Modeling temporal hypergraphs

Homophily, Preferential Attachment or Triadic Closure? How the Topology of Acquaintance Networks is Determined by Processes of Link Creation

Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment

Traditional network generation models aim to replicate global structural properties observed on various real-world networks (power law degree distribution, short average distance, community structure, etc.) through synthetic link formation mechanisms such as triadic closure or preferential attachment. Nowadays the large amount of data available allows to study more precisely the link formation processes and to compare the models to real situations. In this work, we analyse the communication activities of the users of a mobile telephony operator for studying the link formation process. The communications emitted and received by users are strong indicators for understanding the underlying mechanisms of the link formation process and highlighting how user attributes correlate with the formation process in such a network. In a first study conducted at a global level we show that the traditional mechanisms commonly used in network generation models are not the main link formation mechanisms involved in this network. In a second study, we analyse more precisely the link formation process by searching for correlations between some user attributes and the formation of the new links. These results provide valuable guidelines to the design of more realistic generation models taking into account network structure and node attributes.

Preferential attachment models have been widely studied in complex networks, because they can explain the formation of many networks like social networks, citation networks, power grids, and biological networks, to name a few. Motivated by the application of key predistribution in wireless sensor networks (WSN), we initiate the study of preferential attachment with degree bound. Our paper has two important contributions to two different areas. The first is a contribution in the study of complex networks. We propose preferential attachment model with degree bound for the first time. In the normal preferential attachment model, the degree distribution follows a power law, with many nodes of low degree and a few nodes of high degree. In our scheme, the nodes can have a maximum degree $d_{\max}$, where $d_{\max}$ is an integer chosen according to the application. The second is in the security of wireless sensor networks. We propose a new key predistribution scheme based on the above model. The important features of this model are that the network is fully connected, it has fewer keys, has larger size of the giant component and lower average path length compared with traditional key predistribution schemes and comparable resilience to random node attacks. We argue that in many networks like key predistribution and Internet of Things, having nodes of very high degree will be a bottle-neck in communication. Thus, studying preferential attachment model with degree bound will open up new directions in the study of complex networks, and will have many applications in real world scenarios.

Using the dynamic method to evolve a weighted network, a new weighted network with accelerated attachment of newly added edges is proposed. At each time step, the number of newly added edges is proportional to the size of the network. The degree distribution of the network is discussed. The strength distribution is also obtained, and the analytical relation between the strength and the degrees of nodes is presented. The network shows scale-free property in both theoretical analysis and numerical simulations. In addition, regardless of degree preferential attachment or strength preferential attachment in the evolution process, as long as the weight of edge follows a certain probability distribution, the degree distribution always follows the power-law distribution and its exponent is not related with the attachment mechanism and the specific probability distribution.

Social informatics: Beyond emergence

Link prediction algorithms in complex networks, such as social networks, biological networks, drug-drug interactions, communication networks, and so on, assign scores to predict potential links between two nodes. Link prediction (LP) enables researchers to learn unknown, new as well as future interactions among the entities being modeled in the complex networks. In addition to measures like degree distribution, clustering coefficient, centrality, etc., another metric to characterize structural properties is network assortativity which measures the tendency of nodes to connect with similar nodes. In this paper, we explore metrics that effectively predict the links based on the assortativity profiles of the complex networks. To this end, we first propose an approach that generates networks of varying assortativity levels and utilize three sets of link prediction models combining the similarity of neighborhoods and preferential attachment. We carry out experiments to study the LP accuracy (measured in terms of area under the precision-recall curve) of the link predictors individually and in combination with other baseline measures. Our analysis shows that link prediction models that explore a large neighborhood around nodes of interest, such as CH2-L2 and CH2-L3, perform consistently for assortative as well as disassortative networks. While common neighbor-based local measures are effective for assortative networks, our proposed combination of common neighbors with node degree is a good choice for the LP metric in disassortative networks. We discuss how this analysis helps achieve the best-parameterized combination of link prediction models and its significance in the context of link prediction from incomplete social and biological network data.

Consider a preferential attachment model for network evolution that allows both node and edge arrival events: at time t, with probability $$p_t$$ a new node arrives and a new edge is added between the new node and an existing node, and with probability $$1-p_t$$ a new edge is added between two existing nodes. In both cases existing nodes are chosen at random according to preferential attachment, i.e., with probability proportional to their degree. For $$\delta \in 0,1$$, the $$\delta $$-founders of the network at time t is the minimal set of the first nodes to enter the network i.e., founders guaranteeing that the sum of degrees of nodes in the set is at least a $$\delta $$ fraction of the number of edges in the graph at time t. We show that for the common model where $$p_t$$ is constant, i.e., when $$p_t=p$$ for every t and the network is sparse with linear number of edges, the size of the $$\delta $$-founders set is concentrated around $$\delta ^{2/p} n_t$$, and thus is linear in $$n_t$$, the number of nodes at time t. In contrast, we show that for $$p_t=\min \{1,\frac{2a}{\ln t}\}$$ and when the network is dense with super-linear number of edges, the size of the $$\delta $$-founders set is sub-linear in $$n_t$$ and concentrated around $$\tilde{\Theta }n_t^{\eta }$$, where $$\eta =\delta ^{1/a}$$.

