation (WFP) along the manifold and appearance of new contagion clusters (ANC) via long-range ties (Taylor et al., 2014, Mahler, 2018).

3. Dynamical regimes, reinforcement, and pathway statistics

A major line of work studies how local reinforcement and network dimensionality shape pathway statistics. In the SI mutation problem, if a mutation occurs at a uniformly random infected node, the descendant count $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$3 has distribution $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$4. On the complete graph,

$$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$5

and more generally on complete graphs, random graphs, small-world networks, block-structured random networks, and other effectively infinite-dimensional networks,

$$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$6

with exact beta-function forms under linear frontier-growth assumptions (Juul et al., 2019). Because $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$7, the expected mutant component size scales mesoscopically as $$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$8 on such networks (Juul et al., 2019).

Threshold and multiplex models produce qualitatively different phase structures. In weighted multiplex threshold contagion, increasing edge density need not simply eliminate cascades at high connectivity. Instead, when layers differ in density and strength, the system can pass through alternating stable and unstable regions as connectivity increases, yielding reentrant phase transitions of contagion (Unicomb et al., 2019). Under the multiplex OR rule, density skewness alone can sustain high-$$\left(\frac{\beta}{\mu}\right)_c=\frac{1}{\bar{\Lambda}_{\max}},$$9 cascades; under the weighted-sum rule, weight skewness is additionally required; under the AND rule, instability is much more restricted (Unicomb et al., 2019).

Multilayer social reinforcement generates another nonlinear regime. In the two-layer ignorant-spreader-ignorant model with reinforcement across social circles, the extinction state and outbreak state can coexist, producing bistability, an unstable branch, and a hysteresis loop of stationary prevalence (Liu et al., 2020). The diffusion threshold obtained from the Jacobian at the extinction state is uncorrelated with the reinforcement parameter $\bar{\Lambda}_{\max}$0, but the eradication threshold depends on $\bar{\Lambda}_{\max}$1, which is why an already widespread rumor can persist after parameter values have fallen below those needed to initiate spread from rarity (Liu et al., 2020).

Infection-path robustness also depends sharply on contagion class. On a fixed weighted contact network, a variety of simple contagion models—SIR, SEIR, non-Markovian SEIR, and a COVID-like model—yield infection patterns that are extremely robust across models and parameters, whereas complex contagion models show non-trivial dependence on thresholds and on the interplay between pairwise and group contagion (Contreras et al., 2023). In models involving threshold mechanisms, slight parameter changes can significantly impact the spreading paths (Contreras et al., 2023). This suggests that model uncertainty matters much more for higher-order and threshold contagion than for standard simple contagion.

4. Inference from cascades and dynamics-aware representations

A substantial branch of contagion-network research asks what can be inferred from observed cascades. One line treats source detection as maximum-likelihood inference on graph snapshots. In homogeneous SI on infinite regular trees, the exact maximum-likelihood estimator is the rumor center, obtained by maximizing rumor centrality, the number of valid spreading orders consistent with the infected tree (Tan et al., 2023). On trees, rumor center, distance center, and graph centroid coincide (Tan et al., 2023). On finite trees and graphs with cycles, boundary and cycle effects break this equivalence, motivating epidemic centrality and weighted-distance heuristics such as statistical distance centrality (Tan et al., 2023).

A second line asks whether the network itself can be reconstructed from contagion outcomes. In the active query model under the independent cascade process, exact recovery is possible not only for trees but also for two broader sparse families: networks with large girth and low path growth rate, and networks with bounded degree (Supeesun et al., 2017). For the first class, the query complexity remains

$\bar{\Lambda}_{\max}$2

while for maximum degree $\bar{\Lambda}_{\max}$3 the bounded-degree algorithm requires

$\bar{\Lambda}_{\max}$4

queries (Supeesun et al., 2017). The technical mechanism is separation between direct-edge infection signatures and infection via alternate paths.

Another line infers the contagion mechanism rather than the source. Given only one observed spreading instance, the infection order $\bar{\Lambda}_{\max}$5 and local topology yield rank-correlation features such as

$\bar{\Lambda}_{\max}$6

augmented by $\bar{\Lambda}_{\max}$7 and $\bar{\Lambda}_{\max}$8 (Cencetti et al., 2023). On workplace data, $\bar{\Lambda}_{\max}$9 gives AUC $\bar R$0 for classifying threshold vs non-threshold contagion, and a Random Forest on $\bar R$1 yields relative accuracy $\bar R$2 across nine empirical networks for the three-class problem after merging the two higher-order classes (Cencetti et al., 2023).

Dynamics-aware geometric inference appears in the contagion map framework. There, nodes are embedded as vectors of activation times across multiple contagion realizations,

$\bar R$3

and the resulting point cloud is analyzed geometrically, topologically, and dimensionally (Taylor et al., 2014). On noisy geometric networks, WFP-dominated regimes recover latent manifold structure, while ANC-dominated regimes destroy it (Taylor et al., 2014). On torus-embedded networks, the symmetric contagion map in $\bar R$4 recovers the torus’s geometry, topology, and embedding dimension precisely in the WFP regime, with topological comparison performed via calibrated barcode Wasserstein distance (Mahler, 2018).

