Chapter 11: Network Routing Under Endogenous Traffic

Chapter 10 showed that filter bubbles in a recommendation platform appear as boundary equilibria of the coupled actor-critic-law system, and that reward design controls the regime: at the baseline parameters the diverse interior equilibrium attracts and the bubble equilibria repel, while stronger engagement incentives push the platform toward–and across–the threshold where bubble lock-in becomes attracting. The model had five states on a Y-graph, a two-dimensional actor with block-diagonal features, and an engagement-diversity reward tension that created the boundary equilibria.

This chapter applies the same protocol to a different domain: network routing. The starting point is the three-state retail-to-vet routing chain that first appeared in Chapter 2, Section 2.8 and was carried through Chapters 6—7 to illustrate the bridge theorem and the fast-slow reduction. That model had three operational states and showed the generator construction in action, but it was not developed into a full attractor-level study of the traffic configurations the network can settle into.

We extend the three-state chain to a five-state hub-and-spoke network whose equilibrium set contains a balanced interior routing configuration and two spoke lock-in configurations pinned at the actor boundary. At the baseline parameters the balanced configuration attracts and the lock-in configurations repel–a numerical observation we mark as such where it appears–and the design lesson is a regime story: reward levers move the network toward, and across, the threshold at which spoke lock-in becomes attracting. The hub’s role as a reference state for minorization–already visible in the three-state model–becomes the structural anchor that makes the bridge theorem and tracking estimate apply with explicit computable constants.

The Section 2.8 routing model had three states: a retail hub (state 1) where customers enter, a health-discovery funnel (state 2) where customers encounter veterinary content, and a vet-booking page (state 3) where the high-value conversion happens. The routing agent chose between two presentation modes–retail-safe and vet-push–and the softmax mixture of two action-conditioned generators determined how customers flowed through the funnel.

Three structural features made the three-state model work within the Chapter 2 framework:

1. State-independent features. The actor features did not depend on the state: , for all . A single scalar parameter controlled the routing bias at every node simultaneously.

2. Reference-state reachability. The retail hub was reachable from every state at rates bounded below by a positive constant independent of . This gave the minorization condition that Chapter 6 needs.

3. Explicit generator construction. The generators and were built directly from the customer-flow topology, and the softmax mixture inherited the generator property.

All three features survive in a larger network, with one qualification we flag now and prove later. The model we build here has five states instead of three, but the construction follows the same pattern: state-independent features give a scalar policy axis, the hub remains uniformly reachable–though, as the next paragraphs explain, only in two steps from the deepest nodes–and the generators are built from the network’s adjacency structure.

The main difference is that the five-state network has two competing spoke branches instead of a single funnel. The routing agent can direct traffic left (toward the A—C branch) or right (toward the B—D branch), and the equilibrium set includes a configuration for each direction. This is the routing analogue of the recommendation chapter’s filter bubbles: instead of competing content niches, we have competing traffic destinations.

Structural comparison with the three-state chain. The Ch2.8 model and the hub-and-spoke network are related by reinterpretation, not by inclusion. The retail hub (state 1) plays the role of the central hub ; the health-discovery funnel (state 2) corresponds to the primary destination ; the vet-booking page (state 3) corresponds to the secondary destination . The graphs differ in one decisive edge. The Ch2.8 generators keep a direct return edge from the deepest state to the hub: uniformly on the actor box, so every state reaches the hub in a single jump and the one-step minorization hypothesis of Proposition 6.6 holds directly. The hub-and-spoke network deliberately removes that edge: the secondary destination connects only through , the hub is reachable from in two steps but not in one, and Proposition 6.6 no longer applies as stated. Section 11.4 proves the two-step substitute.

