Novel Random Graph Models

• Novel random graph models are advanced frameworks that extend classical random graphs by incorporating hierarchical structures, complex dependencies, and nontrivial dynamics.
• They utilize methodologies such as recursive Erdős–Rényi hierarchies, multilevel Bayesian ERGMs, and randomized reproduction processes to quantify scaling laws and phase transitions.
• These models provide practical insights for null modeling, community detection, and analysis of higher-order network structures with analytical tractability and empirical validation.

Novel random graph models extend classical constructions to encode diverse mesoscopic structures, complex dependencies, and nontrivial dynamics while remaining analytically tractable or computationally feasible. These frameworks span recursive hierarchies, population-level Bayesian models, growth by random reproduction, block agglomeration, hypergraph connectivity, and de-preferential attachment, each introducing key procedural innovations and rigorous characterizations of graph metrics, phase transitions, and scaling laws.

1. Hierarchical Random Graph Models

Random Graph Hierarchy (RGH) and Limited Random Graph Hierarchy (LRGH) present recursive multi-level constructions in which an initial set of $N$ nodes is successively agglomerated using Erdős–Rényi processes at each hierarchy level, with fixed mean degree $c$ per level. In RGH, all clusters (including singletons) at level $h$ are considered nodes at $h+1$, while in LRGH clusters of size one become inactive and do not progress. This mechanism yields a strictly nested hierarchy of cluster aggregations.

At every level, the number of clusters $W_h$ decays exponentially:

• RGH: $W_h \simeq N\,(1-c/2)^h$
• LRGH: $W_h \simeq N\,(1-c/2-e^{-c})^h$

The hierarchical depth $H$ scales logarithmically with $N$, and its dependency on connectivity $c$ is model-specific: $H$ decreases monotonically with $c$ in RGH, but reaches a maximum at $c_{max}$ for LRGH (numerically, $c_{max}\sim 0.8$ for $N=5000$). LRGH further reveals a terminal cluster-size distribution that follows a power-law $P(s)\propto s^{-\alpha}$ with $\alpha\approx 1.25$, reflecting a robust mechanism for generating heavy-tailed cluster sizes (Paluch et al., 2015).

These hierarchy models explicitly encode inclusion relationships via recursive ER linking, not via externally imposed dendrograms. The LRGH's drop-out mechanism also produces nontrivial scaling of hierarchy height and tail broadening, making these frameworks relevant as null models for multi-level community detection, merger/fragmentation modeling in socio-political systems, and as substrates for dynamical processes (e.g., contagion, opinion spreading).

2. Bayesian and Statistical Network Models for Populations

The Bayesian exponential random graph model (multilevel ERGM) introduced in (Lehmann et al., 2021) is designed for populations of networks $(Y_1,\ldots,Y_n)$ where each $Y_i$ is an observed network on a fixed node set (e.g. brain connectivity). Each $Y_i$ is modeled as an ERGM with parameter vector $\theta_i$, but these parameters are coupled via a hierarchical prior:

$$\theta_i \sim \mathcal{N}(\mu, \Sigma),\quad \mu\sim\mathcal{N}(m_0,\Lambda_0),\quad \Sigma\sim \text{Inverse-Wishart}(\nu_0,S_0)$$

Posterior inference is carried out by an "exchange-within-Gibbs" MCMC algorithm that accomplishes correct sampling despite doubly intractable likelihoods, using auxiliary network draws to cancel normalization constants at each update. This framework enables pooling of structural information across the population, precise quantification of within- and between-network variability, and principled assessment of covariate effects (e.g., age or disease status on brain topology).

Compared to SBMs or latent space models, this multilevel ERGM flexibly encodes arbitrary network statistics (including triadic closure, hub formation) and alleviates classic ERGM degeneracy by shrinkage across networks. Algorithmic scalability in $n$ and improved mixing via ancillary–sufficiency interweaving are demonstrated, with empirical applications to fMRI and functional connectivity data.

3. Growth by Random Reproduction Processes

Randomized reproducing graphs (Jordan, 2011) generalize deterministic reproducing graph models by introducing three independent random parameters $(\alpha, \beta, \gamma)$ that govern the probability of forming child-child edges, parent-child links, and child-to-parent's-neighbor links, respectively. The recursive process clones every vertex at each discrete step, and links are randomly assigned per specified Bernoulli rules.

Degree dynamics are governed by a Markov chain whose recurrence properties exhibit sharp phase transitions:

• $(1+\alpha)(1+\gamma)<1$: degree sequence converges to a stationary distribution, usually with a power-law tail exponent determined by parameters $\alpha$ and $\gamma$.
• $(1+\alpha)(1+\gamma)>1$: degree grows without bound; proportion of fixed-degree vertices vanishes.

Degree distributions are explicitly characterized: for subcritical regimes, tails behave as $P(\deg>d)\sim d^{-(p_*+1)},$ via a moment condition on $p_*$. Edge density scaling, densification power laws, and spectral gap scaling for the normalized Laplacian are computed exactly. This model captures evolution by duplication-and-divergence with tunable tail behavior and densification, relevant for biological network modeling.

