Power in Liquid Democracy: A Network Centrality Approach

Abstract: This paper develops a computationally tractable framework for measuring voters' power in collective decision-making platforms that support transitive and suspendable delegations---commonly known as liquid democracy. We employ Random Walk Decay centrality to capture how influence propagates through delegation networks, argue for its intuitive appeal in this context and its advantages over alternatives such as PageRank, and derive a natural axiomatic characterization within the class of power metrics. Moreover, we conceptualize how the framework can be extended into a practical tool for analyzing power distributions in such platforms, and propose and study, both axiomatically and algorithmically, methods for selecting representative slates of influential participants.

• The paper shows that expected voting weight, voting-rights flow, and Random Walk Decay centrality are equivalent, providing an interpretable and efficiently computable power metric for suspendible delegations.
• The analysis finds that delegation-chain power is bounded by p/(1−p), limiting extra influence to at most one vote when p≤0.5, while PageRank fails key fairness properties on delegation cycles.
• The paper proposes a proportional peer-selection rule that is polynomial-time computable on functional delegation graphs, satisfies Clique-Entitlement, and is NP-hard to compute on general graphs.

Motivation and context

Liquid democracy (LD) platforms such as LiquidFeedback allow participants to delegate their voting rights transitively and suspendibly: a delegation activates only when the delegator does not cast a direct vote, and it can be revoked at any time. While this design preserves one-person-one-vote flexibility, it has raised persistent concerns about disproportionate accumulation of power through long delegation chains. Existing formal treatments of power in LD, notably those building on adaptations of the Penrose–Banzhaf index [(Shahin, 2021)-style lineage; see (2608.13188)], are computationally intractable and analytically cumbersome. This paper proposes an alternative: a power metric grounded in network centrality that is both semantically transparent and efficiently computable.

The paper's setting is the model of suspendible delegations, closely mirroring LiquidFeedback's implementation and recently studied game-theoretically by Brill et al. The authors restrict attention primarily to the single-proxy case, formalized as functional delegation graphs—directed graphs where each node has out-degree at most one—and treat the graph as a "potential-delegation graph" reused across issues, with delegations acting as fallbacks.

Expected voting weight as a power metric

The core conceptual move is to augment the delegation graph with a parameter $p \in (0,1)$, the probability that any given agent delegates rather than votes directly. This yields a stochastic delegation matrix, interpretable either as a lottery over functional delegation graphs or as a flow of voting rights along delegation paths. Under the first interpretation, a voter's power is their expected voting weight (EVW): the expectation of $w_u = 1 + |Pred_G(u)|$ when the realized graph is sampled from the lottery. Under the second, power equals one plus the sum of first-passage probabilities from every other voter. The paper proves these two notions coincide (Theorem 3.1) and, further, that both equal Random Walk Decay (RWD) centrality with decay factor $p$ on the induced weighted graph (Theorem 3.2). RWD counts only walks that enter a node for the first time, which is exactly right here: assigning voting power to a voter who has already delegated away would be counterintuitive.

The metric yields several sharp structural observations. As $p \to 1$, EVW converges to nominal weight; as $p \to 0$, it converges to 1, recovering direct democracy. More importantly, a delegation path of length $\ell$ contributes at most $\frac{p}{1-p}$ to expected weight—bounded by one extra vote whenever $p \leq 0.5$—whereas nominal weight grows without bound with chain length. The paper draws from this a substantive claim: once suspendibility is modeled explicitly, transitivity itself acts as a natural brake on power concentration, contradicting the standard criticism that transitive delegation inevitably produces runaway influence (as argued, e.g., by Kahng et al.).

Axiomatic comparison with PageRank

Delegation cycles, though often treated as pathological, occur deliberately in practice: like-minded voters form cycles so that their weight counts whenever at least one member votes directly. On acyclic graphs PageRank and RWD coincide, but on cycles they diverge sharply. The paper introduces two axioms:

• Inverse Distance Monotonicity: the center of a star with $n-1$ leaves should have at least as much power as any member of an $n$-cycle, since direct support outweighs indirect support.
• Cycle Size Monotonicity: members of larger cycles should have strictly more power than members of smaller ones.

