Hedonic

hedonic detects communities in igraph graphs as a hedonic game: wrap a graph in Game and call community_hedonic(). One method returns either a disjoint partition or an overlapping cover.

The detector is a small C core compiled into the package (hedonic._core). The only dependencies are the standard igraph package and NumPy, so hedonic installs next to any igraph 1.x you already use. Python 3.12 or newer.

TL;DR

pip install hedonic              # the library
pip install "hedonic[cli]"       # plus the command line that reproduces the papers

from hedonic import Game

h = Game(g)  # g: an igraph graph (convert NetworkX, edge lists, ... through igraph)
partition = h.community_hedonic(max_memberships=1)  # disjoint
cover = h.community_hedonic(max_memberships=3)      # overlapping

• from hedonic import Game is the public API.
• max_memberships=1 puts each vertex in exactly one community and returns an igraph.VertexClustering.
• max_memberships>1 lets each vertex join up to that many communities and returns an igraph.VertexCover.
• Every call runs until no vertex can certainly improve by moving: the result is an equilibrium of the game, with a certified bound on what any vertex could still gain (equilibrium_slack). There is nothing to tune: no iteration budget, no round cap, no tolerance.
• Every call starts afresh: from singletons, unless you pass initial_membership (or ask for a random start with max_communities, or set a community-count limit). The result is also stored in h.memberships; pass initial_membership=h.memberships to continue from it (see Start states).

What this is

A hedonic game is a coalition-formation game: each vertex is a player, each community is a coalition, and a player prefers the coalition that improves its payoff. Here the payoff is the Constant Potts Model (CPM) score at a resolution γ (by default, the weighted density: the total edge weight over the number of vertex pairs): a vertex gains from its neighbours inside a community and pays γ for every member.

community_hedonic() visits the vertices in random order and lets each one take its best response: move to the community it prefers (disjoint), or choose the set of at most max_memberships communities it prefers (overlapping, where a vertex in k communities takes part in each with intensity 1/√k). After a move it revisits the vertex's neighbours, and it stops only when every vertex has been visited, without moving, since the last move. That is a Nash equilibrium: no vertex can improve its payoff alone. It is the equilibrium this process reaches from its start, not necessarily the partition of highest potential, and another seed can reach another one.

The computation is in floating point, so the core proves each step instead of trusting it: every payoff it compares carries a rigorous bound of its rounding error, and a vertex moves only when its computed gain exceeds that bound. Every move therefore raises the exact CPM potential, and the process always stops. The result reports equilibrium_slack in result._hedonic_provenance: no vertex can raise its exact payoff by more than the slack times the total edge weight (normalized like the quality). For a partition it is exactly 0 whenever no deviation's exact gain can be positive: always when the arithmetic is exact (for instance unit weights at resolution 0, 1/2 or 1), and often otherwise (with integer weights, for instance, when every vertex's best deviation is certainly worse). The result is then an exact Nash equilibrium. A cover's intensities 1/√k are rounded, so its slack stays a tiny positive number, smaller on larger graphs: about 1e-15 on Zachary's karate club and 1e-17 on complete DBLP and Amazon, where partitions get about 1e-21. native/README.md gives the bounds, the termination argument and the proof that the best set of communities is a prefix of the sorted gains.

Game is an igraph.Graph subclass: after wrapping, the usual igraph API (h.vcount(), h.density(), shortest paths, and so on) is still available.

Installation

python -m pip install hedonic

That is the library alone (igraph and numpy). The command line and the paper experiments need the cli extra: python -m pip install "hedonic[cli]".

With uv:

uv add hedonic

Wheels are built for Linux (x86_64, aarch64), macOS (x86_64, arm64) and Windows (AMD64, ARM64). Elsewhere pip builds the C core from the source distribution, which needs a C11 compiler.

The package also installs the experiment command line (hedonic and hedonic-exp, see Command line) and its dependencies (SciPy, pandas, Matplotlib, NetworkX, PyArrow, DuckDB and tqdm). The experiments' robustness audits run on the same C audit as Game.stable_intervals and Game.regrets, so they need nothing more. The command line runs on macOS and Linux; on Windows the package installs the library alone.

