From Relational Data to Graphs: Inferring Significant Links Using Generalized Hypergeometric Ensembles

Advances in data sensing and collection give rise to an increasing volume of data that capture dyadic relations between elements or actors in social, natural, and technical systems. While it is common to apply graph mining and network analysis to such relational data, it is often questionable whether the application of these techniques is actually justified. Consider, for instance, various forms of time series data, which not only tell us which elements of a complex system are related but also when or in which order relations occur. Such data give rise to temporal networks, which question the application of widely used network-based modeling and data mining techniques [13, 24, 26, 27, 30]. Apart from temporal information, we often have access to data that capture multiple types of relations or interactions. The resulting multi-layer network topologies give rise to complications that threaten standard techniques, e.g., to infer and analyze social networks, detect community structures, or to model and control dynamical processes in networked systems [3, 7, 16, 28, 35].

The challenges outlined above are due to the growing availability of additional information – such as time-stamped, sequential or multi-dimensional relational data – which must be incorporated into network-based techniques to model and analyze relational data. However, we are often confronted with situations in which we lack information that is needed to interpret observed relations. Consider, for instance, data sets that capture the simultaneous presence of two users at the same location, the joint expression of two genes in a DNA microarray, or the co-occurrence of two words in the same document. Each of these observed relations can either be due to an underlying social tie, a functional relationship between genes, a semantic link between two words, or it could simply have occurred by mere chance. Rather than naîvely analyzing such data from the perspective of graphs or networks, we should thus treat them as noisy observations that may or may not indicate true relations between a system’s elements.

Principled and efficient methods to solve this network inference problem are of major importance for the modeling and analysis of social networks, the reconstruction of biological networks, and the mining of semantic structures in information systems. The problem has received significant attention from the data mining and machine learning community, as well as from researchers in graph theory and network science. Especially in the latter community, the problem is commonly addressed using statistical ensembles, i.e., generative stochastic models of graphs that can be used for inference, learning and modeling tasks. A common issue of these techniques is that the underlying statistical ensembles are not analytically tractable, thus requiring time-consuming numerical simulations and Monte-Carlo sampling techniques.

To address this problem, in this short paper we propose generalized hypergeometric ensembles (gHypE), a novel framework of statistical ensembles to infer significant links in relational data. The framework can be viewed as generalization of the configuration model, which is commonly used to generate random graph topologies with a given sequence of node degrees. Our framework extends this state-of-the-art graph-theoretic approach in two ways. First, it provides analytically tractable probability spaces of directed and undirected multi-edge graphs, eliminating the need for expensive numerical simulations. Second, it allows to account for known factors that influence the occurrence of interactions, such as known group structures, similarities between elements, or other forms of biases. We demonstrate our framework in two real-world data sets that capture spatio-temporal proximities of actors in a social system. The results show that our framework provides interesting new perspectives for the mining and learning in graphs.

The problem of inferring significant links in relational data has been addressed in a number of works. In the following, we coarsely categorize them into three lines of research.

Applying predictive analytics techniques, a first set of works studied the problem from the perspective of link prediction [17]. In [29], a supervised learning technique is used to predict types of social ties based on unlabeled interactions. The authors of [25] show that tensor factorization techniques allow to infer international relations from data that capture how often two countries co-occur in news reports. In [33], a link-based latent variable model is used to predict friendship relations using data on social interactions.

Using the special characteristics of time-stamped social interactions or geographical co-occurrences, a second line of works has additionally accounted for spatio-temporal information. Studying data on time-stamped proximities of students at MIT campus, the authors of [8] show that the temporal and spatial distribution of proximity events allows to infer social ties with high accuracy. In [5], a model that captures location diversity, regularity, intensity and duration is used to predict social ties based on co-location events. An entropy-based approach taking into account the diversity of interactions’ locations has been used in [22].

