Shareability Graph Overview

• Shareability Graph is a graph-theoretic model that encodes feasible compatibility among entities (e.g., trips, requests) under operational constraints.
• It transforms complex spatio-temporal routing problems into structured matching, hypergraph packing, or assignment tasks for efficient optimization.
• Extensions apply the shareability concept to diverse domains such as textual information flow, quantum steering, and vision, revealing universal network patterns.

Searching arXiv for the provided topic and cited papers to ground the article in published work. A shareability graph is a graph-theoretic representation of feasible sharing relations among entities. Its most explicit and influential use is in transportation, where nodes denote trips, traveller requests, or vehicles and edges encode whether multiple demands can be served jointly under operational constraints; in that setting, the graph converts a large spatio-temporal routing problem into a matching, packing, or assignment problem [(Santi et al., 2013); (Bujak et al., 2022); (Luo et al., 2021)]. Related work extends the same organizing idea to directed information flow in text, to quantum-correlation sharing patterns, to reusable part structures in compositional vision models, to wireless resource sharing, and to static micro-transit zoning, but the underlying role remains the same: a shareability graph records admissible combinations before optimization or inference is performed (Bagrow et al., 2017, Paul et al., 2020, Librino et al., 28 Feb 2025, Hu et al., 14 Sep 2025).

1. Core graph objects and semantic variants

In ride-pooling, the basic shareability graph is built from traveller requests or trips. In the pairwise case, each node is a trip or traveller request, and an edge means the pair can be served together while respecting constraints such as delay, detour, waiting time, or utility [(Santi et al., 2013); (Bujak et al., 2022)]. This is the form introduced in the taxi-sharing literature, where a trip is represented as $T_i=(o_i,d_i,t_i^o,t_i^d)$, and pairwise shareability is defined by the existence of a feasible route in which each origin precedes its destination and no passenger’s delay exceeds the tolerated bound $\Delta$ (Santi et al., 2013).

The concept then branches along two axes. First, several papers distinguish shareability from matching. In this usage, a shareability graph records all attractive or feasible combinations, while a matching graph records only those combinations that are actually selected in an optimal assignment (Bujak et al., 2022). Second, several papers distinguish simple from bipartite representations. A simple network contains only travellers and links them directly; a bipartite network contains both travellers and rides, with edges connecting a traveller to a feasible ride (Bujak et al., 2022).

For higher-capacity pooling, the ordinary graph ceases to be adequate. The mixed-fleet stochastic assignment formulation therefore uses a hypergraph $G=\{S,D,E\}$, where $S$ is the vehicle set, $D$ is the request set, and a hyperedge $e=\{i,J\}_{i\in S,\;J\subseteq D}$ represents one feasible pooled trip consisting of exactly one vehicle and a subset of requests (Luo et al., 2021). In that formulation the shareability object is a weighted hypergraph, because each feasible bundle also carries a value

$$v_e=\sum_{j\in J}u_j+\sum_{j\in J}\tilde{u}_{ij}-c(i,t)\ge 0,$$

so compatibility and operational value are encoded simultaneously (Luo et al., 2021).

A persistent source of confusion is the interpretation of cliques. In the simple shareability network studied with ExMAS, a triangle does not necessarily mean a 3-person ride; it may only mean that all three pairs are mutually shareable. The bipartite representation avoids that information loss by explicitly introducing ride nodes (Bujak et al., 2022). This distinction is central whenever pairwise compatibility is used as a surrogate for higher-order feasibility.

2. Origin in urban ride pooling and matching-based optimization

The transportation literature introduced the shareability network as a scalable reformulation of taxi sharing. In Manhattan, the framework was applied to a dataset of more than 150 million trips in 2011, with the shareability parameter $k$ controlling how many trips may be pooled and the quality-of-service parameter $\Delta$ giving the maximum allowed delay relative to solo travel (Santi et al., 2013). For $k=2$, the graph is standard: nodes are trips, and edges denote feasible shared pairs. For $\Delta$0, the structure becomes a hypergraph, and the paper notes that the problem becomes computationally hard, which is why pairwise sharing is emphasized as the scalable case (Santi et al., 2013).

Once the graph is built, two classical optimization objectives are available. Maximum matching maximizes the number of pooled trips, and weighted maximum matching maximizes aggregate saved travel time (Santi et al., 2013). In the pairwise case, the paper states algorithmic complexities of $\Delta$1 for maximum matching and $\Delta$2 for weighted maximum matching, where $\Delta$3 is the number of nodes (Santi et al., 2013). The operational interpretation is direct: matched edges are pooling assignments, and the weighted formulation is used as a proxy for lower operating cost, fuel use, and emissions (Santi et al., 2013).

