---
title: 'DEMM+: Scalable MR Graph Clustering'
url: https://www.emergentmind.com/topics/demm-ef803277-1253-4d89-8f3d-7156068f1982
type: topic
---

# DEMM+: Scalable MR Graph Clustering

DEMM+ is a scalable method for **multi-relational graph clustering (MRGC)** introduced in “Effective Clustering for Large Multi-Relational Graphs” [2508.17388]. 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 [2508.17388].

## 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 \(Y\in\mathbb{R}^{N\times K}\),
\[
Y_{i,k}= \begin{cases}
\frac{1}{\sqrt{|\mathcal C_k|}}, & \text{if } v_i\in \mathcal C_k,\\
0, & \text{otherwise}.
\end{cases}
\]
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 [2508.17388].

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 [2508.17388].

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 [2605.04093], **LM-DEM** for variational PDEs [2602.07838], and **DCEM-P** in solid mechanics [2302.01538]. 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 [2508.17388].

## 2. Two-stage optimization architecture

The base DEMM and DEMM+ share the same conceptual decomposition. **Stage 1** constructs node features \(H\) by jointly fusing graph relations and attributes through a weighted smoothness objective. **Stage 2** forms an affinity graph from \(H\) and optimizes the clustering result on that graph [2508.17388].

For a feature matrix \(H\in\mathbb{R}^{N\times d}\), the Dirichlet energy on relation graph \(A^{(r)}\) is
\[
\mathcal D(H,A^{(r)}) =
\frac12\sum_{(v_i,v_j)\in E^{(r)}} \left\|
\frac{H_i}{\sqrt{d_i^{(r)}}} -
\frac{H_j}{\sqrt{d_j^{(r)}}}
\right\|_2^2,
\]
which is equivalently
\[
\mathcal D(H,A^{(r)}) = \operatorname{trace}\!\left(H^\top(I-\hat A^{(r)})H\right).
\]
DEMM extends this to the **multi-relational Dirichlet energy**
\[
\mathcal L_{\text{MRDE}} = \sum_{r=1}^R \omega_r \cdot \mathcal D(H,A^{(r)}),
\]
where \(\omega_r\ge 0\) and \(\sum_{r=1}^R \omega_r = 1\) [2508.17388].

Stage 1 then solves
\[
\min_{H\in N,\ \{\omega_r\}}
\|H-X\|_F^2
+\alpha\sum_{r=1}^R \omega_r \mathcal D(H,A^{(r)})
+\beta\sum_{r=1}^R \omega_r \|\hat A^{(r)}\|_F^2
\quad \text{s.t. } \sum_{r=1}^R \omega_r = 1,
\]
where \(N\) 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 [2508.17388].

With fixed \(\omega_r\), the relation layers are fused into the **unified normalized adjacency**
\[
\hat A = \sum_{r=1}^R \omega_r \hat A^{(r)},
\]
and the conditional optimum for \(H\) is
\[
H= \frac{1}{1+\alpha} \left(I-\frac{\alpha}{1+\alpha}\hat A\right)^{-1}X.
\]
With fixed \(H\), define
\[
c_r = \beta\|\hat A^{(r)}\|_F^2 + \alpha\,\operatorname{trace}\!\left(H^\top(I-\hat A^{(r)})H\right),
\]
then the relation-weight update is
\[
\omega_r =
\frac{c_r^{-2}}{\sum_{i=1}^R c_i^{-2}}.
\]
This gives a fully specified alternating-optimization procedure for the base DEMM [2508.17388].

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

| Component | DEMM | DEMM+ |
|---|---|---|
| Stage 1 feature solver | Matrix inverse for \(H\) | FAAO with truncated diffusion and sketch-based weight updates |
| Stage 2 clustering | Dense Gaussian affinity plus eigendecomposition | SSKC with ORF, factorized Sinkhorn-Knopp, and \(K\)-Means |
| Main scalability obstacle addressed | \(N\times N\) 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
\[
|\lambda(\hat A)|\le 1,
\]
which implies
\[
\left|\lambda\!\left(\frac{\alpha}{1+\alpha}\hat A\right)\right|
\le \frac{\alpha}{1+\alpha}<1.
\]
This permits a Neumann-series expansion of the exact solution:
\[
H =
\frac{1}{1+\alpha}
\sum_{\ell=0}^{\infty}
\left(\frac{\alpha}{1+\alpha}\right)^\ell
\hat A^\ell X.
\]
DEMM+ truncates the series at depth \(L\) and approximates the tail by the \(L\)-hop term:
\[
H \approx
\frac{1}{1+\alpha}\sum_{\ell=0}^{L}
\left(\frac{\alpha}{1+\alpha}\right)^\ell \hat A^\ell X
+
\left(\frac{\alpha}{1+\alpha}\right)^{L+1}\hat A^L X.
\]
Operationally, FAAO initializes
\[
\widehat X^{(0)}=\frac{1}{1+\alpha}X,\quad H\gets \widehat X^{(0)},
\]
then iterates
\[
\widehat X^{(\ell)} \gets \frac{\alpha}{1+\alpha}\hat A\widehat X^{(\ell-1)},
\qquad
H\gets H+\widehat X^{(\ell)},
\]
for \(\ell=1,\dots,L\), and finally adds the tail estimator
\[
H\gets H+\alpha\cdot \widehat X^{(L)}.
\]
The feature update thus becomes a sequence of sparse propagations rather than a dense inverse [2508.17388].

