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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 netwo...
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... the sake of illustration of our definitions, let us initially refer to the hypergraph (made of 11 nodes and 3 hyperedges) depicted in Figure 2, and its associated linegraph. For every node i ∈ V of a hypergraph H = (V, H), one can denote by E i the set of hyperedges incident to the node i (in the example of Figure 2, E i = {e 1 }, E j = {e 1 , e 2 } and E k = {e 3 }). ...
Context 2
... the sake of illustration of our definitions, let us initially refer to the hypergraph (made of 11 nodes and 3 hyperedges) depicted in Figure 2, and its associated linegraph. For every node i ∈ V of a hypergraph H = (V, H), one can denote by E i the set of hyperedges incident to the node i (in the example of Figure 2, E i = {e 1 }, E j = {e 1 , e 2 } and E k = {e 3 }). Then, one can define the weighted distance between the pair of nodes ...
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Citations
... Despite these advances, characterising shortest paths and connectivity in systems with higher-order interactions remains an open problem. Recently, efforts have been devoted to characterise the concepts of distance [46] and walks [47] and networks in networks with non-dyadic ties, as well as proposing efficient algorithms [48] to extract shortest paths in hypergraphs, limiting the analysis to static systems. ...
One of the defining features of complex networks is the connectivity properties that we observe emerging from local interactions. Recently, hypergraphs have emerged as a versatile tool to model networks with non-dyadic, higher-order interactions. Nevertheless, the connectivity properties of real-world hypergraphs remain largely understudied. In this work we introduce path size as a measure to characterise higher-order connectivity and quantify the relevance of non-dyadic ties for efficient shortest paths in a diverse set of empirical networks with and without temporal information. By comparing our results with simple randomised null models, our analysis presents a nuanced picture, suggesting that non-dyadic ties are often central and are vital for system connectivity, while dyadic edges remain essential to connect more peripheral nodes, an effect which is particularly pronounced for time-varying systems. Our work contributes to a better understanding of the structural organisation of systems with higher-order interactions.
... To overcome these limitations and, more precisely, describe clustering relationships in online social networks, this paper introduces the mathematical tool of hypergraphs. Hypergraphs extend the edges of traditional graphs to connect multiple nodes, offering an effective method to describe multivalent relationships and complex group interactions [32]. The hypergraph model is particularly suited to depicting group interactions and multiparty information exchanges in online social networks, a capability that stems from its structural characteristics, enabling it to naturally map multivalent relationships and high-order interactions [33,34]. ...
Social networks, functioning as core platforms for modern information dissemination, manifest distinctive user clustering behaviors and state transition mechanisms, thereby presenting new challenges to traditional information propagation models. Based on hypergraph theory, this paper augments the traditional SEIR model by introducing a novel hypernetwork information dissemination SSEIR model specifically designed for online social networks. This model accurately represents complex, multi-user, high-order interactions. It transforms the traditional single susceptible state (S) into active (Sa) and inactive (Si) states. Additionally, it enhances traditional information dissemination mechanisms through reaction process strategies (RP strategies) and formulates refined differential dynamical equations, effectively simulating the dissemination and diffusion processes in online social networks. Employing mean field theory, this paper conducts a comprehensive theoretical derivation of the dissemination mechanisms within the SSEIR model. The effectiveness of the model in various network structures was verified through simulation experiments, and its practicality was further validated by its application on real network datasets. The results show that the SSEIR model excels in data fitting and illustrating the internal mechanisms of information dissemination within hypernetwork structures, further clarifying the dynamic evolutionary patterns of information dissemination in online social hypernetworks. This study not only enriches the theoretical framework of information dissemination but also provides a scientific theoretical foundation for practical applications such as news dissemination, public opinion management, and rumor monitoring in online social networks.
... Notably, this survey focuses on patterns that emerge in real-world hypergraphs and generators designed for reproducing these real-world patterns. Mathematical concepts and generators without explicit validation on real-world hypergraphs (e.g., [134,143]) are not within the scope of this survey. ...
... 2 Estrada C 2 (H) [72] Random walk [74,75] [ 76,77] Vasilyeva [73] Weighted line graph [73] Fig. 4. Diagram of the path between two hypernodes in different cases [73] . ...