The empirical scale is correspondingly large. With $\Delta$4 and $\Delta$5 minutes, the Manhattan shareability network contains more than 150 million nodes and over 100 billion links (Santi et al., 2013). Under the Oracle model, at $\Delta$6 minute about 94.5\% of trips are shareable, whereas in the Online model the shareable fraction is less than 30\%; at $\Delta$7 minutes, the Online model achieves about 32\% total travel time savings, compared with about 40\% in the Oracle model (Santi et al., 2013). The same paper reports that shareability as a function of trip density saturates quickly and is well described by

$\Delta$8

with shareability already near saturation around 100,000 trips/day, roughly 25% of the New York daily average (Santi et al., 2013).

This transportation origin established several enduring principles. First, shareability is defined before assignment. Second, feasibility is encoded locally in graph edges or hyperedges. Third, global pooling is recovered by solving a combinatorial optimization problem on that graph. Later work retains these principles even when it changes the graph type, the uncertainty model, or the operational domain.

3. Quantifying shareability: probabilistic structure, scaling laws, and network-level metrics

Subsequent work turned the shareability graph itself into an object of network analysis. One example uses 147 taxi requests from Manhattan, requested within half an hour in January 2016, and studies four network representations—simple shareability, simple matching, bipartite shareability, and bipartite matching—under a probabilistic pooling model built with ExMAS (Bujak et al., 2022). A shared ride is attractive when each traveller has positive utility gain,

$\Delta$9

and stochastic acceptance is modeled by

$G=\{S,D,E\}$0

The study runs 1000 replications, then constructs weighted graphs whose edge weights equal the number of replications in which a link appears (Bujak et al., 2022). The resulting analysis shows that shareability networks are much denser than matching networks, bipartite shareability networks are large and highly connected, and the transition from shareability to matching removes most potential links (Bujak et al., 2022).

At the city scale, the shareability graph was linked to a universal scaling law. The urban ride-sharing study defines shareability as the fraction of individual trips that can be paired with at least one other trip so that riders accept a detour or delay no larger than a tolerance $G=\{S,D,E\}$1, constructs a shareability network on trips, and computes city-wide shareability from the network’s maximum matching (Tachet et al., 2016). The resulting shareability curve plots shareability $G=\{S,D,E\}$2 against trip density, and especially against a dimensionless rescaled density $G=\{S,D,E\}$3 built from trip density $G=\{S,D,E\}$4, city area $G=\{S,D,E\}$5, average speed $G=\{S,D,E\}$6, and the maximum tolerated delay $G=\{S,D,E\}$7 (Tachet et al., 2016). Empirically, New York City, San Francisco, Singapore, and Vienna exhibit the same fast-saturating curve, and after rescaling by $G=\{S,D,E\}$8 the curves nearly coincide (Tachet et al., 2016).

Under simplifying assumptions—uniform trip generation in time, uniformly distributed origins, and destinations drawn uniformly inside a disk around the origin—the paper derives an explicit universal curve whose supplementary-equation core structure is

$G=\{S,D,E\}$9

The important stated point is that shareability becomes an explicit function of $S$0 alone, with no adjustable parameters (Tachet et al., 2016). Reported $S$1 values are between about 0.91 and 0.98 across the four cities, and with $S$2 minutes the model predicts nearly 100\% shareability for New York, Singapore, and San Francisco and about 83\% for Vienna; the observed taxi-trip densities are reported as roughly 344.12 trips/h/km$S$3 for New York, 24.46 for Singapore, 12.63 for San Francisco, and 0.95 for Vienna (Tachet et al., 2016). The same paper argues that in dense regimes the matching problem can be approximated by the far simpler question of whether a trip has at least one compatible partner, because shareability is strongly correlated with the fraction of non-isolated nodes (Tachet et al., 2016).

A distinct but related metric program defines shareability at the transportation-network level through flow overlap rather than trip pairing. In that framework, flow overlap is “the extent of spatial, and temporal in some cases, commonality between the paths of trips in a (transportation) network,” operationalized as the link-length-weighted average number of other trips sharing the links of a person-trip’s path (JS et al., 2021). The Maximum Network Flow Overlap Problem (MNFLOP) assigns each OD pair to one path so as to maximize network-wide overlap, optionally penalizing detours (JS et al., 2021). From the resulting assignment the paper computes OD-level, node-level, link-level, and network-level metrics, including Trip Overlap, Overlap Distance, Node Overlap Percentage, Average Network Overlap, and Marginal Network Overlap (JS et al., 2021). In the Sioux Falls experiments, the framework differentiates OD demand scenarios not only by demand magnitude but also by dispersion and directionality (JS et al., 2021). This suggests that “shareability graph” can denote either a direct compatibility graph or, more broadly, a network representation of shared path structure.