Addressing scenarios where neither training data nor spatio-temporal information is available, a third line of works is based on generative models for random graphs. Such models can be used as null models for observed dyadic interactions, which help us to assess whether the relations between a given pair of elements occur significantly more often than expected. Existing works in this area typically rely on standard modeling frameworks, such as exponential random graphs [4, 23], or the configuration model for graphs with given degree sequence or distribution [18]. On the one hand, these approaches provide statistically principled network inference and learning methods for general relational data [2, 12, 19, 32]. On the other hand, the underlying generative models are often not analytically tractable, thus requiring expensive numerical simulations [19, 23]. Proposing a framework of analytically tractable generative models for directed and undirected multi-edge graphs, in this work we close this research gap.

In the following we introduce our framework step by step. For this, let us first consider a data set consisting of repeated dyadic interactions (i, j), which have been observed between two nodes i and j. Such a data set can be represented as a multi-edge, or weighted, network \(G=(V,E)\), where V is a set of n nodes, and \(E \subseteq V \times V\) is a multi-set of (directed or undirected) edges. Let us further define an adjacency matrix \(\hat{\mathbf {A}}\), where entries \(\hat{A}_{ij}\in \mathbb N_{0}\) capture the weight of an edge \((i,j)\in V \times V\), i.e., the multiplicity of an edge (i, j) in the multi-set E. For each node \(i \in V\) we further define the (weighted) in-degree \(\hat{k}_{\mathrm {in}}(i) := \sum _{j \in V} \hat{A}_{ji}\) and the (weighted) out-degree \(\hat{k}_{\mathrm {out}}(i) := \sum _{j \in V} \hat{A}_{ij}\).

Rather than directly applying graph mining and learning techniques to such a weighted graph G, in the following we are interested in a crucial question: Which of the links between nodes are significant, i.e., which of the observed weights \(A_{ij}\) go beyond what is expected at random, given (i) the total number of observed interactions, and (ii) the number of times individual nodes engage in interactions? To answer this question, we take the common approach of defining a stochastic model that generates a so-called statistical ensemble, i.e., a probability space of graphs. Different from existing approaches, where link weights are assumed to be continuous (e.g. [1, 6]), we are interested in a statistical ensemble that (i) can handle directed and multi-edge graphs, (ii) is analytically tractable, and (iii) thus allows us to assess the significance of links in a theoretically principled way.

Our construction of a statistical ensemble follows the general idea of the Molloy-Reed configuration model, which is to randomly shuffle the topology of a given network G while preserving the observed node degrees. For this, the configuration model generates edges between randomly sampled pairs of nodes in such a way that the exact observed degrees of all nodes are preserved. Different from this approach, we assume a sampling of m multi-edges such that the sequence of expected degrees of nodes is preserved. For this, for each pair of nodes i and j, we first define the maximum number \(\varXi _{ij}\) of multi-edges that can possibly exist between nodes i and j as \(\varXi _{ij} := \hat{k}_{\mathrm {out}}(i) \hat{k}_{\mathrm {in}}(j)\) (cf. [15, 20]). The maximally possible numbers of links between all pairs of nodes can then be conveniently represented in matrix form as \(\mathbf {\Xi } := \left( \varXi _{ij}\right) _{i,j \in V}\).

Note that for the special case of a uniform dyadic propensity matrix \(\mathbf {\Omega } \equiv \text {const}\), we recover Eq. 1 for the unbiased case, i.e., where all dyadic propensities are identical. We thus obtain a general framework of statistical ensembles which (i) allows to encode arbitrary a priori tendencies of nodes to interact, and (ii) provides an analytical expression for the probability to observe a given number of interactions between any pair of nodes.

In the following, we demonstrate how our framework can be used to infer significant links in two relational data sets: (RM) captures time-stamped proximities between students and faculty at MIT [9] recorded via smart devices. (ZKC) covers frequencies of self-reported encounters between members of a university Karate club collected by Wayne Zachary [34]. We denote the weighted adjacency matrix capturing observed dyadic interactions as \(\mathbf {\hat{A}}\). For a given significance threshold \(\alpha \), we then identify significant links by filtering matrix \(\mathbf {\hat{A}}\) by a threshold \(\Pr (A_{ij} \le \hat{A}_{ij}) > 1 - \alpha \) based on Eq. 3. This can be seen as assigning p-values to dyads (i, j), obtaining a high-pass noise filter for entries in the adjacency matrix.