Preferential attachment drives the evolution of many complex networks. Its analytical studies mostly consider the simplest case of a network that grows uniformly in time despite the accelerating growth of many real networks. Motivated by the observation that the average degree growth of nodes is time invariant in empirical network data, we study the degree dynamics in the relevant class of network models where preferential attachment is combined with heterogeneous node fitness and aging. We propose an analytical framework based on the time invariance of the studied systems and show that it is self-consistent only for two special network growth forms: the uniform and the exponential network growth. Conversely, the breaking of such time invariance explains the winner-takes-all effect in some model settings, revealing the connection between the Bose-Einstein condensation in the Bianconi-Barabási model and similar gelation in superlinear preferential attachment. Aging is necessary to reproduce realistic node degree growth curves and can prevent the winner-takes-all effect under weak conditions. Our results are verified by extensive numerical simulations.

In complex networks, the “rich-get-richer” effect (nodes with high degree at one point in time gain more degree in their future) is commonly observed. In practice this is often studied on a static network snapshot, for example, a preferential attachment model assumed to explain the more highly connected nodes or a rich-club effect that analyses the most highly connected nodes. In this paper, we consider temporal measures of how success (measured here as node degree) propagates across time. By analogy with social mobility (a measure of people moving within a social hierarchy through their life) we define hierarchical mobility to measure how a node’s propensity to gain degree changes over time. We introduce an associated taxonomy of temporal correlation statistics including mobility, philanthropy and community. Mobility measures the extent to which a node’s degree gain in one time period predicts its degree gain in the next. Philanthropy and community measure similar properties related to node neighbourhood. We apply these statistics both to artificial models and to 26 real temporal networks. We find that most of our networks show a tendency for individual nodes and their neighbourhoods to remain in similar hierarchical positions over time, while most networks show low correlative effects between individuals and their neighbourhoods. Moreover, we show that the mobility taxonomy can discriminate between networks from different fields. We also generate artificial network models to gain intuition about the behaviour and expected range of the statistics. The artificial models show that the opposite of the “rich-get-richer” effect requires the existence of inequality of degree in a network. Overall, we show that measuring the hierarchical mobility of a temporal network is an invaluable resource for discovering its underlying structural dynamics.

Explaining the power-law degree distribution in a social commerce network

The importance of network externalities for the development of technology and industry structure has been recognized in evolutionary economic for a long time. However, network externalities are no isolated phenomena. They are based on competing standards in a comprehensive network of technology lines that are based on one another and remain to various degrees interoperable or compatible. As some evidence from the ICT sector inparticular shows, compatibility and tying or bundling of standards may be employed as strategic tools. The present paper investigates the economic role of tied standards for the dynamics of competition between standards. A replicator model operating on an aggregated level is complemented by an agent-based simulation with explicit representation of the network structure among users. A variety of effects are studied, including the role of initial usage share, manipulation of compatibility, expansion of vendors into other segments, as well as the network structure and central or peripheral positioning of agents. The agent-based model contrasts a complete network and a regular ring network with asymmetric network structures derived from Barabàsi and Albert’s preferential attachment mechanism and triadic closure.

Nonlinear growth in weighted networks with neighborhood preferential attachment

The main aim of this paper is to develop algorithms for link prediction based on directed and weighted social network structure. In this paper, the four algorithms such as Modified Common Neighbor (MCN), Modified Jaccard’s Coefficient (MJC), Modified Adamic Adar (MAA) and Modified Preferential Attachment (MPA) has been proposed which is suitable for directed and weighted networks. In our proposed algorithms, the degree of nodes and weightage of each link has been considered. The weightage of each link is assigned using random function. The Modified Common Neighbor (MCN), Modified Jaccard’s Coefficient (MJC), and Modified Adamic Adar (MAA) algorithms are based on an existing Common Neighbor algorithm. The Modified Preferential Attachment (MPA) algorithm depends on the degree of the nodes. The comparative analysis of our proposed algorithms and existing algorithms is performed based on area under receiver operating characteristic values (AUC values), considering different observed links. According to the experimental analysis, it may be concluded that our proposed algorithms provide better performances in comparison to the existing algorithms. Modified Common Neighbor and Modified Adamic Adar results in highest AUC value when twitter dataset and amazon dataset is considered. The proposed algorithms will be applicable in different directed and weighted social network structure for prediction of links between the users.

Dynamic processes in complex network are crucial for better understanding collective behavior in human societies, biological systems, and the Internet. In this article, we first focus on the continuous Markov-based modeling of evolving networks with the birth-death of individuals. A new individual arrives at the group by the Poisson process, while new links are established in the network through either uniform connection or preferential attachment. Moreover, an existing individual has a limited lifespan before leaving the network. We determine stationary topological properties of these networks, including their size and mean degree. To address the effect of the birth-death evolution, we further study the information dynamics in the proposed network model from the random drift and natural selection perspective, based on assumptions of total-stochastic and fitness-driven evolution, respectively. In simulations, we analyze the fixation probability of individual information and find that means of new connections affect the random drift process but do not affect the natural selection process.