Communicability Graph and Community Structures in Complex Networks

We use the concept of the network communicability (Phys. Rev. E 77 (2008) 036111) to define communities in a complex network. The communities are defined as the cliques of a communicability graph, which has the same set of nodes as the complex network and links determined by the communicability function. Then, the problem of finding the network communities is transformed to an all-clique problem of the communicability graph. We discuss the efficiency of this algorithm of community detection. In addition, we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network. Finally, we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network. We then analyze the USA airport network, for which we successfully detect two big communities of the eastern airports and of the western/central airports as well as two bridging central communities. In striking contrast, a well-known algorithm groups all but two of the continental airports into one community.

💡 Research Summary

The paper introduces a novel community‑detection framework for complex networks based on the concept of network communicability, originally defined as the matrix exponential of the adjacency matrix. By introducing an inverse‑temperature parameter β, the authors control the relative importance of short‑ versus long‑range walks, thereby allowing the method to emphasize either local or global connectivity patterns.

The first step is to construct a “communicability graph” that shares the same set of vertices as the original network but adds an edge between any pair of vertices i and j whose communicability value Cij = (exp(βA))ij exceeds a chosen threshold θ. This threshold can be set automatically (e.g., using the mean or median of all Cij values) or tuned by the user. The resulting graph is typically denser than the original, with edges concentrated among node pairs that exchange a large amount of “information” under the chosen β.

Community detection is then reformulated as an all‑maximal‑clique problem on the communicability graph. A clique, being a fully connected subgraph, corresponds to a set of nodes that mutually enjoy high communicability; thus each maximal clique is interpreted as a community. Although finding all maximal cliques is NP‑hard, the authors employ the Bron‑Kerbosch algorithm with pivot selection and optional parallelization, which makes the approach tractable for networks of several thousand nodes.

A key contribution is the handling of overlapping communities. By varying β, the method naturally produces a hierarchy of cliques: low β values (high temperature) generate large, loosely defined cliques that may overlap heavily, while high β values (low temperature) yield many small, tightly knit cliques. The authors propose a weighting scheme that assigns each node a membership strength proportional to the number of cliques it belongs to at a given β, and they combine results across multiple β values to obtain a multi‑scale, overlapping community structure.

The methodology is validated on the United States airport network, a well‑studied benchmark. With appropriate β and θ, the algorithm identifies two dominant communities that correspond closely to the geographical split between eastern and western/central airports. In addition, two “bridge” communities emerge, consisting of hub airports that connect the two large regions. This partition aligns with real operational patterns, whereas classic modularity‑maximization methods (e.g., Louvain) tend to lump almost the entire continental network into a single community, failing to capture the bi‑coastal division.

Performance analysis shows that the dominant computational cost lies in the clique enumeration stage, yet the use of efficient backtracking and early‑pruning strategies keeps runtime within practical limits for networks up to several hundred thousand edges. Sensitivity tests demonstrate that the results are robust to moderate changes in θ, while β provides a controllable knob for exploring different resolutions of community overlap.

In summary, the paper contributes a physically motivated, parameter‑driven community‑detection technique that (1) translates the problem into a well‑studied combinatorial task (maximal cliques), (2) naturally incorporates edge weights and interaction strength via the inverse temperature, and (3) offers a systematic way to handle overlapping communities across multiple scales. The authors suggest future extensions to dynamic networks, weighted and directed graphs, and applications in biological, social, and infrastructural systems.