∑ The algorithm

Eight stages, two model calls, one number.

Every relationship is a probability. A chain of intros multiplies them. The search finds the chain with the largest product, infers the last hop from shared affiliations, and learns from every answer.

O(E log V)
per search, bounded to 3 hops · ~50 ms for 5,000 people
2
model calls per search: model the target, pick the path
4
edge families: observed, co-occurrence, inferred, learned
14
papers wired into the weights and the search
the whole thing in one line

best connector = argmaxu  [ maxchain ≤ 3 ∏ w(edge) ] · [ 1 − ∏k (1 − pk) ] · bridging(u) · φhops−1

0

Observe

persons = ∪ mailboxes (key = SHA-256(salt‖email) | LinkedIn URL | name)
you→u: f = #touches, r = #replies, Δ = days since last, span = last − first
co-occurrence (bipartite projection): s(a,b) = Σ_threads 1/(k−1)
costO(M·k²) over M threads with k people; k ≤ 15, so linear in mail volume

Headers only. Team inboxes, lists, newsletters and mass mails are dropped before they become people.

Neal 2014 · The backbone of bipartite projectionsSocial NetworksGilbert & Karahalios 2009 · Predicting tie strengthCHI
1

Edge probabilities

w(you,u) = 0.35·log(1+f)/log 41 + 0.30·2^(−Δ/120)·d + 0.20·(r/f)·d + 0.15·channel·d, d = min(1, f/3)
decay (Navarro 2017): gap = span/(f−1); if silence > 8.33·gap then w ← w·(1 − ½·min(1, silence/(8.33·gap) − 1))
overlap (Onnela 2007): w ← w + 0.15·Jaccard(N(u), N(you))
co-occurrence (Crandall 2010): w(a,b) = 1 − e^(−0.6·s(a,b)), keep only s ≥ 0.34 (backbone)
LinkedIn floor 0.12 · pending invite 0.05 · non-person 0
costO(E)

Every weight is a probability: the chance u makes a warm intro if asked.

Crandall et al. 2010 · Inferring social ties from geographic coincidencesPNASNavarro et al. 2017 · Temporal patterns behind the strength of persistent tiesEPJ Data ScienceOnnela et al. 2007 · Structure and tie strengths in mobile communication networksPNAS
2

Infer missing edges

employer overlap: w = clamp(1.5/|company|, 0.05, 0.35), 2 ≤ |company| ≤ 60
Adamic-Adar: AA(a,b) = Σ_{z∈N(a)∩N(b)} 1/log(1+deg z); keep AA ≥ 0.9, w = min(0.3, 0.12·AA)
Katz: K(a,b) = Σ_{ℓ≤3} β^ℓ · paths_ℓ(a,b), β = 0.005; w = min(0.25, 400·K)
members: if u has an account, its own observed edges join the graph at 0.9×
referral: a connector's “someone else” → new node, edge 0.6
costAA: O(Σ_z deg(z)²) capped at deg ≤ 40 · Katz: O(V·d³) capped at deg ≤ 60

No model calls here. Pure graph arithmetic, cached per build.

Liben-Nowell & Kleinberg 2007 · The link-prediction problem for social networksJASISTThe Swarm · work & education overlapsproduct docsLinkedIn PYMK · candidate generationengineering blog
3

Model the target

ask → { name?, company, domains, role, keywords, affiliations A = {(org_k, p_k)}, peer roles, industry words }
one structured Claude call, low effort
cost1 model call · ~4 s

Kleinberg: greedy routing only converges when nodes carry coordinates. Affiliations, industry and role are those coordinates.

Kleinberg 2000 · Navigation in a small worldNatureAdamic & Adar 2005 · How to search a social networkSocial Networks
4

Reach(u)

Reach(u) = max over chains you→…→u, ≤ 3 hops, of ∏ w_eff(edge)
w_eff = posterior(w, attempts, successes) · kind_factor, posterior = (4w + successes)/(4 + attempts)
= Dijkstra on cost −log w, bounded to 3 relaxations
costO(E log V) worst case; bounded hops make it O(E) in practice, ~50 ms for 5k people

Chains multiply. Dodds measured ~37% forwarding per hop, so success ≈ p^L and 3 hops is the ceiling.

Dodds, Muhamad & Watts 2003 · An experimental study of search in global social networksSciencePotamias et al. 2010 · k-nearest neighbors in uncertain graphsPVLDB
5

Knows(u, target)

Knows(u) = 1 − ∏_{k∈A(u)} (1 − p_k)
p: is the person 1 · same company & role 0.95 · same company 0.55·s · investor / board / ex-employer / alumni 0.7·strength·s · industry & role 0.25 · industry 0.08
s = 1.25 if the title is senior (C-level, partner, head, director…)
costO(V·|A|)

This is the inferred last hop. It is replaced by an observed edge the moment a member's mailbox shows the tie.

Crandall et al. 2010 (co-affiliation → tie probability)PNAS
6

Score & rank

score(u) = Reach(u) · Knows(u) · bridging(u) · (φ/0.37)^(hops−1)
bridging(u) = 1 + 0.15·exp(−((c−0.45)/0.2)²) (Rajkumar 2022: moderately weak ties transmit the most)
φ = forwarding rate = Beta(3.7, 6.3) prior updated by your answered asks
P(any chain works) = 1 − ∏_{distinct connectors} (1 − score_i)
diversify: ≤ 3 paths per reason; one Claude call picks the best, writes the reason and the ask
costO(V log V) sort · 1 model call

Two model calls per search in total. Everything else is arithmetic.

Rajkumar et al. 2022 · A causal test of the strength of weak tiesScienceAlthoff et al. 2014 · How to ask for a favorICWSM
7

Ask, learn, grow

intro ask if Knows ≥ 0.6, else routing ask: “do you know X, or who is one step closer?”
answer ∈ {yes, no, someone else} → attempts+1, successes+[yes]; w′ = (4w + successes)/(4 + attempts)
someone else → new node + referral edge 0.6 → next search routes one hop closer
every answer page invites the connector to join: each ask recruits a node
costO(1) per answer

Milgram-style forwarding, with incentives from the DARPA challenge and calibration from Flynn & Lake: people say yes about twice as often as askers expect.

Pickard et al. 2011 · Time-critical social mobilization (DARPA balloons)ScienceJøsang & Ismail 2002 · The Beta reputation systemBled eConferenceFlynn & Lake 2008 · If you need help, just askJPSP
10 For a ten-year-old

Everyone you know has friends you don't. Luckgraph looks at who you really talk to, guesses which of their friends know the person you want, and tells you the one friend to ask.