Tutorial

1. Wrap a graph as a Game

Game copies an existing igraph.Graph, or is constructed the same way as an igraph graph. Other libraries convert through igraph first.

import igraph as ig
from hedonic import Game

# Famous example graph (Zachary's karate club)
h = Game(ig.Graph.Famous("Zachary"))

# Equivalent shortcut: Graph class methods return a Game
h = Game.Famous("Zachary")

# Edge list
h = Game(n=4, edges=[(0, 1), (1, 2), (2, 3), (3, 0)])

# Adjacency matrix
h = Game(ig.Graph.Adjacency([[0, 1, 1], [1, 0, 1], [1, 1, 0]], mode="undirected"))

# NetworkX (networkx is not a dependency)
import networkx as nx

h = Game(ig.Graph.from_networkx(nx.karate_club_graph()))

Wrapping preserves the graph's directionality and its vertex, edge and graph attributes, and to_igraph() preserves them on the way back. The detector requires an undirected graph without self-loops, so a directed input or a loop is rejected explicitly rather than silently converted (h.simplify(multiple=False, loops=True) removes the loops). Parallel edges are accepted and count as one edge whose weight is their sum.

2. Detect communities

h = Game.Famous("Zachary")

# Disjoint: each vertex has one community id
partition = h.community_hedonic(max_memberships=1, seed=0)
print(partition.membership[:8])
print(partition.sizes())         # community sizes

# Overlapping: each vertex has a list of community ids (length <= max_memberships)
cover = h.community_hedonic(max_memberships=3, seed=0)
print(cover.membership[:4])

Iterate communities as lists of vertex ids:

for community in partition:          # disjoint clusters
    print(list(community))

for community in cover:              # overlapping clusters
    print(list(community))

Start states

initial_membership	Start state
None (default)	singletons; a random start with max_communities; a feasible start under count limits
one label per vertex	that partition
one list of labels per vertex	that cover

Labels are non-negative integers; a start whose labels have gaps ([5, 9, 5]) is renumbered in label order ([0, 1, 0]).

Every call stores its result in h.memberships, one list of labels per vertex, and never reads it back by itself. To continue from the last equilibrium, pass it:

h.community_hedonic(max_memberships=1, seed=0)
cover = h.community_hedonic(max_memberships=3, initial_membership=h.memberships)

A partition is a valid start for a cover. The reverse works too: when a start gives a vertex more labels than max_memberships, the vertex keeps the ones it prefers (ranked by the payoff of each community on the start, ties to the larger community, then the smaller label) and the game runs to an equilibrium from there. A cover passed to max_memberships=1 therefore starts from the partition in which every vertex keeps its preferred community. _hedonic_provenance["projected_vertices"] counts the vertices that had to choose.

3. Parameters

Parameter	Default	Role
initial_membership	None	The start; see Start states
max_memberships	1	1 = partition; >1 = cover with that per-vertex cap; -1 = no practical per-vertex cap
resolution	weighted density	CPM γ: total edge weight over the number of vertex pairs (the density of an unweighted graph). Smaller γ gives larger communities
allow_isolation	False	Allow a vertex to open a new, empty community
max_communities	None	Random disjoint start over [0, max_communities) when no start is given; without isolation the result then has at most that many communities
edge_weights	unweighted	Edge weights (sequence or edge-attribute name; an attribute called weight is used only when named)
node_weights	unweighted	Vertex weights (sequence or vertex-attribute name) in the crowding term γ w_u w_v
max_total_communities	None	At most K occupied communities, in the result and in every state visited
n_communities	None	Exactly K occupied communities
seed	None	Seed of the visiting order (see below)

Everything after initial_membership is keyword-only.

Reproducible runs. With seed=, the visiting order is drawn from random.Random(seed), so the same graph, parameters, start and seed give the same result:

a = h.community_hedonic(seed=7)
b = h.community_hedonic(seed=7)
assert a.membership == b.membership

Without a seed, the order is drawn from Python's random module, which is also python-igraph's default generator, so random.seed(...) before the call makes it repeatable too.

Community-count limits. max_total_communities=K keeps the number of occupied communities at most K, and n_communities=K keeps it exactly K:

partition = h.community_hedonic(n_communities=2)
cover = h.community_hedonic(max_memberships=3, max_total_communities=4)

Both must be positive integers, and n_communities must not exceed max_total_communities. Without a start state, a deterministic feasible start is built; a supplied start that violates a limit (after its projection to max_memberships) is rejected, never silently repaired. The result is then an equilibrium over the moves the limit allows.

