ρ-Dense Aggregators in Graphs

• ρ-dense aggregators are structured summaries of graphs that cover all maximal cliques using vertex subsets with induced subgraph density at least ρ.
• The method employs a recursive strategy, inspired by Bron–Kerbosch and degeneracy ordering, to efficiently construct dense clusters while reducing redundancy.
• Empirical evaluations show that ρ-dense aggregators significantly lower output size and computation time compared to traditional maximal clique enumeration techniques.

A $\rho$-dense aggregator is a structural summary of a graph that captures its maximal clique structure through a compact family of dense, potentially overlapping clusters. Formally, for a given undirected graph $G=(V, E)$ and a density threshold $\rho \in (0,1]$, a $\rho$-dense aggregator is a family $\mathcal{S} = \{C_1, \ldots, C_k\}$ of vertex subsets such that (i) every maximal clique of $G$ is contained in at least one $C_i$, and (ii) each $C_i$ induces a subgraph of edge density at least $\rho$. This relaxes the notion of maximal clique enumeration—when $\rho = 1$, each $C_i$ must itself be a clique, recovering the standard maximal clique list. For $\rho < 1$, the aggregator provides a much more succinct, overlapping cover of the clique structure while preserving complete coverage of all maximal cliques (Alon et al., 3 Dec 2025).

1. Formal Definition and Basic Properties

Let $G=(V, E)$ be an undirected graph, $\rho \in (0,1]$. For a subset $C \subseteq V$, define $G[C]$ as the induced subgraph, and set density

$$\mathrm{density}(G[C]) = \frac{|E(G[C])|}{\binom{|C|}{2}},$$

where density is set to $1$ if $|C| \le 1$ by convention.

A $\rho$-dense aggregator is a collection $\mathcal{S} = \{C_1, \dots, C_k\}$, with $C_i \subseteq V$, satisfying:

• (Coverage) For every maximal clique $K$ in $G$, there is an $i$ such that $K \subseteq C_i$.
• (Density) For each $i$, $\mathrm{density}(G[C_i]) \ge \rho$.

For $\rho = 1$, each cluster must be a clique, so $\mathcal{S}$ must include every maximal clique. For any strictly smaller $\rho < 1$, the size of $\mathcal{S}$ can be greatly reduced, and each $C_i$ can cover many maximal cliques.

2. Theoretical Size Bounds

A primary motivation for $\rho$-dense aggregators is the reduction in output size compared to maximal clique enumeration. For any constant $\rho < 1$ and $n$-node graph, every $G$ admits a $\rho$-dense aggregator of sub-exponential size: $$k = n^{\frac{1+\log_{1/\rho} n}{2}} = \exp\bigl( O(\log n \cdot \log_{1/\rho} n) \bigr).$$ This follows from a recursive clustering construction that terminates when the induced subgraph of a candidate cluster reaches density $\rho$. For graphs with bounded degeneracy $\alpha$, this bound improves to $O(n\,\alpha^{(1+\log_{1/\rho} \alpha)/2})$, leading to near-linear output size for sparse real-world networks (Alon et al., 3 Dec 2025).

There is also a lower bound showing essential tightness: there exist graphs such that for $\rho \ge c/\log n$ (with fixed $c>0$), every $\rho$-dense aggregator must have size $k = \Omega^*( n^{\log n/\log\log n} )$, where the $\Omega^*$ suppresses sub-polynomial terms. Thus, in some regimes the sub-exponential dependence on $n$ is unavoidable.

$\rho$	Upper Bound on $k$	Lower Bound on $k$
$< 1$ (fixed)	$n^{\frac{1+\log_{1/\rho} n}{2}}$	$\Omega^*( n^{\log n/\log\log n})$
$= 1$	Number of maximal cliques (can be exponential in $n$)	Same

3. Recursive Aggregator Construction Algorithm

The construction of $\rho$-dense aggregators proceeds via a recursive search resembling the Bron–Kerbosch algorithm, but terminating branches that yield clusters with sufficient density.

Algorithm BUILD-AGGREGATOR$(G, \rho)$:

1. Compute a degeneracy ordering of $G$.
2. Initialize with the full vertex set: RECURSE$(H=V(G), C=\emptyset, X=\emptyset)$.
3. Collect and return all clusters $C$ found where $\mathrm{density}(C) \ge \rho$ during recursion.

