Edge Coalition Partition: Models & Applications

• Edge coalition partition is a framework that organizes edge agents or graph edges into coalition structures governed by resource, pricing, or domination constraints.
• In MEC-assisted blockchain networks, overlapping coalition formation enables mobile users to jointly offload mining tasks, optimize nonce allocation, and enhance system utility under pricing and delay constraints.
• In graph theory, an edge coalition partition splits the edge set into dominating and partnering subsets, linking edge domination to vertex domination concepts via the line graph.

Edge coalition partition is a context-dependent term in contemporary research. In MEC-assisted blockchain networks, it denotes the organization of mobile users into overlapping coalitions that jointly offload proof-of-work mining to an edge computing service provider, jointly aggregate transaction sets, and jointly purchase nonce-hash computing resources under pricing and delay constraints (Ye et al., 8 Aug 2025). In graph theory, it denotes a partition of the edge set $\pi=\{E_1,\dots,E_k\}$ such that each part is either a singleton edge-dominating set or a non-edge-dominating set that forms an edge coalition with another part, where neither part is edge-dominating but their union is (Besharati et al., 26 Jul 2025). Adjacent literatures use related language for edge-valued coalition structure generation, weighted graph games, edge partitioning for distributed graph processing, and edge-device coalition formation, but these are not identical notions (Voice et al., 2011, Bachrach et al., 2011, Guerrieri et al., 2014, Cheng et al., 22 Feb 2025).

1. Terminological scope

The phrase has at least three stable usages in the cited literature. This suggests that it should be interpreted contextually rather than as a single standardized object.

Usage	Core object	Representative source
MEC-assisted blockchain	Overlapping coalition cover of mobile users, with membership matrix $\beta_{n,m}$ and nonce allocations $l_{n,m}$	(Ye et al., 8 Aug 2025)
Graph theory	Edge partition $\pi=\{E_1,\dots,E_k\}$ governed by edge domination and coalition partnership	(Besharati et al., 26 Jul 2025)
Collaborative edge inference	Disjoint partition $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$ of mobile devices across edge servers, coupled with DNN layer partitioning	(Cheng et al., 22 Feb 2025)

In the MEC-assisted blockchain formulation, “partition” is not a disjoint partition in the classical set-theoretic sense: each miner may join multiple coalitions, so the resulting structure is a cover of the mobile-user set constrained by a collaboration factor $J$ (Ye et al., 8 Aug 2025). In the graph-theoretic formulation, by contrast, the object is explicitly an edge partition of $E(G)$ into disjoint parts (Besharati et al., 26 Jul 2025). In edge inference, the partition is again disjoint, but the objects being partitioned are users rather than graph edges, and the coalition structure is coupled to model deployment and layer-wise model splitting (Cheng et al., 22 Feb 2025).

2. MEC-assisted blockchain: overlapping edge coalition partition

In the MEC-assisted collaborative blockchain model, the network consists of mobile users $\mathcal{N}=\{1,\dots,N\}$ acting as miners, and an edge computing service provider with computing capacity $f_E$ and unit nonce-hash cost $c$. Each mobile user $\beta_{n,m}$0 collects a transaction set $\beta_{n,m}$1. A coalition $\beta_{n,m}$2 aggregates the transactions of its members into $\beta_{n,m}$3, sorts them by fee, and packs the top $\beta_{n,m}$4 transactions into a block with total fee $\beta_{n,m}$5. If $\beta_{n,m}$6 is user $\beta_{n,m}$7's contributed nonce length to coalition $\beta_{n,m}$8, then $\beta_{n,m}$9, the system-wide nonce length is $l_{n,m}$0, and the coalition reward is

$l_{n,m}$1

User $l_{n,m}$2's reward share in coalition $l_{n,m}$3 is

$l_{n,m}$4

The offloaded mining delay for coalition $l_{n,m}$5 is

$l_{n,m}$6

subject to the feasibility condition $l_{n,m}$7, which induces the upper bound

$l_{n,m}$8

These definitions make coalition strength depend jointly on transaction aggregation, relative nonce length, offloading latency, and orphan probability (Ye et al., 8 Aug 2025).