What the result carries. Besides the communities, a result has _hedonic_provenance (versions, effective parameters, count mode, and equilibrium_slack) and _hedonic_native_quality, the quality of the returned state: (Φ − γ Σ_v n_v² / 2) / W for its exact potential Φ (a sum over pairs of vertices), the vertex weights n_v and the total edge weight W; the constant term is each vertex's own crowding, the same for every state.

4. Scoring against a reference

evaluate_against scores the stored result against a known cover or partition:

h = Game.Famous("Zachary")
h.community_hedonic(seed=0)
truth = [0] * 17 + [1] * 17          # any flat partition, cover or VertexClustering
print(h.evaluate_against(truth, method="f1"))

Methods: f1 (symmetric best match), one_to_one_f1, jaccard, omega, node_micro_f1 and size_weighted_community_f1; the two built on a one-to-one matching (one_to_one_f1, node_micro_f1) need SciPy. omega is exact when the graph has at most 100,000 vertex pairs and otherwise estimated from that many sampled pairs (omega_sample_size, omega_seed). A flat reference may use any hashable labels, but every vertex needs one (None and NaN are refused); a list of lists is read as communities, each a list of vertices of the graph. Communities with the same members count once, on both sides.

A reference that labels only part of the graph (the SNAP ground-truth covers, for instance) is scored against the whole result by default. restrict_to_reference=True first drops the vertices the reference does not contain from every detected community (hedonic.metrics.restrict_cover), the usual convention for partial ground truth:

h.evaluate_against([[0, 1, 2], [33, 32]], method="f1", restrict_to_reference=True)

The underlying functions are in hedonic.metrics (also reachable as hedonic.metrics after import hedonic).

5. Worked example

import igraph as ig
from hedonic import Game

h = Game(ig.Graph.Famous("Zachary"))

partition = h.community_hedonic(max_memberships=1, seed=0)
print(f"{h.vcount()} vertices, {h.ecount()} edges")
print(f"disjoint communities: {len(partition)}, sizes {partition.sizes()}")

cover = h.community_hedonic(max_memberships=3, seed=0)
print(f"overlapping communities: {len(cover)}")
print(f"memberships of the first five vertices: {cover.membership[:5]}")

Command line: reproducing the experiments

pip install "hedonic[cli]" installs two commands. hedonic is the front door for runs; hedonic-exp list names the analysis and utility commands.

hedonic guide                    # where you are, what to do next, and the topics
hedonic show methods             # the nine overlapping methods and whether each can run here
hedonic run exp                  # a one-minute benchmark (a wizard in a terminal; --dry-run plans)

The disjoint (stochastic block model) paper:

hedonic run disjoint --preset smoke             # the whole pipeline in seconds
hedonic run disjoint --preset paper --dry-run   # the paper's plan, time and disk
hedonic run disjoint --preset paper             # durable and resumable, about a day on 8 cores
hedonic run queue disjoint                      # the same, as a queue: the smoke run, then the paper's grid

Its methods are Hedonic (this package's detector), stock igraph's CPM Leiden (community_leiden, the paper's full Leiden), and four references (ground truth, Mirror, one pass, spectral). Each method's results carry their own identity (its call, parameters and the build that computes it), so a store that already holds some methods keeps them and runs only the missing or changed ones; a store made by another build is refused before anything is written, and --rerun METHOD or --reuse-from STORE (which first replays a sample with this build) says what to do with it.

The overlapping-communities paper is one queue of six runs: the Hedonic calibration, the baseline calibration, the synthetic networks, then, pair by pair, the ground-truth start curve, the robustness spectrum and the benchmark, on eight SNAP graph/cover pairs (the top5000 covers of DBLP, Amazon, YouTube, Friendster and LiveJournal, then the all covers of DBLP, Amazon and YouTube). The robustness spectrum and the start curve read each pair's calibrated resolution from the Hedonic calibration:

hedonic setup --all --dry-run    # the pairs, the eight baselines and the system tools they need
hedonic setup --all              # download the pairs; build and install the baselines
hedonic run queue paper --dry-run
hedonic run queue paper          # detached and resumable; days on a laptop
hedonic run attach paper-queue   # live view (q or Ctrl-C closes the view, never the run)
hedonic run queue paper --report # what is done, run by run and pair by pair

hedonic setup needs git, a C and C++ compiler, CMake and make, a C++ compiler with OpenMP for FOX (Homebrew g++ on macOS) and uv for the locked CDlib environment of ANGEL and Ego-Splitting; --dry-run says which are missing. Starting the queue again continues it: every run decides from its files what is already done. The queue writes under --root DIR (default ~/Databases/Hedonic/Overlap/paper-queue).

