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Vector centrality in hypergraphs
Abstract
Identifying the most influential nodes in networked systems is of vital importance to optimize their function and control. Several scalar metrics have been proposed to that effect, but the recent shift in focus towards network structures which go beyond a simple collection of dyadic interactions has rendered them void of performance guarantees. We here introduce a new measure of node's centrality, which is no longer a scalar value, but a vector with dimension one lower than the highest order of interaction in a hypergraph. Such a vectorial measure is linked to the eigenvector centrality for networks containing only dyadic interactions, but it has a significant added value in all other situations where interactions occur at higher-orders. In particular, it is able to unveil different roles which may be played by the same node at different orders of interactions – information that is otherwise impossible to retrieve by single scalar measures. We demonstrate the efficacy of our measure with applications to synthetic networks and to three real world hypergraphs, and compare our results with those obtained by applying other scalar measures of centrality proposed in the literature.
... As a result, traditional network modeling techniques only provide a limited representation of complex systems. In contrast, higher-order network models (HONs) that better capture many-body interactions can improve the analysis of various network analysis tasks [11][12][13][14][15][16][17]. Recent work has developed four different lines of modeling approaches to embed higher-order dependencies into HON models, including hypergraph models [11,[18][19][20][21][22], simplicial complex models [16,[23][24][25][26][27][28][29][30], motif-based higher-order models [31-38] and higher-order Markov models [5,6,9,[39][40][41][42][43]. ...
... In contrast, higher-order network models (HONs) that better capture many-body interactions can improve the analysis of various network analysis tasks [11][12][13][14][15][16][17]. Recent work has developed four different lines of modeling approaches to embed higher-order dependencies into HON models, including hypergraph models [11,[18][19][20][21][22], simplicial complex models [16,[23][24][25][26][27][28][29][30], motif-based higher-order models [31-38] and higher-order Markov models [5,6,9,[39][40][41][42][43]. ...
... where SD (p random (ϕ)) is the standard deviation σθ of transition probabilities of k-order dependencies in simulation datasets. Then we could derive a confidence interval with a confidence level 1 − α based on the population mean θ of transition probabilities of k-order dependencies in simulation datasets, as shown in equation (11) [52,54,55] ...
Higher-order networks (HONs), which go beyond the limitations of pairwise relation modeling by graphs, capture higher-order dependencies involving three or more components for various systems. As the number of potential higher-order dependencies increases exponentially with both network size and the order of dependency, it is of particular importance for HON models to balance their representation power against model complexity. In this study, we propose a method, significant k -order dependencies mining (S k DM), based on hypothesis testing and the Markov chain Monte Carlo (MCMC), to identify significant higher-order dependencies in real systems. Through synthetic clickstreams with elaborately designed higher-order dependencies, S k DM shows a powerful ability to correctly identify all significant dependencies at preset significance levels of α = 0.01, 0.05, 0.10, performing as the only method, in comparison to the state of the arts, that can robustly maintain the Type I error rate, and without generating any Type II error across all the experimental settings. We further apply the S k DM method to various empirical networks, including journal citations, air traffic, and email communications. Empirical results show that among those tested networks, only 6.03%, 1.47%, and 1.28% of all potential dependencies are of statistical significance (α = 0.01). The proposed S k DM method, therefore, provides an efficient tool for higher-order network analysis tasks at reduced computational complexity.
... Finally, higher-order matrix representations of higher-order networks have also been introduced in Refs. [28,30,35,36,[44][45][46]. A common formalism is to define one N × · · · × N k times adjacency hyper-matrix A (k) (H) for each cardinality k of the hyperedges, so that ...
... Also in this case, the same considerations about existence and uniqueness apply. More recently, a new measure of node centrality in hypergraphs has been introduced, which is no longer a scalar value, but a vector with dimension equal to the rank of the hypergraph minus one [46]. The procedure to calculate this vector centrality starts with constructing the corresponding line graph and calculating the classic eigenvector centrality c (h) for each of its nodes h, which correspond to the hyperedges of the original hypergraph. ...
... In other words, the kth component of the vector centrality of node i is the sum of the eigenvector centralities of all the nodes of the line graph of the hypergraph that correspond to hyperedges of size k that contain i. Ref. [46] shows rigorously that this quantity is related to the classic eigenvector centrality in graphs, but it introduces a significant added value in higher-order networks. In particular, it is able to discriminate between the roles that the same node may play at different orders of interactions, providing information that cannot be inferred from scalar measures. ...
... A similar idea has been explored for pairwise networks in [26]. Recent work utilizing subsets of hypergraphs has examined degree-degree mixing [16,27] and degree centrality [28]. In [29], the authors count the number of motifs of different sizes to describe the structure of datasets. ...
... This has implications for sub-fields of network science, such as community detection, dynamics on networks, and structural measures, among other topics. We believe that our approach unifies studies that have indirectly examined the effect of size on structure and dynamics of higher-order datasets [15,27,28,[39][40][41] and will be a fruitful area of research in the future. ...
Many complex systems often contain interactions between more than two nodes, known as higher-order interactions, which can change the structure of these systems in significant ways. Researchers often assume that all interactions paint a consistent picture of a higher-order dataset's structure. In contrast, the connection patterns of individuals or entities in empirical systems are often stratified by interaction size. Ignoring this fact can aggregate connection patterns that exist only at certain scales of interaction. To isolate these scale-dependent patterns, we present an approach for analyzing higher-order datasets by filtering interactions by their size. We apply this framework to several empirical datasets from three domains to demonstrate that data practitioners can gain valuable information from this approach.
... So far, scholars have conducted related research on the task of mining important nodes in hypergraphs, and have proposed some classical methods such as hyperdegree centrality [19] (HDC), closeness centrality [20] (CC), betweenness centrality [21] (BC), and vector centrality [22] (VC), which have provided new perspectives and methods for subsequent research on identifying important nodes in hypergraphs, especially regarding research on entropy. Chen et al. [23] developed the notion of entropy for hypergraphs by using the probability distribution of the generalized singular values of the Laplacian tensor of uniform hypergraphs. ...
... where N denotes the total number of nodes in the hypergraph, d ij is the shortest distance between node v i and node v j , one of common algorithms in solving shortest path problem is Dijkstra algorithm. VC: The vector centrality [22] of hypergraphs is a vector measure related to the eigenvector centrality in ordinary graphs. First, we project the hypergraph H into a 1-line graph L 1 (H) and calculate the eigenvector centrality of each node in L 1 (H) (hyperedge in H); let c e j be the eigenvector centrality of any hyperedge e j ∈ E in H. ...
Hypergraphs have become an accurate and natural expression of high-order coupling relationships in complex systems. However, applying high-order information from networks to vital node identification tasks still poses significant challenges. This paper proposes a von Neumann entropy-based hypergraph vital node identification method (HVC) that integrates high-order information as well as its optimized version (semi-SAVC). HVC is based on the high-order line graph structure of hypergraphs and measures changes in network complexity using von Neumann entropy. It integrates s-line graph information to quantify node importance in the hypergraph by mapping hyperedges to nodes. In contrast, semi-SAVC uses a quadratic approximation of von Neumann entropy to measure network complexity and considers only half of the maximum order of the hypergraph’s s-line graph to balance accuracy and efficiency. Compared to the baseline methods of hyperdegree centrality, closeness centrality, vector centrality, and sub-hypergraph centrality, the new methods demonstrated superior identification of vital nodes that promote the maximum influence and maintain network connectivity in empirical hypergraph data, considering the influence and robustness factors. The correlation and monotonicity of the identification results were quantitatively analyzed and comprehensive experimental results demonstrate the superiority of the new methods. At the same time, a key non-trivial phenomenon was discovered: influence does not increase linearly as the s-line graph orders increase. We call this the saturation effect of high-order line graph information in hypergraph node identification. When the order reaches its saturation value, the addition of high-order information often acts as noise and affects propagation.