The overlapping structure is encoded by binary variables $l_{n,m}$9, where $\pi=\{E_1,\dots,E_k\}$0 iff user $\pi=\{E_1,\dots,E_k\}$1, together with the collaboration-factor constraint

$\pi=\{E_1,\dots,E_k\}$2

When $\pi=\{E_1,\dots,E_k\}$3, the model reduces to a non-overlapping single-coalition regime. When $\pi=\{E_1,\dots,E_k\}$4, each miner may join multiple coalitions simultaneously. In this sense, the edge coalition partition is a coalition structure $\pi=\{E_1,\dots,E_k\}$5 plus the membership matrix $\pi=\{E_1,\dots,E_k\}$6, such that every mobile user belongs to at least one coalition, some coalitions may share members, and the delay and resource constraints remain feasible. The pair $\pi=\{E_1,\dots,E_k\}$7 determines not only which coalitions exist, but also their hashing power and their demand for edge resources (Ye et al., 8 Aug 2025).

3. Stackelberg, OCF, and ERC mechanisms

The blockchain formulation is solved as a two-stage Stackelberg game. In Stage I, the edge computing service provider chooses the price $\pi=\{E_1,\dots,E_k\}$8 for computing resources. In Stage II, mobile users form coalitions and coalitions choose edge nonce length. User $\pi=\{E_1,\dots,E_k\}$9's utility from coalition $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$0 is

$\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$1

and total user utility is $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$2. Because the joint optimization over membership variables $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$3 and nonce allocations $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$4 is coupled and mixed discrete–integer, Stage II is decomposed into two nested games: an overlapping coalition formation game for membership choice and an edge resource competition game for coalition-level nonce purchases (Ye et al., 8 Aug 2025).

In the ERC game, the players are the coalitions $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$5, the strategy of coalition $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$6 is $\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$7, and the utility is

$\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$8

Writing

$\mathcal{F}=\{\mathcal{F}_1,\dots,\mathcal{F}_M\}$9

the relaxed best response is

$J$0

The strategy sets are convex and compact and each $J$1 is continuous and concave in $J$2, so a Nash equilibrium exists. The relaxed equilibrium admits the closed form

$J$3

with integer equilibrium obtained by rounding to the better of $J$4 and $J$5, and with rounding loss bounded by

$J$6

Within each coalition, the paper uses equal splitting,

$J$7

which affects payoff sharing but not coalition best responses (Ye et al., 8 Aug 2025).

Coalition formation is governed by individual stability. A stable structure $J$8 is one in which no mobile user can change memberships and strictly improve its utility, subject to acceptance conditions when joining existing coalitions. The local moves are five atomic strategies: Merge-preferred A, Merge-preferred B, Split-preferred A, Split-preferred B, and Leave-preferred. These respectively cover joining without leaving, swapping one coalition for another, creating an extra singleton coalition, leaving one coalition to form a singleton, and unilateral leaving. The OCF-based alternating algorithm nests this inner coalition-adjustment process inside an outer price search over $J$9, repeatedly computing the ERC equilibrium for each candidate structure. Because the number of coalition structures is finite and accepted moves never revisit an old structure, the coalition-formation procedure terminates in finite steps at an individually stable structure (Ye et al., 8 Aug 2025).

4. Performance profile and modeling assumptions in the MEC setting

The blockchain paper evaluates coalition partition through the average number of members per coalition,

$E(G)$0

together with system utility, total nonce length, and provider utility. In multi-coalition mode, for example $E(G)$1, users can belong to several coalitions, which increases $E(G)$2 relative to the single-coalition case $E(G)$3. As the number of mobile users $E(G)$4 increases, $E(G)$5 also increases, and overlapping structures become richer because users have stronger incentives to collaborate under reward and resource competition. Price acts as a direct structural control: at low $E(G)$6, individual mining remains profitable and coalitions are smaller; at moderate $E(G)$7, cooperation becomes more attractive and overlap intensifies; at high $E(G)$8, mining becomes too expensive, so nonce investments shrink and coalitions contract or disappear (Ye et al., 8 Aug 2025).