The benchmark runs a deterministic method (CoDeSEG at one thread, CliquePerc, FOX, BigCLAM, NcGame, NEO-K-Means) once per pair, at the lowest seed of the plan, since its other seeds would repeat the same cover; Hedonic, ANGEL and Ego-Splitting run at every seed. --replication all runs every method at every seed, to check that determinism; both settings reuse the same records.

Runs execute detached (tmux when available), so closing the terminal does not stop them: hedonic run list, status, attach, logs, stop NAME and resume NAME manage them. Data goes to ~/Databases/Hedonic (HEDONIC_DATA_DIR; SNAP files under Networks/, or HEDONIC_NETWORKS_DIR), overlapping runs to ~/Databases/Hedonic/Overlap (HEDONIC_OUTPUT_DIR), disjoint runs to ~/Databases/Hedonic/Disjoint (HEDONIC_DISJOINT_OUTPUT_DIR), and built method runtimes to ~/.cache/hedonic/codeseg (HEDONIC_CODESEG_CACHE, or hedonic config set cache_dir PATH). hedonic config set output_dir PATH and hedonic config set network_root PATH change them for every run; hedonic config example prints every setting.

Interactive explainer

The web/ directory contains an educational website that explains the disjoint hedonic game visually: friends and strangers on a balance scale, the resolution parameter, the Familiarity Index, the decision tree, the metagraph of all partitions of a four-vertex graph, and better-response walks to a stable equilibrium. It also hosts documentation generated from this package's source. It is a small browser-side teaching model, not a replacement for the detector. From the repository root:

npm install
npm run dev

The GitHub Actions workflow .github/workflows/pages.yml tests, builds and publishes it to GitHub Pages; see web/README.md for details.

Stability of a fixed cover

from hedonic.stability import stable_fraction, stable_measures, nash_window

intervals = h.stable_intervals(max_memberships=3, allow_isolation=True)
regret, tolerance = h.regrets([0.0, h.density(), 0.1, 1.0], max_memberships=3,
                            allow_isolation=True)
stable = regret <= tolerance
linear_share, log_share = stable_measures(intervals)  # log floor: 1e-4
fractions = stable_fraction(intervals, [0.0, 0.1, 1.0])
window = nash_window(intervals)

Both methods default to h.memberships, preserving its community labels. Pass memberships= to audit another cover, or vertices=range(start, stop) to split work across processes. Rows are copied, repeated labels count once, and label gaps are renumbered in numeric order. Every vertex needs a label; complete uncovered vertices explicitly with private labels. The cap defaults to the largest row and cannot be smaller.

The C audit finds closed stable intervals over resolution [0, 1] using the paper's tolerance 1e-10 + 1e-9 * max(1, abs(current_utility)). This differs from the detector's rigorous equilibrium certificate. Audits support unit node weights only, finite non-negative edge weights, and undirected graphs without self-loops. edge_weights=None means ones, as in community_hedonic; pass edge_weights="weight" to audit with an edge attribute. The native implementation computes each vertex's supports on demand, avoiding a global vertex-by-label support table. See the core arithmetic contract.

Development

git clone https://github.com/lucaslopes/hedonic.git
cd hedonic
python -m pip install -e ".[cli]" pytest
python -m pytest
make -C native test            # the C core's own tests (sanitizers)
python tools/native_tests.py   # bit-level pins, as the wheel builds run them

The C core lives in native/ (include/hedonic_core.h is its API, native/README.md its design) and is compiled by setup.py into hedonic._core. Rounding is fixed by the source (explicit fused multiply-adds, computed by the instruction or the core's own correctly rounded soft fma, never the C library's; no contraction), so results are identical across platforms; the wheel workflow (.github/workflows/wheels.yml) checks this on every platform it builds for.

License

This project is distributed under the GNU General Public License, version 3 or later. See LICENSE. The C core was written for this package and contains no code from igraph or other libraries; the binding implements the MT19937 generator step (Matsumoto and Nishimura) as CPython's random module does, so that seeded runs draw exactly what random.Random would.