On the action side, Ch2.8’s vet-push action (which drives traffic deeper into the funnel) corresponds to Ch11’s left-bias (which drives traffic into the — branch); the Ch2.8 retail-safe action (which keeps mass at the hub) is replaced in Ch11 by a competing right-bias that drives traffic into a mirror funnel . The Ch2.8 reward asymmetry between core retail and vet booking generalizes to the Ch11 reward asymmetry between matching and mismatching routing modes at primary and secondary destinations (, , and the mirror inequalities at ).

The structural promotion is therefore twofold: the single funnel becomes one of two competing mirror branches, and the binary action set is reinterpreted from “advance versus hold” to “advance left versus advance right”. A single funnel has only one direction in which traffic can self-reinforce. The two-branch network supports two distinct self-reinforcing directions, one for each spoke, and the boundary equilibria at are the configurations each direction selects. Whether those configurations attract is a parameter question, and Sections 11.5—11.6 answer it.

We execute the full Model Specification Protocol.

Step 1. State space. Five nodes on a hub-and-spoke network:

• : Central hub. The primary entry point where traffic arrives. Moderate revenue is earned passively from hub traffic.
• : Primary destination A. A service node connected to the hub. Moderate service-specific revenue.
• : Primary destination B. A second service node, also connected to the hub. Symmetric to in the baseline model.
• : Secondary destination via A. A high-value service node reachable only through . The highest revenue in the left branch.
• : Secondary destination via B. A high-value service node reachable only through . The highest revenue in the right branch.

The network topology is a diamond with tails: . There are no direct connections between the two branches ( and are connected only through ), and the secondary destinations are accessible only through their primary nodes.

Step 2. Action space. Two routing modes:

• (Left-bias): route traffic preferentially toward the — branch.
• (Right-bias): route traffic preferentially toward the — branch.

Step 3. Actor features. Scalar actor () with state-independent features, matching the Section 2.8 pattern:

A positive favors right-bias at every node. A negative favors left-bias. The actor box is .

Step 4. Critic features. Scalar critic () with matching state-independent features:

The critic tracks a single signal: the reward advantage of right-bias over left-bias.

Step 5. Rewards. The reward table reflects the network’s value structure: secondary destinations carry the highest-value services, and the rewards are asymmetric across branches to create competing incentives.

• At the hub : and .
• At state : and .
• At state : and .
• At state : and .
• At state : and .

At the hub, both routing modes earn the same moderate reward. At primary destinations, the matching routing mode earns more: left-bias at earns 0.5 versus 0.2 for right-bias, and vice versa at . At secondary destinations, the advantage is extreme: left-bias at earns 1.2 versus 0.1 for right-bias. This encodes the domain logic: the high-value service at is designed for left-routed traffic, and the service at is designed for right-routed traffic. Routing mismatches earn little.

The reward asymmetry between branches creates competing incentives. If traffic concentrates at and , the left-bias rewards dominate and the gradient pushes negative. If traffic concentrates at and , the right-bias rewards dominate and the gradient pushes positive. At the hub, the rewards are balanced and do not favor either direction.

Parameters. Temperature , critic regularization , actor box radius .

We construct two action-conditioned generators from the network’s adjacency graph, following the method of Chapter 9.4.

Left-bias generator : Under left-bias routing, traffic flows preferentially from the hub toward and from toward . Return rates to the hub are maintained at moderate levels. The right branch has reduced forward rates.

The rows are indexed by . The columns are indexed in the same order. Reading the entries:

• Hub : Exits at total rate 1.6. Transitions to at rate 1.2 (fast, left-biased) and to at rate 0.4 (slow).
• Primary : Returns to hub at rate 0.4, advances to at rate 0.6.
• Primary : Returns to hub at rate 0.6 (faster return from non-preferred branch), advances to at rate 0.2 (slow forward under left-bias).
• Secondary : Returns to at rate 0.8. High-value destination with significant residence time.
• Secondary : Returns to at rate 0.6. Lower residence time at the non-preferred secondary.

Right-bias generator : Under right-bias routing, the pattern reverses: traffic flows toward and .

