DEMM+: Scalable MR Graph Clustering

• DEMM+ is a scalable multi-relational graph clustering method that partitions graph nodes by adaptively fusing structure and attribute information.
• It employs a two-stage optimization pipeline: Stage 1 uses FAAO for efficient node feature learning, and Stage 2 utilizes SSKC for effective clustering.
• The method replaces costly matrix inversions and dense affinity computations with scalable approximations, achieving significant runtime improvements on large-scale graphs.

DEMM+ is a scalable method for multi-relational graph clustering (MRGC) introduced in “Effective Clustering for Large Multi-Relational Graphs” (Lin et al., 24 Aug 2025). It is the optimized version of DEMM, a Dirichlet-energy-minimization-based approach that partitions the nodes of a multi-relational graph into $K$ disjoint clusters while jointly exploiting heterogeneous graph structure and, when available, node attributes. Its defining design is a two-stage optimization pipeline: Stage 1 learns node features $H$ by optimizing a multi-relational Dirichlet energy (MRDE) objective with adaptive relation weights, and Stage 2 clusters those features by minimizing the Dirichlet energy of cluster assignments on an affinity graph. DEMM+ differs from the base DEMM by replacing matrix inversion, dense affinity construction, and eigendecomposition with a suite of scalable approximations, chiefly FAAO for feature construction and SSKC for clustering (Lin et al., 24 Aug 2025).

1. Problem setting and terminological scope

In the DEMM+ formulation, the input is a multi-relational graph

$G=(V,\{E^{(r)}\}_{r=1}^R),$

with $N=|V|$ nodes, multiple relation-specific edge sets, and optionally a node attribute matrix $X\in\mathbb{R}^{N\times D}$. For relation type $r$, the adjacency matrix is $A^{(r)}\in\{0,1\}^{N\times N}$, the diagonal degree matrix is $D^{(r)}$, and the normalized adjacency is

$$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$

The clustering result is represented by a node-cluster indicator matrix $Y\in\mathbb{R}^{N\times K}$,

$H$0

The objective is to place nodes in the same cluster when they are structurally close across relation types and, when attributes are present, attribute-similar (Lin et al., 24 Aug 2025).

The method is motivated by two claimed shortcomings in prior MRGC work. First, existing approaches often yield “severely compromised result quality by ineffective fusion of heterogeneous graph structures and attributes.” Second, many methods do not cope well with “sizable MRGs with millions of nodes and billions of edges” because they depend on sophisticated deep models, dense affinity matrices, or expensive matrix operations (Lin et al., 24 Aug 2025).

The exact term DEMM+ is used in this MRGC setting. Similar strings in other areas denote different objects: the Decision Evidence Maturity Model in agentic AI (Solozobov, 29 Apr 2026), LM-DEM for variational PDEs (Wang et al., 8 Feb 2026), and DCEM-P in solid mechanics (Wang et al., 2023). A common misconception is therefore to treat DEMM+ as a generic “DEM-plus” label across fields; in the clustering literature, it refers specifically to the scalable MRGC method of (Lin et al., 24 Aug 2025).

2. Two-stage optimization architecture

The base DEMM and DEMM+ share the same conceptual decomposition. Stage 1 constructs node features $H$1 by jointly fusing graph relations and attributes through a weighted smoothness objective. Stage 2 forms an affinity graph from $H$2 and optimizes the clustering result on that graph (Lin et al., 24 Aug 2025).

For a feature matrix $H$3, the Dirichlet energy on relation graph $H$4 is

$H$5

which is equivalently

$H$6

DEMM extends this to the multi-relational Dirichlet energy

$H$7

where $H$8 and $H$9 (Lin et al., 24 Aug 2025).

Stage 1 then solves

$G=(V,\{E^{(r)}\}_{r=1}^R),$0

where $G=(V,\{E^{(r)}\}_{r=1}^R),$1 denotes the set of row-normalized matrices. The three terms respectively preserve closeness to attributes, impose multi-relational graph smoothness, and regularize relation weights to avoid pathological emphasis on relation layers with very different graph volumes or densities (Lin et al., 24 Aug 2025).

With fixed $G=(V,\{E^{(r)}\}_{r=1}^R),$2, the relation layers are fused into the unified normalized adjacency

$G=(V,\{E^{(r)}\}_{r=1}^R),$3

and the conditional optimum for $G=(V,\{E^{(r)}\}_{r=1}^R),$4 is

$G=(V,\{E^{(r)}\}_{r=1}^R),$5

With fixed $G=(V,\{E^{(r)}\}_{r=1}^R),$6, define

$G=(V,\{E^{(r)}\}_{r=1}^R),$7

then the relation-weight update is

$G=(V,\{E^{(r)}\}_{r=1}^R),$8

This gives a fully specified alternating-optimization procedure for the base DEMM (Lin et al., 24 Aug 2025).

A compact comparison between the baseline and the scalable version is as follows.

