Graph Modularity Research Articles

Modularity based community detection in hypergraphs

Abstract In this paper, we propose a scalable community detection algorithm using hypergraph modularity function, h–Louvain. It is an adaptation of the classical Louvain algorithm in the context of hypergraphs. We observe that a direct application of the Louvain algorithm to optimize the hypergraph modularity function often fails to find meaningful communities. We propose a solution to this issue by adjusting the initial stage of the algorithm via carefully and dynamically tuned linear combination of the graph modularity function of the corresponding two-section graph and the desired hypergraph modularity function. The process is guided by Bayesian optimization of the hyper-parameters of the proposed procedure. Various experiments on synthetic as well as real-world networks are performed showing that this process yields improved results in various regimes.

Graph analysis uncovers an opposing impact of methylphenidate on connectivity patterns within default mode network sub-divisions

BackgroundThe Default Mode Network (DMN) is a central neural network, with recent evidence indicating that it is composed of functionally distinct sub-networks. Methylphenidate (MPH) administration has been shown before to modulate impulsive behavior, though it is not yet clear whether these effects relate to MPH-induced changes in DMN connectivity. To address this gap, we assessed the impact of MPH administration on functional connectivity patterns within and between distinct DMN sub-networks and tested putative relations to variability in sub-scales of impulsivity.MethodsFifty-five right-handed healthy adults underwent two resting-state functional MRI (rs-fMRI) scans, following acute administration of either MPH (20 mg) or placebo, via a randomized double-blind placebo-controlled design. Graph modularity analysis was implemented to fractionate the DMN into distinct sub-networks based on the impact of MPH (vs. placebo) on DMN connectivity patterns with other neural networks.ResultsMPH administration led to an overall decreased DMN connectivity, particularly with the auditory, cinguloopercular, and somatomotor networks, and increased connectivity with the parietomedial network. Graph analysis revealed that the DMN could be fractionated into two distinct sub-networks, with one exhibiting MPH-induced increased connectivity and the other decreased connectivity. Decreased connectivity of the DMN sub-network with the cinguloopercular network following MPH administration was associated with elevated impulsivity and non-planning impulsiveness.ConclusionCurrent findings highlight the intricate effects of MPH administration on DMN rs-fMRI connectivity, uncovering its opposing impact on distinct DMN sub-divisions. MPH-induced dynamics in DMN connectivity patterns with other neural networks may account for some of the effects of MPH administration on impulsive behavior.

Seeing Is Believing: Brain-Inspired Modular Training for Mechanistic Interpretability.

We introduce Brain-Inspired Modular Training (BIMT), a method for making neural networks more modular and interpretable. Inspired by brains, BIMT embeds neurons in a geometric space and augments the loss function with a cost proportional to the length of each neuron connection. This is inspired by the idea of minimum connection cost in evolutionary biology, but we are the first the combine this idea with training neural networks with gradient descent for interpretability. We demonstrate that BIMT discovers useful modular neural networks for many simple tasks, revealing compositional structures in symbolic formulas, interpretable decision boundaries and features for classification, and mathematical structure in algorithmic datasets. Qualitatively, BIMT-trained networks have modules readily identifiable by the naked eye, but regularly trained networks seem much more complicated. Quantitatively, we use Newman's method to compute the modularity of network graphs; BIMT achieves the highest modularity for all our test problems. A promising and ambitious future direction is to apply the proposed method to understand large models for vision, language, and science.

Analyzing teacher-student interactions through graph theory applied to hyperscanning fNIRS data.

Teacher-student relationships have been found consistently important for student school effectiveness in mathematics in the last three decades. Although this observation is generally made from the teacher's perspective, neuroscience can provide new insights by establishing the neurobiological underpinning of social interactions. This paper further develops this line of research by utilizing graph theory to represent interactions between teachers and students at the neural level. Through hyperscanning with functional near-infrared spectroscopy (fNIRS), we collected data from the prefrontal cortex and the temporoparietal junction of 24 dyads composed of a teacher and a student. Each dyad used a board game to perform a programming logic class that consisted of three steps: independent activities (control), presentation of concepts, and interactive exercises. Graph theory provides results regarding the strength of teacher-student interaction and the main channels involved in these interactions. We combined graph modularity and bootstrap to measure pair coactivation, thus establishing the strength of teacher-student interaction. Also, graph centrality detects the main brain channels involved during this interaction. In general, the teacher's most relevant nodes rely on the regions related to language and number processing, spatial cognition, and attention. Also, the students' most relevant nodes rely on the regions related to task management.