The entries are the left-right mirror of : the rate is now 1.2 (fast), is 0.4 (slow), and the B-branch forward rates are high while the A-branch forward rates are low.

Verification of the generator property. Both matrices have nonnegative off-diagonal entries and rows that sum to zero (check each row). The softmax mixture

is formed state by state. Because the features are state-independent, the softmax weights and are the same at every state. Therefore

a global convex combination. The mixture inherits the generator property and is smooth in .

Hub reachability. The hub is reachable from every state in at most two steps:

• From : direct, at rate .
• From : direct, at rate .
• From : two steps, at rates and .
• From : two steps, at rates and .

All lower bounds are independent of because the softmax assigns positive weight to both actions and both generators have positive return rates to the hub. This establishes the reference-state reachability condition.

The reference-state structure identified in Section 11.3 feeds the Chapter 6 minorization criterion, but with one structural caveat. Proposition 6.6 (Chapter 6, Section 6.5) postulates a uniform lower bound on the direct one-step jump rate for every . In our hub-and-spoke generators and , the secondary destinations and have and reach the hub only via the primary nodes, so the literal Proposition 6.6 hypothesis fails. A two-step version of the minorization condition does hold: the propagator at a fixed time –rather than the infinitesimal generator–has every column dominating a uniform multiple of . Chapter 10 met the same obstruction on the Y-graph–its deep-engagement states reach the diverse root only through the niche-discovery states–and resolved it with the two-step minorization of Section 10.4 (“Reference-state structure”). We state and prove the analogous lemma here with this chain’s constants, then explain why the same Dobrushin decomposition that drives Proposition 6.6 carries the same downstream conclusions.

Lemma 11.1 (Two-step reference-state minorization). There exist constants T_{\min} > 0$ and \alpha_2 > 0$ such that for every \theta \in [-1,1]$ and every i \in \{H, A, B, C, D\}$,

Proof. For states with a direct edge to the hub, Proposition 6.6’s one-step argument applies with the rate lower bound

For itself, mass leaves only at total rate (read off the entry, which is in both and ), so by Gronwall for all .

For states and , the chain reaches the hub only by first jumping to a primary node. The relevant rate is

Set (the time scale for the or step) and (for the or step), so .

By the same integrating-factor argument used in Proposition 6.6’s proof, the -component of the propagator started at , evaluated at , satisfies

where is the global maximum exit rate. (Proposition 6.6’s integrating-factor argument gives the bound at time ; we record the weaker form obtained from with .) Numerically, .

Once mass has reached , apply Proposition 6.6’s one-step argument from to with rate at least and propagator running for an additional . The mass at at time , starting from , is at least the product

The same bound applies to the path by the L-R symmetry of the rate lower bounds. For the directly connected states and for itself, the same integrating-factor estimate at time yields a lower bound of similar order. Taking as the minimum over all five starting states gives a uniform constant; the bound from and is the binding one, so we record .

The two-step minorization furnishes exactly what Proposition 6.6’s Dobrushin argument needs: a column-domination bound on a fixed-time propagator , uniform in . The proof of Proposition 6.6 (Chapter 6, Section 6.5) decomposes with column-stochastic–we write for the remainder matrix of that proof, keeping for the actor box radius–then iterates on the zero-mass subspace . The same decomposition with replaced by delivers uniform exponential mixing.

Proposition 11.2 (Mixing and Lipschitz invariant law). Under Lemma 11.1, the frozen chain mixes uniformly exponentially: there exist C_{\mathrm{mix}}, \gamma > 0$ independent of \theta$ such that

where \mathcal Z$ is the zero-mass subspace. Consequently, for each \theta \in [-1,1]$ the generator Q_\theta$ has a unique invariant law \mu_\theta \in \Delta_5$, and the map \theta \mapsto \mu_\theta$ is Lipschitz with constant L_\mu \leq L_Q \cdot C_{\mathrm{mix}}/\gamma$, where L_Q = \sup_\theta \|dQ_\theta^*/d\theta\|_{1\to 1}$ is the Lipschitz constant of the adjoint generator in the operator norm induced by the \ell^1$ norm on laws, as in Assumption 2.8.

