Higher-Order Homophily in Complex Networks

• Higher-order homophily is the extension of traditional homophily to multi-node interactions, capturing the arrangement of attributes in hyperedges, cliques, and motifs.
• It is measured using hypergraphs and simplicial complexes to reveal scale-dependent mixing patterns that influence polarization, contagion, and information flow.
• Emerging frameworks employ maximum-entropy models, affinity baselines, and message-passing algorithms to quantify and leverage higher-order homophily in network analysis.

Searching arXiv for papers on higher-order homophily and closely related frameworks. Higher-order homophily is the extension of homophily from dyadic links to group interactions, where association patterns are defined not only by whether two nodes are similar but by how attributes are arranged within hyperedges, cliques, simplices, motifs, or multi-hop interaction structures. In this setting, homophily is no longer a single scalar attached to a graph: it can vary with group size, group composition, message-passing depth, and the null model used for comparison. Recent work treats higher-order homophily as a property of hypergraphs and related higher-order representations, develops size- and type-resolved measures, shows that pairwise projections can erase essential structure, and demonstrates that higher-order homophily materially alters polarization, contagion, information access, structural balance, and representation learning (Veldt et al., 2021, Rizi et al., 2024, Laber et al., 1 Jun 2026).

1. Conceptual scope and relation to pairwise homophily

Traditional homophily metrics operate at the pairwise level and distinguish only within-group from between-group edges. Higher-order homophily begins where that binary distinction breaks down: a group interaction can contain any mixture of attributes, and the relevant question becomes whether particular group compositions are overrepresented or underrepresented relative to a baseline. Hypergraphs and simplicial complexes are therefore natural formalisms, because they encode multi-way interactions directly rather than as projections of dyads (Veldt et al., 2021, Saxena et al., 22 Sep 2025).

A central consequence is that homophily becomes scale-dependent. In the maximum-entropy clique framework, homophily is parameterized for all clique sizes $c$, not only dyads, allowing local homophily within tightly connected small groups and weaker or different mixing patterns at larger scales. The paper explicitly decomposes overall homophily as

$$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$

so that dyadic mixing and larger-group mixing are analytically separable (Rizi et al., 2024).

This scale dependence is empirically consequential. The same network may be heterophilic at the dyadic level but homophilic within larger social circles, or vice versa. The Pokec social network is described as heterophilic between males and females at the dyadic level, while Facebook university networks show that larger cliques tend to be more homophilic than dyadic connections. This suggests that “the homophily of a network” is often an underspecified object unless the interaction scale is stated explicitly (Rizi et al., 2024).

A related but distinct line of work defines higher-order homophily locally in ordinary graphs. In heterophilous graphs, higher-hop neighborhoods can exhibit more homophily than immediate neighbors, motivating definitions based on paths, motifs, or $k$-hop neighborhoods rather than only direct edges. This use of “higher-order” is topological rather than hypergraphic, but it serves the same purpose: it captures similarity patterns that are invisible to first-order adjacency alone (Zhou et al., 2023, Wang et al., 2021).

2. Formal definitions and measurement frameworks

One influential hypergraph formalization uses type-$t$ affinity scores in a $k$-uniform hypergraph. For a class $X$, the affinity score is

$h_t(X) = \frac{D_t(X)}{D(X)},$

where $D_t(X)$ counts participations of nodes in $X$ in groups with exactly $t$ members from $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$0, and $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$1 is total group participations by $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$2. The associated baseline is

$$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$3

and the ratio score is $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$4. This framework supports notions such as simple homophily, order-$$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$5 majority homophily, strict majority homophily, strict monotonic homophily, and the indices MaHI and MoHI (Veldt et al., 2021).

A complementary framework is maximum entropy over clique types. For clique size $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$6, the clique-type distribution is

$$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$7

where $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$8 is the Coleman Homophily Index for a $$h_c = \sum_{i=0}^{c} h_{c,i} F_{c,i}, \qquad h = \sum_{c \geq 2} \alpha_c h_c,$$9-clique with $k$0 red nodes. This construction yields one homophily parameter per group size and treats the stochastic block model as a special case of the dyadic layer (Rizi et al., 2024).

Hyperedge-centric measurement is represented by the Perplexity-Homophily Index. For a hyperedge $k$1, interaction perplexity is

$k$2

the diversity gap is $k$3, the normalized hyperedge score is

$k$4

and the global score is

$k$5

This framework is explicitly hyperedge-size-resolved and compares observed diversity with a degree-preserving random baseline (Kumar et al., 24 Nov 2025).

In hypergraph neural-network analysis, homophily has also been defined through message passing. For a hyperedge $k$6, the 0-level score is

$k$7

and higher-order homophily is updated recursively by alternating node and hyperedge aggregation. The stability of homophily across propagation steps is summarized by

$k$8

This makes homophily an explicitly dynamical property of a message-passing process rather than only a static composition statistic (Telyatnikov et al., 2023).