RWD satisfies both; PageRank violates both. Concretely, PageRank assigns every node of any cycle the same value $w_u = 1 + |Pred_G(u)|$0 regardless of cycle size, and for $w_u = 1 + |Pred_G(u)|$1 cycle members outrank the star center despite receiving only indirect support. Given that PageRank has been repeatedly proposed for weighting votes in delegation systems, this is a pointed negative result for that literature.

Axiomatic characterization

Beyond the comparison, the paper characterizes RWD uniquely among all power metrics on functional graphs via five axioms: Endpoint Reassignment (redirecting a delegation between endpoints redistributes power locally with no spillover), Origin of Endpoint Power (an endpoint delegating to a fresh endpoint transfers a fixed fraction $w_u = 1 + |Pred_G(u)|$2 of its power), Power of the Single User, Lack-of-Self-Impact (a voter's power does not depend on whom they delegate to—a strategyproofness-flavored property), and Locality (power depends only on the connected component). The characterization adapts Was et al.'s earlier result to functional graphs, dispensing with the Random Walk Property axiom and handling the restricted edge operations the functional setting imposes. Notably, PageRank fails Lack-of-Self-Impact, reinforcing its unsuitability.

Proportional peer selection

As an application, the paper addresses the peer selection problem: choosing a slate of $w_u = 1 + |Pred_G(u)|$3 participants that proportionally represents the delegation structure. Existing mechanisms return hierarchical solutions—no node is selected unless all its successors are—which the authors argue is counterintuitive here, since a selected participant "absorbs" the delegations pointing to them. They propose a max-min rule: select $w_u = 1 + |Pred_G(u)|$4 maximizing $w_u = 1 + |Pred_G(u)|$5, i.e., the least-supported selected node after removing outgoing edges of all selected nodes. With $w_u = 1 + |Pred_G(u)|$6 taken as PageRank with decay approaching 1, this becomes the proposed mechanism.

On paths, the rule selects nodes evenly spaced at positions $w_u = 1 + |Pred_G(u)|$7, whereas hierarchical rules pick the last $w_u = 1 + |Pred_G(u)|$8 nodes—an intuitive illustration of the representational difference. Three results anchor the analysis:

Result	Content
Polynomial tractability	Computable in polynomial time on functional graphs, via tree-partitioning algorithms and dynamic programming over components
Clique-Entitlement	Satisfied by the proposed rule; violated by Top-RWD and Top-PageRank
NP-hardness	Computing the rule is NP-hard on general graphs

Clique-Entitlement, imported from multiwinner proportionality, requires that a clique component of size $w_u = 1 + |Pred_G(u)|$9 receive at least $p$0 seats. The proof establishes exact PageRank values for clique members under edge removal ($p$1 for selected clique nodes) and bounds for outsiders, showing the minimum can always be improved until proportional representation of the clique is achieved. The NP-hardness reduction from Independent Set uses gadgets attaching $p$2 new nodes per edge, forcing selections to correspond to independent sets.

Limitations and open questions

The framework's scope is deliberately narrow in places. The axiomatic characterization holds for functional graphs; while RWD extends to general, node- and edge-weighted graphs—relevant for DAO governance with stake-based weights and ranked delegations—the authors concede it is unclear whether all five axioms retain intuitive interpretations beyond this class. Heterogeneous voting probabilities across participants would require a generalized RWD with node-dependent decay, which is not developed. Algorithmically, the polynomial-time result for slate selection does not survive the move to general graphs, where the problem is NP-hard; whether a mechanism preserving the rule's fairness properties while running in polynomial time exists remains open. Finally, the equivalence results rely on the assumption of a uniform delegation probability $p$3, whose empirical calibration on real platform data is not addressed.

Conclusion

The paper replaces intractable game-theoretic power indices for liquid democracy with a single, well-behaved quantity—expected voting weight—shown to coincide with Random Walk Decay centrality and uniquely characterized by domain-appropriate axioms. It delivers a concrete negative verdict on PageRank for this application, demonstrates that suspendibility tames power accumulation along delegation chains, and contributes a proportionally fair, axiomatically justified slate-selection method that is efficient on the graphs arising in practice. The main gaps—general graphs, heterogeneous probabilities, and calibrated parameters—define the immediate agenda for extending this framework.