Identification of Overlapping Communities by Locally Calculating Community-Changing Resolution Levels

An algorithm for the detection of overlapping natural communities in networks was proposed by Lancichinetti, Fortunato, and Kertesz (LFK) last year. The LFK algorithm constructs natural communities of (in principle) all nodes of a graph by maximising the local fitness of communities. The resulting modules can overlap. The generation of communities can easily be repeated for many values of resolution; thus allowing different views on the network at different resolutions. We implemented the main idea of the LFK algorithm—to generate natural communities of each node of a network—in a different way. We start with a value of the resolution parameter that is high enough for each node to be its own natural community. As soon as the resolution is reduced, each node acquires other nodes as members of its community, i.e. natural communities grow. For each community found at a certain resolution level we calculate the next lower resolution where a node is added. After adding a node to a community of a seed node we check whether it is also the natural community of a node that we have already analysed. In this case, we can stop expanding the seed node’s community. We tested our algorithm on a small benchmark graph and on a network of about 500 papers in information science (weighted with the Salton index of bibliographic coupling). In our tests, this approach results in characteristic ranges of resolution where a large resolution change does not lead to a growth of the natural community. Such stable modules were also obtained by applying the LFK algorithm but since we determine communities for all resolution values in one run, our approach is faster than the LFK reference. And our algorithm reveals the hierarchical structure of the graph more easily.

💡 Research Summary

The paper presents MONC (Merging Overlapping Natural Communities), a novel algorithm for detecting overlapping communities in networks that builds on the core idea of the Lancichinetti‑Fortunato‑Kertesz (LFK) method but overcomes its major computational limitation.
LFK constructs a “natural community” for every node by locally maximizing a fitness function that depends on a resolution parameter α. To explore the full hierarchy of covers, LFK must be run repeatedly for many α values, which becomes prohibitive for large graphs. MONC instead starts at a very high resolution where each node is its own community and then gradually lowers α. At each step it adds the neighbor that would allow the community to remain optimal up to the highest possible α (denoted α_incl). This α_incl is derived analytically (log‑ratio formula) from the change in internal degree (k_in) and total degree (k_tot) when a candidate node is added.
Key differences from LFK:

1. No exclusion step – MONC never removes nodes once they have been added, preserving the locality principle and avoiding the paradox where a seed node could be expelled from its own community.
2. Modified fitness – a constant “+1” is added to the numerator of the fitness function to prevent the trivial zero fitness of single‑node communities.
3. Clique‑based seeding – to avoid early expansion into low‑density regions, MONC initializes seeds from maximal cliques and then prunes the weakest members (using an α_excl measure) until a tightly‑bound sub‑clique remains. This yields more robust starting points than single nodes.
4. Duplicate detection – after each growth step the algorithm checks whether the newly formed community coincides with any already discovered community. If so, the corresponding seed’s further expansion is halted, dramatically reducing redundant computation.
   The algorithm proceeds iteratively: (i) initialise each node (or optimal clique) as a community G, (ii) compute α_incl for all neighbors in N(G), (iii) add the neighbor with maximal α_incl, (iv) update G and its neighborhood, (v) repeat until N(G) is empty, and (vi) perform duplicate checks.
   The authors evaluate MONC on two datasets. First, the classic Zachary karate‑club network (34 nodes, unweighted). MONC reproduces the same overlapping cover reported by LFK in the resolution interval α≈0.76–0.84 and, in some resolution ranges, yields a more consistent overlap than the random‑seed version of LFK. Second, a bibliographic‑coupling network of 492 information‑science papers (2008 volume) where edge weights are Salton cosine similarities of reference lists. MONC discovers stable modules—intervals of α where community composition does not change—indicating robust thematic clusters. Because MONC computes communities for the entire α spectrum in a single run, its runtime is substantially lower than LFK’s repeated scans.
   Overall, MONC offers four major advantages: (1) full‑resolution exploration in one pass, (2) strict locality by omitting node exclusion, (3) improved seed selection via optimized cliques, and (4) early termination through duplicate detection. These features make MONC a practical tool for large‑scale networks where overlapping and hierarchical community structure are of interest, such as citation, co‑authorship, or biological interaction graphs.