Framework	Core object	Main emphasis
Affinity / baseline ratios	Group composition counts by size and type	Over- or under-representation relative to random mixing
Maximum-entropy clique model	Clique-type distributions $k$9	Decomposition within and across group sizes
Perplexity-Homophily Index	Hyperedge diversity via perplexity	Hyperedge-centric, size-resolved homophily
Message-passing homophily	Node/hyperedge scores over propagation steps	Homophily as an iterative structural signal

These frameworks are not interchangeable. Some are class-specific, some hyperedge-specific, some global, and some recursive. The survey literature emphasizes that higher-order homophily is high-dimensional: instead of a single number, one often obtains a family of scores indexed by class, group size, composition, or propagation depth (Laber et al., 1 Jun 2026).

3. Generative models, null models, and inference

Measurement depends critically on the baseline against which observed group compositions are compared. In higher-order settings, baselines may preserve hyperedge sizes, node degrees, joint degree distributions, or nesting relations, and these choices change the interpretation of any homophily score (Saxena et al., 22 Sep 2025, Laber et al., 1 Jun 2026).

The $t$0 model is a generative hypergraph model with hyperedge homophily, a hyperedge size-dependent property, and tunable degree distribution. Nodes carry binary group labels $t$1, hyperedges have size $t$2 and type $t$3, and the model controls the exact counts $t$4 of hyperedges of each size and type. The associated size- and type-specific homophily score is

$t$5

where $t$6 is the affinity score and $t$7 is the random baseline conditioned on at least one node from group $t$8 being present. Because $t$9 is directly specified, the model can realize neutral, homophilous, heterophilous, or mixed homophily patterns across hyperedge sizes (Laber et al., 30 May 2025).

Directed hypergraph null models make the baseline more stringent. The Directed Hypergraph Configuration Model preserves node in- and out-degree sequences together with hyperedge head and tail size sequences, while the Directed Hypergraph JOINT Model preserves the full joint tensor of in-/out-degrees and hyperedge size characteristics. These ensembles are sampled with NuDHy-Degs and NuDHy-JOINT, and were used to analyze group affinity in the US Congress. In that application, each bill is a directed hyperedge with a single sponsor in the head and multiple cosponsors in the tail, and the simplified affinity is

$k$0

Observed affinity is then compared to the null-model mean to obtain an affinity ratio (Preti et al., 2024).

Maximum-entropy methods address a different inferential problem. Rather than preserving fine-grained hypergraph structure, they fit the least-biased clique-type distribution that matches empirical class proportions and scale-specific homophily targets. This is particularly useful when the question is whether homophily varies across clique sizes even when aggregate assortativity appears uniform (Rizi et al., 2024).

Even in pairwise graphs, the distinction between choice homophily and outcome homophily requires a principled ensemble. The statistical network-ensemble approach of “Unveiling homophily beyond the pool of opportunities” fixes group sizes and activity disparities and estimates homophily beyond the pool of opportunities. A plausible implication for higher-order work is that analogous structural controls are indispensable whenever group-size imbalance or participation heterogeneity could induce apparent homophily without a corresponding choice bias (Sajjadi et al., 2024).

4. Dynamical consequences in social systems

Higher-order homophily changes collective dynamics because group composition affects who is exposed to whom, how dissent enters a local interaction, and which scales of social organization dominate contagion or coordination.

In a dynamical framework on hypergraphs, opinions evolve according to

$k$1

with higher-order homophilic weights

$k$2

Within this model, sparse higher-order structures amplify polarization by limiting exposure to dissenting views, whereas densifying group interactions suppresses polarization by increasing cross-cutting memberships. The exposure metric

$k$3

shows that sparse structures produce many agents with $k$4, while full connectivity raises the critical homophily threshold $k$5 required for stable polarization, especially as group size increases (Pérez-Martínez et al., 16 Jul 2025).

Higher-order homophily also shapes inequality in information access. The $k$6 model combined with the naSI contagion process introduces group-dependent asymmetric transmission with rate

$k$7

Stochastic simulations and empirical hypergraphs show that homophily together with asymmetric contagion yields the highest inequality in time to receive information and ability to spread information; which hyperedge sizes are homophilous or heterophilous matters, especially under nonlinear contagion. The paper further reports that even adding a small proportion of neutral or mixed hyperedges at a key size can sharply reduce information-access disparities (Laber et al., 30 May 2025).

In structural balance dynamics, higher-order homophily appears through similarity across many underlying attributes rather than through hyperedges. Agents carry $k$8 Boolean attributes and link weights are

$k$9

A triad-updating rule changes one attribute of one agent per update, so local attempts to balance one triad create accidental changes elsewhere. In the large-$X$0 limit, the link-weight distribution obeys a Fokker-Planck equation, and the fully positive “paradise state” requires $X$1 and $X$2. The stated conclusion is that homophily based on few attributes can impede structural balance (Górski et al., 2020).

