A nonlinear spectral core–periphery detection method for multiplex networks

The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multilayer networks, i.e., networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers and between activities and degrees. We show that real-world networks exhibit indeed nontrivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

The robustness and resilience of complex systems are crucial for maintaining functionality amid disruptions or intentional attacks. Many such systems can be modeled as networks, where identifying structurally central nodes is essential for assessing their robustness and susceptibility to failure. Traditional centrality metrics often face challenges in identifying structurally important nodes in networks exhibiting heterogeneity at the network scale, with multilayer networks being a prime example of such networks. These metrics typically fail to balance the trade-off between capturing local layer-specific structures and integrating global multiplex connectivity. In this study, we extend DomiRank centrality, a metric that has been shown to effectively assess nodal importance across diverse monoplex topologies, to multiplex networks. Our approach combines layer-specific DomiRank calculations with a global contextualization step, incorporating multiplex-wide DomiRank scores to combine rankings. Through synthetic and real-world network studies, we demonstrate that our generalized DomiRank framework significantly improves the identification of key nodes in highly heterogeneous multiplex networks. This work advances centrality-based robustness assessments by addressing the fundamental trade-off between layer adaptability and multiplex-wide coherence.

Centrality ranking in multiplex networks using topologically biased random walks

Learning low-dimensional representations of graphs has facilitated the use of traditional machine learning techniques to solving classic network analysis tasks such as link prediction, node classification, community detection, etc. However, to date, the vast majority of these learning tasks are focused on traditional single-layer/unimodal networks and largely ignore the case of multiplex networks. A multiplex network is a suitable structure to model multi-dimensional real-world complex systems. It consists of multiple layers where each layer represents a different relationship among the network nodes. In this work, we propose MUNEM, a novel approach for learning a low-dimensional representation of a multiplex network using a triplet loss objective function. In our approach, we preserve the global structure of each layer, while at the same time fusing knowledge among different layers during the learning process. We evaluate the effectiveness of our proposed method by testing and comparing on real-world multiplex networks from different domains, such as collaboration network, protein-protein interaction network, online social network. Finally, in order to deliberately examine the effect of our model’s parameters we conduct extensive experiments on synthetic multiplex networks.

In many complex systems, the interactions between objects span multiple aspects. Multiplex networks are accurate paradigms to model such systems, where each edge is associated with a type. A key graph mining primitive is extracting dense subgraphs, and this has led to interesting notions such as k-cores, known as building blocks of complex networks. Despite recent attempts to extend the notion of core to multiplex networks, existing studies suffer from a subset of the following limitations: They (1) force all nodes to exhibit their high degree in the same set of relation types while in multiplex networks some connection types can be noisy for some nodes, (2) either require high computational cost or miss the complex information of multiplex networks, and (3) assume the same importance for all relation types. We introduce Score, a novel and unifying family of dense structures in multiplex networks that uses a function S(.) to summarize the degree vector of each node. We then discuss how one can choose a proper S(.) from the data. To demonstrate the usefulness of Scores, we focus on finding the densest subgraph as well as modeling user engagement in multiplex networks. We present a new density measure in multiplex networks and discuss its advantages over existing density measures. We show that the problem of finding the densest subgraph in multiplex networks is NP-hard and design an efficient approximation algorithm based on Scores. Finally, we present a new mathematical model of user engagement in the presence of different relation types. Our experiments shows the efficiency and effectiveness of our algorithms and supports the proposed mathematical model of user engagement.

Multiplex network community detection algorithm based on motif awareness

At rest, human brain functional networks display striking modular architecture in which coherent clusters of brain regions are activated. The modular account of brain function is pervasive, reliable, and reproducible. Yet, a complementary perspective posits a core–periphery or rich-club account of brain function, where hubs are densely interconnected with one another, allowing for integrative processing. Unifying these two perspectives has remained difficult due to the fact that the methodological tools to identify modules are entirely distinct from the methodological tools to identify core–periphery structure. Here, we leverage a recently-developed model-based approach—the weighted stochastic block model—that simultaneously uncovers modular and core–periphery structure, and we apply it to functional magnetic resonance imaging data acquired at rest in 872 youth of the Philadelphia Neurodevelopmental Cohort. We demonstrate that functional brain networks display rich mesoscale organization beyond that sought by modularity maximization techniques. Moreover, we show that this mesoscale organization changes appreciably over the course of neurodevelopment, and that individual differences in this organization predict individual differences in cognition more accurately than module organization alone. Broadly, our study provides a unified assessment of modular and core–periphery structure in functional brain networks, offering novel insights into their development and implications for behavior.

Multiplex networks describe a large number of systems ranging from social networks to the brain. These multilayer structure encode information in their structure. This information can be extracted by measuring the correlations present in the multiplex networks structure, such as the overlap of the links in different layers. Many multiplex networks are also weighted, and the weights of the links can be strongly correlated with the structural properties of the multiplex network. For example, in multiplex network formed by the citation and collaboration networks between PRE scientists it was found that the statistical properties of citations to coauthors differ from the one of citations to noncoauthors, i.e., the weights depend on the overlap of the links. Here we present a theoretical framework for modeling multiplex weighted networks with different types of correlations between weights and overlap. To this end, we use the framework of canonical network ensembles, and the recently introduced concept of multilinks, showing that null models of a large variety of network structures can be constructed in this way. In order to provide a concrete example of how this framework apply to real data we consider a multiplex constructed from gene expression data of healthy and cancer tissues.