Relative to a non-cooperative baseline, single-coalition collaboration improves system utility by about $E(G)$9, while multi-coalition collaboration, such as $\mathcal{N}=\{1,\dots,N\}$0, improves it by about $\mathcal{N}=\{1,\dots,N\}$1. The multi-coalition regime also raises the edge provider’s utility, with increases up to $\mathcal{N}=\{1,\dots,N\}$2 in some settings, and enlarges both total nonce length and average coalition membership. The reported gains are most pronounced when the block reward $\mathcal{N}=\{1,\dots,N\}$3 or transaction volume $\mathcal{N}=\{1,\dots,N\}$4 is high, edge capacity $\mathcal{N}=\{1,\dots,N\}$5 is sufficient, and the price $\mathcal{N}=\{1,\dots,N\}$6 is moderate rather than negligible or prohibitive (Ye et al., 8 Aug 2025).

The same study makes several restrictive assumptions explicit. It is a static snapshot model with quasi-static users and parameters; dynamic joins, leaves, and mobility are not represented. It assumes perfect information in the sense that users can evaluate payoffs for alternative coalition structures through message passing. Intra-coalition sharing follows a uniform equal-splitting rule rather than bargaining or fairness-aware allocation. Security and energy costs are not modeled; the focus is mining performance and resource pricing. These assumptions delimit the meaning of edge coalition partition in this setting: it is a joint mechanism for coalition overlap, edge-resource competition, and pricing, rather than a full model of mobile blockchain behavior (Ye et al., 8 Aug 2025).

5. Edge coalition partition in graph theory

In graph theory, an edge coalition is defined on a simple graph $\mathcal{N}=\{1,\dots,N\}$7 in terms of edge domination. A set $\mathcal{N}=\{1,\dots,N\}$8 is edge-dominating if every edge not in $\mathcal{N}=\{1,\dots,N\}$9 is adjacent to some edge in $f_E$0. Two disjoint edge sets $f_E$1 form an edge coalition if neither $f_E$2 nor $f_E$3 is edge-dominating, but $f_E$4 is edge-dominating. An edge coalition partition, or $f_E$5-partition, is then a partition $f_E$6 of the edge set such that every part is either a singleton edge-dominating set or a non-edge-dominating part that forms an edge coalition with another non-edge-dominating part. The edge coalition number $f_E$7 is the maximum possible cardinality of such a partition. The line-graph identity $f_E$8 links edge domination in $f_E$9 to vertex domination in the line graph $c$0, and underlies the transfer of several coalition-theoretic ideas from vertices to edges (Besharati et al., 26 Jul 2025).

Every graph admits an $c$1-partition, so $c$2 is always well-defined, and the general bound

$c$3

holds for a connected graph of order $c$4 and size $c$5. Several classes admit exact formulas. For stars,

$c$6

For double stars $c$7,

$c$8

For paths $c$9, the values are

$\beta_{n,m}$00

and for cycles $\beta_{n,m}$01,

$\beta_{n,m}$02

At the extremal dense end,

$\beta_{n,m}$03

if and only if $\beta_{n,m}$04. The theory also introduces the edge coalition graph $\beta_{n,m}$05, whose vertices are the parts of $\beta_{n,m}$06 and where adjacency records coalition partnership; this captures the interaction pattern induced by a fixed $\beta_{n,m}$07-partition (Besharati et al., 26 Jul 2025).

