Higher-order clustering in networks

Hyperedge overlap drives explosive transitions in systems with higher-order interactions

Recent studies have shown that novel collective behaviors emerge in complex systems due to the presence of higher-order interactions. However, how the collective behavior of a system is influenced by the microscopic organization of its higher-order interactions is not fully understood. In this work, we introduce a way to quantify the overlap among the hyperedges of a higher-order network, and we show that real-world systems exhibit different levels of intra-order hyperedge overlap. We then study two types of dynamical processes on higher-order networks, namely complex contagion and synchronization, finding that intra-order hyperedge overlap plays a universal role in determining the collective behavior in a variety of systems. Our results demonstrate that the presence of higher-order interactions alone does not guarantee abrupt transitions. Rather, explosivity and bistability require a microscopic organization of the structure with a low value of intra-order hyperedge overlap.

Node features of chromosome structure networks and their connections to genome annotation

The 3D conformations of chromosomes can encode biological significance, and the implications of such structures have been increasingly appreciated recently. Certain chromosome structural features, such as A/B compartmentalization, are frequently extracted from Hi-C pairwise genome contact information (physical association between different regions of the genome) and compared with linear annotations of the genome, such as histone modifications and lamina association. We investigate how additional properties of chromosome structure can be deduced using an abstract graph representation of the contact heatmap, and describe specific network properties that can have a strong connection with some of these biological annotations. We constructed chromosome structure networks (CSNs) from bulk Hi-C data and calculated a set of site-resolved (node-based) network properties. These properties are useful for characterizing certain aspects of chromosomal structure. We examined the ability of network properties to differentiate several scenarios, such as haploid vs diploid cells, partially inverted nuclei vs conventional architecture, depletion of chromosome architectural proteins, and structural changes during cell development. We also examined the connection between network properties and a series of other linear annotations, such as histone modifications and chromatin states including poised promoter and enhancer labels. We found that semi-local network properties exhibit greater capability in characterizing genome annotations compared to diffusive or ultra-local node features. For example, the local square clustering coefficient can be a strong classifier of lamina-associated domains. We demonstrated that network properties can be useful for highlighting large-scale chromosome structure differences that emerge in different biological situations.

Clustering coefficients for networks with higher order interactions

We introduce a clustering coefficient for nondirected and directed hypergraphs, which we call the quad clustering coefficient. We determine the average quad clustering coefficient and its distribution in real-world hypergraphs and compare its value with those of random hypergraphs drawn from the configuration model. We find that real-world hypergraphs exhibit a nonnegligible fraction of nodes with a maximal value of the quad clustering coefficient, while we do not find such nodes in random hypergraphs. Interestingly, these highly clustered nodes can have large degrees and can be incident to hyperedges of large cardinality. Moreover, highly clustered nodes are not observed in an analysis based on the pairwise clustering coefficient of the associated projected graph that has binary interactions, and hence higher order interactions are required to identify nodes with a large quad clustering coefficient.

Graph Motif Entropy for Understanding Time-Evolving Networks

The structure of networks can be efficiently represented using motifs, which are those subgraphs that recur most frequently. One route to understanding the motif structure of a network is to study the distribution of subgraphs using statistical mechanics. In this article, we address the use of motifs as network primitives using the cluster expansion from statistical physics. By mapping the network motifs to clusters in the gas model, we derive the partition function for a network, and this allows us to calculate global thermodynamic quantities, such as energy and entropy. We present analytical expressions for the number of certain types of motifs, and compute their associated entropy. We conduct numerical experiments for synthetic and real-world data sets and evaluate the qualitative and quantitative characterizations of the motif entropy derived from the partition function. We find that the motif entropy for real-world networks, such as financial stock market networks, is sensitive to the variance in network structure. This is in line with recent evidence that network motifs can be regarded as basic elements with well-defined information-processing functions.

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

Locating the Principal Sectors for Carbon Emission Reduction on the Global Supply Chains by the Methods of Complex Network and Susceptible–Infective Model

How to locate the reasonable targets for carbon emission reduction in the complex global supply chain remains a big challenge for policy makers. This paper proposed a novel framework for finding more accurate carbon emission reduction targets, combining multi-regional input-output analysis, complex network approach and an improved susceptible–infective model called the influence spreading model. The results showed that the global embodied carbon emission flow network had the characteristic of being significantly scale-free, and there were a few important industrial sectors in the network with different capabilities, including strength-out, closeness-out, betweenness and clustering coefficient. The simulation results of the influence spreading process showed that the effective infection thresholds were relatively low, which were between 0 and 0.005 due to the significant scale-free characteristic of the global embodied carbon emission flow network. With the change of the infection thresholds, the proportion of the infected sectors significantly decreased from about 0.95 to 0.10 on average, and spread time also decreased from about three rounds to about eight rounds. In the aspects of the spreading scope and spreading speed, the industrial sectors with high closeness-out and betweenness had better performance than the ones with high strength-out. This indicated that the spreading capabilities of industrial sectors which exported significant carbon emissions, such as petroleum, chemicals and non-metallic mineral products in China, were commonly weaker than industrial sectors which occupied the most important positions in the entire supply chain, such as transport equipment in Germany. Hence, the industrial sectors with high global spreading capability and media capability were important for global carbon emission reduction. Such information suggested that the policies for carbon emission reduction should be made based on a global perspective of the supply chain system. This work proved that the policies for carbon emission reduction should be based on a global perspective of supply chain system.

