The effect of the initial network configuration on preferential attachment

The classical preferential attachment model is sensitive to the choice of the initial configuration of the network. As the number of initial nodes and their degree grow, so does the time needed for an equilibrium degree distribution to be established. We study this phenomenon, provide estimates of the equilibration time, and characterize the degree distribution cutoff observed at finite times. When the initial network is dense and exceeds a certain small size, there is no equilibration and a suitable statistical test can always discern the produced degree distribution from the equilibrium one. As a by-product, the weighted Kolmogorov-Smirnov statistic is demonstrated to be more suitable for statistical analysis of power-law distributions with cutoff when the data is ampl

Preferential attachment during the evolution of a potential energy landscape

It has previously been shown that the network of connected minima on a potential energy landscape is scale-free, and that this reflects a power-law distribution for the areas of the basins of attraction surrounding the minima. Here, we set out to understand more about the physical origins of these puzzling properties by examining how the potential energy landscape of a 13-atom cluster evolves with the range of the potential. In particular, on decreasing the range of the potential the number of stationary points increases and thus the landscape becomes rougher and the network gets larger. Thus, we are able to follow the evolution of the potential energy landscape from one with just a single minimum to a complex landscape with many minima and a scale-free pattern of connections. We find that during this growth process, new edges in the network of connected minima preferentially attach to more highly-connected minima, thus leading to the scale-free character. Furthermore, minima that appear when the range of the potential is shorter and the network is larger have smaller basins of attraction. As there are many of these smaller basins because the network grows exponentially, the observed growth process thus also gives rise to a power-law distribution for the hyperareas of the basins.Comment: 10 pages, 10 figure

On the Stability of Community Detection Algorithms on Longitudinal Citation Data

There are fundamental differences between citation networks and other classes of graphs. In particular, given that citation networks are directed and acyclic, methods developed primarily for use with undirected social network data may face obstacles. This is particularly true for the dynamic development of community structure in citation networks. Namely, it is neither clear when it is appropriate to employ existing community detection approaches nor is it clear how to choose among existing approaches. Using simulated data, we attempt to clarify the conditions under which one should use existing methods and which of these algorithms is appropriate in a given context. We hope this paper will serve as both a useful guidepost and an encouragement to those interested in the development of more targeted approaches for use with longitudinal citation data.Comment: 17 pages, 7 figures, presenting at Applications of Social Network Analysis 2009, ETH Zurich Edit, August 17, 2009: updated abstract, figures, text clarification

Coevolution of Glauber-like Ising dynamics on typical networks

We consider coevolution of site status and link structures from two different initial networks: a one dimensional Ising chain and a scale free network. The dynamics is governed by a preassigned stability parameter $S$, and a rewiring factor $\phi$, that determines whether the Ising spin at the chosen site flips or whether the node gets rewired to another node in the system. This dynamics has also been studied with Ising spins distributed randomly among nodes which lie on a network with preferential attachment. We have observed the steady state average stability and magnetisation for both kinds of systems to have an idea about the effect of initial network topology. Although the average stability shows almost similar behaviour, the magnetisation depends on the initial condition we start from. Apart from the local dynamics, the global effect on the dynamics has also been studied. These parameters show interesting variations for different values of $S$ and $\phi$, which helps in determining the steady-state condition for a given substrate.Comment: 8 pages, 10 figure

Growing Attributed Networks through Local Processes

This paper proposes an attributed network growth model. Despite the knowledge that individuals use limited resources to form connections to similar others, we lack an understanding of how local and resource-constrained mechanisms explain the emergence of rich structural properties found in real-world networks. We make three contributions. First, we propose a parsimonious and accurate model of attributed network growth that jointly explains the emergence of in-degree distributions, local clustering, clustering-degree relationship and attribute mixing patterns. Second, our model is based on biased random walks and uses local processes to form edges without recourse to global network information. Third, we account for multiple sociological phenomena: bounded rationality, structural constraints, triadic closure, attribute homophily, and preferential attachment. Our experiments indicate that the proposed Attributed Random Walk (ARW) model accurately preserves network structure and attribute mixing patterns of six real-world networks; it improves upon the performance of eight state-of-the-art models by a statistically significant margin of 2.5-10x.Comment: 11 pages, 13 figure

Towards realistic artificial benchmark for community detection algorithms evaluation

Assessing the partitioning performance of community detection algorithms is one of the most important issues in complex network analysis. Artificially generated networks are often used as benchmarks for this purpose. However, previous studies showed their level of realism have a significant effect on the algorithms performance. In this study, we adopt a thorough experimental approach to tackle this problem and investigate this effect. To assess the level of realism, we use consensual network topological properties. Based on the LFR method, the most realistic generative method to date, we propose two alternative random models to replace the Configuration Model originally used in this algorithm, in order to increase its realism. Experimental results show both modifications allow generating collections of community-structured artificial networks whose topological properties are closer to those encountered in real-world networks. Moreover, the results obtained with eleven popular community identification algorithms on these benchmarks show their performance decrease on more realistic networks

A self-consistent approach to measure preferential attachment in networks and its application to an inherent structure network

Preferential attachment is one possible way to obtain a scale-free network. We develop a self-consistent method to determine whether preferential attachment occurs during the growth of a network, and to extract the preferential attachment rule using time-dependent data. Model networks are grown with known preferential attachment rules to test the method, which is seen to be robust. The method is then applied to a scale-free inherent structure network, which represents the connections between minima via transition states on a potential energy landscape. Even though this network is static, we can examine the growth of the network as a function of a threshold energy (rather than time), where only those transition states with energies lower than the threshold energy contribute to the network.For these networks we are able to detect the presence of preferential attachment, and this helps to explain the ubiquity of funnels on energy landscapes. However, the scale-free degree distribution shows some differences from that of a model network grown using the obtained preferential attachment rules, implying that other factors are also important in the growth process.Comment: 8 pages, 8 figure

Socioeconomic Networks with Long-Range Interactions

We study a modified version of a model previously proposed by Jackson and Wolinsky to account for communicating information and allocating goods in socioeconomic networks. In the model, the utility function of each node is given by a weighted sum of contributions from all accessible nodes. The weights, parameterized by the variable $\delta$, decrease with distance. We introduce a growth mechanism where new nodes attach to the existing network preferentially by utility. By increasing $\delta$, the network structure evolves from a power-law to an exponential degree distribution, passing through a regime characterised by shorter average path length, lower degree assortativity and higher central point dominance. In the second part of the paper we compare different network structures in terms of the average utility received by each node. We show that power-law networks provide higher average utility than Poisson random networks. This provides a possible justification for the ubiquitousness of scale-free networks in the real world.Comment: 11 pages, 8 figures, minor correction