A benchmarking tool for the generation of bipartite network models with overlapping communities

Abstract

Many real-world networks display hidden community structures with important potential implications in their dynamics. Many algorithms highly relevant to network analysis have been introduced to unveil community structures. Accurate assessment and comparison of alternative solutions are typically approached by benchmarking the target algorithm(s) on a set of diverse networks that exhibit a broad range of controlled features, ensuring the assessment contemplates multiple representative properties. Tools have been developed to synthesize bipartite networks, but none of the previous solutions address the issue of generating networks with overlapping community structures. This is the motivation for the BNOC tool introduced in this paper. It allows synthesizing bipartite networks that mimic a wide range of features from real-world networks, including overlapping community structures. Multiple parameters ensure flexibility in controlling the scale and topological properties of the networks and embedded communities. BNOC’s applicability is illustrated assessing and comparing two popular overlapping community detection algorithms on bipartite networks, namely HLC and OSLOM. Results reveal interesting features of the algorithms in this scenario and confirm the relevant role played by a suitable benchmarking tool. Finally, to validate our approach, we present results comparing networks synthesized with BNOC with those obtained with an existing benchmarking tool and with already established sets of synthetic networks, in two different scenarios.

Appendix A HNOC: Extension to k-partite and heterogeneous networks

As in the case of bipartite networks there is a lack of benchmarking tools to create k-partite and heterogeneous networks for assessing community detection and other algorithms. k-partite networks have k vertex types, rather than just two, with edges connecting only vertices of different types. In heterogeneous networks, this second restriction is dropped, i.e., edges can occur between vertices of the same type. Since they are direct generalizations of bipartite networks, we extended BNOC to synthesize general heterogeneous networks. This extension, called HNOC, can be useful to generate HIN models to support development and validation of new methods. HNOC inherits BNOC’s major features as a flexible and robust resource to synthesize a variety of benchmarking networks with distinct properties in reasonable times.

A heterogeneous information network (HIN) [48] consists of m disjoint subsets of vertices of different types (called layers) and edges connecting these elements. A network $\mathcal {X}$ with m vertex types can be partitioned into subsets $X_i = \{ x_{i,1}, \dots , x_{i,n_i} \}$ for each type i. A HIN is represented as a graph $G = (V, E, W)$, where $V = \bigcup _{i=1}^m X_i$ with $m > 2$, E is the set of edges, and W is the set of edge weights.

HINs have a inherently complex structure that can be difficult to handle and visualize. The structure of connections can be described by a network schema [48], which defines a meta-template for the network that describes its vertex and connection types. Given a graph G, its network schema, denoted $T_G(\mathcal {A}, \mathcal {R})$, is a directed graph defined over element types $\mathcal {A}$ with edges as relations from $\mathcal {R}$, obtained via mapping functions $\varphi : V \rightarrow \mathcal {A}$ and $\psi : E \rightarrow \mathcal {R}$, respectively. Figure 18 exemplifies network schemas for a bipartite network, a k-partite network and an heterogeneous network.

Fig. 18

Network schemas: a bipartite network schema; bk-partite network schema; and c heterogeneous network schema

The schema explicitly identifies the m vertex types and the r connection types. Each connection type is described by the types of the two endpoints and the connection meaning, since pairs of entities can admit multiple types of connections, as in the case of multi-relational networks [56]. Thus, a HIN can be interpreted as a composition of r bipartite (or homogeneous, if both endpoints of the connections are of the same type) networks, as illustrated in Fig. 19.

Fig. 19

A HIN described as r bipartite/homogeneous networks

Following this interpretation, we extended BNOC to generate synthetic heterogeneous networks. The extension iterates over each pair of vertex and connection types specified in the network schema, employing similar steps to build the communities in each iteration:

• 1. Execute m iterations of BNOC’s Steps 1 and 2 to build each layer i with $V_i$ vertices, set a single community on each layer and introduce the overlapping structures.

• 2. For each pair of layers specified in the schema, execute BNOC’s Steps 3, 4 and 5 in order to establish the specified connections, weights, density and noise levels.

Since heterogeneous networks have multiple layers and multiple connection types, the extension required modifying some BNOC parameters and introducing a few additional parameters, as described in Table 3. The following parameters were added: the number of layers m and the set of connected layers, henceforth called “schema”, e. Furthermore, the dispersion and noise parameters must be defined for each schema, since each iteration handles a pair of connected layers.

Table 3 Modified and additional parameters in HNOC

Figure 20 illustrates two networks created with distinct parameter combinations. Unless informed otherwise, the parameter settings correspond to the default values informed in Tables 1 and 3. Figure 20a depicts a 4-partite network with some community overlapping.

Fig. 20

Heterogeneous networks generated with HNOC presenting distinct topological structures and properties: red squares depict overlapping vertices and colored circles indicate non-overlapping vertices and their assigned community; line widths reflect the corresponding edge weights. a a 4-partite network with $v=[15,15,15,15]$, $e=[(0,1), (1,2), (2,3), (3,1)]$ and $x=[8,3,0,1]$; b a heterogeneous network obtained with settings $v=[40,25,15]$, $e=[(0,1), (1,2), (2,2)]$, $c=[2,2,4]$, and $d=[0.45, 0.85, 0.15, 0.15]$. The network drawings were obtained based on the technique described by [50] (color figure online)

The network was built so that all layers have the same number of communities, the probability set to produce balanced communities, different numbers of overlapping vertices in each layer, and the same dispersion (edge density) d in all pairs of connected layers. Figure 20b illustrates a 3-partite network with heterogeneous structure and edges between vertices of the same type in one of the layers. The example can describe a hypothetical author-paper-term network, in which authors are connected with their papers, terms are connected with their neighboring terms in the text and with the papers in which they appear. The upper left, central and upper right layers, represent, respectively, the Term, Paper and Author entities. The network has been created so that the connections between the different pairs of entities display different patterns, e.g., there is a dense topology of connections between terms and papers, whereas connections between terms and terms, or between authors and papers are sparser. The schema is inspired in the largely used real-world data of DBLP¹² (Digital Bibliography & Library Project), a computer science bibliographic dataset that relates documents, authors and terms.