Component	DEMM	DEMM+
Stage 1 feature solver	Matrix inverse for $G=(V,\{E^{(r)}\}_{r=1}^R),$9	FAAO with truncated diffusion and sketch-based weight updates
Stage 2 clustering	Dense Gaussian affinity plus eigendecomposition	SSKC with ORF, factorized Sinkhorn-Knopp, and $N=|V|$0-Means
Main scalability obstacle addressed	$N=|V|$1 inverse and dense affinity	Avoids explicit inverse and explicit dense affinity

3. Stage 1 in DEMM+: FAAO

DEMM+ replaces the exact Stage 1 solver with FAAO. The starting point is the eigenvalue bound

$N=|V|$2

which implies

$N=|V|$3

This permits a Neumann-series expansion of the exact solution: $N=|V|$4 DEMM+ truncates the series at depth $N=|V|$5 and approximates the tail by the $N=|V|$6-hop term: $N=|V|$7 Operationally, FAAO initializes

$N=|V|$8

then iterates

$N=|V|$9

for $X\in\mathbb{R}^{N\times D}$0, and finally adds the tail estimator

$X\in\mathbb{R}^{N\times D}$1

The feature update thus becomes a sequence of sparse propagations rather than a dense inverse (Lin et al., 24 Aug 2025).

The second Stage 1 optimization in DEMM+ concerns the relation weights. The exact update depends on

$X\in\mathbb{R}^{N\times D}$2

The paper rewrites this using a normalized incidence matrix $X\in\mathbb{R}^{N\times D}$3 so that

$X\in\mathbb{R}^{N\times D}$4

DEMM+ then applies CountSketch to obtain $X\in\mathbb{R}^{N\times D}$5, $X\in\mathbb{R}^{N\times D}$6, and substitutes

$X\in\mathbb{R}^{N\times D}$7

The resulting approximate weight update is

$X\in\mathbb{R}^{N\times D}$8

This keeps the optimization coupled to relation-specific smoothness while avoiding full-cost trace computation (Lin et al., 24 Aug 2025).

4. Stage 2 in DEMM+: SSKC

In the base method, Stage 2 constructs a dense Gaussian affinity matrix

$X\in\mathbb{R}^{N\times D}$9

after row normalization with mean-centering,

$r$0

The clustering objective is

$r$1

The paper states that

$r$2

and under the stated assumptions this connects to normalized cut (Lin et al., 24 Aug 2025).

DEMM+ avoids explicit construction of $r$3 through SSKC. Its key theorem is that if $r$4 is doubly stochastic and factorizes as

$r$5

then minimizing the clustering objective is equivalent to the $r$6-Means problem

$r$7

This converts spectral clustering on a dense affinity graph into low-dimensional $r$8-Means, provided that a suitable factor $r$9 can be produced (Lin et al., 24 Aug 2025).

To approximate the Gaussian kernel implicitly, DEMM+ uses Orthogonal Random Features (ORF). With a random orthogonal matrix $A^{(r)}\in\{0,1\}^{N\times N}$0,

$A^{(r)}\in\{0,1\}^{N\times N}$1

and

$A^{(r)}\in\{0,1\}^{N\times N}$2

Then

$A^{(r)}\in\{0,1\}^{N\times N}$3

with output dimension $A^{(r)}\in\{0,1\}^{N\times N}$4. This replaces $A^{(r)}\in\{0,1\}^{N\times N}$5 kernel materialization by an explicit feature map (Lin et al., 24 Aug 2025).

The paper then applies a factorized form of Sinkhorn-Knopp normalization. Initializing

$A^{(r)}\in\{0,1\}^{N\times N}$6

it alternates rescaling steps so that $A^{(r)}\in\{0,1\}^{N\times N}$7 becomes doubly stochastic without ever materializing that matrix. At convergence, the paper proves that the product is doubly stochastic and that $A^{(r)}\in\{0,1\}^{N\times N}$8. The final stage is simply $A^{(r)}\in\{0,1\}^{N\times N}$9-Means on the rows of the normalized factor (Lin et al., 24 Aug 2025).

5. Guarantees, complexity, and the attribute-less extension

DEMM+ is not presented as a heuristic speedup alone; the paper provides several formal guarantees. For Stage 1, if $D^{(r)}$0 denotes the exact infinite-series solution, then the approximation error satisfies

$D^{(r)}$1

and is further upper bounded by

$D^{(r)}$2

For the sketched weight update, with sufficiently large sketch size $D^{(r)}$3,

$D^{(r)}$4

with probability at least $D^{(r)}$5. These statements tie the feature approximation to graph mixing behavior and the weight approximation to standard sketching preservation guarantees (Lin et al., 24 Aug 2025).

