§8 Network coupling

Mean-field · networked foreclosure · cascade · percolation · cohort amplification

The single-user apparatus of §0–§7 extends to networked populations via mean-field limit. Real platforms are networks: users connected by friendship graphs, follower–followee relations, recommendation cascades. The framework's empirical predictions about mobilization–dispersion dynamics, cohort propagation, and intermediary-institution roles depend on the network extension.

§8.1 The networked closed-loop model

The single-user setup extends naturally to a population of $N$ users connected by a weighted directed graph $G_{N}$ . Per-user dynamics, observation, and policy on each user's state. Networked Hawkes intensities couple users through the graph adjacency $W_{ij}^{N}$ .

Population. Users indexed by $i \in [N]$ . User $i$ 's state $u_{t}^{i} \in U$ decomposed into the four registers as in §0.1.

Graph. Weighted directed graph $G_{N}$ with row-stochastic adjacency $W^{N} = (W_{ij}^{N})$ .

Per-user dynamics, observation, and networked policy.

$u_{t + 1}^{i} \sim T (\cdot ∣ u_{t}^{i}, s_{t}^{i})$
$y_{t}^{i} \sim O (\cdot ∣ u_{t}^{i})$
$s_{t}^{i} \sim π_{θ} (\cdot ∣ y_{1 : t}^{i}, {y_{1 : t}^{j}}_{j \in [N]}, W^{N})$

The networked Hawkes intensity for user $i$ , register $k$ :

λ^{i, k} (t) = λ_{0}^{i, k} + j = 1 \sum N W_{ij}^{N} l \sum α_{k l} \int_{0}^{t} κ (t - s) d N_{s}^{j, l} .

in wordsUser $i$ 's firing rate on register $k$ now draws on the recent events of every other user $j$ , weighted by the graph edge $W_{ij}$ , on top of the within-user coupling $α_{k l}$ . Two channels of contagion run at once: across registers inside one user, and across users along the graph.

The off-diagonal $α_{k l}$ generates within-user libidinal routing (§3.4); the off-diagonal $W_{ij}^{N}$ for $i \neq = j$ generates cross-user engagement cascades.

§8.2 Theorem 10 — mean-field convergence

As $N \to \infty$ with the graph converging to a graphon $W : [0, 1]^{2} \to [0, 1]$ , the population's performative stable point converges to the continuum performative stable point $θ_{\infty}^{*}$ . The single-user framework is recovered in the mean-field limit; the framework's claims survive the network extension.

The mean-field limit takes $N \to \infty$ with $G_{N}$ converging to a graphon $W : [0, 1]^{2} \to [0, 1]$ — the continuum analog of the adjacency matrix. The population-level performative operator $Φ_{N}$ converges to a continuum operator $Φ_{\infty}$ ; the population's performative stable point converges to $θ_{\infty}^{*}$ .

Theorem 10 (mean-field convergence)

Under (R1), (R2) extended to the networked setting, plus square-integrability of $W$ and Lipschitz dependence of $D$ on the population measure flow $μ$ :

θ_{N}^{*} N \to \infty θ_{\infty}^{*},

in wordsAs the population grows, the platform's stable point settles onto a single continuum stable point. The network does not change the kind of object the framework studies — the closure still has a well-defined fixed point in the large-population limit, so every single-user result has a networked counterpart.

where $θ_{\infty}^{*}$ is the continuum performative stable point — the $θ$ maximizing

F^{\infty} (θ) = \int_{0}^{1} E_{μ (x; θ)} [\hat{R}^{x}] d x

in wordsThe continuum platform score is the per-user reward averaged over a continuum of users indexed by $x \in [0, 1]$ . The finite friend graph is replaced by a graphon — a density of connections — and the sum over users becomes an integral. Otherwise this is the same performative value $F$ from §0, lifted to the population.

under the continuum-limit consistency.

Proof sketch

Standard mean-field convergence (Carmona & Delarue 2018, Probabilistic Theory of Mean Field Games, Theorem 1.2, adapted): graphon convergence + Lipschitz dependence + uniqueness of the continuum stable point give convergence in distribution of the discrete dynamics to the continuum. Continuity of the argmax operator (under (R2)) gives convergence of the stable points.

§8.3 Networked Lemma 1

Foreclosure survives the network extension. Under positive lower density of the graphon $W (x, y) \geq w_{m i n} > 0$ , the average per-user survival at $θ_{ref}$ vanishes; uniform per-user bound is $(1 + (c_{v}^{*})^{2}) / ((θ_{ref} / δ_{m i n})^{2} \cdot w_{m i n}^{2})$ — the graph-density penalty.