4. Structure-aware algorithms and optimization on shareability graphs

Once shareability is encoded as a graph or hypergraph, several algorithmic programs exploit that structure directly. ExMAS first identifies pairwise shareable trips, then extends them to triples, quadruples, and higher-order rides until no more feasible cliques can be found; the resulting attractive ride set is then fed into a bipartite assignment problem in which travellers are uniquely assigned to rides while total distance traveled is minimized (Bujak et al., 2022). In this sense, ExMAS separates shareability generation from matching, and the probabilistic extension shows how repeated stochastic realizations induce weighted shareability graphs (Bujak et al., 2022).

The stochastic mixed-fleet problem makes that separation explicit. There, the platform first chooses which augmented vehicles to activate, then, after a demand scenario $S$4 is realized, solves a second-stage hypergraph assignment problem

$S$5

subject to the usual disjointness constraints that each request is covered at most once and each vehicle is used at most once (Luo et al., 2021). The paper rewrites the second stage as a $S$6-set packing problem, proves that the objective is not submodular, and gives two approximation algorithms: LSLPR for mid-capacity fleets, yielding the exact approximation ratio

$S$7

and MMO for high-capacity fleets, yielding

$S$8

where $S$9 (Luo et al., 2021). Here the shareability graph is not merely descriptive; it is the central combinatorial object of the optimization problem.

A more explicitly structure-aware formulation is StructRide, a batch-based framework for the Batched Dynamic Ridesharing Problem. It defines the shareability graph as $D$0, with one node per request and an edge $D$1 when there exists at least one feasible vehicle schedule serving them together (Zhan et al., 2024). The framework builds the graph batch by batch with a Dynamic Shareability Graph Builder, uses spatial indexes and an angle pruning strategy, and exploits two observations: the degree of a node reflects its shareability, and any feasible group of $D$2 requests corresponds to a $D$3-clique in the shareability graph (Zhan et al., 2024). Its shareability loss

$D$4

measures how much selecting a group damages the remaining graph, and the two-phase proposal-acceptance algorithm SARD chooses groups of minimum shareability loss (Zhan et al., 2024). Reported results state that SARD can run up to 72.68 times faster and serve up to 50\% more requests than state-of-the-art algorithms, with additional gains from angle pruning (Zhan et al., 2024).

The same compatibility-before-scheduling idea appears in dense smart-factory URLLC. That work first constructs a bipartite graph between channels and devices for orthogonal allocation, then augments it with a device-level shareability graph $D$5 in which an edge means two devices can profitably share resource units under SIC (Librino et al., 28 Feb 2025). Pairings are selected by maximum matching and replaced by equivalent virtual devices, after which the ordinary graph-based allocator is reused unchanged (Librino et al., 28 Feb 2025). The paper reports an increase in spectral efficiency of as much as 50\% with respect to the best benchmark, together with a Jain fairness index around 0.9987 and served-user fractions of 0.4525 for FSA, 0.7570 for BCA, 0.8274 for GBA, and 0.9474 for GBA+SIC (Librino et al., 28 Feb 2025).

A static planning adaptation appears in optimal micro-transit zoning. There the shareability graph is redefined from trip-level feasibility to node-level spatial shareability: two spatial units are shareable when their shortest-path distance is within a maximum diameter $D$6, and a valid zone is a $D$7-bounded clique

$D$8

Candidate zones are generated by CliqueGen with convex-hull extension, then selected by an ILP for a Weighted Maximum Coverage Problem (Hu et al., 14 Sep 2025). On Chattanooga data the framework serves 27.03\% more demand than the baseline, and on synthetic data up to 49.5\% more (Hu et al., 14 Sep 2025). The paper explicitly describes this as an adaptation of the shareability graph from dynamic trip assignment to static zoning.

5. Extensions beyond transportation

The phrase “shareability graph” is also used outside mobility, although the semantics of a node and an edge change with the domain. In the quoter model, the graph $D$9 is a directed social network whose nodes are individuals generating symbolic time series and whose weighted directed edges carry quote probabilities $e=\{i,J\}_{i\in S,\;J\subseteq D}$0; an edge $e=\{i,J\}_{i\in S,\;J\subseteq D}$1 means that text can flow from node $e=\{i,J\}_{i\in S,\;J\subseteq D}$2 to node $e=\{i,J\}_{i\in S,\;J\subseteq D}$3 through quoting (Bagrow et al., 2017). Information propagation is then quantified through entropy-rate and cross-entropy estimators such as

$e=\{i,J\}_{i\in S,\;J\subseteq D}$4

The model is used to distinguish true interactions from spurious interactions caused by common sources, shared vocabularies, or topology (Bagrow et al., 2017). Here the shareability graph is a directed weighted network of textual reusability.