The second Stage 1 optimization in DEMM+ concerns the relation weights. The exact update depends on
\[
\operatorname{trace}\!\left(H^\top(I-\hat A^{(r)})H\right).
\]
The paper rewrites this using a normalized incidence matrix \(\hat E^{(r)}\) so that
\[
\operatorname{trace}\!\left(H^\top(I-\hat A^{(r)})H\right)
=
\|H^\top \hat E^{(r)}\|_F^2.
\]
DEMM+ then applies **CountSketch** to obtain \(\tilde E^{(r)}\in\mathbb{R}^{N\times m}\), \(m\ll M^{(r)}\), and substitutes
\[
\|H^\top \hat E^{(r)}\|_F^2
\approx
\|H^\top \tilde E^{(r)}\|_F^2.
\]
The resulting approximate weight update is
\[
\omega_r =
\frac{
\left( \beta\|\hat A^{(r)}\|_F^2 + \alpha\|H^\top \tilde E^{(r)}\|_F^2 \right)^{-2}
}{
\sum_{i=1}^R
\left( \beta\|\hat A^{(i)}\|_F^2 + \alpha\|H^\top \tilde E^{(i)}\|_F^2 \right)^{-2}
}.
\]
This keeps the optimization coupled to relation-specific smoothness while avoiding full-cost trace computation [2508.17388].

## 4. Stage 2 in DEMM+: SSKC

In the base method, Stage 2 constructs a dense Gaussian affinity matrix
\[
S_{i,j} = \exp\left( -\frac{\|H_i-H_j\|_2^2}{\sigma} \right),
\]
after row normalization with mean-centering,
\[
H_i \leftarrow \frac{H_i-\bar h_i}{\|H_i-\bar h_i\|_2},
\qquad
\bar h_i=\frac1d\sum_{\ell=1}^d H_{i,\ell}.
\]
The clustering objective is
\[
\min_{\mathcal C_1,\dots,\mathcal C_K}\mathcal D(Y,S).
\]
The paper states that
\[
\min_Y \mathcal D(Y,S)
\iff
\max_Y \operatorname{trace}(Y^\top S Y),
\]
and under the stated assumptions this connects to normalized cut [2508.17388].

DEMM+ avoids explicit construction of \(S\) through **SSKC**. Its key theorem is that if \(S\) is **doubly stochastic** and factorizes as
\[
S=ZZ^\top,
\]
then minimizing the clustering objective is equivalent to the \(K\)-Means problem
\[
\min_{C_1,\ldots,C_K}
\sum_{k=1}^K
\sum_{v_i\in C_k}\|Z_i-\mathbf c^{(k)}\|_2^2,
\qquad
\mathbf c^{(k)}=\frac{1}{|C_k|}\sum_{v_j\in C_k} Z_j.
\]
This converts spectral clustering on a dense affinity graph into low-dimensional \(K\)-Means, provided that a suitable factor \(Z\) can be produced [2508.17388].

To approximate the Gaussian kernel implicitly, DEMM+ uses **Orthogonal Random Features (ORF)**. With a random orthogonal matrix \(Q\in\mathbb{R}^{d\times d}\),
\[
\tilde H = H Q^\top,
\]
and
\[
Z^\circ
=
\frac{1}{\sqrt d}
\big(\sin(\tilde H)\ \Vert\ \cos(\tilde H)\big).
\]
Then
\[
Z^\circ {Z^\circ}^\top \approx S,
\]
with output dimension \(z=2d\ll N\). This replaces \(O(N^2)\) kernel materialization by an explicit feature map [2508.17388].

The paper then applies a factorized form of **Sinkhorn-Knopp normalization**. Initializing
\[
\overleftarrow Z = Z^\circ,\qquad \overrightarrow Z = Z^\circ,
\]
it alternates rescaling steps so that \(\overleftarrow Z\overrightarrow Z^\top\) becomes doubly stochastic without ever materializing that matrix. At convergence, the paper proves that the product is doubly stochastic and that \(\overleftarrow Z=\overrightarrow Z\). The final stage is simply \(K\)-Means on the rows of the normalized factor [2508.17388].

