Games & Graphs

Snapshot-Relative Diffusion (PageRank / PPR)

Influence defined as a PageRank/PPR fixed point evaluated relative to a committed epoch snapshot, with protocol-defined market-relative teleport.

Local Protocol defines influence using a diffusion score that is a fixed point over a committed epoch snapshot. Practically, this is PageRank / Personalized PageRank (PPR) semantics, evaluated relative to the snapshot, not continuously recomputed as a global “ledger fact”.

1) Transition operator

Let the global transaction graph at epoch be a weighted, directed graph:

Define a row-stochastic transition matrix derived from outgoing weights:

For dangling nodes (no outgoing edges), the protocol redirects mass according to the teleport distribution (standard PageRank handling).

Intuition

You can read

as a “next hop” rule: if you are at

, then

is the chance you move to

next. Rows sum to 1 because this is a Markov chain.

Related work

Markov chains and stationary distributions are the standard lens for PageRank-style scores; see e.g. Levin–Peres–Wilmer.

2) Teleport distribution (protocol-defined)

Local Protocol uses Personalized PageRank (PPR) to anchor diffusion to a protocol-defined set of trusted starting points (users do not get to choose personalization; that would be instantly gameable).

Why market-relative teleport?

In real marketplaces, trust is often local to a market context (a naturally fragmented city / vertical can be real yet weakly connected to global anchors). A single global seed set can accidentally treat a legitimate, fragmented market as “low influence” simply because diffusion cannot reach it.

Market-relative teleport addresses this: diffusion (and claims derived from it) are evaluated in a market context marketId = m.

Formally:

teleport distribution per market:
market-relative diffusion score:

The protocol commits to each epoch (users do not choose it).

Intuition

Teleport is the protocol’s “source of ground truth”: where trust starts. Market-relative teleport keeps that rule intact while preventing “fragmented-but-real” markets from being unfairly treated as low influence.

Related work

Personalization vectors in Personalized PageRank, topic/context-sensitive ranking variants, and seed-set anchoring like TrustRank.

3) Fixed point definition

For a given market context , the protocol defines a market-scoped transition operator (outgoing edges filtered to marketId = m, plus any explicitly-global edges the protocol defines), and a market-relative teleport distribution .

Market-scoped walks (important)

Claims are verified against market-scoped walks: the walk uses the market-scoped operator

and teleports according to the market’s committed seed table

. This prevents reinterpreting the same transcript under a different market context.

Where marketId comes from

marketId is derived from execution: it must match the registered MarketContext that emitted the interaction record, and the market must be ACTIVE. See: Market Registry.

The market-relative diffusion score is defined as:

Where is the restart probability (teleport rate).

This fixed point exists and is unique for .

4) Random-walk interpretation

Sample a random walk:

start from a teleport sample
at each step: with probability restart from , otherwise follow a market-scoped outgoing edge proportional to weights

Then is the stationary probability of being at node (in market context ).

5) Why snapshot-relative?

Diffusion is defined relative to a committed snapshot:

the ledger commits to via
claims derived from diffusion must specify which snapshot they reference
economic outputs are computed with respect to that snapshot

Diffusion is a global fixed point and is not composable by one-way merging of independently computed partition-local vectors.

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