In quantum information, several papers motivate a natural shareability-graph picture rather than an operational assignment graph. For three-qubit steering under the three-settings CJWR inequality, vertices are qubits and an edge indicates that the corresponding reduced bipartite state is steerable under the chosen test (Paul et al., 2020). With

$e=\{i,J\}_{i\in S,\;J\subseteq D}$5

as the $e=\{i,J\}_{i\in S,\;J\subseteq D}$6-steering criterion, the paper proves that for any three-qubit state at most two of the three reduced states can violate the inequality, because $e=\{i,J\}_{i\in S,\;J\subseteq D}$7 (Paul et al., 2020). In graph language, the shareability graph can be empty, have one edge, or have two edges, but never a triangle (Paul et al., 2020). A later optical experiment on W-like states demonstrates reduced-subsystem steering shareability directly, including states in which one party is simultaneously steered by two others and one case in which all reduced bipartite directional steering parameters are below the threshold $e=\{i,J\}_{i\in S,\;J\subseteq D}$8 in the measured pattern (Hao et al., 2021). In a many-body spin system with two- and three-body interactions, the graph is only implicit, but the paper interprets pairwise quantities $e=\{i,J\}_{i\in S,\;J\subseteq D}$9 as edge-like correlations and the monogamy score

$$v_e=\sum_{j\in J}u_j+\sum_{j\in J}\tilde{u}_{ij}-c(i,t)\ge 0,$$0

as a node-wise correlation budget (Kiran et al., 2023). It reports that discord and work deficit exhibit a higher percentage of non-monogamous states than concurrence or logarithmic negativity over the sampled parameter ranges (Kiran et al., 2023).

A different non-transport use appears in CHOP, where part shareability is realized through repeated subgraph structure across images. Parts are modeled as random graphs, object structure is represented by object graphs, pairwise spatial relations are learned by Minimum Conditional Entropy Clustering, and descriptive substructures are selected by an MDL criterion

$$v_e=\sum_{j\in J}u_j+\sum_{j\in J}\tilde{u}_{ij}-c(i,t)\ge 0,$$1

The paper states that vocabulary size grows sub-linearly with the number of categories, objects, or views, and attributes this to high part shareability in early layers; test inference time is reported as approximately 0.5–3 seconds per image (Aktas et al., 2015). In this setting, the “shareability graph” is effectively the learned graph vocabulary together with the indexing structure that records repeated subgraphs.

6. Interpretive issues, limitations, and enduring significance

Several limitations recur across the literature. A shareability graph is generally a graph of potential compatibility, not the realized operational plan. The ride-pooling literature is explicit that shareability and matching are different objects, and that matching networks are much sparser because unique assignment removes many feasible links (Bujak et al., 2022). Likewise, pairwise graph density does not by itself resolve higher-order grouping, vehicle routing, or scenario uncertainty; those tasks require bipartite formulations, hypergraph packing, or stochastic optimization (Luo et al., 2021).

A second limitation concerns aggregation. The universal urban scaling law is derived under simplifying assumptions—uniform trip generation in time, origins uniformly distributed in the city, and destinations drawn uniformly inside a disk around the origin (Tachet et al., 2016). The close empirical collapse of city curves suggests substantial robustness, but the paper’s own explanation is that city-specific details “largely drop out” only after reduction to the dimensionless group $$v_e=\sum_{j\in J}u_j+\sum_{j\in J}\tilde{u}_{ij}-c(i,t)\ge 0,$$2 (Tachet et al., 2016). A plausible implication is that the scaling law is best understood as a portable estimator of ride-sharing potential rather than as a complete model of operational dispatch.

A third limitation is semantic drift across disciplines. In transportation, the graph usually encodes directly testable joint service feasibility. In quantum steering, by contrast, the graph is a compact picture of how steering correlations are distributed across reduced states; in the quoter model it is a weighted directed network of possible copying; in CHOP it is a representation of part reuse; in micro-transit zoning it is a purely spatial feasibility graph (Bagrow et al., 2017, Paul et al., 2020, Hu et al., 14 Sep 2025). The common abstraction is real, but the edge predicate is domain-specific.

Despite that variation, a stable methodological pattern emerges. The graph is used to expose compatibility structure that would otherwise remain buried inside routing, temporal, statistical, or physical constraints. Once that structure is made explicit, standard combinatorial primitives—maximum matching, weighted matching, $$v_e=\sum_{j\in J}u_j+\sum_{j\in J}\tilde{u}_{ij}-c(i,t)\ge 0,$$3-set packing, clique generation, maximum coverage, or network-topology measures—become applicable [(Santi et al., 2013); (Luo et al., 2021); (Zhan et al., 2024)]. This suggests that the lasting importance of the shareability graph lies less in any single formal definition than in its role as a reusable abstraction for converting high-dimensional feasibility relations into analyzable network objects.