4. Agglomerative Inhomogeneous Random Graph Models

Agglomerated random graph models (Kang et al., 2013) construct an inhomogeneous graph by starting with $G(n,p)$ and mapping $n$ vertices into $N$ blocks of prescribed sizes $(s_1,\ldots,s_N)$. The induced super-graph is defined so that a pair of super-vertices is connected iff there is at least one edge between their corresponding constituent nodes.

Connection probability between blocks $i$ and $j$ is

$P_{ij} = 1 - (1-p)^{s_i s_j}$

The resulting super-graph exhibits classic phase transitions:

• Connectivity threshold at $p = [\ln k_{i^*} + c]/[n-i^*]$, with $i^*$ maximizing $k_i(n-i)$.
• Giant component: threshold determined by $c\sigma^2$ (with $\sigma^2$ the size-bias average).
• Degree distribution: empirical convergence to a mixed Poisson law, $$P(E=k) = \sum_{i=1}^r p_i \text{Poisson}(c\,i/\sigma^2)$$, where $p_i$ is the frequency of block size $i$.

Crucially, power-law degree tails emerge if the block size distribution is heavy-tailed; the degree distribution inherits the same exponent $\alpha$ from the block sizes. This constructive mapping from homogeneous $G(n,p)$ to a block-induced inhomogeneous structure supports explicit community modeling and emergent scale-free phenomena.

5. Random Connection Hypergraphs and Higher-Order Connectivity

The Random Connection Hypergraph Model (RCHM) (Brun et al., 23 Jul 2024) generalizes random connection models to hypergraphs, constructed via marked Poisson point processes on both sides of a bipartite graph. Each vertex (author or paper) carries a mark, and bipartite connection is governed by an inhomogeneous geometric rule

$|x-z| \leq t\,u^{-\alpha} w^{-\alpha'}$

Dowker's construction is then used to define author–author simplices (hyperedges) witnessed by shared paper co-occurrences.

Analytic theorems include:

• Degree distributions of $m$-simplices exhibit power-law tails: $\log P(\deg \geq k)/\log k = m - (m+1)/\alpha$.
• Betti numbers of the Dowker complex obey CLTs in the subcritical tail regime $\alpha < 1/4$, $\alpha' < 1/[4(m+1)]$.
• Simplex counts exhibit stable-law versus normal fluctuations depending on $\alpha$; the transition is driven by infinite-variance tails.

Empirical application to arXiv collaboration networks demonstrates matching of marginal degree laws, but deviations in higher-order (simplicial) structure. This suggests that RCHM is effective for null modeling of collaboration topology, but realistic clustering requires further refinements.

6. Asymptotic De-Preferential Attachment Models

Generalized de-preferential models (Bandyopadhyay et al., 13 Dec 2025) introduce attachment kernels that penalize high-degree vertices with either linear or inverse-power terms (possibly with shifts), countering the standard "rich-get-richer" effect. For linear de-preferential models (kernel $\propto \theta-\frac{\alpha d_i}{2n-1}$), all limiting degree distributions and scaling orders remain geometric and grow as $\Theta(\log n)$, invariant under the shift.

Inverse-power-law de-preferential models (kernel $\propto (\delta+d_i)^{-\alpha}$) yield degrees that grow more slowly, with scaling $\Theta((\log n)^{1/(1+\alpha)})$ and empirical distributions decaying faster than exponential (factorial decay), despite regularly varying attachment. No power law is observed in the degree distribution, even though the attachment kernel is heavy-tailed. These results clarify that not all sublinear or regular-varying attachment mechanisms induce scale-free graphs; detailed branching process analysis is required.

7. Comparative Features and Applications

Across these novel random graph models, several shared themes arise:

Model class	Key mechanisms	Tail behaviors / transitions
Hierarchical (RGH/LRGH)	Recursive Erdős–Rényi clustering, drop-outs	Exponential cluster decay, power-law sizes
Population Bayesian ERGM	Hierarchical likelihood and exchange MCMC	Shrinkage, multilevel inference
Random reproduction	Bernoulli edge formation under duplication	Stationary vs. growing degree, power laws
Agglomerated ER	Super-vertex mapping, block-induced inhomogeneity	Mixed-Poisson, inherited power laws
Connection hypergraph	Mark-dependent bipartite geometric connections	Scale-free higher order, Betti transitions
De-preferential models	Inverse/linear repulsion kernels	Geometric or super-exponential tails

Novelty arises through explicit construction principles (recursive hierarchies, block agglomeration, hybrid kernel dynamics), theoretical advances in the characterization of scaling behaviors, and the capacity for these frameworks to serve both as null models for real systems and as testbeds for algorithmic and statistical network methods.

These models enable practical inference for multi-level, heterogeneous, temporally-evolving, or higher-order network datasets, support identification of fundamental phase transitions, and clarify the mechanisms underpinning observed heavy-tails or clustering in complex networks.