ON THE DEGREE DISTRIBUTION OF A BIPARTITE NETWORK WITH REWIRING DYNAMICS

An important step in unveiling the relation between network structure and dynamics defined on networks is to detect communities, and numerous methods have been developed separately to identify community structure in different classes of networks, such as unipartite networks, bipartite networks, and directed networks. Here, we show that the finding of communities in such networks can be unified in a general framework-detection of community structure in bipartite networks. Moreover, we propose an evolutionary method for efficiently identifying communities in bipartite networks. To this end, we show that both unipartite and directed networks can be represented as bipartite networks, and their modularity is completely consistent with that for bipartite networks, the detection of modular structure on which can be reformulated as modularity maximization. To optimize the bipartite modularity, we develop a modified adaptive genetic algorithm (MAGA), which is shown to be especially efficient for community structure detection. The high efficiency of the MAGA is based on the following three improvements we make. First, we introduce a different measure for the informativeness of a locus instead of the standard deviation, which can exactly determine which loci mutate. This measure is the bias between the distribution of a locus over the current population and the uniform distribution of the locus, i.e., the Kullback-Leibler divergence between them. Second, we develop a reassignment technique for differentiating the informative state a locus has attained from the random state in the initial phase. Third, we present a modified mutation rule which by incorporating related operations can guarantee the convergence of the MAGA to the global optimum and can speed up the convergence process. Experimental results show that the MAGA outperforms existing methods in terms of modularity for both bipartite and unipartite networks.

On the degree distribution of projected networks mapped from bipartite networks

Recent years have witnessed a widespread increase of interest in network representation learning (NRL). By far most research efforts have focused on NRL for homogeneous networks like social networks where vertices are of the same type, or heterogeneous networks like knowledge graphs where vertices (and/or edges) are of different types. There has been relatively little research dedicated to NRL for bipartite networks. Arguably, generic network embedding methods like node2vec and LINE can also be applied to learn vertex embeddings for bipartite networks by ignoring the vertex type information. However, these methods are suboptimal in doing so, since real-world bipartite networks concern the relationship between two types of entities, which usually exhibit different properties and patterns from other types of network data. For example, E-Commerce recommender systems need to capture the collaborative filtering patterns between customers and products, and search engines need to consider the matching signals between queries and webpages. This work addresses the research gap of learning vertex representations for bipartite networks. We present a new solution BiNE, short for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bi</b> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">partite</i> <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</b> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">etwork</i> <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</b> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mbedding</i> , which accounts for two special properties of bipartite networks: long-tail distribution of vertex degrees and implicit connectivity relations between vertices of the same type. Technically speaking, we make three contributions: (1) We design a biased random walk generator to generate vertex sequences that preserve the long-tail distribution of vertices; (2) We propose a new optimization framework by simultaneously modeling the explicit relations (i.e., observed links) and implicit relations (i.e., unobserved but transitive links); (3) We explore the theoretical foundations of BiNE to shed light on how it works, proving that BiNE can be interpreted as factorizing multiple matrices. We perform extensive experiments on five real datasets covering the tasks of link prediction (classification) and recommendation (ranking), empirically verifying the effectiveness and rationality of BiNE. Our experiment codes are available at: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/clhchtcjj/BiNE</uri> .

A key challenge in the social network community is the problem of network generation -- that is, how can we create synthetic networks that match characteristics traditionally found in most real world networks? Important characteristics that are present in social networks include a power law degree distribution, small diameter, and large amounts of clustering. However, most current network generators, such as the Chung Lu and Kronecker models, largely ignore the clustering present in a graph and focus on preserving other network statistics, such as the power law distribution. Models such as the exponential random graph model have a transitivity parameter that can capture clustering, but they are computationally difficult to learn, making scaling to large real world networks intractable. In this work, we propose an extension to the Chung Lu random graph model, the Transitive Chung Lu (TCL) model, which incorporates the notion transitive edges. Specifically, it combines the standard Chung Lu model with edges that are formed through transitive closure (e.g., by connecting a 'friend of a friend'). We prove TCL's expected degree distribution is equal to the degree distribution of the original input graph, while still providing the ability to capture the clustering in the network. The single parameter required by our model can be learned in seconds on graphs with millions of edges, networks can be generated in time that is linear in the number of edges. We demonstrate the performance of TCL on four real-world social networks, including an email dataset with hundreds of thousands of nodes and millions of edges, showing TCL generates graphs that match the degree distribution, clustering coefficients and hop plots of the original networks.