Percolation in random graphs with higher-order clustering

Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally treelike and does not contain short-range loops between neighbors. In this paper we use the generating function formulation to examine clustered networks that contain simple cycles and cliques of any order. We use the natural generalization to the Molloy-Reed criterion for these networks to describe their critical properties and derive an approximate analytical description of the size of the giant component, providing solutions for Poisson and power-law networks. We find that networks comprising larger simple cycles behave increasingly more treelike. Conversely, clustering composed of larger cliques increasingly deviate from the treelike solution, although the behavior is strongly dependent on the degree-assortativity.

Random graphs with arbitrary clustering and their applications

The structure of many real networks is not locally treelike and, hence, network analysis fails to characterize their bond percolation properties. In a recent paper [P. Mann, V. A. Smith, J. B. O. Mitchell, and S. Dobson, arXiv:2006.06744], we developed analytical solutions to the percolation properties of random networks with homogeneous clustering (clusters whose nodes are degree equivalent). In this paper, we extend this model to investigate networks that contain clusters whose nodes are not degree equivalent, including multilayer networks. Through numerical examples, we show how this method can be used to investigate the properties of random complex networks with arbitrary clustering, extending the applicability of the configuration model and generating function formulation.

Competitive Pi-Stacking and H-Bond Piling Increase Solubility of Heterocyclic Redoxmers

Redoxmers are organic molecules that carry electric charge in flow batteries. In many instances, they consist of heteroaromatic moieties modified with appended groups to prevent stacking of the planar cores and increase solubility in liquid electrolytes. This higher solubility is desired as it potentially allows achieving greater energy density in the battery. However, the present synthetic strategies often yield bulky molecules with low molarity even when they are neat and still lower molarity in liquid solutions. Fortunately, there are exceptions to this rule. Here, we examine one well-studied redoxmer, 2,1,3-benzothiadiazole, which has solubility ∼5.7 M in acetonitrile at 25 °C. We show computationally and prove experimentally that the competition between two packing motifs, face-to-face π-stacking and random N-H bond piling, introduces frustration that confounds nucleation in crowded solutions. Our findings and examples from related systems suggest a complementary strategy for the molecular design of redoxmers for high energy density redox flow cells.

Dismantling complex networks based on the principal eigenvalue of the adjacency matrix

The connectivity of complex networks is usually determined by a small fraction of key nodes. Earlier works successfully identify an influential single node, yet have some problems for the case of multiple ones. In this paper, based on the matrix spectral theory, we propose the collective influence of multiple nodes. An interesting finding is that some traditionally influential nodes have strong internal coupling interactions that reduce their collective influence. We then propose a greedy algorithm to dismantle complex networks by optimizing the collective influence of multiple nodes. Experimental results show that our proposed method outperforms the state of the art methods in terms of the principal eigenvalue and the giant component of the remaining networks.

Quantification of node importance in rain gauge network: influence of temporal resolution and rain gauge density

Rain gauge network is important for collecting rainfall information effectively and efficiently. Rain gauge networks have been studied for several decades from a range of hydrological perspectives, where rain gauges with unique or non-repeating information are considered as important. However, the problem of quantification of node importance and subsequent identification of the most important nodes in rain gauge networks have not yet been extensively addressed in the literature. In this study, we use the concept of the complex networks to evaluate the Indian Meteorological Department (IMD) monitored 692 rain gauge in the Ganga River Basin. We consider the complex network theory-based Degree Centrality (DC), Clustering Coefficient (CC) and Mutual Information (MI) as the parameters to quantify the rainfall variability associated with all the rain gauges in the network. Multiple rain gauge network scenario with varying rain gauge density (i.e. Network Size (NS) = 173, 344, 519, and 692) and Temporal Resolution (i.e. TR = 3 hours, 1 day, and 1 month) are introduced to study the effect of rain gauge density, gauge location and temporal resolution on the node importance quantification. Proxy validation of the methodology was done using a hydrological model. Our results indicate that the network density and temporal resolution strongly influence a node's importance in rain gauge network. In addition, we concluded that the degree centrality along with clustering coefficient is the preferred parameter than the mutual information for the node importance quantification. Furthermore, we observed that the network properties (spatial distribution, DC, Collapse Correlation Threshold (CCT), CC Range distributions) associated with TR = 3 hours and 1 day are comparable whereas TR = 1 month exhibit completely different trends. We also found that the rain gauges situated at high elevated areas are extremely important irrespective of the NS and TR. The encouraging results for the quantification of nodes importance in this study seem to indicate that the approach has the potential to be used in extreme rainfall forecasting, in studying changing rainfall patterns and in filling gaps in spatial data. The technique can be further helpful in the ground-based observation network design of a wide range of meteorological parameters with spatial correlation.

A framework for second-order eigenvector centralities and clustering coefficients

We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron-Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.

Evolution of resilience in protein interactomes across the tree of life

Phenotype robustness to environmental fluctuations is a common biological phenomenon. Although most phenotypes involve multiple proteins that interact with each other, the basic principles of how such interactome networks respond to environmental unpredictability and change during evolution are largely unknown. Here we study interactomes of 1,840 species across the tree of life involving a total of 8,762,166 protein-protein interactions. Our study focuses on the resilience of interactomes to network failures and finds that interactomes become more resilient during evolution, meaning that interactomes become more robust to network failures over time. In bacteria, we find that a more resilient interactome is in turn associated with the greater ability of the organism to survive in a more complex, variable, and competitive environment. We find that at the protein family level proteins exhibit a coordinated rewiring of interactions over time and that a resilient interactome arises through gradual change of the network topology. Our findings have implications for understanding molecular network structure in the context of both evolution and environment.