A Network-Control Approach to a
Learning-Optimal Brain State

Dhruv Singh

Work in progress · a personal project · July 2026 · code at github.com/DhruvSingh0905/cognitive-substrate-map

📌 Archived snapshot — v1 · 10 July 2026. This page is frozen; the current version may differ → latest version.

Status — work in progress. This is a personal learning project, not a finished or peer-reviewed paper. The knowledge-graph substrate, the propagation engine, and the controllability analysis are built and tested; the objective, the uncertainty pass, and the optimizer are laid out below but not yet run. I'm sharing the approach and the parts that work — feedback very welcome.

Overview. This project treats a biomedical knowledge graph as a signed, directed control system and asks a simple question with a hard answer: which genes would you nudge, and by how much, to move the brain toward a state that's good for learning — without disturbing everything else? Starting from PrimeKG[1] (129,375 nodes), I scope down to a 280-node "brain core" (2,871 signed edges, with a 200-node feedback loop), push perturbations through it with a random-walk-with-restart[4], and use control theory[9] to ask what the network can even be steered to. The scoring objective, the uncertainty analysis[12], and the optimizer[15] are written out below but not yet run — that's the next phase. What is built is checked by comparing its predicted signs against curated regulatory edges (8/8). The eventual output is a ranking of interventions and a trade-off map — not fold-change predictions, and not a protocol.

1. Substrate

The base graph is PrimeKG[1] — 129,375 nodes and about 4 million edges across ten types (genes, drugs, diseases, pathways, and so on). It comes from the Hetionet / Rephetio line of work[2], which scores paths through a graph like this to repurpose drugs.

Most of that graph has nothing to do with learning, so I keep only genes that are mostly expressed in brain tissue. The score for a gene $v$ is the fraction of the tissues it appears in that are brain tissue, over PrimeKG's anatomy_protein_present edges $E_a$:

$$e(v)=\frac{\bigl|\{a:(v,a)\in E_a,\ a\in A_{\mathrm{brain}}\}\bigr|}{\bigl|\{a:(v,a)\in E_a\}\bigr|},\qquad \text{keep if } e(v)\ge 0.15 \ \wedge\ \deg_a(v)\ge 5 \tag{1}$$

In words — keep a gene only if a decent share of the tissues it's active in are brain, and only if it's active in enough places for that share to mean something.

After pruning: 280 nodes in four layers — 70 intervention "knobs", 15 systemic inputs, 6 disease readouts, 2 off-limits, and 187 one-hop neighbors — with 3,912 directed edges (2,871 signed: 2,253 activating, 618 inhibiting) and a single 200-node feedback loop.

substrate
Figure 1. The 280-node substrate, colored by layer; node size grows with degree. The dense core is the feedback machinery.

2. The propagation operator

Put the edges in a matrix $W$: the entry $W_{ij}$ is $+1$, $-1$, or $0$ for the effect of gene $j$ on gene $i$ (oriented so $(Wx)_i$ is what flows into $i$). A hub gene with many out-edges would swamp everything, so I divide each gene's column by its out-degree — the same trick as Google's PageRank[3] ($d^{\mathrm{out}}_j=\sum_i|W_{ij}|$):

$$\hat{W}=W\,D_{\mathrm{out}}^{-1},\qquad \hat{W}_{ij}=\frac{W_{ij}}{\sum_k|W_{kj}|} \tag{2}$$

In words — spread each gene's outgoing effect evenly across the genes it points at, so one gene that talks to everyone can't dominate.

Each column now sums to 1 in absolute value, which forces the largest eigenvalue to be $\le 1$ (Perron–Frobenius) and keeps the next step stable (measured: $\rho(\hat W)=0.544$). Dividing by out-degree is the right move for a directed graph, where the problem is out-hubs; the symmetric GCN version $D^{-1/2}\tilde W D^{-1/2}$[6] is for undirected graphs.

out-degree distribution
Figure 2. Out-degree across the 280 genes is heavy-tailed — most point at a handful of others, while a few hubs point at dozens (RELA at 114). That tail is why the normalization matters.

3. Propagation by random-walk-with-restart

To see what a nudge does, push a signal $p$ into one gene and let it spread — but pull a fraction $\alpha$ back to the source at each step, so it stays near where it started[5],[4]:

$$x^{(t+1)}=(1-\alpha)\,\hat{W}x^{(t)}+\alpha\,p \tag{3}$$

In words — at each step, spread the signal one hop, but yank a fixed slice $\alpha$ of it back to the gene you started from.

Solving for the value it settles to gives a closed form (a sum over every path length $k$):

$$x^{*}=\alpha\bigl(I-(1-\alpha)\hat{W}\bigr)^{-1}p=\alpha\sum_{k\ge0}(1-\alpha)^{k}\hat{W}^{k}p \tag{4}$$

In words — the settled response; each extra hop is discounted by $(1-\alpha)^k$, so far-away genes barely feel it.