We describe a class of new algorithms to construct bipartite networks that preserves a prescribed degree and joint-degree (degree-degree) distribution of the nodes. Bipartite networks are graphs that can represent real-world interactions between two disjoint sets, such as actor-movie networks, author-article networks, co-occurrence networks, and heterosexual partnership networks. Often there is a strong correlation between the degree of a node and the degrees of the neighbors of that node that must be preserved when generating a network that reflects the structure of the underling system. Our bipartite 2K (B2K) algorithms generate an ensemble of networks that preserve prescribed degree sequences for the two disjoint set of nodes in the bipartite network, and the joint-degree distribution that is the distribution of the degrees of all neighbors of nodes with the same degree. We illustrate the effectiveness of the algorithms on a romance network using the NetworkX software environment to compare other properties of a target network that are not directly enforced by the B2K algorithms. We observe that when average degree of nodes is low, as is the case for romance and heterosexual partnership networks, then the B2K networks tend to preserve additional properties, such as the cluster coefficients, than algorithms that do not preserve the joint-degree distribution of the original network.

An evolving model of online bipartite networks

We study how the community structure of bipartite mutualistic networks changes in a dynamic context. First, we consider a real mutualistic network and introduce extinction events according to several scenarios. We model extinctions as node or interaction removals. For node removal, we consider random, directed and sequential extinctions; for interaction removal, we consider random extinctions. The bipartite network reorganizes showing an increase of the effective modularity and a fast decrease of the persistence of the species in the original communities with increasing number of extinction events. Second, we compare extinctions in a real mutualistic network with the growth of a bipartite network model. The modularity reaches a stationary value and nodes remain in the same community after joining the network. Our results show that perturbations and disruptive events affect the connectivity pattern of mutualistic networks at the mesoscale level. The increase of the effective modularity observed in some scenarios could provide some protection to the remaining ecosystem.

The assembly of plant pathogenic communities is one of the central topics in community ecology. Based on collected metadata, a summary host-pathogen bipartite network was constructed, and network metrics such as connectance, node degree distribution, nestedness and modularity were employed to verify the non-random nature of host-pathogen interactions. The selected hosts (Quercus robur, Q. cerris, Q. frainetto, Q. pubescens, Q. petraea, Fagus sylvatica and Picea abies) are tree species characteristic for European forest ecosystems, covering large areas of Central and South-Eastern Europe including Romania. The selected tree species (based on their dominance in forest ecosystems) establish 109 forest ecosystem types in Romanian forest ecosystems’ typology. 105 species of wood pathogens and sapro-pathogens, covering whole area of Europe, were retrieved from published papers. The established bipartite network of wood pathogens and sapro-pathogens and their tree hosts was significantly modular, displaying four modules (Qnetwork = 0.24) and significantly nested (Nnetwork = 0.67), presented power law degree distribution, a common topology with mutualistic networks. Network topological characteristics generally encountered in parasitic networks described in literature, confirm the non-random assembly of the wood and bark pathogenic communities. Non-metric Multidimensional Scaling (NMDS) performed on wood pathogens ‘matrix and forest types’ matrix showed high resemblance in the positioning pattern of trees in ordination space. Mantel test confirmed the NMDS result - a significant correlation between similarities in pathogen share and similarities in tree composition of the forest ecosystems in Romanian forest classification. The network architecture resembles other published networks depicting symbiotic interactions. As tree species assemble in distinct forest types, the transmission of emerging or invasive pathogens is expected to continue in the future due to modular nature of the networks, triggered by major climatic changes.

The data deluge is changing the operating environment of many sensing systems from data-poor to data-rich-so data-rich that we are in jeopardy of being overwhelmed. Managing and exploiting the data deluge require a reinvention of sensor system design and signal processing theory. The potential pay-offs are huge, as the resulting sensor systems will enable radically new information technologies and powerful new tools for scientific discovery.

Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree-degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree-degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree-degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree-degree distance distribution better represents the scale-free property of a complex network.