Complex networks recently opened new ways for investigating how language use is influenced by the mental representation of word similarities. This work adopts the framework of multiplex lexical networks for investigating lexical retrieval from memory. The focus is on priming, i.e., exposure to a given stimulus facilitating or inhibiting retrieval of a given lexical item. Supported by recent findings of network distance influencing lexical retrieval, the multiplex network approach tests how the layout of hundreds of thousands of word‐word similarities in the mental lexicon can lead to priming effects on multiple combined semantic and phonological levels. Results provide quantitative evidence that phonological priming effects are encoded directly in the multiplex structure of the mental representation of words sharing phonemes either in their onsets (cohort priming) or at their ends (rhyme priming). By comparison with randomised null models, both cohort and rhyming effects are found to be emerging properties of the mental lexicon arising from its multiplexity. These priming effects are absent on individual layers but become prominent on the combined multiplex structure. The emergence of priming effects is displayed both when only semantic layers are considered, an approximated representation of the so‐called semantic memory, and when semantics is enriched with phonological similarities, an approximated representation of the lexical‐auditory nature of the mental lexicon. Multiplex lexical networks can account for connections between semantic and phonological information in the mental lexicon and hence represent a promising modelling route for shedding light on the interplay between multiple aspects of language and human cognition in synergy with experimental psycholinguistic data.

Structural correlation between communities and core-periphery structures in social networks: Evidence from Twitter data

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor's semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.

We extend coordinate descent to manifold domains and provide convergence analyses for geodesically convex and nonconvex smooth objective functions. Our key insight is to draw an analogy between coordinate blocks in Euclidean space and tangent subspaces of a manifold. Hence, our method is called tangent subspace descent (TSD). The core principle behind ensuring convergence of TSD is the appropriate choice of subspace at each iteration. To this end, we propose two novel conditions, the (C, r)-norm and C-randomized norm conditions on deterministic and randomized modes of subspace selection, respectively, that promise convergence for smooth functions and that are satisfied in practical contexts. We propose two subspace selection rules, one deterministic and another randomized, of particular practical interest on the Stiefel manifold. Our proof-of-concept numerical experiments on the sparse principal component analysis problem demonstrate TSD’s efficacy. Funding: This work was supported by the National Science Foundation [Grant 1740707] and the Defense Advanced Research Projects Agency Lagrange Program [Grant N660011824020].

Community detection in social networks is a promising research area. However, network exploration with multiple types of relationships remains an underexplored yet crucial domain. To address the challenge of community detection in multiplex networks, we propose SFCSA, a smart flattening-based Cuckoo Search Algorithm, designed to address the complexities of such networks effectively. SFCSA unfolds in three integral steps. Firstly, we employ DeepWalk and node2vec algorithms to extract node embeddings, which are merged to construct the multiplex network node embeddings, minimizing information loss. Secondly, a flattened network is built by merging the network layers and adopting the Pearson correlation coefficient as a weight to capture node similarity. Thirdly, we employ a community detection approach that unveils the community structure through a cuckoo search algorithm applied to the multiplex node embeddings to extract the best set of cluster centres by maximizing the modularity function. Experimental evaluations across synthetic and real-world multiplex networks attest to the efficiency and effectiveness of our approach, outperforming state-of-the-art algorithms.

In this paper we propose an iterative algorithm for solving the problem of extracting a sparse source signal when a reference signal for the desired source signal is available. In the proposed algorithm, a nonconvex objective function is used for measuring the diversity (antisparsity) of the desired source signal. The nonconvex function is locally replaced by a quadratic convex function. This results in a simple iterative algorithm. The proposed algorithm has two different versions, depending on the measure of closeness between the extracted source signal and the reference signal. The proposed algorithm has useful applications to EEG/MEG signal processing. This is demonstrated by an example in which eye blink artifacts are automatically removed from a real EEG data.

Multiplex networks have been widely used to describe the multi-type connections of entities in the real world. However, researches for multiplex networks visualization unilaterally focus on the presentation of topological structure, lacking of specific high-level information presentation for quantitative comparison of interlayer structure. Users cannot participate in the exploration and freely choose the layers (or sub-graphs, regions, etc.) of interest for structural comparison. Contraposing the layer-edge patterns visual analysis tasks of multiplex networks, this paper puts forward a novel solution for exploration and analysis that tightly couples topological structure and high-level patterns. It mainly contains a multi-force directed model to realize the balanced layout of nodes in multi-layer topology, as well as two kinds of high-level patterns of which the visual representations are, respectively, designed by a familiar metaphor—that is, the similar pattern representation based on the area-proportional Venn diagrams and the interaction pattern representation based on the directed arrows. Furthermore, views association is implemented through underlying data sharing and multiple interactions which can be used to gain insights through the creation of selections of interest and produce high-level infographic-style overviews simultaneously. The experiments on real-world data demonstrate the support of the proposed method for layer-edge patterns analysis tasks in multiplex networks and the effectiveness for analyzing the multi-layer structure of multiplex networks.