Because far hops are discounted, the effect stays local — most of the signal is gone within a few hops. The 200-node feedback loop is why we solve for a settled value instead of taking a single push.

decay vs hop distance
Figure 3. Measured on the real graph: perturb CREB1 and the response $|x^*|$ drops about 10× per hop. Straight line on a log axis = exponential decay, exactly the $(1-\alpha)^k$ term.

4. Controllability

Can a few inputs steer the whole thing? Written as a linear system with input map $B$ (the genes we can drive), $\dot{x}=Wx+Bu$, it's fully steerable when the controllability matrix has full rank (Kalman[7]):

$$\operatorname{rank}\mathcal{C}=N,\qquad \mathcal{C}=[\,B,\ WB,\ W^{2}B,\ \dots,\ W^{N-1}B\,] \tag{5}$$

In words — you can reach any state exactly when your inputs, pushed through more and more hops, together cover all $N$ directions.

Testing that directly is impractical at this size, so I use structural controllability (Lin[8]; Liu–Slotine–Barabási[9]): the fewest genes you must control is the number left unmatched after pairing up the network as well as possible,

$$N_{D}=\max\bigl(N-|M^{*}|,\ 1\bigr) \tag{6}$$

In words — match each gene to one it directly drives; whatever's left over is what you have to control by hand.

The matching is found with Hopcroft–Karp[10]. The result: all 280 genes are reachable, but full control would need 81 driver genes and only 37 of those are druggable — so you can't push it to any state you like. You can steer the readouts you actually care about (target control[11]), which takes fewer inputs, and that's the setting the optimizer works in.

driver nodes
Figure 4. The 81 driver genes on the substrate (orange), of which 37 are druggable (green). The rest of the network (grey) is steered indirectly through them.
Table 1. Controllability of the substrate.
quantityvalue
druggable inputs $|\mathcal D|$172
reachable $|R|/N$280 / 280
min drivers $N_D$ (full control)81
drivers that are druggable37 / 81
feedback component (SCC)200

5. Target state and objective

The target is a vector $d$ of desired changes. The key fact: most brain variables aren't "more is better" — they have a sweet spot, an inverted-U (Yerkes–Dodson[18]; Arnsten[19]): dopamine, arousal, cortisol, excitation/inhibition balance, mTOR. Benefit is how much closer to the target you get:

$$b_i=\lvert d_i\rvert-\lvert x^{*}_i-d_i\rvert \tag{7}$$

In words — positive if the nudge moved a gene toward its ideal level, negative if it pushed past it.

Total benefit weights each gene by how sure we are of its target (readouts count for nothing); cost is how hard you push plus how much you disturb genes outside the target set:

$$B(p)=\sum_i w_i\,b_i,\qquad C(p)=\lVert p\rVert_1+\gamma\!\!\sum_{j\,\notin\,T}\!\!\lvert x^{*}_j\rvert \tag{8}$$

In words — add up the good across all genes, then subtract the cost: effort plus collateral movement elsewhere.

The current target vector covers 94 genes: 43 set-point, 30 up, 12 down.

target set
Figure 5. The 94-gene target set grouped by system; color is the wanted direction — green up, red down, gold set-point.

6. Uncertainty quantification

Many edge strengths are known only by sign, so a single number would be overconfident. The plan: run the model many times over that uncertainty (Monte-Carlo[12]) and report a range instead of a point,

$$\widehat{\mathrm{CI}}_{90\%}=\bigl[\,Y_{(\lceil0.05M\rceil)},\ Y_{(\lfloor0.95M\rfloor)}\,\bigr] \tag{9}$$

In words — run it many times, sort the scores, and report the middle 90% as an honest range.

Then use Sobol indices[13],[14] to see which unknown is driving the range:

$$S_i=\frac{\operatorname{Var}_{\theta_i}\!\bigl(\mathbb{E}[Y\mid\theta_i]\bigr)}{\operatorname{Var}(Y)}, \qquad S_{T_i}=\frac{\mathbb{E}\bigl[\operatorname{Var}(Y\mid\theta_{\sim i})\bigr]}{\operatorname{Var}(Y)} \tag{10}$$

In words — of all the wobble in the score, how much comes from each unknown — so you know which fact to go check first.

7. Optimization

There's no single best answer — more benefit usually costs more — so the output is a trade-off curve (a Pareto front). I trace it with the ε-constraint method[15]: get the most benefit while keeping cost under a cap $\varepsilon$, then slide the cap:

$$\max_p\ B(p)\quad\text{s.t.}\quad C(p)\le\varepsilon \tag{11}$$

In words — squeeze out the most benefit at each cost budget, then vary the budget to draw the whole curve.

A plain weighted sum, $\max_p B(p)-\lambda C(p)$, would quietly skip part of the curve when it bends the wrong way (a non-convex front)[16],[17]; ε-constraint doesn't, and the small size makes it cheap to run.