Proof sketch. Apply the Dobrushin decomposition of Proposition 6.6 to using the column-domination bound from Lemma 11.1. The argument is the same as that of Proposition 6.6 with replaced by and by , and yields

The Lipschitz consequence for follows by differentiating the stationary equation along and using the spectral-gap inverse from ; see Chapter 6, Theorem 6.4.

The propagation chain with explicit numbers. Lemma 11.1 gives and . Hence

The Lipschitz constant of is bounded, using , by

with and (the adjoint acts on laws, so the relevant norm is the maximum row-sum of , attained at row ), giving . Hence

These constants are not sharp–the column-domination bound from a two-step path through a slow link is intentionally pessimistic–but they are explicit. The propagation chain runs from the minorization pair to the exponential mixing rate , then to the Lipschitz constant , and finally into the singular-limit tracking estimate of Theorem 7.3 with all constants in place.

Pathwise contraction and tracking. Proposition 7.6 of Chapter 7 is the pathwise (non-autonomous) analogue of Proposition 6.6 and again requires a direct one-step jump rate to the reference state. The same two-step modification applies: replacing the one-block bound of Proposition 7.6 by the two-step bound of Lemma 11.1 gives the pathwise contraction estimate with constants above. This is legitimate because Lemma 11.1’s proof uses only jump-rate bounds that hold for every –, , exit rates at most –so each Gronwall and integrating-factor step, and the Chapman—Kolmogorov factorization, apply verbatim to the evolution family generated along any measurable actor path ; the two-step column-domination bound therefore holds pathwise with the same pair . Pathwise contraction in turn lets Theorem 7.3 deliver the finite-time tracking estimate and Corollary 7.5 the upper-semicontinuity of attractors as .

Phase space. The system lives on , with dimension . Compactness of the long-time dynamics is derived, not assumed: the Chapter 4 estimates construct the compact absorbing set, and the exact-system theorem of Chapter 7, Section 7.1 turns it into the global attractor for the generator-law model. The forcing bound is , so with the Chapter 4 radius gives the compact forward-invariant absorbing set , which is what Corollary 11.4’s invocation of the Chapter 7 exact-system theorem uses. (This model’s constant coercivity gives a sharper ball: and yield whenever , so is already forward invariant and absorbing. We keep the framework radius so the constants match the general construction.)

Occupancy and critic coefficients. Because the features are state-independent with , , the score function reduces to , with centered-feature average , where and are state-independent. Therefore

using in the final equalities. The sanity check holds as required. The formulas match the analogous Chapter 0 expression (Section 0.8), with the sign convention reversed because here the right-bias action carries the positive feature.

The critic matrix simplifies to

which is constant–a consequence of the state-independent features with for all . The critic equation is

where . The critic equilibrium at fixed is .

Actor drift. The raw drift is

Substituting the score formulas and , and using state-independence of and , the action sum at each state collapses:

where the cross terms and factor as times the difference of the bracketed expressions. Summing over states gives the closed form

since at every state. With and , the drift simplifies further:

This matches the Section 2.8 routing drift term for term: a population-weighted reward differential, a critic term from the critic features, and the entropy pull . Chapter 0’s shows the same structure one model earlier, with two differences: its reward differential is the constant , and its critic term is rather than because the Chapter 0 critic features take values in rather than . The damped drift is .

The full system:

Equilibria. Three classes of equilibria have domain meaning.

Left-spoke equilibrium (\theta = -1$). The policy strongly favors left-bias. Traffic flows through the hub to and then to , where the high-value left-routing service earns reward 1.2. The frozen invariant law concentrates mass on the — branch. The hub retains some mass (it is the entry point), but the right branch is underutilized.