... Recent work relating subsets of a hypergraph has examined degree-degree mixing (16,27) and degree centrality (28). In Ref. (29), the authors count the number of motifs of different sizes as a way to describe the structure of datasets. ...
... This has implications for sub-fields of network science such as community detection, dynamics on networks, and structural measures, among other topics. We believe that our approach unifies studies that have indirectly examined the effect of size on structure and dynamics of higher-order datasets (15,27,28,(39)(40)(41) and will be a fruitful area of research in the future. ...
Many complex systems often contain interactions between more than two nodes, known as higher-order interactions, which can change the structure of these systems in significant ways. Researchers often assume that all interactions paint a consistent picture of a higher-order dataset's structure. In contrast, the connection patterns of individuals or entities in empirical systems are often stratified by interaction size. Ignoring this fact can aggregate connection patterns that exist only at certain scales of interaction. To isolate these scale-dependent patterns, we present an approach for analyzing higher-order datasets by filtering interactions by their size. We apply this framework to several empirical datasets from three domains to demonstrate that data practitioners can gain valuable information from this approach.
... Thus, as such objects or systems are present at the same time in different interaction networks (layers), these layers are interconnected. Moreover, the existence of interactions of different nature and simultaneous interactions between nodes and edges (group collaborations, chemical reactions in which more than two components interact, etc.) have shown that hypergraphs and multilayer networks are very suitable structures for this type of studies [9,[13][14][15][16]. Moreover, the latest advances in modern linguistics are based on the treatment of a language as a system or a complex network, to which mathematical tools, statistical measures, and procedures of this branch of science can be applied to obtain a new, efficient, and effective approach to the study of language [3,[17][18][19][20][21][22][23][24][25][26][27][28]. ...
... h 14 We show numerically that the information of edge correlation between layer offers substantial insight in the complex multiplex structure thus contributing to simplified and faster analysis and design of multiplex graph with desired consensus or synchronization properties. h 15 Although weak subcriticality sets the necessary condition for a freezing out dynamics as we crossover a secondary bifurcation, we find out an unexpected critical behavior in regard of the standard Kibble-Zurek prediction. h 16 Although our system can work at a realtime frame rate for the considered video resolution, its tracking accuracy is affected by factors like the scene illumination condition, the contrast of the targets with respect to the background, the velocity of each target, and the frame rate of the video. ...
Forensic linguistics and stylometry have in the exploration of linguistic patterns one of their fundamental tools. Mathematical structures such as complex multilayer networks and hypergraphs provide remarkable resources to represent and analyze texts. In this paper, we present a model that includes some specific mesoscopic relations between the different types of words in a corpus (lexical words, verbs, linking words, other words) according to the sentences or paragraphs in which they appear. This model is supported by various mathematical structures such as partial multiline graphs, multilayer hypergraphs, and their derivative graphs. The methodology proposed from this new point of view is of singular help to find meaningful sentences from any text to set up an automatic summary of the text and, eventually, to determine its linguistic level.
... Benson 23 proposed an eigenvector centrality that is applicable to uniform hypergraphs whose hyperedges are in uniform size. Kovalenko et al. 24 defined a vector centrality that describes the importance of nodes in different sizes of hyperedges. Aksoy et al. 25 defined the s-closeness centrality and s-eccentricity of a hyperedge in a hypergraph. ...
... {|e m |} corresponds to the maximum cardinality of hyperedges. 24 It first projects a hypergraph into a line graph. 37 Then, it calculates the eigenvector centralities of all hyperedges, which is denoted as c(m), m = 1, . . . ...
Hypergraphs that can depict interactions beyond pairwise edges have emerged as an appropriate representation for modeling polyadic relations in complex systems. With the recent surge of interest in researching hypergraphs, the centrality problem has attracted much attention due to the challenge of how to utilize higher-order structure for the definition of centrality metrics. In this paper, we propose a new centrality method (HGC) on the basis of the gravity model as well as a semi-local HGC, which can achieve a balance between accuracy and computational complexity. Meanwhile, two comprehensive evaluation metrics, i.e., a complex contagion model in hypergraphs, which mimics the group influence during the spreading process and network [Formula: see text]-efficiency based on the higher-order distance between nodes, are first proposed to evaluate the effectiveness of our methods. The results show that our methods can filter out nodes that have fast spreading ability and are vital in terms of hypergraph connectivity.
... Motivated by these advances, several approaches have been introduced to study the structure of higher-order networks. Some of these mainly extended the traditional network approaches to include group (higher-order) interactions, such as community detection methods based on generalized modularity, 33,34 spectral clustering 35 , bayesian statistics approaches, 36 centrality metrics, 35,37,38 clustering coefficient, 39,40 and k-core decomposition methods. 41 On the other hand, new approaches were appositely proposed to characterize group interactions. ...
Higher-order networks effectively represent complex systems with group interactions. Existing methods usually overlook the relative contribution of group interactions (hyperlinks) of different sizes to the overall network structure. Yet, this has many important applications, especially when the network has meaningful node labels. In this work, we propose a comprehensive methodology to precisely measure the contribution of different orders to the overall network structure. First, we propose the order contribution measure, which quantifies the contribution of hyperlinks of different orders to the link weights (local scale), number of triangles (mesoscale) and size of the largest connected component (global scale) of the pairwise weighted network. Second, we propose the measure of order relevance, which gives insights in how hyperlinks of different orders contribute to the considered network property. Most interestingly, it enables an assessment of whether this contribution is synergistic or redundant with respect to that of hyperlinks of other orders. Third, to account for labels, we propose a metric of label group balance to assess how hyperlinks of different orders connect label-induced groups of nodes. We applied these metrics to a large-scale board interlock network and scientific collaboration network, in which node labels correspond to geographical location of the nodes. Experiments including a comparison with randomized null models reveal how from the global level perspective, we observe synergistic contributions of orders in the board interlock network, whereas in the collaboration network there is more redundancy. The findings shed new light on social scientific debates on the role of busy directors in global business networks and the connective effects of large author teams in scientific collaboration networks.
... To address these shortcomings, higher-order networks have been introduced, offering more accurate representations of multi-agent interactions. These models facilitate link prediction, the identification of influential nodes, and the exploration of complex trade relationships [8,9]. ...
World trade networks are exhaustively described by pairwise interactions, and overlook higher-order structure from the outcome of collective interactions at the level of groups of nodes like multilateral trade agreements. To address this limitation, we collect multiplex world trade networks, including the bilateral regional trade agreement network, which represents pairwise interactions; the multilateral regional trade agreement network, which naturally represents a higher-order network structure; and the import and export trade network, which represents pairwise interactions and additional complexities. The analysis of simplicial centrality, including degree, closeness, and subgraph at 0, 1, and 2-simplex levels, reveals that intra-level correlations are high, while inter-levels may exhibit significant disparities. Nodes with low centrality at higher-order levels could influence network robustness due to the diversity of interactions and higher-order dependencies. Simplicial centrality on robustness of multiplex world trade networks under random and targeted attacks reveals that the complex connectivity of higher-order levels renders them more vulnerable post-attack. An optimization strategy of the rebalancing of network centrality is proposed to enhance the robustness, and the simulation shows risks posed to central nodes are minimized and opportunities for peripheral nodes to partake in global trade are broadened.
