Preferential Attachment in Online Networks: Measurement and Explanations

We perform an empirical study of the preferential attachment phenomenon in temporal networks and show that on the Web, networks follow a nonlinear preferential attachment model in which the exponent depends on the type of network considered. The classical preferential attachment model for networks by Barab\'asi and Albert (1999) assumes a linear relationship between the number of neighbors of a node in a network and the probability of attachment. Although this assumption is widely made in Web Science and related fields, the underlying linearity is rarely measured. To fill this gap, this paper performs an empirical longitudinal (time-based) study on forty-seven diverse Web network datasets from seven network categories and including directed, undirected and bipartite networks. We show that contrary to the usual assumption, preferential attachment is nonlinear in the networks under consideration. Furthermore, we observe that the deviation from linearity is dependent on the type of network, giving sublinear attachment in certain types of networks, and superlinear attachment in others. Thus, we introduce the preferential attachment exponent $\beta$ as a novel numerical network measure that can be used to discriminate different types of networks. We propose explanations for the behavior of that network measure, based on the mechanisms that underly the growth of the network in question.Comment: 10 pages, 5 figures, Accepted for the WebSci'13 Conference, Paris, 201

The Magic of Networks Grown by Redirection

We highlight intriguing features of complex networks that are grown by \emph{redirection}. In this mechanism, a target node is chosen uniformly at random from the pre-existing network nodes and the new node attaches either to this initial target or to a neighbor of this target. This exceedingly simple algorithm generates preferential attachment networks in an algorithmic time that is linear in the number of network nodes $N$. Even though preferential attachment ostensibly requires \emph{global knowledge} of the network, redirection requires only \emph{local knowledge}. We also show that changing just a \emph{single} attachment rate in linear preferential attachment leads to a non-universal degree distribution. Finally, we present unexpected consequences of redirection in networks with undirected links, where highly modular and non-sparse networks arise.Comment: 12 pages, 7 figures. For a special issue on "Statistical Physics and Complex Systems" in the Indian Journal of Physics. Version 2: small changes and additional references include

Emergent phenomena and fluctuations in cooperative systems

We explore the role of cooperativity and large deviations on a set of fundamental non-equilibrium many-body systems. In the cooperative asymmetric exclusion process, particles hop to the right at a constant rate only when the right neighboring site is vacant and hop at a faster rate when the left neighbor is occupied. In this model, a host of new heterogeneous density profile evolutions arise, including inverted shock waves and continuous compression waves. Cooperativity also drives the growth of complex networks via preferential attachment, where well-connected nodes are more likely to attract future connections. We introduce the mechanism of hindered redirection and show that it leads to network evolution by sublinear preferential attachment. We further show that no local growth rule can recreate superlinear preferential attachment. We also introduce enhanced redirection and show that the rule leads to networks with three unusual properties: (i) many macrohubs -- nodes whose degree is a finite fraction of the number of nodes in the network, (ii) a non-extensive degree distribution, and (iii) large fluctuations between different realizations of the growth process. We next examine large deviations in the diffusive capture model, where N diffusing predators initially all located at L 'chase' a diffusing prey initially at x<L. The prey survives if it reaches a haven at the origin without meeting any predator. We reduce the stochastic movement of the many predators to a deterministic trajectory of a single effective predator. Using optimized Monte Carlo techniques, we simulate up to 10^500 predators to confirm our analytic prediction that the prey survival probability S ~ N^-z^2, where z=x/L. Last, we quantify `survival of the scarcer' in two-species competition. In this model, individuals of two distinct species reproduce and engage in both intra-species and inter-species competition. Here a well-mixed population typically reaches a quasi steady state. We show that in this quasi-steady state the situation may arise where species A is less abundant than B but rare fluctuations make it more likely that species B first becomes extinct

Spatial preferential attachment networks: Power laws and clustering coefficients

We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The egalitarian effect of search engines

Search engines have become key media for our scientific, economic, and social activities by enabling people to access information on the Web in spite of its size and complexity. On the down side, search engines bias the traffic of users according to their page-ranking strategies, and some have argued that they create a vicious cycle that amplifies the dominance of established and already popular sites. We show that, contrary to these prior claims and our own intuition, the use of search engines actually has an egalitarian effect. We reconcile theoretical arguments with empirical evidence showing that the combination of retrieval by search engines and search behavior by users mitigates the attraction of popular pages, directing more traffic toward less popular sites, even in comparison to what would be expected from users randomly surfing the Web.Comment: 9 pages, 8 figures, 2 appendices. The final version of this e-print has been published on the Proc. Natl. Acad. Sci. USA 103(34), 12684-12689 (2006), http://www.pnas.org/cgi/content/abstract/103/34/1268

Growing Networks Through Random Walks Without Restarts

International audienceNetwork growth and evolution is a fundamental theme that has puzzled scientists for the past decades. A number of models have been proposed to capture important properties of real networks. In an attempt to better describe reality, more recent growth models embody local rules of attachment, however they still require a primitive to randomly select an existing network node and then some kind of global knowledge about the network (at least the set of nodes and how to reach them). We propose a purely local network growth model that makes no use of global sampling across the nodes. The model is based on a continuously moving random walk that after $s$ steps connects a new node to its current location, but never restarts. Through extensive simulations and theoretical arguments, we analyze the behavior of the model finding a fundamental dependency on the parity of $s$, where networks with either exponential or a conditional power law degree distribution can emerge. As $s$ increases parity dependency diminishes and the model recovers the degree distribution of Barabási-Albert preferential attachment model. The proposed purely local model indicates that networks can grow to exhibit interesting properties even in the absence of any global rule, such as global node sampling