Scale-free networks via attaching to random neighbors

Preferential attachment is considered one of the key factors in the formation of scale-free networks. However, complete random attachment without a preferential mechanism can also generate scale-free networks in nature, such as protein interaction networks in cells. This article presents a new scale-free network model that applies the following general mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach to random neighbors of random vertices that are already well connected. The proposed model does not require global-based preferential strategies and utilizes only the random attachment method. Theoretical analysis and numerical simulation results denote that the proposed model has steady scale-free network characteristics, and random attachment without a preferential mechanism may generate scale-free networks.     

Error and attack tolerance of evolving networks with local preferential attachment

Networks generated by local-world evolving network model display a transition from exponential network to power-law network with respect to connectivity distribution. We investigate statistical properties of the evolving networks and the responses of these networks under random errors and intentional attacks. It has been found that local world size M has great effect on the network's heterogeneity, thus leading to transitional behaviors in network's robustness against errors and attacks. Numerical results show that networks constructed with local preferential attachment mechanism can maintain the robustness of scale-free networks under random errors and concurrently improve reliance against targeted attacks on highly connected nodes.     

Scale-free networks from varying vertex intrinsic fitness

A new mechanism leading to scale-free networks is proposed in this Letter. It is shown that, in many cases of interest, the connectivity power-law behavior is neither related to dynamical properties nor to preferential attachment. Assigning a quenched fitness value x(i) to every vertex, and drawing links among vertices with a probability depending on the fitnesses of the two involved sites, gives rise to what we call a good-get-richer mechanism, in which sites with larger fitness are more likely to become hubs (i.e., to be highly connected).     

Clique percolation in random networks

The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdos-Rényi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p(c) (k) = [(k - 1)N](-1/(k - 1)). At the transition point the scaling of the giant component with N is highly nontrivial and depends on k. We discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks.     

Modeling dynamics of information networks

We propose an information-based model for network dynamics in which imperfect information leads to networks where the different vertices have widely different numbers of edges to other vertices, and where the topology has hierarchical features. The possibility to observe scale-free networks is linked to a minimally connected system where hubs remain dynamic.     

Modeling Aggregation Processes of Lennard-Jones particles Via Stochastic Networks

We model an isothermal aggregation process of particles/atoms interacting according to the Lennard-Jones pair potential by mapping the energy landscapes of each cluster size N onto stochastic networks, computing transition probabilities from the network for an N-particle cluster to the one for \(N+1\), and connecting these networks into a single joint network. The attachment rate is a control parameter. The resulting network representing the aggregation of up to 14 particles contains 6427 vertices. It is not only time-irreversible but also reducible. To analyze its transient dynamics, we introduce the sequence of the expected initial and pre-attachment distributions and compute them for a wide range of attachment rates and three values of temperature. As a result, we find the configurations most likely to be observed in the process of aggregation for each cluster size. We examine the attachment process and conduct a structural analysis of the sets of local energy minima for every cluster size. We show that both processes taking place in the network, attachment and relaxation, lead to the dominance of icosahedral packing in small (up to 14 atom) clusters.     

增长及非增长无标度网络的成因解析

分析两类无标度网络的形成原因,提出一个无标度网络演化模型并进行一系列数值实验.基于分析和实验得到推论:只要保持足够低的网络密度,通过基于度的偏好连接就可形成长期稳定的无标度网络.规模增长和点边增删既是客观存在,又起到了控制网络密度的作用,足够低的网络密度和基于度的偏好连接是所有无标度网络共同的必要条件.推论可同时解释增长和非增长无标度网络的形成原因.研究结果有助于理解各种真实无标度网络和建立相应的模型.     

Detecting hierarchical and overlapping network communities using locally optimal modularity changes

Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the preferred edge that edge for which the modularity change is maximal. When an edge is preferred by both vertices that it links, it appears to be the optimal choice from the local viewpoint. We use the locally optimal edges to define the algorithm: simultaneously merge all pairs of communities that are connected by locally optimal edges that would increase the modularity, redetermining the locally optimal edges after each step and continuing so long as the modularity can be further increased. We apply the algorithm to model and empirical networks, demonstrating that it can efficiently produce high-quality community solutions. We relate the performance and implementation details to the structure of the resulting community hierarchies. We additionally consider a complementary local clustering algorithm, describing how to identify overlapping communities based on the local optimality condition.     

An Information-Theoretic Bound on p-Values for Detecting Communities Shared between Weighted Labeled Graphs

Extraction of subsets of highly connected nodes (“communities” or modules) is a standard step in the analysis of complex social and biological networks. We here consider the problem of finding a relatively small set of nodes in two labeled weighted graphs that is highly connected in both. While many scoring functions and algorithms tackle the problem, the typically high computational cost of permutation testing required to establish the p-value for the observed pattern presents a major practical obstacle. To address this problem, we here extend the recently proposed CTD (“Connect the Dots”) approach to establish information-theoretic upper bounds on the p-values and lower bounds on the size and connectedness of communities that are detectable. This is an innovation on the applicability of CTD, broadening its use to pairs of graphs.     

Evolution of a Modified Binomial Random Graph by Agglomeration

In the classical Erd?s–Rényi random graph G(n, p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n, p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erd?s-Rényi random graph G(n, p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n, p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n, p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real-world applications.     

Fuzzy analysis of community detection in complex networks

A snowball algorithm is proposed to find community structures in complex networks by introducing the definition of community core and some quantitative conditions. A community core is first constructed, and then its neighbors, satisfying the quantitative conditions, will be tied to this core until no node can be added. Subsequently, one by one, all communities in the network are obtained by repeating this process. The use of the local information in the proposed algorithm directly leads to the reduction of complexity. The algorithm runs in O(n+m) time for a general network and O(n) for a sparse network, where n is the number of vertices and m is the number of edges in a network. The algorithm fast produces the desired results when applied to search for communities in a benchmark and five classical real-world networks, which are widely used to test algorithms of community detection in the complex network. Furthermore, unlike existing methods, neither global modularity nor local modularity is utilized in the proposal. By converting the considered problem into a graph, the proposed algorithm can also be applied to solve other cluster problems in data mining.     