RECURSE$(H, C, X)$ operates as follows:

• If there is $x \in X$ with $H \subseteq N(x)$, prune the branch (all cliques covered already).
• If $\mathrm{density}(C \cup H) \ge \rho$, emit $C \cup H$ as a cluster.
• Otherwise, select $v \in H$ of degree at most $\rho|H|$, and make recursive calls on:
  • (extend-branch): $C \cup \{v\},\,H \cap N(v),\,X \cap N(v)$
  • (omit-branch): $C,\,H \setminus \{v\},\,X \cup \{v\}$

This strategy ensures both coverage and the density property. By employing degeneracy ordering and pruning via already-encountered nodes ($X$), the search avoids redundant clusters and unnecessary work. In practice, efficient bit-matrix representations and branch pruning inspired by maximal clique search optimize for real-world graphs (Alon et al., 3 Dec 2025).

Graph Property	Aggregator Size	Runtime Complexity
Arbitrary $G$	$n^{\frac{1+\log_{1/\rho} n}{2}}$	$O\big((m+n)\, n^{(1+\log_{1/\rho} n)/2}\big)$
Degeneracy $\alpha$	$O(n\,\alpha^{\frac{1+\log_{1/\rho} \alpha}{2}})$	$$O(m\,\alpha+n\,\alpha^{\frac{3+\log_{1/\rho} \alpha}{2}})$$

4. Empirical Evaluation and Performance

Empirical analysis on 16 real-world networks (ranging from $10^3$ to $2 \times 10^7$ vertices and degeneracy up to $2,\!200$), shows pronounced benefits for $\rho$-dense aggregators:

• Output Reduction: For $\rho = 0.1$, aggregator size averaged approximately $1/20$ the number of maximal cliques; even for $\rho = 0.9$, the aggregator contained roughly half as many clusters as maximal cliques.
• Runtime: For $\rho = 0.1$, median speed-up over QuickCliques was $6\times$ (up to $>300\times$ in extreme cases), and $2\times$ for $\rho = 0.9$ (completing in minutes where clique enumeration exceeded a 5-hour threshold).
• Redundancy Reduction: Maximum per-vertex overlap (number of clusters containing a vertex) decreased by up to $66\times$ for $\rho = 0.1$ and $5.2\times$ for $\rho = 0.9$.
• Overlapping vs. Non-overlapping Clustering: Non-overlapping clustering methods such as Louvain, LambdaCC, and AGM, even when tuned for density, covered only 20–30% of maximal cliques, emphasizing the necessity of overlapping aggregators for full coverage (Alon et al., 3 Dec 2025).

5. Applications and Implications

$\rho$-dense aggregators provide a succinct, information-preserving summary of clique structure with practical and theoretical advantages:

• Community Detection: They enable the discovery of dense "super-communities" that collectively cover all smaller clique-based subcommunities.
• Bioinformatics: Useful for representing all protein complexes (full or partial), which are typically modeled as cliques in protein–protein interaction networks.
• Network Summarization: Serve as compact bases for visualization and downstream graph mining, reducing redundancy while guaranteeing maximal clique coverage.
• Scalability: Real-world graphs, with bounded or low degeneracy, allow near-linear size and runtime, making these summarizations feasible for very large networks (Alon et al., 3 Dec 2025).

6. Open Problems and Future Directions

Several open and emerging research directions are highlighted:

• Special Classes of Graphs: Graphs with specific structural properties (e.g., $c$-closed graphs) may admit polynomial-size aggregators for fixed $\rho < 1$.
• Parallel and Distributed Algorithms: Designing scalable versions to handle even larger inputs remains an open challenge.
• Lossy Aggregation: Developing versions that approximately cover "most" maximal cliques with even fewer clusters, quantifying the coverage–size tradeoff.
• Overlap Minimization: Further study of minimizing per-vertex or per-edge overlap in the aggregator to improve interpretability and efficiency.

A plausible implication is that advances in these areas will further lower the computational and memory profiles needed to analyze and summarize clique structures in massive networks, and extend the utility of $\rho$-dense aggregators in emerging applications where clique enumeration is infeasible (Alon et al., 3 Dec 2025).