This edge-based theory extends a vertex-based coalition-partition literature. In that predecessor line, a coalition consists of two disjoint vertex sets whose union is dominating although neither set is dominating individually, and a coalition partition is a partition of $\beta_{n,m}$08 in which every non-dominating part participates in such a coalition. Recent work on cubic graphs shows that the general upper bound $\beta_{n,m}$09 is sharp in the 3-regular case by constructing infinite families of cubic graphs with coalition number $\beta_{n,m}$10 (Dobrynin et al., 2024). The edge-coalition notion should therefore be read as an edge-domination analogue of an established vertex-domination framework, not as a variant of edge cut or clique partition.

6. Neighboring frameworks and non-equivalent uses

Several adjacent literatures study partitions or coalition structures that are edge-based but not edge coalition partitions in the domination-theoretic sense. Graph coalition structure generation over graphs asks for a partition of vertices into connected coalitions maximizing $\beta_{n,m}$11, usually under valuations independent of disconnected members; the edge-sum subclass uses

$\beta_{n,m}$12

This problem is NP-complete in general, remains NP-complete for planar graphs, admits polynomial-time algorithms on trees and on $\beta_{n,m}$13- and $\beta_{n,m}$14-minor-free graphs, and is linear-time on bounded-treewidth graphs (Voice et al., 2011, Voice et al., 2014). Weighted Graph Games adopt the same internal-edge valuation

$\beta_{n,m}$15

and study optimal coalition structures under positive and negative edge weights; the resulting optimization is NP-complete and inapproximable in general, NP-complete on planar graphs, but has constant-factor approximations on planar, minor-free, and bounded-degree graphs (Bachrach et al., 2011).

Distributed graph processing uses yet another notion of edge partition. In the etsch framework, an edge partition is a disjoint decomposition $\beta_{n,m}$16, each subgraph $\beta_{n,m}$17 is processed locally, and communication is mediated by frontier vertices with cost

$\beta_{n,m}$18

The distributed funding-based partitioner d-fep constructs connected edge partitions by propagating funding from vertices to edges and buying edges round by round (Guerrieri et al., 2014). In balanced graph edge partitioning, the objective is instead to minimize the replication factor

$\beta_{n,m}$19

under an $\beta_{n,m}$20-balance constraint on edge counts, and recent local-search algorithms use adjustable edges and blocks to improve existing partitions (Guo et al., 2020). At much larger scale, StreamCPI combines streaming partitioners with run-length compression of block assignments; the framework reports that a graph with $\beta_{n,m}$21 billion nodes and $\beta_{n,m}$22 trillion edges can be partitioned on a Raspberry Pi (Chhabra et al., 2024).

Machine-learning and edge-systems papers introduce still different meanings. A variational edge partition model for supervised graph representation learning decomposes each observed edge into community-specific weighted edges $\beta_{n,m}$23 satisfying

$\beta_{n,m}$24

and uses those weighted partitions to define community-specific GNN channels (He et al., 2022). Privacy-aware collaborative edge inference defines an edge coalition partition as a disjoint user–server coalition structure

$\beta_{n,m}$25

and jointly optimizes user–server association, model deployment, and DNN layer partitioning under a Lyapunov-controlled privacy budget (Cheng et al., 22 Feb 2025). Finally, edge clique partition studies whether $\beta_{n,m}$26 can be partitioned into cliques. In the above-$\beta_{n,m}$27 regime, ECP/$\beta_{n,m}$28 is fixed-parameter tractable, whereas ECC/$\beta_{n,m}$29 is NP-complete for every fixed $\beta_{n,m}$30 but polynomial-time solvable for $\beta_{n,m}$31 (Fomin et al., 26 Jun 2025).

Taken together, these strands indicate that edge coalition partition is best understood as a family of structurally related but non-equivalent formulations. In networked systems it typically denotes the organization of edge-side agents, tasks, or resources into coalitions under pricing, privacy, or deployment constraints. In graph theory it denotes an edge partition governed by domination and coalition partnership. In optimization and learning it often refers more broadly to decompositions whose objective or representation is determined by internal edges.