Localized multiple kernel learning using graph modularity

• A graph-based heuristic approach for multiple kernel learning (MKL). • Assigns sample-specific kernel weights based on contribution to graph modularity . • Classification performance comparable to state-of-the-art MKL algorithms. • Outperforms other heuristic approaches and has low time complexity. Multiple kernel learning (MKL) algorithms exploit information from multiple feature representations by assigning weights to each representation in the kernel space, and later combining them. However, this ignores the fact that data points exhibit locally varying characteristics. To address this problem, localized MKL algorithms learn locality-specific kernel weights which determine each base kernel’s influence in the locality under consideration. Here, we relate the problem of determining the relevance of base kernels for classification to that of quantifying community structure in graphs. Next, we derive sample-specific kernel weights using graph modularity. Through experiments on publicly available datasets, we show that the proposed approach offers a viable alternative to state-of-the-art MKL approaches while being computationally inexpensive.

The modularity of random graphs on the hyperbolic plane

Abstract Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this article, we consider this quantity on a probabilistic model of complex networks introduced by Krioukov et al. (2010, Phys. Rev. E, 82, 036106). This model views a complex network as an expression of hidden popularity hierarchies (i.e. nodes higher up in the hierarchies have more global reach), encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to one in probability.

MetaProb 2: Metagenomic Reads Binning Based on Assembly Using Minimizers and K-Mers Statistics.

Current technologies allow the sequencing of microbial communities directly from the environment without prior culturing. One of the major problems when analyzing a microbial sample is to taxonomically annotate its reads to identify the species it contains. The major difficulties of taxonomic analysis are the lack of taxonomically related genomes in existing reference databases, the uneven abundance ratio of species, and sequencing errors. Microbial communities can be studied with reads clustering, a process referred to as genome binning. In this study, we present MetaProb 2 an unsupervised genome binning method based on reads assembly and probabilistic k-mers statistics. The novelties of MetaProb 2 are the use of minimizers to efficiently assemble reads into unitigs and a community detection algorithm based on graph modularity to cluster unitigs and to detect representative unitigs. The effectiveness of MetaProb 2 is demonstrated in both simulated and real datasets in comparison with state-of-art binning tools such as MetaProb, AbundanceBin, Bimeta, and MetaCluster. On real datasets, it is the only one capable of producing promising results while being parsimonious with computational resources.

Modularity in Biological Networks.

Network modeling, from the ecological to the molecular scale has become an essential tool for studying the structure, dynamics and complex behavior of living systems. Graph representations of the relationships between biological components open up a wide variety of methods for discovering the mechanistic and functional properties of biological systems. Many biological networks are organized into a modular structure, so methods to discover such modules are essential if we are to understand the biological system as a whole. However, most of the methods used in biology to this end, have a limited applicability, as they are very specific to the system they were developed for. Conversely, from the statistical physics and network science perspective, graph modularity has been theoretically studied and several methods of a very general nature have been developed. It is our perspective that in particular for the modularity detection problem, biology and theoretical physics/network science are less connected than they should. The central goal of this review is to provide the necessary background and present the most applicable and pertinent methods for community detection in a way that motivates their further usage in biological research.

Molecular Graph Modularity as a Descriptor for Property Estimation—Application to the Viscosity of Biomass-Derived Molecules

Molecular Graph Modularity as a Descriptor for Property Estimation—Application to the Viscosity of Biomass-Derived Molecules

Exact modularity of line graphs of complete graphs

Modularity is designed to measure the strength of a division of a network into clusters. For n∈ℕ, consider a set S of n elements. Let V be the set of all subsets of S of size r. Consider the graph G(n,r,s) with vertices V and edges that connect two vertices if and only if their intersection has size s. In this article, we find the exact modularity of G(n,2,1) for n≥5.