... Other centralities which provide insight into a nodes interaction with the edges to which it belongs could also be considered for application to a PCH. In particular, subgraph centrality [43], which assigns a node ranking based on closed walks originating from the node, and vector centrality [44], which gives a vector communicating a nodes interaction with edges of different cardinality. ...
The use of graph centrality measures applied to biological networks, such as protein interaction networks, underpins much research into identifying key players within biological processes. This approach however is restricted to dyadic interactions and it is well-known that in many instances interactions are polyadic. In this study we illustrate the merit of using hypergraph centrality applied to a hypernetwork as an alternative. Specifically, we review and propose an extension to a recently introduced node and edge nonlinear hypergraph centrality model which provides mutually dependent node and edge centralities. A Saccharomyces Cerevisiae protein complex hypernetwork is used as an example application with nodes representing proteins and hyperedges representing protein complexes. The resulting rankings of the nodes and edges are considered to see if they provide insight into the essentiality of the proteins and complexes. We find that certain variations of the model predict essentiality more accurately and that the degree-based variation illustrates that the centrality-lethality rule extends to a hypergraph setting. In particular, through exploitation of the models flexibility, we identify small sets of proteins densely populated with essential proteins. One of the key advantages of applying this model to a protein complex hypernetwork is that it also provides a classification method for protein complexes, unlike previous approaches which are only concerned with classifying proteins.
... Indeed, several dynamical processes, including contagion dynamics, synchronization phenomena and consensus formation, exhibit richer and more complex dynamics when defined on higher-order networks, with important differences with respect to the dynamics occurring on pairwise networks, such as changes in the nature of the phase transitions observed [15,20,21,23]. Despite the relevance of such higher-order effects, tools to characterize hypergraphs at various scales have only recently been proposed: for example, efforts have been devoted to defining explicitly higher-order centrality measures, accounting for information otherwise impossible to retrieve by pairwise measures [15,24]; moreover, a few techniques and methods have been developed to identify relevant higher-order substructures in hypergraphs [15,[25][26][27]. Among them, the hyper-core decomposition [26,27] identifies a doubly nested hierarchy of mesoscopic subhypergraphs, the hyper-cores, composed of nodes progressively more densely connected to each other through interactions of increasing size. ...
The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the behaviour of the system itself. An adequate description of these systems is offered by temporal hypergraphs, that integrate these features within the same framework. However, tools for their temporal and topological characterization are still scarce. Here we develop a series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales. Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution of the hyper-cores through time, characterizing the hypergraph structure and its temporal dynamics at different topological scales, and quantifying the multi-scale structural stability of the system. We also define two static hypercoreness centrality measures that provide an overall description of the nodes aggregated structural behaviour. We apply the characterization methods to several data sets, establishing connections between structural properties and specific activities within the systems. Finally, we show how the proposed method can be used as a model-validation tool for synthetic temporal hypergraphs, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution. Our work opens several research directions, from the understanding of dynamic processes on temporal higher-order networks to the design of new models of time-varying hypergraphs.
... Other centralities which provide insight into a nodes interaction with the edges to which it belongs could also be considered for application to a PCH. In particular, subgraph centrality [42], which assigns a node ranking based on closed walks originating from the node, and vector centrality [43], which gives a vector communicating a nodes interaction with edges of different cardinality. ...
The use of graph centrality measures applied to biological networks, such as protein interaction networks, underpins much research into identifying key players within biological processes. This approach however is restricted to dyadic interactions and it is well-known that in many instances interactions are polyadic. In this study we illustrate the merit of using hypergraph centrality applied to a hypernetwork as an alternative. Specifically, we review and propose an extension to a recently introduced node and edge nonlinear hypergraph centrality model which provides mutually dependent node and edge centralities. A Saccharomyces Cerevisiae protein complex hypernetwork is used as an example application with nodes representing proteins and hyperedges representing protein complexes. The resulting rankings of the nodes and edges are considered to see if they provide insight into the essentiality of the proteins and complexes. We find that certain variations of the model predict essentiality more accurately and that the degree-based variation illustrates that the centrality-lethality rule extends to a hypergraph setting. In particular, through exploitation of the models flexibility, we identify small sets of proteins densely populated with essential proteins.
... This method strikes a balance between efficiency and precision, while it is effective for the hypergraph threshold model only. Although many approaches [42], [43], [44] have been proposed for IM problems in hypergraphs, they perform less effectively due to the complexity of propagation dynamics and the strong coupling characteristics between nodes and hyperedges. ...
Influence maximization (IM) is a crucial optimization task related to analyzing complex networks in the real world, such as social networks, disease propagation networks, and marketing networks. Publications to date about the IM problem focus mainly on graphs, which fail to capture high-order interaction relationships from the real world. Therefore, the use of hypergraphs for addressing the IM problem has been receiving increasing attention. However, identifying the most influential nodes in hypergraphs remains challenging, mainly because nodes and hyperedges are often strongly coupled and correlated. In this paper, to effectively identify the most influential nodes, we first propose a novel hypergraph-independent cascade model that integrates the influences of both node and hyperedge failures. Afterward, we introduce genetic algorithms (GA) to identify the most influential nodes that leverage hypergraph collective influences. In the GA-based method, the hypergraph collective influence is effectively used to initialize the population, thereby enhancing the quality of initial candidate solutions. The designed fitness function considers the joint influences of both nodes and hyperedges. This ensures the optimal set of nodes with the best influence on both nodes and hyperedges to be evaluated accurately. Moreover, a new mutation operator is designed by introducing factors, i.e., the collective influence and overlapping effects of nodes in hypergraphs, to breed high-quality offspring. In the experiments, several simulations on both synthetic and real hypergraphs have been conducted, and the results demonstrate that the proposed method outperforms the compared methods.
... In addition, methods relying on higher-order matrix representations for capturing higher-order interactions are described in Refs. [36,[132][133][134] ...
... We utilize six centrality measures [2,21,[27][28][29][30] ...
Hypernetwork is a useful way to depict multiple connections between nodes, making it an ideal tool for representing complex relationships in network science. In recent years, there has been a marked increase in studies on hypernetworks, however, the comparison of the difference between two hypernetworks has been given less attention. This paper proposes a hyper-distance-based method (HD) for comparing hypernetworks. This method takes into account high-order information, such as the high-order distance between nodes. The experiments carried out on synthetic hypernetworks have shown that HD is capable of distinguishing between hypernetworks generated with different parameters, and it is successful in the classification of hypernetworks. Furthermore, HD outperforms current state-of-the-art baselines to distinguish empirical hypernetworks when hyperedges are disrupted.
... However, considering the clique projection, one obviously loses very important information about the structure of such higher-order interactions. This was brightly demonstrated in the series of recent studies, where it was shown that more complex generalizations of topological measures were needed in order to provide significant and meaningful information about the hypergraph structure [6][7][8][9][10][11]. In our study, we introduce a new concept of distance among nodes in a hypergraph. ...
We explore the metric structure of networks with higher-order interactions and introduce a novel definition of distance for hypergraphs that extends the classic methods reported in the literature. The new metric incorporates two critical factors: (1) the inter-node distance within each hyperedge, and (2) the distance between hyperedges in the network. As such, it involves the computation of distances in a weighted line graph of the hypergraph. The approach is illustrated with several ad hoc synthetic hypergraphs, where the structural information unveiled by the novel metric is highlighted. Moreover, the method’s performance and effectiveness are shown through computations on large real-world hypergraphs, which indeed reveal new insights into the structural features of networks beyond pairwise interactions. Namely, using the new distance measure, we generalize the definitions of efficiency, closeness and betweenness centrality for the case of hypergraphs. Comparing the values of these generalized measures with their analogs calculated for the hypergraph clique projections, we show that our measures provide significantly different assessments on the characteristics (and roles) of the nodes from the information-transferability point of view. The difference is brighter for hypergraphs in which hyperedges of large sizes are frequent, and nodes relating to these hyperedges are rarely connected by other hyperedges of smaller sizes.