8. Validation

The engine is test-driven (11 tests). The main check: push one gene by $+1$ and confirm the response sign matches the edge sign in the database — not the model. All eight match ($\alpha=0.15$, $\rho(\hat W)=0.544$).

sign check
Figure 6. Perturb a gene, read the response at a target: green is up, red is down, and each bar's label is the edge sign it should match. All eight agree, including two-hop sign flips.
Table 2. Sign consistency: unit perturbation vs. curated edge sign.
perturb→ targetedge$x^{*}$check
CREB1 ↑BDNF+1+0.0030✓
CREB1 ↑JUN−1−0.0027✓ flip
CREB1 ↑NTRK2+1→+1+0.0080✓ net
GSK3B ↑CREB1−1−0.0041✓ inhib
AKT1 ↑NFKB1+1→−1−0.0008✓ net-flip

One nice check on the operator: two-hop NTRK2 moves more than one-hop BDNF, because CREB1's ~40 out-edges split its signal while BDNF's single out-edge passes it straight through — exactly what equation (2) predicts.

9. Groundedness and assumptions

The layers rest on different evidence: brain-scoping is computed, edge signs are curated from causal databases, edge strengths are mostly sign-only, and the target vector is hand-built. And $\hat W$ is a linear (first-order) stand-in for nonlinear biology, so the results hold for small nudges near the resting state, not big ones. So the output is an ordered list of interventions and a trade-off map — not exact fold-changes, and not medical advice.

10. Status & roadmap

Built and tested so far: the brain-scoped substrate (§1), the propagation operator and RWR engine (§2–3), and the structural-controllability analysis (§4). Written out here but not yet run: the set-point objective (§5), the Monte-Carlo uncertainty pass (§6), and the ε-constraint optimizer (§7) — this is the active work. Further out: fitting real edge strengths where data exists, and a nonlinear cross-check of the linear propagation. This is a living write-up; it will change as those pieces land.

References

  1. Chandak, P., Huang, K., Zitnik, M. Building a knowledge graph to enable precision medicine. Scientific Data 10, 67 (2023).
  2. Himmelstein, D.S. et al. Systematic integration of biomedical knowledge prioritizes drugs for repurposing. eLife 6, e26726 (2017).
  3. Page, L., Brin, S., Motwani, R., Winograd, T. The PageRank Citation Ranking: Bringing Order to the Web. Stanford InfoLab (1999).
  4. Gasteiger, J., Bojchevski, A., Günnemann, S. Predict then Propagate: Graph Neural Networks meet Personalized PageRank. ICLR (2019).
  5. Tong, H., Faloutsos, C., Pan, J-Y. Fast Random Walk with Restart and Its Applications. ICDM (2006).
  6. Kipf, T.N., Welling, M. Semi-Supervised Classification with Graph Convolutional Networks. ICLR (2017).
  7. Kalman, R.E. Mathematical Description of Linear Dynamical Systems. J. SIAM Control 1(2), 152–192 (1963).
  8. Lin, C-T. Structural Controllability. IEEE Trans. Automatic Control 19(3), 201–208 (1974).
  9. Liu, Y-Y., Slotine, J-J., Barabási, A-L. Controllability of complex networks. Nature 473, 167–173 (2011).
  10. Hopcroft, J.E., Karp, R.M. An $n^{5/2}$ Algorithm for Maximum Matchings in Bipartite Graphs. SIAM J. Comput. 2(4), 225–231 (1973).
  11. Gao, J., Liu, Y-Y., D'Souza, R.M., Barabási, A-L. Target control of complex networks. Nature Communications 5, 5415 (2014).
  12. JCGM 101:2008. Evaluation of measurement data — Supplement 1 to the GUM — Propagation of distributions using a Monte Carlo method. BIPM (2008).
  13. Sobol, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55, 271–280 (2001).
  14. Saltelli, A. et al. Global Sensitivity Analysis: The Primer. Wiley (2008).
  15. Haimes, Y.Y., Lasdon, L.S., Wismer, D.A. On a bicriterion formulation … the ε-constraint method. IEEE Trans. Syst. Man Cybern. 1(3), 296–297 (1971).
  16. Boyd, S., Vandenberghe, L. Convex Optimization (§4.7, scalarization & Pareto optimality). Cambridge Univ. Press (2004).
  17. Marler, R.T., Arora, J.S. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26, 369–395 (2004).
  18. Yerkes, R.M., Dodson, J.D. The relation of strength of stimulus to rapidity of habit-formation. J. Comp. Neurol. Psychol. 18, 459–482 (1908).
  19. Arnsten, A.F.T. Stress signalling pathways that impair prefrontal cortex structure and function. Nat. Rev. Neurosci. 10, 410–422 (2009).

Project overview · source code · archived v1 · a personal learning project — not medical advice.