Right-spoke equilibrium (\theta = +1$). By the reward symmetry, the mirror configuration: traffic concentrates on —, and the high-value right-routing service earns the dominant reward.

Interior equilibrium. On the reduced manifold (with and ), the closed form makes the sign-of-advantage argument direct. The reduced advantage is

with the bracket function

The reward table is invariant under the involution : , , , , . The generators satisfy the matching mirror identity under the same state permutation. Combined with , the frozen invariant law obeys , where is the mirror map , , fixed (we reserve for the sigmoid). Each summand of is therefore odd in :

So . Continuity of on together with oddness forces , giving an interior zero at . The prefactor is nonzero, hence ; the damping factor equals at and does not vanish, so the full reduced drift vanishes at .

Proposition 11.3. The hub-and-spoke routing system has at least three equilibria: two boundary equilibria at \theta = \pm 1$ and one interior equilibrium at \theta = 0$.

Proof. The boundary equilibria exist because , and the critic and distribution equations have unique fixed points at frozen by constant coercivity together with the uniform exponential mixing of Proposition 11.2. For the interior equilibrium, the bracket function defined above is odd in by the L-R reward and generator symmetry, so ; since the prefactor is strictly positive, this gives . The critic and distribution components again have unique fixed points at , with by the same odd-symmetry argument and the unique invariant law of . Hence is the interior equilibrium.

Remark. The L-R reward symmetry forces the interior equilibrium to be at exactly. Asymmetric reward tables would shift the interior zero away from the origin and would generally require an explicit intermediate-value-theorem argument (sign of at versus ). The reward sweep in Exercise 11.7 scales and together, which preserves the symmetry; the interior zero stays pinned at throughout the sweep.

Corollary 11.4. The hub-and-spoke routing system has a compact global attractor \mathcal{A}$ containing all three equilibria.

Proof. The model data satisfy Assumption 2.7 and the generator regularity assumption 2.8. The law equation is in generator form, so the prescribed-closure attractor theorem of Chapter 5 is not the theorem being invoked directly. Instead, apply the exact-system theorem of Chapter 7, Section 7.1 with . It gives a continuous semiflow on , the compact absorbing set , and the unique compact global attractor . Equilibria are invariant, hence contained in .

Scope of this section. The theorems proved in Sections 11.4—11.5 give: (i) existence of at least three equilibria (, per Proposition 11.3); (ii) existence of a compact global attractor that contains all three (Corollary 11.4); (iii) a Lipschitz frozen-law map and a singular-limit tracking estimate as (Proposition 11.2 and Theorem 7.3). The stability type of the equilibria and the basin structure are not proved in these notes. For those questions we report numerical computations, marked as such, and the design discussion at the end of the section rests on the numerical regime picture.

Balanced operation (numerical). At the interior equilibrium , the routing policy is symmetric: , so traffic is sent to both branches with equal preference. The frozen invariant law is the unique solution of on the simplex; solving the linear system numerically (Exercise 11.1 carries out the computation) gives

By the L-R symmetry of the law is exactly mirror-symmetric across the two branches, and the operational reading is balance rather than hub dominance: the hub holds about 17% of the traffic–less than either primary node–while the branches split the remainder equally and the high-value services at and both run at moderate utilization. No branch is starved.

Spoke lock-in (numerical). At the boundary equilibria , the routing policy is fully committed: at and the mirror at . The frozen invariant law at , computed numerically (Exercise 11.2 carries out the computation),

places about 67% of the traffic on the — branch and only 18% on —: the high-value left service runs hot while the right branch is starved. This is the routing analogue of the filter-bubble configuration in Chapter 10. The boundary equilibrium is created by parameter saturation ( at ), not by the vanishing of the advantage signal: is generally nonzero, but the damping factor kills the actor velocity. The gradient interpretation is that the policy is pinned at the boundary by the actor box, not chosen by the system.