... Different positional encodings are used for different message-passing directions. • Vector Centrality [32]: A vectorial measure of the roles of each node at different orders of interactions. The dimension of this measure is one less than the maximum hyperedge size. ...
A hypergraph is a data structure composed of nodes and hyperedges, where each hyperedge is an any-sized subset of nodes. Due to the flexibility in hyperedge size, hypergraphs represent group interactions (e.g., co-authorship by more than two authors) more naturally and accurately than ordinary graphs. Interestingly, many real-world systems modeled as hypergraphs contain edge-dependent node labels, i.e., node labels that vary depending on hyperedges. For example, on co-authorship datasets, the same author (i.e., a node) can be the primary author in a paper (i.e., a hyperedge) but the corresponding author in another paper (i.e., another hyperedge).
In this work, we introduce a classification of edge-dependent node labels as a new problem. This problem can be used as a benchmark task for hypergraph neural networks, which recently have attracted great attention, and also the usefulness of edge-dependent node labels has been verified in various applications. To tackle this problem, we propose WHATsNet, a novel hypergraph neural network that represents the same node differently depending on the hyperedges it participates in by reflecting its varying importance in the hyperedges. To this end, WHATsNet models the relations between nodes within each hyperedge, using their relative centrality as positional encodings. In our experiments, we demonstrate that WHATsNet significantly and consistently outperforms ten competitors on six real-world hypergraphs, and we also show successful applications of WHATsNet to (a) ranking aggregation, (b) node clustering, and (c) product return prediction.
... If a network is modeled by graph ( which is a 2-uniform hypergraph) then all the relationships in the networks are binary and there are numerous definitions, techniques, and results for centrality (see [4,5,7,12,15,16,18]). There are very few studies on centrality with beyond-binary paradigm where the underlying architecture of the network is hypergraph(see [6,14,17]). Our study reveals some facts about the centralities in hypergraphs that cannot be experienced in the case of graphs. ...
To keep track of the importance of each vertex and each hyperedge of a hypergraph, we introduce some positive valued functions on the set of vertices and hyperedges, respectively. The terms vertex centrality and hyperedge centrality, respectively, refer to these functions on the set of vertices and hyperedges. The concepts of centrality vary according to the sense of importance. We introduce and investigate two eigenvector-based vertex centralities and one eigenvector-based hyperedge centrality here. We introduce some distance-based hypergraph centrality on the set of vertices by employing a pseudo metric called unit-distance. We also explore random walk-based hypergraph centralities. Our study shows that these centrality functions become constant on some clusters of vertices and thus provide some clusters of equally important vertices. Some of these clusters appear only in networks with multi-body interactions and disappear in networks with only pair-wise interactions.
Hypergraphs, which belong to the family of higher-order networks, are a natural and powerful choice for modeling group interactions in the real world. For example, when modeling collaboration networks, which may involve not just two but three or more people, the use of hypergraphs allows us to explore beyond pairwise (dyadic) patterns and capture groupwise (polyadic) patterns. The mathematical complexity of hypergraphs offers both opportunities and challenges for hypergraph mining. The goal of hypergraph mining is to find structural properties recurring in real-world hypergraphs across different domains, which we call patterns. To find patterns, we need tools. We divide hypergraph mining tools into three categories: (1) null models (which help test the significance of observed patterns), (2) structural elements (i.e., substructures in a hypergraph such as open and closed triangles), and (3) structural quantities (i.e., numerical tools for computing hypergraph patterns such as transitivity). There are also hypergraph generators, whose objective is to produce synthetic hypergraphs that are a faithful representation of real-world hypergraphs. In this survey, we provide a comprehensive overview of the current landscape of hypergraph mining, covering patterns, tools, and generators. We provide comprehensive taxonomies for each and offer in-depth discussions for future research on hypergraph mining.
The identification of central nodes within networks constitutes a task of fundamental importance in various disciplines, and it is an extensively explored problem within the scientific community. Several scalar metrics have been proposed for classic networks with dyadic connections, and many of them have later been extended to networks with higher-order interactions. We here introduce two novel measures for annotated hypergraphs: that of matrix centrality and that of role centrality. These concepts are formulated for hypergraphs where the roles of nodes within hyper-edges are explicitly delineated. Matrix centrality entails the assignment of a matrix to each node, whose dimensions are determined by the size of the largest hyper-edge in the hypergraph and the number of roles defined by the annotated hypergraph’s labeling function. This formulation facilitates the simultaneous ranking of nodes based on both hyper-edge size and role type. The second concept, role centrality, involves assigning a vector to each node, the dimension of which equals the number of roles specified. This metric enables the identification of pivotal nodes across different roles without distinguishing hyper-edge sizes. Through the application of these novel centrality measures to a range of synthetic and real-world examples, we demonstrate their efficacy in providing enhanced insights into the structural characteristics of the systems under consideration.
Hypernetwork is a useful way to depict multiple connections between nodes, making it an ideal tool for representing complex relationships in network science. In recent years, there has been a marked increase in studies on hypernetworks; however, the comparison of the difference between two hypernetworks has received less attention. This paper proposes a hyper-distance (HD)-based method for comparing hypernetworks. The method is based on higher-order information, i.e, the higher-order distance between nodes and Jensen–Shannon divergence. Experiments carried out on synthetic hypernetworks have shown that HD is capable of distinguishing between hypernetworks generated with different parameters, and it is successful in the classification of hypernetworks. Furthermore, HD outperforms current state-of-the-art baselines to distinguish empirical hypernetworks when hyperedges are randomly perturbed.
Higher-order networks reveal intricate structural connections within neural brain models by exposing relationships between multiple nodes. Within human brain networks, information transmission among neurons invariably incurs a time delay, resulting in abnormal discharge of membrane potential. In this study, we analyze correlation visualizations of the Laplacian matrix derived from both general networks and higher-order networks, highlighting eigenvector distributions. The presence of time delay significantly influences the dynamical behavior of the system. Therefore, we examine the phase diagram, identifying firing patterns such as fast-spiking and mixed-mode oscillations. Ultimately, we demonstrate that both time delay and diffusion can alter the dynamical behavior of the FitzHugh–Nagumo model. Notably, within higher-order networks, the phenomenon of Turing instability becomes more pronounced, offering an analogous framework for studying associated diseases. This instability emerges as neuronal states approach abnormal levels, leading to the onset of related diseases. Numerical simulation and data also verify these results.
Identifying the most influential nodes in spreading on the network is an important step to control the speed and range of spreading, which can be applied to accelerate the spread of beneficial information such as healthy behaviors, innovations and suppress the spread of epidemics, rumors, fake news. Existing researches on identification of influential spreaders are mostly based on low-order complex networks with pairwise interactions. However, interactions between individuals occur not only between pairwise nodes but also in groups of three or more nodes, which introduces more complex mechanism of reinforcement and indirect influence. The high-order networks such as simplicial complexes and hypergraphs, can describe features of interactions that go beyond the limitation of pairwise interactions. Currently, there are relatively few works in identifying the most spreading influential nodes in higher-order networks. Some centralities of nodes such as higher-order degree centrality and eigenvector centrality are proposed, but they mostly consider only the network structure. As for identification of influential spreaders, the spreading influence of a node is closely related to the spreading process. In this paper, we work on identification of influential spreaders on the simplicial complexes by taking both network structure and dynamical process into consideration. Firstly, we quantitatively describe the dynamics of disease spreading on simplicial complexes using the Susceptible-Infected-Recovered microscopic Markov equations. Next, we calculate the probability of nodes being infected in the spreading process with the microscopic Markov equations, which is defined as the spreading centrality (SC) of nodes. This spreading centrality involves both the structure of simplicial complex and the dynamical process on it, and is then used to rank the spreading influence of nodes. Simulation results on two types of synthetic simplicial complexes and four real simplicial complexes show that compared with the existing centralities on high-order networks and the optimal centralities of collective influence and nonbacktracking centrality in simple networks, the proposed spreading centrality can more accurately identify the most influential spreaders in the simplicial complexes. In addition, we find that the probability of nodes being infected is highly positively correlated with its influence, which is because disease preferentially reaches nodes with many contacts, who can in turn infect their many neighbors and become influential spreaders.
Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron−Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.
Hypergraphs are a natural modeling paradigm for networked systems with multiway interactions. A standard task in network analysis is the identification of closely related or densely interconnected nodes. We propose a probabilistic generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum likelihood inference in this model leads to a clustering objective that generalizes the popular modularity objective for graphs. From this, we derive an inference algorithm that generalizes the Louvain graph community detection method, and a faster, specialized variant in which edges are expected to lie fully within clusters. Using synthetic and empirical data, we demonstrate that the specialized method is highly scalable and can detect clusters where graph-based methods fail. We also use our model to find interpretable higher-order structure in school contact networks, U.S. congressional bill cosponsorship and committees, product categories in copurchasing behavior, and hotel locations from web browsing sessions.
We live and cooperate in networks. However, links in networks only allow for pairwise interactions, thus making the framework suitable for dyadic games, but not for games that are played in larger groups. Here, we study the evolutionary dynamics of a public goods game in social systems with higher-order interactions. First, we show that the game on uniform hypergraphs corresponds to the replicator dynamics in the well-mixed limit, providing a formal theoretical foundation to study cooperation in networked groups. Second, we unveil how the presence of hubs and the coexistence of interactions in groups of different sizes affects the evolution of cooperation. Finally, we apply the proposed framework to extract the actual dependence of the synergy factor on the size of a group from real-world collaboration data in science and technology. Our work provides a way to implement informed actions to boost cooperation in social groups.
Many real systems are strongly characterized by collective cooperative phenomena whose existence and properties still need a satisfactory explanation. Coherently with their collective nature, they call for new and more accurate descriptions going beyond pairwise models, such as graphs, in which all the interactions are considered as involving only two individuals at a time. Hypergraphs respond to this need, providing a mathematical representation of a system allowing from pairs to larger groups. In this work, through the use of different hypergraphs, we study how group interactions influence the evolution of cooperation in a structured population, by analyzing the evolutionary dynamics of the public goods game. Here we show that, likewise to network reciprocity, group interactions also promote cooperation. More importantly, by means of an invasion analysis in which the conditions for a strategy to survive are studied, we show how, in heterogeneously-structured populations, reciprocity among players is expected to grow with the increasing of the order of the interactions. This is due to the heterogeneity of connections and, particularly, to the presence of individuals standing out as hubs in the population. Our analysis represents a first step towards the study of evolutionary dynamics through higher-order interactions, and gives insights into why cooperation in heterogeneous higher-order structures is enhanced. Lastly, it also gives clues about the co-existence of cooperative and non-cooperative behaviors related to the structural properties of the interaction patterns.
The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review both the measures designed to characterize the structure of these systems, and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We then introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher-order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers.
In the past 20 years network science has proven its strength in modeling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Nevertheless, in many relevant cases, interactions are not pairwise but involve larger sets of nodes at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multibody interactions. Here we propose and study a class of random walks defined on such higher-order structures and grounded on a microscopic physical model where multibody proximity is associated with highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterization of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalized random-walk Laplace operator that reduces to the standard random-walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have full control of the high-order structures and on real-world networks where higher-order interactions are at play. As the first application of the method, we compare the behavior of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As the second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unraveling the effect of higher-order interactions on diffusive processes in higher-order networks, shedding light on mechanisms at the heart of biased information spreading in complex networked systems.
Prediction and diagnosis of complex disease may not always be possible with a small number of biomarkers. Modern 'omics' technologies make it possible to cheaply and quantitatively assay hundreds of molecules generating large amounts of data from individual samples. In this study, we describe a parenclitic network-based approach to disease classification using a synthetic data set modelled on data from the United Kingdom Collaborative Trial of Ovarian Cancer Screening (UKCTOCS) and serological assay data from a nested set of samples from the same study. This approach allows us to integrate quantitative proteomic and categorical metadata into a single network, and then use network topologies to construct logistic regression models for disease classification. In this study of ovarian cancer, comprising of 30 controls and cases with samples taken < 14 months to diagnosis (n = 30) and/or > 34 months to diagnosis (n = 29), we were able to classify cases with a sensitivity of 80.3% within 14 months of diagnosis and 18.9% in samples exceeding 34 months to diagnosis at a specificity of 98%. Furthermore, we use the networks to make observations about proteins within the cohort and identify GZMH and FGFBP1 as changing in cases (in relation to controls) at time points most distal to diagnosis. We conclude that network-based approaches may offer a solution to the problem of complex disease classification that can be used in personalised medicine and to describe the underlying biology of cancer progression at a system level.
Networks provide a powerful formalism for modeling complex systems, by representing the underlying set of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to-person, collaboration among a team rather than a pair of co-authors, or biological interaction between a set of molecules rather than just two. We refer to these type of simultaneous interactions on sets of more than two nodes as higher-order interactions; they are ubiquitous, but the empirical study of them has lacked a general framework for evaluating higher-order models. Here we introduce such a framework, based on link prediction, a fundamental problem in network analysis. The traditional link prediction problem seeks to predict the appearance of new links in a network, and here we adapt it to predict which (larger) sets of elements will have future interactions. We study the temporal evolution of 19 datasets from a variety of domains, and use our higher-order formulation of link prediction to assess the types of structural features that are most predictive of new multi-way interactions. Among our results, we find that different domains vary considerably in their distribution of higher-order structural parameters, and that the higher-order link prediction problem exhibits some fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.
Encoding brain regions and their connections as a network of nodes and edges captures many of the possible paths along which information can be transmitted as humans process and perform complex behaviors. Because cognitive processes involve large, distributed networks of brain areas, principled examinations of multi-node routes within larger connection patterns can offer fundamental insights into the complexities of brain function. Here, we investigate both densely connected groups of nodes that could perform local computations as well as larger patterns of interactions that would allow for parallel processing. Finding such structures necessitates that we move from considering exclusively pairwise interactions to capturing higher order relations, concepts naturally expressed in the language of algebraic topology. These tools can be used to study mesoscale network structures that arise from the arrangement of densely connected substructures called cliques in otherwise sparsely connected brain networks. We detect cliques (all-to-all connected sets of brain regions) in the average structural connectomes of 8 healthy adults scanned in triplicate and discover the presence of more large cliques than expected in null networks constructed via wiring minimization, providing architecture through which brain network can perform rapid, local processing. We then locate topological cavities of different dimensions, around which information may flow in either diverging or converging patterns. These cavities exist consistently across subjects, differ from those observed in null model networks, and – importantly – link regions of early and late evolutionary origin in long loops, underscoring their unique role in controlling brain function. These results offer a first demonstration that techniques from algebraic topology offer a novel perspective on structural connectomics, highlighting loop-like paths as crucial features in the human brain’s structural architecture.