5. Control, intervention design, and empirical applications

Control problems depend strongly on substrate type. For time-varying networks in the activity-driven class, three immunization strategies admit analytic thresholds: random immunization, targeted immunization by activity, and egocentric sampling (Liu et al., 2013). Targeting high-activity nodes is the most effective; the paper reports that immunizing only about $\bar R$5 of the highest-activity nodes can stop the epidemic in synthetic examples (Liu et al., 2013). Egocentric sampling, which uses only local observations from probe nodes, performs substantially better than random immunization and approaches targeted performance (Liu et al., 2013).

Experimental design and estimation can also exploit contact-network structure. In clustered randomized studies with SI transmission on contact networks, augmented generalized estimating equations using network covariates improve efficiency without meaningful bias (Wang et al., 2016). Across 16 simulated scenarios, the number of affected nodes in the same component at baseline,

$\bar R$6

produced a $\bar R$7 RMSE reduction and power $\bar R$8, while stepwise adjustment reached power $\bar R$9 with essentially zero bias (Wang et al., 2016). This reframes component structure, baseline infected neighborhoods, and path-based proximity as design variables for contagion experiments rather than post hoc descriptive covariates.

Coupled contagions show that behavioral diffusion can restructure the substrate for pathogen spread. When negative vaccination sentiment spreads by complex contagion rather than simple contagion, the resulting susceptible network contains fewer but larger susceptible communities, higher path redundancy, and more favorable topology for disease transmission (Campbell et al., 2012). Under the reported simulations, outbreaks are larger and more frequent in the complex-contagion case, and outbreaks still occur at $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$0 vaccination coverage, where they disappear under simple contagion (Campbell et al., 2012).

Multiplex field data reveal that not all social layers contribute equally to diffusion pathways. In 110 rural Honduran villages, network torque

$$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$1

quantifies how much layer $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$2 contributes non-redundant shortest paths in the composite multiplex network (Shi et al., 21 Oct 2025). The closest friend layer has the highest average torque, $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$3, implying that removing closest-friend ties would disrupt at least $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$4 of shortest pathways; its torque is $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$5 times that of free time and $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$6 times that of personal and private (Shi et al., 21 Oct 2025). In the counterfactual diffusion analysis, removing closest-friend pathways reduces adoption most strongly, including for correct knowledge about newborn feeding with “chupones” (Shi et al., 21 Oct 2025).

6. Limitations, controversies, and emerging extensions

The literature is methodologically rich but assumption-sensitive. Many exact results rely on SI-like monotone dynamics, static networks, single-parent infection trees, or locally tree-like structure (Juul et al., 2019, Supeesun et al., 2017). Multiplex theory often assumes identity coupling between replicas and homogeneous ratio conditions such as $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$7 (Cozzo et al., 2013). Higher-order contagion theory remains ahead of empirical validation: while discontinuous transitions, hysteresis, bistability, multistability, intermittency, and localization have been derived for many higher-order models, empirical validation through data or experiments remains scant (Arruda et al., 2024).

There are also substantive controversies about model reduction. One concerns aggregation: the multiplex contagion literature argues that collapsing layers into a single graph is generally inaccurate because it changes spectra, thresholds, and prevalence (Cozzo et al., 2013). Another concerns universality: infection patterns appear extremely robust within simple contagion, but not within threshold or higher-order contagion, where slight parameter changes can strongly alter spreading routes (Contreras et al., 2023). Mechanism inference is therefore inherently topology-dependent; the classifier trained on one network does not transfer universally to another (Cencetti et al., 2023).

A recent extension carries contagion networks beyond human and infrastructural systems into multi-agent AI. In Contagion Networks, agents are nodes, strategy distributions are the transmitted states, and pairwise evaluator effects populate the Cross-Agent Contagion Matrix

$$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$8

The spectral radius $$\dfrac{d {Y_i}}{dt} = {-\delta Y_i + \lambda \sum_{j:v_i \in e_j} \lambda^\ast(|e_j|) X_i f_j^i (\{ Y\})},$$9 defines suppression, persistence, and cascade regimes (Liu, 18 Jun 2026). In the reported three-agent DeepSeek-chat experiment, evaluator biases propagate with off-diagonal coefficients in the range $\mathcal H$0–$\mathcal H$1 in the detailed table, while the abstract reports $\mathcal H$2–$\mathcal H$3, a discrepancy the paper does not resolve (Liu, 18 Jun 2026). The same work finds that increasing evaluator committee size from $\mathcal H$4 to $\mathcal H$5 reduces effective contagion by $\mathcal H$6 (Liu, 18 Jun 2026). This suggests that the core language of contagion networks—pathway dependence, spectral regimes, non-redundant influence channels, and mitigation by structural redesign—now applies to closed-loop evaluative systems as well as to epidemics, rumors, finance, and behavioral diffusion.