research/tools/simulators/blend/pd

pd — peering-degree Monte-Carlo graph simulator

Quantifies how a node's peering degree trades off, in the Blend network:

• propagation speed — the full delay (ms) of a message: a random sender routes it along a blend_hops-relay Blend path (each relay a free-running timed-release mix node) and the last relay floods the whole network;
• adversary exposure — with a fraction f_adv of adversarial nodes, how many honest nodes are peered with ≥1 adversary (observed) and how many are fully surrounded (eclipsed);
• deanonymization — tying propagation to the adversary: how often a message's whole blend path is adversarial (deanonymization — the adversary owns the cascade end-to-end) and how often the honest sender is additionally peered with an adversary (full deanonymization — the message is tied back to its originator); and
• reliability under churn — with a fraction unresponsive_frac of nodes that relay nothing, the message success-delivery-rate (fraction of messages that survive the whole blend cascade to a responsive final relay) and the flood coverage of those that do.

The peer graph is a seeded random d-regular graph (exactly degree symmetric peers, identical for everyone from one global seed). This is static-graph analysis — no consensus — so it is far lighter than the TSI simulators and scales to 10⁶ nodes (sparse CSR + sampled Dijkstra; the adversary metrics are exact at every N).

Model (all delays in ms)

• Link delay: geographic base (metro 15 → antipodal 200 ms) + exponential transport jitter.
• Processing lag: each node draws a fixed lag from a categorical distribution (default {10, 50, 100} ms at {0.5, 0.4, 0.1}), incurred every time it relays.
• Blend mixing: each relay releases on a free-running clock whose successive intervals are Uniform{0…max_blend_delay} whole seconds; a held message waits for the relay's next release (the renewal residual). Mixing happens only at the blend_hops relays; the final flood is plain.
• Unresponsive nodes: a random unresponsive_frac of the population relays nothing (its outgoing edges are removed). Relays are drawn from the whole node list blind to responsiveness, so a message dies if any relay on its path is unresponsive — the delivery-rate then tracks (1−unresponsive_frac)^blend_hops. Unresponsive nodes still receive, but they are routing holes, so a delivered flood can strand pockets; a higher peering degree supplies redundant paths that keep coverage high. This axis affects propagation only, not the adversary metrics.
• Deanonymization: relays are picked blind to who is adversarial, so P(the whole blend path is adversarial) is the exact hypergeometric C(n_adv, blend_hops) / C(N−1, blend_hops) ≈ f_adv^blend_hops (deanon_rate) — placement-independent, driven by path length, not degree. Multiplying by the fraction of honest nodes with ≥1 adversary peer (observed_frac, which the worst-case-coverage placement maximizes) gives full_deanon_rate — the honest sender is also directly exposed, so the message is tied to its originator. Lengthening the blend path is the dominant defence; a higher degree speeds propagation but raises the chance a sender directly touches the adversary. Both are exact at every N (no Monte-Carlo), like the other adversary metrics.
• Messaging redundancy: redundancy R sends each emission over R independent blend cascades. A node receives the message from whichever cascade reaches it first (arrival times are combined element-wise), so it is delivered if any cascade delivers (delivery = 1−(1−(1−u)^blend_hops)^R) and captured if any cascade is whole-path-adversarial (deanon = 1−(1−f_adv^blend_hops)^R) — the same 1−(1−x)^R law, so redundancy trades reliability against anonymity. It buys no extra coverage: a cascade only delivers if the sender could route to its relay, so every delivered cascade floods the sender's own component. R = 1 is the plain single-cascade model (default), to which the whole aggregation reduces exactly.
• Churn percolation: the flood only crosses responsive nodes, so it lives on the responsive sub-graph — site percolation on a d-regular graph, whose giant component survives only while the responsive fraction exceeds 1/(degree−1). A network tolerates churn up to u_c = 1 − 1/(degree−1) (degree 3 → 0.5, degree 6 → 0.8, degree 16 → 0.93) and shatters above it; configs/percolation.yaml walks u across the threshold and make verify checks it.
• Linkability over time (pd.linkability): given the rates above and an emission cadence (one node emits per 30 s slot, chosen ∝ stake), the module derives the time to link an emitter (≈ 30 s·ln(1/(1−α))/(stake·q), inverse in stake) and the time to learn its stake to a threshold from the count of attributable observations. See configs/redundancy.yaml and the report.

Quick start

make install                 # or reuse a sibling venv: PYTHONPATH=src <python> -m pd.sweep ...
make smoke                   # fast end-to-end (every code path + 19 of 21 figure builders)
make verify                  # analytic checks (closed forms + graph invariants)
make test                    # unit tests
make sweep                   # configs/default.yaml (N up to 1e5, both adversary modes)
make sweep-fullscale         # configs/fullscale.yaml (N up to 1e6, random-mode exact)
make redundancy              # configs/redundancy.yaml (R=1..4: delivery vs deanonymization)
make percolation             # configs/percolation.yaml (churn threshold u_c = 1-1/(degree-1))
make figures RUN=runs/<dir>

Outputs

Three parquets per run: propagation.parquet (full_delay_ms_*, delivery_rate, frac_reached, coverN_ms vs degree / blend_hops / N / unresponsive_frac / redundancy), adversary.parquet (observed_frac / eclipsed_frac vs degree / f_adv / mode, random + worst-case envelope), and deanon.parquet (deanon_rate / full_deanon_rate vs degree / blend_hops / redundancy / f_adv / mode — propagation paths crossed with the adversary set). Figures render all three: delay vs degree / path length / N, observation and eclipse vs f_adv and degree, delivery and coverage vs the unresponsive fraction (with the u_c = 1-1/(degree-1) threshold), the deanonymization rates, and the time-to-link / stake-inference / redundancy curves derived from them by linkability.

Layout

src/pd/: graph (matching-union CSR d-regular), propagation (Blend cascade), mixclock (release-clock residual), adversary (exact observation/eclipse + deanonymization + placement), config/engine/sweep/metrics, plotting. configs/ sweeps, tests/, scripts/ shims. Reports of record live outside the sim at reports/blend/pd/.