A number of centrality measures are available to determine the relative importance of a node in a complex network, and betweenness is prominent among them. However, the existing centrality measures are not adequate in network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of disease over a network of towns) because they do not account for the changing percolation states of individual nodes. We propose a new measure, percolation centrality, that quantifies relative impact of nodes based on their topological connectivity, as well as their percolation states. The measure can be extended to include random walk based definitions, and its computational complexity is shown to be of the same order as that of betweenness centrality. We demonstrate the usage of percolation centrality by applying it to a canonical network as well as simulated and real world scale-free and random networks.
In recent years, the application of network analysis to neuroimaging data has provided useful insights about the brain’s functional and structural organization in both health and disease. This has proven a significant paradigm shift from the study of individual brain regions in isolation. Graph-based models of the brain consist of vertices, which represent distinct brain areas, and edges which encode the presence (or absence) of a structural or functional relationship between each pair of vertices. By definition, any graph metric will be defined upon this dyadic representation of the brain activity. It is however unclear to what extent these dyadic relationships can capture the brain’s complex functional architecture and the encoding of information in distributed networks. Moreover, because network representations of global brain activity are derived from measures that have a continuous response (i.e. interregional BOLD signals), it is methodologically complex to characterize the architecture of functional networks using traditional graph-based approaches. In the present study, we investigate the relationship between standard network metrics computed from dyadic interactions in a functional network, and a metric defined on the persistence homological scaffold of the network, which is a summary of the persistent homology structure of resting-state fMRI data. The persistence homological scaffold is a summary network that differs in important ways from the standard network representations of functional neuroimaging data: i) it is constructed using the information from all edge weights comprised in the original network without applying an ad hoc threshold and ii) as a summary of persistent homology, it considers the contributions of simplicial structures to the network organization rather than dyadic edge-vertices interactions. We investigated the information domain captured by the persistence homological scaffold by computing the strength of each node in the scaffold and comparing it to local graph metrics traditionally employed in neuroimaging studies. We conclude that the persistence scaffold enables the identification of network elements that may support the functional integration of information across distributed brain networks.
One of the most important problems in complex network’s theory is the location of the entities that are essential or have a main role within the network. For this purpose, the use of dissimilarity measures (specific to theory of classification and data mining) to enrich the centrality measures in complex networks is proposed. The centrality method used is the eigencentrality which is based on the heuristic that the centrality of a node depends on how central are the nodes in the immediate neighbourhood (like rich get richer phenomenon). This can be described by an eigenvalues problem, however the information of the neighbourhood and the connections between neighbours is not taken in account, neglecting their relevance when is one evaluates the centrality/importance/influence of a node. The contribution calculated by the dissimilarity measure is parameter independent, making the proposed method is also parameter independent. Finally, we perform a comparative study of our method versus other methods reported in the literature, obtaining more accurate and less expensive computational results in most cases.
The whole frame of interconnections in complex networks hinges on a specific
set of structural nodes, much smaller than the total size, which, if activated,
would cause the spread of information to the whole network [1]; or, if
immunized, would prevent the diffusion of a large scale epidemic [2,3].
Localizing this optimal, i.e. minimal, set of structural nodes, called
influencers, is one of the most important problems in network science [4,5].
Despite the vast use of heuristic strategies to identify influential spreaders
[6-14], the problem remains unsolved. Here, we map the problem onto optimal
percolation in random networks to identify the minimal set of influencers,
which arises by minimizing the energy of a many-body system, where the form of
the interactions is fixed by the non-backtracking matrix [15] of the network.
Big data analyses reveal that the set of optimal influencers is much smaller
than the one predicted by previous heuristic centralities. Remarkably, a large
number of previously neglected weakly-connected nodes emerges among the optimal
influencers. These are topologically tagged as low-degree nodes surrounded by
hierarchical coronas of hubs, and are uncovered only through the optimal
collective interplay of all the influencers in the network. Eventually, the
present theoretical framework may hold a larger degree of universality, being
applicable to other hard optimization problems exhibiting a continuous
transition from a known phase [16].
We make use of ideas from the theory of complex networks to implement a machine learning classification of human DNA methylation data, that carry signatures of cancer development. The data were obtained from patients with various kinds of cancers and represented as parenclictic networks, wherein nodes correspond to genes, and edges are weighted according to pairwise variation from control group subjects. We demonstrate that for the 10 types of cancer under study, it is possible to obtain a high performance of binary classification between cancer-positive and negative samples based on network measures. Remarkably, an accuracy as high as 93−99% is achieved with only 12 network topology indices, in a dramatic reduction of complexity from the original 15295 gene methylation levels. Moreover, it was found that the parenclictic networks are scale-free in cancer-negative subjects, and deviate from the power-law node degree distribution in cancer. The node centrality ranking and arising modular structure could provide insights into the systems biology of cancer.
Given their importance in shaping social networks and determining how
information or diseases propagate in a population, human interactions are the
subject of many data collection efforts. To this aim, different methods are
commonly used, from diaries and surveys to wearable sensors. These methods show
advantages and limitations but are rarely compared in a given setting. As
surveys targeting friendship relations might suffer less from memory biases
than contact diaries, it is also interesting to explore how daily contact
patterns compare with friendship relations and with online social links. Here
we make progresses in these directions by leveraging data from a French high
school: face-to-face contacts measured by two concurrent methods, sensors and
diaries; self-reported friendship surveys; Facebook links. We compare the data
sets and find that most short contacts are not reported in diaries while long
contacts have larger reporting probability, with a general tendency to
overestimate durations. Measured contacts corresponding to reported friendship
can have durations of any length but all long contacts correspond to reported
friendships. Online links not associated to reported friendships correspond to
short face-to-face contacts, highlighting the different nature of reported
friendships and online links. Diaries and surveys suffer from a low sampling
rate, showing the higher acceptability of sensor-based platform. Despite the
biases, we found that the overall structure of the contact network, i.e., the
mixing patterns between classes, is correctly captured by both self-reported
contacts and friendships networks. Overall, diaries and surveys tend to yield a
correct picture of the structural organization of the contact network, albeit
with much less links, and give access to a sort of backbone of the contact
network corresponding to the strongest links in terms of cumulative durations.
Random walks on networks is the standard tool for modelling spreading processes in social and biological systems. This first-order Markov approach is used in conventional community detection, ranking and spreading analysis, although it ignores a potentially important feature of the dynamics: where flow moves to may depend on where it comes from. Here we analyse pathways from different systems, and although we only observe marginal consequences for disease spreading, we show that ignoring the effects of second-order Markov dynamics has important consequences for community detection, ranking and information spreading. For example, capturing dynamics with a second-order Markov model allows us to reveal actual travel patterns in air traffic and to uncover multidisciplinary journals in scientific communication. These findings were achieved only by using more available data and making no additional assumptions, and therefore suggest that accounting for higher-order memory in network flows can help us better understand how real systems are organized and function.
We introduce a novel method to represent time independent, scalar data sets as complex networks. We apply our method to investigate gene expression in the response to osmotic stress of Arabidopsis thaliana. In the proposed network representation, the most important genes for the plant response turn out to be the nodes with highest centrality in appropriately reconstructed networks. We also performed a target experiment, in which the predicted genes were artificially induced one by one, and the growth of the corresponding phenotypes compared to that of the wild-type. The joint application of the network reconstruction method and of the in vivo experiments allowed identifying 15 previously unknown key genes, and provided models of their mutual relationships. This novel representation extends the use of graph theory to data sets hitherto considered outside of the realm of its application, vastly simplifying the characterization of their underlying structure.