The complexity reductions are central to the identity of DEMM+. For Stage 1, the paper states CountSketch preprocessing over all relations costs $D^{(r)}$6. Each alternating iteration then incurs $D^{(r)}$7 for sparse propagations and $D^{(r)}$8 for the sketched weight update. With small constant $D^{(r)}$9, $$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$0, and iteration count, the total Stage 1 complexity is summarized as

$$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$1

time and

$$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$2

space. For Stage 2, the paper gives $$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$3 for ORF, $$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$4 per Sinkhorn iteration, and $$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$5 per $$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$6-Means iteration, yielding total time

$$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$7

and space

$$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$8

A plausible implication is that DEMM+ is designed so that both stages remain near-linear in the graph size when $$\hat A^{(r)} = {D^{(r)}}^{-\frac12}A^{(r)}{D^{(r)}}^{-\frac12}.$$9, $Y\in\mathbb{R}^{N\times K}$0, and iteration counts are moderate (Lin et al., 24 Aug 2025).

The method is also extended to attribute-less MRGs. When $Y\in\mathbb{R}^{N\times K}$1, the original feature-fitting term disappears, so the paper adds the orthogonality constraint

$Y\in\mathbb{R}^{N\times K}$2

The feature update then becomes equivalent to

$Y\in\mathbb{R}^{N\times K}$3

whose solution is given by the top $Y\in\mathbb{R}^{N\times K}$4 eigenvectors of the unified adjacency $Y\in\mathbb{R}^{N\times K}$5. The attribute-less scalable version is stated to have complexity

$Y\in\mathbb{R}^{N\times K}$6

with space

$Y\in\mathbb{R}^{N\times K}$7

This extension is described as “non-trivial,” because the main attribute-bearing formulation relies on a reconstruction term $Y\in\mathbb{R}^{N\times K}$8 that is absent when attributes do not exist (Lin et al., 24 Aug 2025).

6. Empirical performance, ablations, and limitations

The empirical evaluation compares DEMM+ against 20 baselines on 11 real MRG datasets spanning academic, e-commerce, movie, biological, and risk domains. Reported scales range from a few thousand nodes to 11.9M nodes and 0.78B edges. Clustering quality is measured by ACC, NMI, and ARI against ground-truth labels (Lin et al., 24 Aug 2025).

On small datasets, DEMM+ is reported as top or near-top. Representative scores include ACM with ACC $Y\in\mathbb{R}^{N\times K}$9, NMI $H$00, ARI $H$01; DBLP with ACC $H$02, NMI $H$03, ARI $H$04; Yelp with ACC $H$05, NMI $H$06, ARI $H$07; and IMDB with ACC $H$08, NMI $H$09, ARI $H$10 (Lin et al., 24 Aug 2025). On large datasets, the paper emphasizes both quality and scale. For OAG-ENG, DEMM+ achieves ACC $H$11, NMI $H$12, ARI $H$13, with gains of $H$14, $H$15, and $H$16 over the best baseline. For OAG-CS, it reports ACC $H$17, NMI $H$18, ARI $H$19, improving by $H$20, $H$21, and $H$22. For RCDD, it reports ACC $H$23, NMI $H$24, ARI $H$25, improving by $H$26, $H$27, and $H$28 (Lin et al., 24 Aug 2025).

Runtime results are a second major claim. Relative to the best quality baselines, the paper reports speedups such as 62.5\times on ACM, 23.9\times on DBLP, 25.6\times on ACM2, 21.4\times on Yelp, 67.6\times on IMDB, over 139\times on OAG-CS, and over 53\times on OAG-ENG. On RCDD, DEMM+ is stated to finish in under 30 minutes and to be nearly $H$29 faster than the best viable baseline. The direct DEMM-to-DEMM+ comparison reports an overall speedup of 3,252× on MAG (Lin et al., 24 Aug 2025).

Ablation studies identify three ingredients as important: adaptive relation weights $H$30, the tail estimator $H$31, and the regularization term $H$32. Removing any of them degrades performance, and the regularizer is described as especially important on large datasets with highly imbalanced relation volumes. Parameter studies over $H$33 indicate that larger $H$34 often helps, $H$35 need only be modest, $H$36 improves results up to around $H$37 before saturating, and small sketch size $H$38 is usually sufficient (Lin et al., 24 Aug 2025).

The paper also delineates several limitations. DEMM+ uses an approximation stack—truncated diffusion, sketching, random features, and implicit Sinkhorn normalization—so its scalability depends on approximate rather than exact realization of the original optimization. The main version depends on attributes, while the attribute-less case requires a structurally different extension. Hyperparameter choice for $H$39 remains consequential. Finally, the conclusion states that the method is mainly for static MRGs, not dynamic ones (Lin et al., 24 Aug 2025).

Within the arXiv literature covered here, DEMM+ should therefore be understood as a specific large-scale MRGC method built around adaptive multi-relational smoothing, sketched relation-weight updates, and factorized affinity clustering, rather than as a generic label for “enhanced DEM” methods in other domains (Lin et al., 24 Aug 2025).