Microscopic face-to-face group formation shows another mechanism. In the Group Attractiveness Model, a group’s attractiveness is

$X$3

and higher-order homophily is parameterized by matrices $X$4, $X$5, and their extensions. The model reproduces the observed distribution of group sizes, the correlation in group participations across size, and the persistence of groups in empirical face-to-face data, while also showing that pairwise homophily is insufficient to match triplet-level mixing patterns (Gallo et al., 2024).

5. Higher-order homophily in machine learning and network representation

In graph and hypergraph learning, higher-order homophily is treated both as a measurable property of data and as a signal that architectures should exploit explicitly rather than approximate through deeper first-order propagation.

PathMLP is motivated by the empirical observation that heterophilous graphs often contain higher-order neighborhoods with higher homophily. It defines edge homophily as

$X$6

and samples smooth paths by feature similarity so that path aggregation captures semantically coherent higher-order information. The model aggregates sampled path encodings through learnable weights and is reported to outperform baselines on 16 out of 20 datasets, while being immune to over-smoothing because it does not rely on stacked message passing (Zhou et al., 2023).

HOG-GCN modifies propagation directly by learning a homophily degree matrix

$X$7

where $X$8 is attribute-based and $X$9 is topology-based, both defined over $h_t(X) = \frac{D_t(X)}{D(X)},$0-hop neighborhoods. The propagation rule is

$h_t(X) = \frac{D_t(X)}{D(X)},$1

Theoretical analysis shows that this minimizes a similarity constraint

$h_t(X) = \frac{D_t(X)}{D(X)},$2

so representation similarity is explicitly modulated by learned homophily degree. On heterophilic datasets, using $h_t(X) = \frac{D_t(X)}{D(X)},$3 improves performance substantially relative to $h_t(X) = \frac{D_t(X)}{D(X)},$4, illustrating the value of higher-order homophily (Wang et al., 2021).

Motif-based transformation offers another route. H$h_t(X) = \frac{D_t(X)}{D(X)},$5NT defines a motif adjacency matrix $h_t(X) = \frac{D_t(X)}{D(X)},$6, a heterophily matrix $h_t(X) = \frac{D_t(X)}{D(X)},$7, and a unified proximity

$h_t(X) = \frac{D_t(X)}{D(X)},$8

Micro-level walk paths capture homophily and macro-level walk paths capture heterophily, so the transformed network can be fed to existing embedding methods. The paper reports that DeepWalk-based H$h_t(X) = \frac{D_t(X)}{D(X)},$9NT achieves 24% improvement in terms of precision on motif prediction while reducing 46% computational time compared to the original DeepWalk (Ge et al., 2020).

For hypergraphs, message-passing homophily informs architecture design directly. MultiSet and MultiSetMixer maintain hyperedge-dependent node representations rather than collapsing all hyperedge messages into one state at each layer. The tutorial-style conclusion is that datasets where homophily is preserved across message-passing steps are precisely those where hypergraph neural networks benefit most from higher-order structure, and that clique-expanded homophily is often a weaker predictor than recursive message-passing homophily (Telyatnikov et al., 2023).

6. Combinatorial limits, paradoxes, and interpretive cautions

Higher-order homophily is constrained not only by behavior but by combinatorics. In two-class $D_t(X)$0-uniform hypergraphs, it is impossible except under trivial or degenerate groupings for both classes to simultaneously exhibit strict majority homophily or strict monotonic homophily in the natural sense defined by affinity-over-baseline ratios. For odd $D_t(X)$1, both classes cannot have strict monotonic homophily; for even $D_t(X)$2, the tie case prevents the analogous simultaneous condition. The implication stated in the paper is that differences in higher-order homophily between classes may be unavoidable consequences of combinatorial constraints rather than direct evidence of asymmetric preferences (Veldt et al., 2021).

This interpretive warning extends beyond hypergraphs. In ordinary two-type graphs, the paradox of second-order homophily shows that if there is homophily diversity among red nodes, then red friends of red nodes are on average more homophilous than red friends of blue nodes. The red gap is

$D_t(X)$3

and in a random-graph analysis with degrees held constant,

$D_t(X)$4

This gap holds in both globally homophilous and heterophilous networks, so neighborhood-level homophily comparisons can contain intrinsic structural bias (Evtushenko et al., 2021).

A further caution concerns projection and aggregation. Tutorials on higher-order homophily emphasize that clique projection can erase hyperedge size information and can make strong heterophily at one scale and strong homophily at another appear neutral in aggregate. Likewise, a single global homophily parameter can obscure group-level and scale-dependent variation that materially affects percolation, epidemic thresholds, and intervention design (Saxena et al., 22 Sep 2025, Laber et al., 1 Jun 2026).

The accumulated literature therefore converges on a consistent methodological point. Higher-order homophily should be interpreted relative to a stated interaction representation, a stated scale, and a stated null model. Without those choices, claims about “more homophily” or “less homophily” are often incomplete; with them, higher-order homophily becomes a precise language for analyzing how group composition shapes social organization, dynamical processes, and learning systems (Rizi et al., 2024, Preti et al., 2024).