We extend the concept of eigenvector centrality to multiplex networks, and introduce several alternative parameters that quantify the importance of nodes in a multi-layered networked system, including the definition of vectorial-type centralities. In addition, we rigorously show that, under reasonable conditions, such centrality measures exist and are unique. Computer experiments and simulations demonstrate that the proposed measures provide substantially different results when applied to the same multiplex structure, and highlight the non-trivial relationships between the different measures of centrality introduced.
Most real and engineered systems include multiple subsystems and layers of
connectivity, and it is important to take such features into account to try to
improve our understanding of these systems. It is thus necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community detection,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.
A Family of new measures of point and graph centrality based on early intuitions of Bavelas (1948) is introduced. These measures define centrality in terms of the degree to which a point falls on the shortest path between others and therefore has a potential for control of communication. They may be used to index centrality in any large or small network of symmetrical relations, whether connected or unconnected.
Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be proposed. Many of the classical complex networks centralities are based on the degree of a node, so in order to define degree centrality measures for simplices (which would characterise the relevance of a simplicial community in a simplicial network), a different definition of adjacency between simplices is required, since, contrarily to what happens in the vertex case (where there is only upper adjacency), simplices might also have other types of adjacency. The aim of these notes is threefold: first we will use the recently introduced notions of higher order simplicial degrees to propose new degree based centrality measures in simplicial complexes. These theoretical centrality measures, such as the simplicial degree centrality or the eigenvector centrality would allow not only to study the relevance of a simplicial community and the quality of its higher-order connections in a simplicial network, but also they might help to elucidate topological and dynamical properties of simplicial networks; sencond, we define notions of walks and distances in simplicial complexes in order to study connectivity of simplicial networks and to generalise, to the simplicial case, the well known closeness and betweenness centralities (needed for instance to study the relevance of a simplicial community in terms of its ability of transmitting information); third, we propose a new clustering coefficient for simplices in a simplicial network, different from the one knows so far and which generalises the standard graph clustering of a vertex. This measure should be essential to know the density of a simplicial network in terms of its simplicial communities.
PageRank has been widely used to measure the authority or the influence of a user in social networks. However, conventional PageRank only makes use of edge-based relations, which represent first-order relations between two connected nodes. It ignores higher-order relations that may exist between nodes. In this paper, we propose a novel framework, motif-based PageRank (MPR), to incorporate higher-order relations into the conventional PageRank computation. Motifs are subgraphs consisting of a small number of nodes. We use motifs to capture higher-order relations between nodes in a network and introduce two methods, one linear and one non-linear, to combine PageRank with higher-order relations. We conduct extensive experiments on three real-world networks, namely, DBLP, Epinions, and Ciao. We study different types of motifs, including 3-node simple and anchor motifs, 4-node and 5-node motifs. Besides using single motif, we also run MPR with ensemble of multiple motifs. We also design a learning task to evaluate the abilities of authority prediction with motif-based features. All experimental results demonstrate that MPR can significantly improve the performance of user ranking in social networks compared to the baseline methods.
Rich data are revealing that complex dependencies between the nodes of a network may not be captured by models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems. Rich data are revealing that complex dependencies between the nodes of a network may not be captured by models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems.
Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend the concept of node centrality to that of simplicial centrality and study several mathematical properties of degree, closeness, betweenness, eigenvector, Katz, and subgraph centrality for simplicial complexes. We study the degree distributions of these centralities at the different levels. We also compare and describe the differences between the centralities at the different levels. Using these centralities we study a method for detecting essential proteins in PPI networks of cells and explain the varying abilities of the centrality measures at the different levels in identifying these essential proteins.
We introduce a framework for the modeling of sequential data capturing pathways of varying lengths observed in a network. Such data are important, e.g., when studying click streams in the Web, travel patterns in transportation systems, information cascades in social networks, biological pathways, or time-stamped social interactions. While it is common to apply graph analytics and network analysis to such data, recent works have shown that temporal correlations can invalidate the results of such methods. This raises a fundamental question: When is a network abstraction of sequential data justified?Addressing this open question, we propose a framework that combines Markov chains of multiple, higher orders into a multi-layer graphical model that captures temporal correlations in pathways at multiple length scales simultaneously. We develop a model selection technique to infer the optimal number of layers of such a model and show that it outperforms baseline Markov order detection techniques. An application to eight real-world data sets on pathways and temporal networks shows that it allows to infer graphical models that capture both topological and temporal characteristics of such data. Our work highlights fallacies of network abstractions and provides a principled answer to the open question when they are justified. Generalizing network representations to multi-order graphical models, it opens perspectives for new data mining and knowledge discovery algorithms.
The tremendous diversity of species in ecological communities has motivated a century of research into the mechanisms that maintain biodiversity. However, much of this work examines the coexistence of just pairs of competitors. This approach ignores those mechanisms of coexistence that emerge only in diverse competitive networks. Despite the potential for these mechanisms to create conditions under which the loss of one competitor triggers the loss of others, we lack the knowledge needed to judge their importance for coexistence in nature. Progress requires borrowing insight from the study of multitrophic interaction networks, and coupling empirical data to models of competition.
We introduce a framework for the modeling of sequential data capturing pathways of varying lengths observed in a network. Such data are important, e.g., when studying click streams in information networks, travel patterns in transportation systems, information cascades in social networks, biological pathways or time-stamped social interactions. While it is common to apply graph analytics and network analysis to such data, recent works have shown that temporal correlations can invalidate the results of such methods. This raises a fundamental question: when is a network abstraction of sequential data justified? Addressing this open question, we propose a framework which combines Markov chains of multiple, higher orders into a multi-layer graphical model that captures temporal correlations in pathways at multiple length scales simultaneously. We develop a model selection technique to infer the optimal number of layers of such a model and show that it outperforms previously used Markov order detection techniques. An application to eight real-world data sets on pathways and temporal networks shows that it allows to infer graphical models which capture both topological and temporal characteristics of such data. Our work highlights fallacies of network abstractions and provides a principled answer to the open question when they are justified. Generalizing network representations to multi-order graphical models, it opens perspectives for new data mining and knowledge discovery algorithms.
Real networks exhibit heterogeneous nature with nodes playing far different roles in structure and function. To identify vital nodes is thus very significant, allowing us to control the outbreak of epidemics, to conduct advertisements for e-commercial products, to predict popular scientific publications, and so on. The vital nodes identification attracts increasing attentions from both computer science and physical societies, with algorithms ranging from simply counting the immediate neighbors to complicated machine learning and message passing approaches. In this review, we clarify the concepts and metrics, classify the problems and methods, as well as review the important progresses and describe the state of the art. Furthermore, we provide extensive empirical analyses to compare well-known methods on disparate real networks, and highlight the future directions. In despite of the emphasis on physics-rooted approaches, the unification of the language and comparison with cross-domain methods would trigger interdisciplinary solutions in the near future.
Here, we show a method to reconstruct connectivity hypermatrices of a general
hypergraph (without any self loop or multiple edge) using tensor. We also study
the different spectral properties of these hypermatrices and find that these
properties are similar for graphs and uniform hypergraphs. The representation
of a connectivity hypermatrix that is proposed here can be very useful for the
further development in spectral hypergraph theory.
Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mech-anisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
In the past years, network theory has successfully characterized the
interaction among the constituents of a variety of complex systems, ranging
from biological to technological, and social systems. However, up until
recently, attention was almost exclusively given to networks in which all
components were treated on equivalent footing, while neglecting all the extra
information about the temporal- or context-related properties of the
interactions under study. Only in the last years, taking advantage of the
enhanced resolution in real data sets, network scientists have directed their
interest to the multiplex character of real-world systems, and explicitly
considered the time-varying and multilayer nature of networks. We offer here a
comprehensive review on both structural and dynamical organization of graphs
made of diverse relationships (layers) between its constituents, and cover
several relevant issues, from a full redefinition of the basic structural
measures, to understanding how the multilayer nature of the network affects
processes and dynamics.