The networked extension of Lemma 1 gives both an average claim (foreclosure holds on average across the population) and a uniform per-user claim (no individual user escapes). The uniform claim incurs a graph-density penalty $w_{m i n}^{- 2}$ : in sparser-graph regions, per-user firing rates are weaker, hence per-user foreclosure is weaker.

Networked Lemma 1

Let $W$ be a graphon with positive lower density $W (x, y) \geq w_{m i n} > 0$ for a.e. $(x, y)$ . Under (R1), (R2), (P-I)–(P-III) extended to the networked setting, at the continuum stable point $θ_{\infty}^{*}$ :

(a) Average survival. $lim_{N \to \infty} S^{\infty} (θ_{ref}; θ_{N}^{*}) = 0$ .

(b) Uniform per-user bound.

x \in [0, 1] sup S (θ_{ref}; θ_{\infty}^{*}, x) \leq \frac{1 + ( c _{v}^{*} ) ^{2}}{( θ _{ref} / δ _{m i n} ) ^{2} \cdot w _{m i n}^{2}} .

in wordsForeclosure holds uniformly: the $sup$ over all users $x$ is bounded, so even the most weakly-connected user is foreclosed. The bound is the single-user one inflated by $1/ w_{m i n}^{2}$ , the graph-density penalty: the most weakly-connected user is foreclosed least, but even they do not escape.

Proof sketch

Same three-step structure as Lemma 1, with each step extended to the networked setting. The factor $w_{m i n}^{- 2}$ is the graph-density penalty: lower connectivity gives weaker per-user firing rates, hence weaker per-user foreclosure.

§8.4 Engagement cascade theorem

The networked multi-mark Hawkes process is stable iff $ρ (W) ρ (α) \int κ < 1$ . Below threshold, cascade descendants are finite; the cascade amplification factor $A$ is given by the geometric series of the branching ratio.

The networked Hawkes branching ratio multiplies the within-user spectral radius $ρ (α)$ (the framework's within-user libidinal routing strength) by the cross-user spectral radius $ρ (W)$ (the graph's connectivity) by the kernel integral $\int κ$ . Stability requires the product to be below 1.

The platform's incentive structure pushes $ρ (W) \cdot ρ (α) \cdot \int κ$ toward 1 from below. Each factor is a quantity the platform's optimization can influence. $ρ (W)$ — graph connectivity — is influenced by recommendation surfaces that increase exposure across users (the “people you may know” widget, the cross-user re-share architecture, the algorithmic-feed broadcast pattern). $ρ (α)$ , the within-user cross-register coupling, is influenced by content that activates multiple registers simultaneously: rage-bait that is somatic and political and cognitive, aspirational lifestyle content that is kinaesthetic and cognitive. $\int κ$ — the engagement-memory kernel — is influenced by interface design that extends the user's attentional dwell. The closure's incentive structure aligns all three factors with the engagement-monetization objective. Approach to the boundary $ρ (W) ρ (α) \int κ \to 1^{-}$ is what produces viral cascades and the platform-spanning attention spikes the cascade literature documents.

Theorem (engagement cascade stability)

Let $ρ (W)$ be the graphon's spectral radius (the largest eigenvalue of $f \mapsto \int W (x, y) f (y) d y$ on $L^{2} ([0, 1])$ ). Let $ρ (α)$ be the spectral radius of the cross-register coupling $α = (α_{k l})$ . The networked multi-mark Hawkes process is stable if and only if

ρ (W) \cdot ρ (α) \cdot \int_{0}^{\infty} κ (s) d s < 1.

in wordsThe cascade stays finite exactly when one event begets, on average, fewer than one further event — the branching ratio below one. It is a product of three gains: user-to-user reach $ρ (W)$ , register-to-register reach $ρ (α)$ , and how long each event keeps firing $\int κ$ . Lift any one and the network drifts toward criticality.

The cascade amplification factor — expected cross-user events triggered by a single event — is

A (W, α, κ) = \frac{ρ ( W ) ρ ( α ) \int κ}{1 - ρ ( W ) ρ ( α ) \int κ} .

in wordsThe expected number of cross-user events a single event sets off, summed over all generations — a geometric series in the branching ratio. As the branching ratio climbs toward one, the denominator goes to zero and the amplification diverges: one post triggers an unbounded expected cascade. This is the arithmetic of “going viral.”