## 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 \(H^\ast\) denotes the exact infinite-series solution, then the approximation error satisfies
\[
\|H-H^\ast\|_F \le
\sum_{\ell=L+1}^{\infty}
\frac{\alpha^\ell}{(1+\alpha)^{\ell+1}}
\|\hat A^\ell-\hat A^L\|_2 \|X\|_F,
\]
and is further upper bounded by
\[
\|H-H^\ast\|_F \le
\left(\frac{\alpha}{1+\alpha}\right)^{L+1}
\|X\|_F
\max_{\ell\ge1}\mu_{L,L+\ell}.
\]
For the sketched weight update, with sufficiently large sketch size \(m\),
\[
\|H^\top \tilde E^{(r)}\|_F^2
=
(1\pm\epsilon)^2
\operatorname{trace}\!\left(H^\top(I-\hat A^{(r)})H\right)
\]
with probability at least \(1-\delta\). These statements tie the feature approximation to graph mixing behavior and the weight approximation to standard sketching preservation guarantees [2508.17388].

The complexity reductions are central to the identity of DEMM+. For Stage 1, the paper states **CountSketch preprocessing over all relations** costs \(O(M)\). Each alternating iteration then incurs \(O(MLd)\) for sparse propagations and \(O(NdmR)\) for the sketched weight update. With small constant \(L\), \(m\), and iteration count, the total Stage 1 complexity is summarized as
\[
O(Md + NdR)
\]
time and
\[
O(M+Nd)
\]
space. For Stage 2, the paper gives \(O(Nd^2)\) for ORF, \(O(Nd)\) per Sinkhorn iteration, and \(O(NK)\) per \(K\)-Means iteration, yielding total time
\[
O(Nd^2 + NK)
\]
and space
\[
O(Nd).
\]
A plausible implication is that DEMM+ is designed so that both stages remain near-linear in the graph size when \(d\), \(K\), and iteration counts are moderate [2508.17388].

The method is also extended to **attribute-less MRGs**. When \(X=\mathbf 0\), the original feature-fitting term disappears, so the paper adds the orthogonality constraint
\[
H^\top H = I.
\]
The feature update then becomes equivalent to
\[
\max_{H^\top H=I} \operatorname{trace}(H^\top \hat A H),
\]
whose solution is given by the top \(d\) eigenvectors of the unified adjacency \(\hat A\). The attribute-less scalable version is stated to have complexity
\[
O(Md + N(d^2+dR+K))
\]
with space
\[
O(M+N(d+K)).
\]
This extension is described as “non-trivial,” because the main attribute-bearing formulation relies on a reconstruction term \(\|H-X\|_F^2\) that is absent when attributes do not exist [2508.17388].

## 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 [2508.17388].

On small datasets, DEMM+ is reported as top or near-top. Representative scores include **ACM** with ACC \(93.6\), NMI \(77.2\), ARI \(81.9\); **DBLP** with ACC \(93.7\), NMI \(79.6\), ARI \(84.8\); **Yelp** with ACC \(92.7\), NMI \(72.6\), ARI \(77.7\); and **IMDB** with ACC \(67.6\), NMI \(24.4\), ARI \(26.5\) [2508.17388]. On large datasets, the paper emphasizes both quality and scale. For **OAG-ENG**, DEMM+ achieves ACC \(42.3\), NMI \(41.8\), ARI \(24.8\), with gains of \(16.6\%\), \(17.3\%\), and \(11.0\%\) over the best baseline. For **OAG-CS**, it reports ACC \(40.1\), NMI \(42.7\), ARI \(24.1\), improving by \(4.7\%\), \(4.2\%\), and \(2.7\%\). For **RCDD**, it reports ACC \(83.4\), NMI \(18.6\), ARI \(29.0\), improving by \(8.3\%\), \(5.4\%\), and \(12.1\%\) [2508.17388].

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 \(2\times\) faster than the best viable baseline. The direct DEMM-to-DEMM+ comparison reports an overall speedup of **3,252× on MAG** [2508.17388].

Ablation studies identify three ingredients as important: adaptive relation weights \(\{\omega_r\}\), the tail estimator \(\alpha\widehat X^{(L)}\), and the regularization term \(\mathcal L_{\text{reg}}\). 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 \(\alpha,\beta,L,d,m\) indicate that larger \(\alpha\) often helps, \(L\) need only be modest, \(d\) improves results up to around \(128\) before saturating, and small sketch size \(m\) is usually sufficient [2508.17388].

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 \(\alpha,\beta,L,d,m\) remains consequential. Finally, the conclusion states that the method is mainly for **static MRGs**, not dynamic ones [2508.17388].

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 [2508.17388].

Source: https://www.emergentmind.com/topics/demm-ef803277-1253-4d89-8f3d-7156068f1982