Virtually all domains of cognitive function require the integration of distributed neural activity. Network analysis of human brain connectivity has consistently identified sets of regions that are critically important for enabling efficient neuronal signaling and communication. The central embedding of these candidate 'brain hubs' in anatomical networks supports their diverse functional roles across a broad range of cognitive tasks and widespread dynamic coupling within and across functional networks. The high level of centrality of brain hubs also renders them points of vulnerability that are susceptible to disconnection and dysfunction in brain disorders. Combining data from numerous empirical and computational studies, network approaches strongly suggest that brain hubs play important roles in information integration underpinning numerous aspects of complex cognitive function.
Given a network G, it is known that the Bonacich centrality of the bipartite graph B(G) associated with G can be obtained in terms of the centralities of the line graph L(G) associated with G and the centrality of the network G+gr (whose adjacency matrix is obtained by adding to the adjacency matrix A(G) the diagonal matrix D=B ij , where B ii is the degree of node i in G) and conversely. In this contribution, we use the centrality of G to estimate the centrality of G+gr and show that the error committed is bounded by some measure of the irregularity of G. This estimate gives an analytical comparison of the eigenvector centrality of G with the centrality of L(G) in terms of some irregularity measure of G.
If we wish to be “scientific” in our approach to the Social Sciences then, in some way, we must want to make these disciplines more like the Physical Sciences. But here we meet the immediate snag that our intuition runs away from the thought that, as individual people, we can be adequately described as so many electrons and protons. This also leads to the fear that perhaps Physics is not formulated in such a manner as to show us the heart of its methods, as opposed to the fruits of its labours.
This paper is a review of a personal search over many years to find a formulation for physical science which would not do too much violence to accepted theories and yet show a way for its extension and generalization into fields of social science. Since the latter seems to require a language which mathematicians would call “combinatorial”, being concerned with finite sets, it became a search for a mathematical language which would describe certain key properties of the familiar continuum but which would carry these over when that continuum should be replaced by a finite set of points.
The language which exhibits these properties is the twentieth-century development of algebraic topology (what used to be called combinatorial topology), and some of its basic concepts are referred to in the first half of the paper.
From this point of view an indication of the role of the cocycle in physics is first developed (although many of the intriguing details are given elsewhere), and this is replaced by the same idea but referred to a relation between finite sets. The notion of the simplicial complex is then developed as the vehicle for that sense of structure which is inherent in either the laws of physics or the behaviour of social systems. The result is that one finds, when applying it to some specific field of enquiry, that it is peculiarly powerful for the representation of social or human activities. It leads too to a view of data which is structural (in a multi-dimensional space) in a way which the statistical view is unable to penetrate. Hopefully the method will help to develop new techniques for understanding, the data of relations.
We give here conditions that two graphs be congruent and some theorems on the connectivity of graphs, and we conclude with some applications to dual graphs. These last theorems might also be proved by topological methods. The definitions and results of a paper by the author on “Non-separable and planar graphs,” † will be made use of constantly. We shall refer to this paper as N. For convenience, we shall say two arcs touch if they have a common vertex.
The representation of complex systems as networks is inappropriate for the study of certain problems. We show several examples of social, biological, ecological and technological systems where the use of complex networks gives very limited information about the structure of the system. Consequently, we extend the concepts of subgraph centrality and clustering for complex networks represented by hypergraphs: complex hyper-networks. The first parameter characterizes the node participation in different sub-hypergraphs and the second one characterizes the transitivity in the hyper-network through the proportion of hyper-triangles to paths of length two. Another measure characterizing the formation of triples of mutually adjacent groups in the hyper-network is also introduced. All of these characteristics are studied in three different hyper-networks: a scientific collaboration hyper-network, an ecological competition hyper-network and the hyper-network formed by the American corporate elite in 1999.
In the US House and Senate, each piece of legislation is sponsored by a unique legislator. In addition, legislators can publicly express support for a piece of legislation by cosponsoring it. The network of sponsors and cosponsors provides information about the underlying social networks among legislators. I use a number of statistics to describe the cosponsorship network in order to show that it behaves much differently than other large social networks that have been recently studied. In particular, the cosponsorship network is much denser than other networks and aggregate features of the network appear to be influenced by institutional arrangements and strategic incentives. I also demonstrate that a weighted closeness centrality measure that I call ‘connectedness’ can be used to identify influential legislators.
Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
Betweenness is a measure of the centrality of a node in a network, and is normally calculated as the fraction of shortest paths between node pairs that pass through the node of interest. Betweenness is, in some sense, a measure of the influence a node has over the spread of information through the network. By counting only shortest paths, however, the conventional definition implicitly assumes that information spreads only along those shortest paths. Here, we propose a betweenness measure that relaxes this assumption, including contributions from essentially all paths between nodes, not just the shortest, although it still gives more weight to short paths. The measure is based on random walks, counting how often a node is traversed by a random walk between two other nodes. We show how our measure can be calculated using matrix methods, and give some examples of its application to particular networks.
The intuitive background for measures of structural centrality in social networks is reviewed and existing measures are evaluated in terms of their consistency with intuitions and their interpretability.Three distinct intuitive conceptions of centrality are uncovered and existing measures are refined to embody these conceptions. Three measures are developed for each concept, one absolute and one relative measure of the centrality of positions in a network, and one reflecting the degree of centralization of the entire network. The implications of these measures for the experimental study of small groups is examined.
We define the k-line graph of a hypergraph H as the graph whose vertices are the edges of H, two vertices being joined if the edges they represent intersect in at least k elements. In this paper we show that for any integer k and any graph G there exists a partial hypergraph H of some complete h-partite hypergraph Khh x N such that G is the k-line graph of H. We also prove that, for any integer p, there exist graphs which are not the (h - p)-line graph of some h-uniform hypergraph. As a corollary we answer a problem of C. Cook. Further we show that it is not possible to characterize the (h - 1)-line graphs by excluding a finite number of forbidden induced subgraphs.
The centrality and efficiency measures of a network G are strongly related to the respective measures on the dual G⋆ and the bipartite B(G) associated networks. We show some relationships between the Bonacich centralities c(G), c(G⋆) and c(B(G)) and between the efficiencies E(G) and E(G⋆) and we compute the behavior of these parameters in some examples.
A great variety of systems in nature, society and technology -- from the web
of sexual contacts to the Internet, from the nervous system to power grids --
can be modeled as graphs of vertices coupled by edges. The network structure,
describing how the graph is wired, helps us understand, predict and optimize
the behavior of dynamical systems. In many cases, however, the edges are not
continuously active. As an example, in networks of communication via email,
text messages, or phone calls, edges represent sequences of instantaneous or
practically instantaneous contacts. In some cases, edges are active for
non-negligible periods of time: e.g., the proximity patterns of inpatients at
hospitals can be represented by a graph where an edge between two individuals
is on throughout the time they are at the same ward. Like network topology, the
temporal structure of edge activations can affect dynamics of systems
interacting through the network, from disease contagion on the network of
patients to information diffusion over an e-mail network. In this review, we
present the emergent field of temporal networks, and discuss methods for
analyzing topological and temporal structure and models for elucidating their
relation to the behavior of dynamical systems. In the light of traditional
network theory, one can see this framework as moving the information of when
things happen from the dynamical system on the network, to the network itself.
Since fundamental properties, such as the transitivity of edges, do not
necessarily hold in temporal networks, many of these methods need to be quite
different from those for static networks.