Perhaps the most recent controversial topic in network science research is to determine whether real-world complex networks are scale-free or not. Recently, Broido and Clauset [A.D. Broido, A. Clauset, Nature Communication, 10, 1017 (2019)] asserted that the degree distributions of real-world networks are rarely power law under statistical tests. Such complex networks, including social, biological, information, temporal, and brain networks, are often heavy-tailed where the assumption on the scale-free nature of real-world heavy-tailed networks become insignificant as the complex system evolves over time. The failure of power law distribution in fitting the degree distribution data is mainly due to the presence of an identifiable non-linearity within the entire degree distribution in a log-log scale of a complex heavy-tailed network. In this study, we attempt to address this issue by proposing a new class of heavy-tailed probability distributions for modeling the entire degree distributions of complex networks. We introduce a new family of generalized Lomax models (GLM) to capture the non-linearity of these heavy-tailed networks. These newly introduced GLM-type distributions provide better fitting and greater flexibility to the entire node degree distribution of complex networks. Several statistical properties of the proposed model, such as extreme value and inferential statistical properties, are derived into this context. Interestingly, the GLM family belongs to the basin of attraction of Frechet distribution, a heavy-tailed extreme value distribution. Rigorous experimental analysis showcases the excellent performance of the proposed family of distributions while fitting the heavy-tailed real-world complex networks over fifty real-world datasets in comparison with benchmark probability models. Our results show that GLM-type distributions are not rare, able to model almost 90% of the tested networks accurately compared to benchmark probability models.

Bipartite networks that characterize complex relationships among data arise in various domains. The existing bipartite network models are mainly based on a type of relationship between objects, and cannot effectively describe multiple relationships in the real world. In this paper, we propose a multi-relationship bipartite network (MBN) model, which can describe multiple relationships between two types of objects, and realizes simple weighted bipartite network reconstruction. Our model contains three major modules, namely multi-relationship bipartite network modeling (MBNM), multi-relationship aggregation module (MAM) and network reconstruction module (NRM). In MBNM, a multi-relationship bipartite network is proposed to describe multiple relationships between two types of objects. In the MAM, considering that different relationships have different information for the model, we introduce a novel relationship-level attention mechanism, and the aggregation of multiple relationships is carried out through the importance of each relationship. Based on the learning framework, the NRM can learn the potential representations of nodes after multi-relationship aggregation, and design a nonlinear fusion mechanism to reconstruct weighted bipartite network. We conducted extensive experiments on three real-world datasets and the results show that multi-relationship aggregation can effectively improve the performance of the model. In addition, experiments also show that our model can outperform existing competitive baseline method.

Many networks generated by nature have two generic properties: they are formed in the process of preferential attachment and they are scale-free. Considering these features, by interfering with mechanism of the preferential attachment, we propose a generalisation of the Barabási–Albert model—the ’Fractional Preferential Attachment’ (FPA) scale-free network model—that generates networks with time-independent degree distributions with degree exponent (where corresponds to the typical value of the BA model). In the FPA model, the element controlling the network properties is the f parameter, where . Depending on the different values of f parameter, we study the statistical properties of the numerically generated networks. We investigate the topological properties of FPA networks such as degree distribution, degree correlation (network assortativity), clustering coefficient, average node degree, network diameter, average shortest path length and features of fractality. We compare the obtained values with the results for various synthetic and real-world networks. It is found that, depending on f, the FPA model generates networks with parameters similar to the real-world networks. Furthermore, it is shown that f parameter has a significant impact on, among others, degree distribution and degree correlation of generated networks. Therefore, the FPA scale-free network model can be an interesting alternative to existing network models. In addition, it turns out that, regardless of the value of f, FPA networks are not fractal.

In this paper we study the epidemic spread in adaptive networks with multitype agents. We assume that the infection probability as well as the rewiring probability between agents is type-dependent. Under this assumption we analyze a system of M-type networks with mean-field approximation. These general expressions apply to networks with various choices of infection mechanisms and rewiring rules. We explicitly evaluate the infection level for two-type networks with intratype/intertype infecting and rewiring and investigate their impacts on the epidemic threshold. By plotting the bifurcation diagram for various parameters, we find a bistability region where both the disease-free state and the endemic state co-exist for appropriate rewiring dynamics. The area of each phase depends on the corresponding interaction modes. We show that consistency between infecting and rewiring modes speeds up the disease spread, while inconsistency contributes to halting the outbreak.

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical textsf {NP}-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta )-approximation for Minimum Vertex Cover and Minimum Dominating Set, and a Omega (Delta )-approximation for Maximum Independent Set. We prove that the PLB-U property with beta >2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.