... 2 Estrada C 2 (H) [72] Random walk [74,75] [ 76,77] Vasilyeva [73] Weighted line graph [73] Fig. 4. Diagram of the path between two hypernodes in different cases [73] . ...
... 2 Estrada C 2 (H) [72] Random walk [74,75] [ 76,77] Vasilyeva [73] Weighted line graph [73] Fig. 4. Diagram of the path between two hypernodes in different cases [73] . ...
Complex networks serve as indispensable instruments for characterizing and comprehending intricate real-world systems. Recently, researchers have delved into the realm of higher-order networks, seeking to delineate interactions within these networks with greater precision or analyzing traditional pairwise networks from a higher-dimensional perspective. This endeavor has unearthed novel phenomena distinct from those observed in conventional pairwise networks. However, despite the significance of higher-order networks, research in this area remains comparatively sparse. Furthermore, the intricacy of higher-order interactions has led to a dearth of standardized definitions for their structural statistical measures, posing additional challenges in their investigation. In recognition of these challenges, this paper presents a comprehensive survey of commonly employed statistics and their underlying physical significance in two prevalent types of higher-order networks:hypergraphs and simplicial complex networks. It not only outlines the specific calculation methods and application scenarios of these statistical indicators but also provides a glimpse into future research trends. This comprehensive overview serves as a valuable resource for beginners or cross-disciplinary researchers interested in higher-order networks, enabling them to swiftly grasp the fundamental statistics pertaining to these advanced structures. By fostering a deeper understanding of higher-order networks, this paper facilitates quantitative analysis of their structural characteristics and serves as a guide for researchers aiming to develop novel statistical methods tailored to higher-order networks.
... Recently, both the properties that define smallworldness in a network; namely clustering coefficient [23] and average path length [24] have been separately studied taking higher order interactions into account, prompting the question of the existence of small-world phenomena in such systems. To our knowledge, the small-world property of higher-order networks has not been studied yet. ...
Most real-world networks are endowed with the small-world property, by means of which the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. The evidence sparkled a wealth of studies trying to reveal possible mechanisms through which the pairwise interactions amongst the units of a network are structured in a way to determine such observed
regularity. Here we show that smallworldness occurs also when interactions are of higher order. Namely, by considering Q-uniform hypergraphs and a process through which connections can be randomly rewired with given probability p, we find that
such systems may exhibit prominent clustering properties in connection with small average path lengths for a wide range of p values, in analogy to the case of dyadic interactions. The nature of small-world transition remains the same at different orders
Q (=2, 3, 4, 5, 6) of the interactions, however, the increase in the hyperedge order reduces the range of rewiring probability for which smallworldness emerge.
... Currently, most of the complex network modeling and reconstruction methods are based on traditional FONs [10,13]; however, in real life, higher-order interactions and higher-order dependencies are ubiquitous [26]. Collaboration among more than two authors of a paper [27], online social networks with group relationships among more than two people [28], etc., are all examples of higher-order interactions, which can generally be modeled as hypergraph or simplicial complex [29][30][31][32][33]. Networks that take these higher-order relationships into account are generally referred to as higher-order networks. ...
In the era of the popularization of the Internet of Things (IOT), analyzing people’s daily life behavior through the data collected by devices is an important method to mine potential daily requirements. The network method is an important means to analyze the relationship between people’s daily behaviors, while the mainstream first-order network (FON) method ignores the high-order dependencies between daily behaviors. A higher-order dependency network (HON) can more accurately mine the requirements by considering higher-order dependencies. Firstly, our work adopts indoor daily behavior sequences obtained by video behavior detection, extracts higher-order dependency rules from behavior sequences, and rewires an HON. Secondly, an HON is used for the RandomWalk algorithm. On this basis, research on vital node identification and community detection is carried out. Finally, results on behavioral datasets show that, compared with FONs, HONs can significantly improve the accuracy of random walk, improve the identification of vital nodes, and we find that a node can belong to multiple communities. Our work improves the performance of user behavior analysis and thus benefits the mining of user requirements, which can be used to personalized recommendations and product improvements, and eventually achieve higher commercial profits.