Proof sketch

The networked multi-mark Hawkes process is a multivariate Hawkes on a population × register product space. The branching ratio is $ρ (W) ρ (α) \int κ$ (Bacry, Mastromatteo, Muzy 2015, Hawkes Processes in Finance, §2.2). Standard stability theorem gives the condition; the geometric-series formula gives $A$ .

§8.5 Percolation threshold

The critical connectivity $ρ_{c} := 1/ (ρ (α) \int κ)$ separates local cascades from platform-wide cascades. At threshold, cascade dynamics are critical with power-law decay and approximate scale-invariance. The framework's mobilization–dispersion claim depends on $ρ (W) > ρ_{c}$ for major platforms.

The cascade stability condition rewrites as a phase transition in $ρ (W)$ : at fixed within-user coupling $ρ (α) \int κ$ , the cascade is sub-critical below $ρ_{c}$ and super-critical above. The fourth-cell mode (regulatory intervention on the observation channel) and other graph-structure interventions can in principle lower $ρ (W)$ below $ρ_{c}$ , converting platform-wide cascade dynamics back to local.

Theorem (percolation threshold)

Define the critical connectivity

ρ_{c} := \frac{1}{ρ ( α ) \int _{0}^{\infty} κ ( s ) d s} .

in wordsThe critical connectivity: the graph reach $ρ (W)$ at which cascades flip from local to platform-wide. Below $ρ_{c}$ an event's influence dies out in a neighborhood; above it, the cascade percolates across the whole network. Regulating the observation channel or the graph structure to hold $ρ (W)$ under $ρ_{c}$ keeps cascades local — a concrete handle for the fourth-cell mode.

(i) Below threshold ( $ρ (W) < ρ_{c}$ ). Engagement cascades remain bounded; expected cascade descendants are finite; platform reach is confined to local neighborhoods.

(ii) Above threshold ( $ρ (W) > ρ_{c}$ ). Engagement cascades are super-critical at the population level; events propagate platform-wide before dispersing.

(iii) At threshold ( $ρ (W) = ρ_{c}$ ). Critical cascade dynamics with power-law decay; long-range temporal correlations; approximate scale- invariance.

Proof sketch

Branching-process percolation theorem (Athreya & Ney 1972, Branching Processes, Ch. I) with branching ratio $ρ (W) ρ (α) \int κ$ .

The framework's substantive empirical claim — that mobilization–dispersion patterns are characteristic of contemporary platforms — depends on $ρ (W) > ρ_{c}$ for major platforms. A regulatory intervention that lowers $ρ (W)$ below threshold would convert super-critical cascade dynamics into local cascades.

§8.6 Cohort graph-amplification

Network clustering amplifies the per-user cohort gradient of Proposition 8. When users in similar developmental cohorts cluster in the graph, the local-average exposure $\overset{τ}{ˉ}_{exp} (x)$ replaces the per-user $τ_{exp}$ in the cohort gradient bound. Peer-environment exposure compounds with individual exposure.

The native cohort's platform-fluency is structurally amplified by its peers' platform- fluency. If two users with similar developmental exposures are graph-clustered ( $w_{intra} > w_{inter}$ ), their effective exposure to the platform-induced target $π_{\hat{R}}^{*}$ reflects both their own developmental history and the local-average history of their peers.

Theorem (cohort graph-amplification)

Suppose users in similar developmental cohorts are clustered in the graph: intra-community density $w_{intra} > w_{inter}$ (inter-community density). Let $\overset{τ}{ˉ}_{exp} (x)$ denote the local-average developmental exposure in a neighborhood of label $x$ . Then:

D_{K L} (ν_{u}^{x, init} ∥ π_{\hat{R}}^{*}) \leq C \cdot e^{- 2 K_{glob} \overset{τ}{ˉ}_{exp} (x)} .

in wordsThe cohort gradient (Proposition 8) read across the network. A user's distance from the platform target is bounded by the exponential decay in their neighborhood's average exposure $\overset{τ}{ˉ}_{exp} (x)$ — a collective quantity, governed by the community, not by the user's own exposure alone. Developmentally-saturated communities pull their members toward the target faster: the cohort effect compounds through the graph.

Remark

Network clustering amplifies the per-user cohort gradient of Proposition 8. Peer-environment exposure adds to individual exposure. The native cohort's platform-fluency is structurally amplified by its peers' platform-fluency.

Cross-references

Underlying single-user results: Lemma 1 (foreclosure) and §3.5 (Hawkes)
Cohort gradient amplified here: Proposition 8
Intervention site for graph-structure regulation: the fourth-cell mode at §5
Required: §0 axioms (R1, R2, P-I, P-II, P-III) extended to the networked setting