Community-Aware Graph Signal Processing: Modularity Defines New Ways of Processing Graph Signals

The emerging field of graph signal processing (GSP) allows one to transpose classical signal processing operations (e.g., filtering) to signals on graphs. The GSP framework is generally built upon the graph Laplacian, which plays a crucial role in studying graph properties and measuring graph signal smoothness. Here, instead, we propose the graph modularity matrix as the centerpiece of GSP to incorporate knowledge about graph community structure when processing signals on the graph but without the need for community detection. We study this approach in several generic settings, such as filtering, optimal sampling and reconstruction, surrogate data generation, and denoising. Feasibility is illustrated by a small-scale example and a transportation network data set as well as one application in human neuroimaging where community-aware GSP reveals relationships between behavior and brain features that are not shown by Laplacian-based GSP. This work demonstrates how concepts from network science can lead to new, meaningful operations on graph signals.

Modularity of Some Distance Graphs

New bounds on the modularity of distance graphs were obtained and the exact value of modularity was calculated for G(n, 2, 1) graphs.

Sleep EEG signal analysis based on correlation graph similarity coupled with an ensemble extreme machine learning algorithm

BackgroundSleep plays an essential role in repairing and healing human mental and physical health. Developing an efficient method for scoring electroencephalogram (EEG) sleep stages is expected to help medical specialists in the early diagnosis of sleep disorders. MethodIn this paper, a novel technique is proposed for classifying sleep stages EEG signals using correlation graphs. First, each 30 s EEG segment is divided into a set of sub-segments. The dimensionality of each sub-segment is reduced by using a statistical model. Second, each EEG segment is transferred into a graph considering each sub-segment as a node in a graph, and a link between each pair of nodes is calculated based on their correlation coefficient. Graph's modularity is used as input features into an ensemble classifier. ResultsDifferent community detection algorithm based correlation graph are investigated to discern the most effective features to reveal the differences between EEG sleep stages. A combination of various classification techniques: a least square vector machine (LS-SVM), k-means, Naïve Bayes, Fuzzy C-means, k-nearest, and logistic regression are tested using multi criteria decision making (MCDM) to design an ensemble classifier. Based on the results of the MCDM, the best four: LS-SVM, Naïve Bayes, logistic regression and k-nearest are integrated, to finally utilise as an ensemble classifier to categorise the graph's characteristics. The results obtained from the ensemble classifier are compared with those from the individual classifiers. The performance of the proposed method is compared with state of the art of sleep stages classification. The experimental results showed that the EEG sleep classification based on correlation graphs are able to achieve better recognition results than the existing state of the art techniques.

Large-Scale Dynamic Social Network Directed Graph K-In&Out-Degree Anonymity Algorithm for Protecting Community Structure

Social network data publishing is dynamic, and attackers can perform association attacks based on social network directed graph data at different times. The existing social network privacy protection technology has low performance in dealing with large-scale dynamic social network directed graph data, and anonymous data publishing does not meet the needs of community structure analysis. A Dynamic Social Network Directed Graph K-In&Out-Degree Anonymity (DSNDG-KIODA) method to protect community structure is proposed. The method is based on the dynamic grouping anonymity rule to anonymize the dynamic K-in&out-degree sequence, and the virtual node distribution is added in parallel to construct an anonymous graph. The node information is transmitted based on the GraphX, and the virtual node pairs are selected and deleted according to the change of the directed graph modularity to reduce information loss. The experimental results show that the DSNDG-KIODA method improves the efficiency of processing large-scale dynamic social network directed graph data, and ensures the availability of community structure analysis when data is released.

Graph configuration model based evaluation of the education-occupation match.

To study education—occupation matchings we developed a bipartite network model of education to work transition and a graph configuration model based metric. We studied the career paths of 15 thousand Hungarian students based on the integrated database of the National Tax Administration, the National Health Insurance Fund, and the higher education information system of the Hungarian Government. A brief analysis of gender pay gap and the spatial distribution of over-education is presented to demonstrate the background of the research and the resulted open dataset. We highlighted the hierarchical and clustered structure of the career paths based on the multi-resolution analysis of the graph modularity. The results of the cluster analysis can support policymakers to fine-tune the fragmented program structure of higher education.

Modularity of regular and treelike graphs

Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the modularity of two kinds of graphs. For $r$-regular graphs with a given number of vertices, we investigate the minimum possible modularity, the typical modularity, and the maximum possible modularity. In particular, we see that for random cubic graphs the modularity is usually in the interval $(0.666, 0.804)$, and for random $r$-regular graphs with large $r$ it usually is of order $1/\sqrt{r}$. These results help to establish baselines for statistical tests on regular graphs. The modularity of cycles and low degree trees is known to be close to 1: we extend these results to `treelike' graphs, where the product of treewidth and maximum degree is much less than the number of edges. This yields for example the (deterministic) lower bound $0.666$ mentioned above on the modularity of random cubic graphs.