Open image in new window

To illustrate our approach, Fig. 1(a) shows the entries of the (original) adjacency matrix \(\mathbf {A}\) for (RM). The high-pass noise filter resulting from our methodology (using \(\alpha =0.01\)) is shown in Fig. 1(b), where black entries correspond to pairs of nodes with non-significant links. The application of this filter to the original matrix yields the noise-filtered matrix shown in Fig. 1(c). While in the full network there are 721, 889 observed multi-edges amounting to 2, 952 distinct links, after filtering there are 626 \((21.2\%)\) significant links left (617, 069 multi-edges, \(85.5\%\) of the original). We validate the benefit of filtering the original interactions in (RM) by comparing the output of a standard community detection algorithm – the degree-corrected block model [21] – in (i) the original, unfiltered graph shown in Fig. 1(d), and (ii) the filtered, significant graph shown in Fig. 1(f). Using known classes of students and affiliations of staff members as ground truth allows us to compare the quality of the community detection. Figure 1(e) shows the set overlaps between the ground truth labels (middle column) and detected partitions in the unfiltered (left column) and filtered graph (right column). Due to the high number of non-significant links in the unfiltered graph, the algorithm only detects three partitions, each spanning multiple labs and classes. In contrast, applying the algorithm to the filtered graph yields six partitions that better capture the ground truth lab and class structure (cf. Fig. 1(e)). As expected, detected partitions do not perfectly correspond to the ground truth, since labs and classes are likely not the only driving force behind observed proximities.

Open image in new window

A major advantage of gHypEs is that, by specifying a non-uniform matrix \(\mathbf {\Omega }\), we can additionally encode known factors that influence the occurrence of interactions between nodes, while still obtaining an analytically tractable ensemble. In our second illustrative example, we use this to encode the known structure of two separate Karate classes in the (ZKC) data. These two classes naturally influence the frequency of encounters between actors beyond what would be expected “at random”. We incorporate this prior knowledge via a block matrix \(\mathbf {\Omega }\) that assigns higher dyadic propensities to pairs of actors in the same class (cf. [3]). This approach allows to establish a “random baseline” accounting both (i) for combinatorial effects due to heterogeneous node degrees, and (ii) the known group structure in the data. Using a significance threshold of \(\alpha =0.01\), for (ZKC) this yields the striking result that only 8 out of 78 observed links are significant (\(\sim 90\%\) of 231 observed multi-edges are filtered out, cf. Fig. 2). In other words, taking into account the partitioning of members in two classes for (ZKC) almost all encounters between club members can simply be explained by random effects. Figure 2 compares the original weighted network, illustrated in Fig. 2(a), and the filtered network, in Fig. 2(b).

In this short paper we introduce gHypEs, a broad class of statistical ensembles of graphs that can be used to infer significant links from noisy data. Our work makes three important contributions: First, we provide an analytically tractable statistical model of directed and undirected multi-edge graphs that can be used for inference and learning tasks. Second, the formulation of our ensemble highlights a – to the best of our knowledge – previously unknown relation between random graph theory and Wallenius‘non-central hypergeometric distribution. And finally, different from existing statistical ensembles such as, e.g., the configuration model, our framework can be used to encode prior knowledge on factors that influence the formation of relations. This flexible approach allows for a tuning of the “random baseline”, opening perspectives for a statistically principled network inference that accounts for effects that are not purely random. We thus argue that our work advances the theoretical foundation for the mining of relational data on social systems. It further highlights that principled model selection and hypothesis testing are crucial prerequisites that should precede the application of network-based data mining and modeling techniques.