Stability at baseline (numerical). Whether the lock-in configurations attract is decided by the bracket function on the reduced manifold. The closed-form drift makes the competition visible: when concentrates on the left branch, pulls toward and the critic term reinforces the pull, while the entropy term –equal to at –pulls back toward balance. Computing numerically along at the baseline rewards shows the entropy term winning everywhere:

with a single zero at and slope . On the reduced flow every interior trajectory moves toward the balanced equilibrium, and the lock-in equilibria repel: at the reward feedback is no match for the entropy pull . The full system’s linearization agrees. Because the damping factor vanishes at the boundary, the -row of the Jacobian at has zero entries in the and columns, so its diagonal entry– at , and the same value at by the mirror symmetry–is an exact eigenvalue, positive and independent of . At the baseline parameters the network is monostable: balance attracts, lock-in repels. These are numerical statements; the sign structure of and the eigenvalue are computed, not proved.

The lock-in regime (numerical). The baseline conclusion is not the end of the design story, because the reward scale is a lever. Scale the two secondary matching rewards together, , keeping the rest of the table fixed. The scaling preserves the L-R mirror symmetry, so the interior equilibrium stays at throughout (the Remark after Proposition 11.3), and recomputing along the sweep locates two thresholds. At the slope turns positive: the balanced equilibrium loses stability, and a mirror pair of attracting interior equilibria splits off–a pitchfork bifurcation, in dynamical-systems language–and drifts outward as grows. At the boundary values and change sign: the attracting pair reaches the actor box, and beyond this value the lock-in equilibria attract while repels. The network is then bistable, and which spoke wins depends on the initial routing bias. The domain lesson is the regime story: the high-value services need only dominate the baseline reward by roughly a factor of three before self-reinforcing traffic concentration becomes the stable outcome. Both thresholds are numerical observations about this model, computed from the reduced bracket function; Appendix B records the sweep values and describes how to reproduce the corresponding phase-portrait computations.

The fast-slow reading. When network equilibration is fast (traffic redistributes quickly in response to routing changes, small ), Theorem 7.3 supplies the finite-time tracking estimate and Corollary 7.5 the upper-semicontinuity of attractors as . The reduced system is two-dimensional in on the invariant-law graph and is more tractable than the full six-dimensional system. This is a proved consequence of the framework–not a numerical observation–modulo the caveat that the rate constants and of Section 11.4 are explicit but pessimistic.

Design principle (numerical). A routing operator who wants balanced utilization should keep the network on the monostable side of the regime thresholds, and the drift formula identifies which levers actually move them. The lever that matters is the secondary reward asymmetry: and are what feed once traffic concentrates on a spoke, so keeping the secondary matching rewards below the pitchfork threshold ( in the baseline geometry) keeps balance attracting. A lever that does not work: raising the common hub reward leaves unchanged at every , because the hub contributes ; it enters only through the critic forcing , and in our sweeps it slightly weakens the balanced equilibrium’s attraction ( rises from at to at ). A structural lever that does work: raising the return rates to the hub in both generators re-mixes traffic faster, weakens the dependence of on , and strengthens balance (doubling the and rates moves from to and shrinks from to ). All of these are numerical observations about this model, not theorems; the framework supplies the attractor’s existence and the equilibrium count, and the thresholds come from the computations reported in Appendix B.

This chapter extended the Section 2.8 three-state routing chain to a five-state hub-and-spoke network and showed that the same Model Specification Protocol that produced the recommendation model in Chapter 10 produces a structurally parallel attractor study for a routing network.

The boundary equilibria of the routing model are spoke lock-in traffic configurations rather than content-niche filter bubbles, but the mathematical mechanism is the same: self-reinforcing feedback between the policy and the population distribution creates the lock-in equilibria, and reward design decides the regime. At the baseline parameters the balanced interior configuration attracts and lock-in repels (numerically), while scaling the secondary matching rewards past a numerically located threshold makes lock-in the stable outcome.