Graph modularity maximization as an effective method for co-clustering text data

In this paper we show how the modularity measure can serve as a useful criterion for co-clustering document-term matrices. We present and investigate the performance of CoClus, a novel, effective block-diagonal co-clustering algorithm which directly maximizes this modularity measure. The maximization is performed using an iterative alternating optimization procedure, in contrast to algorithms that use spectral relaxations of the discrete optimization problems. Extensive comparative experiments performed on various document-term datasets demonstrate that this approach is very effective, stable, and outperforms other block-diagonal co-clustering algorithms devoted to the same task. Another important advantage of using modularity in the co-clustering context is that it provides a novel, simple way of determining the appropriate number of co-clusters.Availability: an implementation of CoClus is available as part of the recently released coclust Python package which is available at: https://pypi.python.org/pypi/coclust

Semantic Concept Co-Occurrence Patterns for Image Annotation and Retrieval.

Describing visual image contents by semantic concepts is an effective and straightforward way to facilitate various high level applications. Inferring semantic concepts from low-level pictorial feature analysis is challenging due to the semantic gap problem, while manually labeling concepts is unwise because of a large number of images in both online and offline collections. In this paper, we present a novel approach to automatically generate intermediate image descriptors by exploiting concept co-occurrence patterns in the pre-labeled training set that renders it possible to depict complex scene images semantically. Our work is motivated by the fact that multiple concepts that frequently co-occur across images form patterns which could provide contextual cues for individual concept inference. We discover the co-occurrence patterns as hierarchical communities by graph modularity maximization in a network with nodes and edges representing concepts and co-occurrence relationships separately. A random walk process working on the inferred concept probabilities with the discovered co-occurrence patterns is applied to acquire the refined concept signature representation. Through experiments in automatic image annotation and semantic image retrieval on several challenging datasets, we demonstrate the effectiveness of the proposed concept co-occurrence patterns as well as the concept signature representation in comparison with state-of-the-art approaches.

Identification of segregated regions in the functional brain connectome of autistic patients by a combination of fuzzy spectral clustering and entropy analysis.

Several neuroimaging studies support the model of abnormal development of brain connectivity in patients with autism-spectrum disorders (ASD). In this study, we aimed to test the hypothesis of reduced functional network segregation in autistic patients compared with controls. Functional MRI data from children acquired under a resting-state protocol (Autism Brain Imaging Data Exchange [ABIDE]) were submitted to both fuzzy spectral clustering (FSC) with entropy analysis and graph modularity analysis. We included data from 814 children in our analysis. We identified 5 regions of interest comprising the motor, temporal and occipitotemporal cortices with increased entropy (p < 0.05) in the clustering structure (i.e., more segregation in the controls). Moreover, we noticed a statistically reduced modularity (p < 0.001) in the autistic patients compared with the controls. Significantly reduced eigenvector centrality values (p < 0.05) in the patients were observed in the same regions that were identified in the FSC analysis. There is considerable heterogeneity in the fMRI acquisition protocols among the sites that contributed to the ABIDE data set (e.g., scanner type, pulse sequence, duration of scan and resting-state protocol). Moreover, the sites differed in many variables related to sample characterization (e.g., age, IQ and ASD diagnostic criteria). Therefore, we cannot rule out the possibility that additional differences in functional network organization would be found in a more homogeneous data sample of individuals with ASD. Our results suggest that the organization of the whole-brain functional network in patients with ASD is different from that observed in controls, which implies a reduced modularity of the brain functional networks involved in sensorimotor, social, affective and cognitive processing.

Some Results on More Flexible Versions of Graph Motif

The problems studied in this paper originate from Graph Motif, a problem introduced in 2006 in the context of biological networks. Informally speaking, it consists in deciding if a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Due to the high rate of noise in the biological data, more flexible definitions of the problem have been outlined. We present in this paper two inapproximability results for two different optimization variants of Graph Motif: one where the size of the solution is maximized, the other when the number of substitutions of colors to obtain the motif from the solution is minimized. We also study a decision version of Graph Motif where the connectivity constraint is replaced by the well known notion of graph modularity. While the problem remains N P-complete, it allows algorithms in F P T for biologically relevant parameterizations.