Phase transitions and overlapping modules in complex networks

Complex systems can be described in terms of networks capturing the intricate web of connections among the units they are made of. Here we review two aspects of the possible organization of such networks. First, we provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. In our studies we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure. Next, we address a question of great current interest which is about the modular structure of networks. We describe, how to interpret the global organization as the coexistence of structural sub-units (modules or communities) associated with more highly interconnected parts. The existing deterministic methods used for large networks find separated communities, while most of the actual networks are made of highly overlapping cohesive groups of nodes. We describe a recently introduced an approach to analyze the main statistical features of the interwoven sets of overlapping communities making a step towards the uncovering of the modular structure of complex systems. Our approach is based on defining communities as clusters of percolating complete subgraphs called k-cliques. We present the basic features of the associated percolation transition of overlapping k-cliques. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique to explore overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks.     

Detecting community structure from coherent oscillation of excitable systems

Many networks are proved to have community structures. On the basis of the fact that the dynamics on networks are intensively affected by the related topology, in this paper the dynamics of excitable systems on networks and a corresponding approach for detecting communities are discussed. Dynamical networks are formed by interacting neurons; each neuron is described using the FHN model. For noisy disturbance and appropriate coupling strength, neurons may oscillate coherently and their behavior is tightly related to the community structure. Synchronization between nodes is measured in terms of a correlation coefficient based on long time series. The correlation coefficient matrix can be used to project network topology onto a vector space. Then by the K-means cluster method, the communities can be detected. Experiments demonstrate that our algorithm is effective at discovering community structure in artificial networks and real networks, especially for directed networks. The results also provide us with a deep understanding of the relationship of function and structure for dynamical networks.     

Diffusion and networks: A powerful combination!

Over the last decade, an enormous interest and activity in complex networks have been witnessed within the physics community. On the other hand, diffusion and its theory have equipped the toolbox of the physicist for decades. In this paper, we will demonstrate how to combine these two seemingly different topics in a fruitful manner. In particular, we will review and develop further an auxiliary diffusive process on weighted networks that represents a powerful concept and tool for studying network (community) structures. The working principle of the method is the observation that the relaxation of the diffusive process toward the stationary state is non-local and fastest in the highly connected regions of the network. This can be used to acquire non-trivial information about the structure of clustered and non-clustered networks.     

Maximum modular graphs

Modularity has been explored as an important quantitative metric for community and cluster detection in networks. Finding the maximum modularity of a given graph has been proven to be NP-complete and therefore, several heuristic algorithms have been proposed. We investigate the problem of finding the maximum modularity of classes of graphs that have the same number of links and/or nodes and determine analytical upper bounds. Moreover, from the set of all connected graphs with a fixed number of links and/or number of nodes, we construct graphs that can attain maximum modularity, named maximum modular graphs. The maximum modularity is shown to depend on the residue obtained when the number of links is divided by the number of communities. Two applications in transportation networks and data-centers design that can benefit of maximum modular partitioning are proposed.     

Network formed by traces of random walks

We propose a model of time evolving networks in which a kind of transport between vertices generates new edges in the graph. We call the model “Network formed by traces of random walks”, because the transports are represented abstractly by random walks. Our numerical calculations yield several important properties observed commonly in complex networks, although the graph at initial time is only a one-dimensional lattice. For example, the distribution of vertex degree exhibits various behaviors such as exponential, power law like, and bi-modal distribution according to change of probability of extinction of edges. Another property such as strong clustering structure and small mean vertex–vertex distance can also be found. The transports represented by random walks in a framework of strong links between regular lattice is a new mechanisms which yields biased acquisition of links for vertices.     

The BA model with finite-precision preferential attachment

Preferential attachment is an indispensable ingredient of the BA model and its variants. In this paper, we modify the BA model by considering the effect of finite-precision preferential attachment, which exists in many real networks. Finite-precision preferential attachment refers to existing nodes with preferential probability Π varying within a certain interval, which is determined by the value of a given precision, being considered to have an equal chance of capturing a new link. The new model reveals a transition from exponential scaling to a power-law distribution along with the increase of the precision. Epidemic dynamics and immunization on the new network are investigated and it is found that the finite-precision effect should be considered in tasks such as infection rate prediction or immunization policy making.     

Ordered community structure in networks

Community structure in networks is often a consequence of homophily, or assortative mixing, based on some attribute of the vertices. For example, researchers may be grouped into communities corresponding to their research topic. This is possible if vertex attributes have unordered discrete values, but many networks exhibit assortative mixing by some ordered (discrete or continuous) attribute, such as age or geographical location. In such cases, the identification of discrete communities may be difficult or impossible. We consider how the notion of community structure can be generalized to networks that have assortative mixing by ordered attributes. We propose a method of generating synthetic networks with ordered communities and investigate the effect of ordered community structure on the spread of infectious diseases. We also show that current community detection algorithms fail to recover community structure in ordered networks, and evaluate an alternative method using a layout algorithm to recover the ordering.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

An efficient agent-based algorithm for overlapping community detection using nodes’ closeness

Communities are groups of nodes forming tightly connected units in networks. Some nodes can be shared between different communities of a network. The presence of overlapping nodes and their associated membership diversity is a common characteristic of social networks. Analyzing these overlapping structures can reveal valuable information about the intrinsic features of realistic complex networks, especially social networks.