Three domain-specific insights came from the analysis. First, the hub’s role as a reference state for minorization is structural, but the topology charges a price: removing the direct secondary-to-hub edge forces the two-step minorization of Lemma 11.1, with explicit computable constants. Second, spoke lock-in equilibria are boundary artifacts created by parameter saturation, not by incentive alignment–and whether they attract is a reward-regime question, not a topological one. Third, the fast-slow reduction gives a practical simplification: when traffic equilibrates quickly, the reduced actor-critic system on the invariant-law graph governs the routing outcome.

Appendix B records the numerical sweep data for the routing model and describes how to compute phase portraits for both the recommendation and routing models.

Exercise 11.1 (Verify: frozen invariant law at the hub). Compute the frozen invariant law at by solving with the simplex constraint. At , the policy is balanced: , so . What fraction of traffic is at the hub? Is the distribution symmetric between the two branches? Compare with the values reported in Section 11.6.

Exercise 11.2 (Compute: left-spoke equilibrium). At :

(a) Compute the softmax weights: , .

(b) Form the mixture generator and solve for the invariant law .

(c) What fraction of traffic is at the secondary destination ?

(d) Compute the critic equilibrium . Is positive or negative, and what does its sign mean in domain terms?

Exercise 11.3 (Verify: minorization constants). Compute the two-step transition probability from state to state under the mixture generator : the path with rates from . Verify that this probability is positive and compute an explicit lower bound.

Exercise 11.4 (Break: dead-end topology). Remove the edge from to in both generators: set and adjust the diagonal entries to maintain row-sum-zero. Now state is absorbing: under every policy, traffic that reaches never leaves. (Removing the edge in alone would not break anything–the softmax floor keeps the mixed rate on the actor box; check this first.)

(a) Does the hub minorization of Lemma 11.1 still hold? From which starting state does it fail?

(b) What happens to the invariant law of the mixture generator at ? Compute it: where does all the mass end up?

(c) Is the bridge theorem (Chapter 6) actually broken? Check whether uniform exponential mixing (Assumption 6.2) survives with in the role of reference state, and reconcile your answer with (a): minorization at a chosen reference state is sufficient for mixing, not necessary. What is broken here is the routing model, not the theorem.

Exercise 11.5 (Connect: three-state versus five-state mixing). Compute the spectral gap of at for both the Section 2.8 three-state chain and the Chapter 11 five-state chain. Which chain mixes faster? Give a topological reason: how does the diameter of the state graph relate to the spectral gap?

Exercise 11.6 (Extend: adding a shortcut). Add a direct shortcut between and under left-bias: set (and correspondingly under right-bias), adjusting the diagonals to keep row sums zero. Reconstruct and with the new edges.

(a) How does the left-spoke equilibrium change? Does more traffic reach ?

(b) Does the reference-state minorization improve? Show that now has a direct route to the hub with mixed rate , so qualifies for the one-step argument of Proposition 6.6; compare the resulting one-step constant with the two-step constant of Lemma 11.1.

Exercise 11.7 (Connect: reward design and the lock-in regime). The routing operator wants the balanced interior equilibrium to attract all initial conditions.

(a) Explain why convergence cannot hold literally from every point of the phase space as long as the boundary equilibria exist (equilibria are invariant), and why the operationally meaningful question is whether the balanced equilibrium attracts the interior.

(b) Scale the secondary matching rewards together by setting the C-left and D-right rewards equal to a common value rho, and recompute the bracket function B(theta) of Section 11.5 numerically along a grid in rho. Locate the two regime thresholds reported in Section 11.6: the pitchfork value rho about 2.58, where the derivative B’(0) changes sign, and the lock-in value rho about 3.71, where the two boundary values B(-1) and B(+1) change sign.

(c) Explain in domain terms why raising the common hub reward is not a substitute lever. The hub rewards for L and R are kept equal, so show that the hub contribution cancels in Delta r(mu) for every mu.