text: We provide a framework for modeling social network formation through conditional multinomial logit models from discrete choice and random utility theory, in which each new edge is viewed as a “choice” made by a node to connect to another node, based on (generic) features of the other nodes available to make a connection. This perspective on network formation unifies existing models such as preferential attachment, triadic closure, and node fitness, which are all special cases, and thereby provides a flexible means for conceptualizing, estimating, and comparing models. The lens of discrete choice theory also provides several new tools for analyzing social network formation; for example, the significance of node features can be evaluated in a statistically rigorous manner, and mixtures of existing models can be estimated by adapting known expectation-maximization algorithms. We demonstrate the flexibility of our framework through examples that analyze a number of synthetic and real-world datasets. For example, we provide rigorous methods for estimating preferential attachment models and show how to separate the effects of preferential attachment and triadic closure. Non-parametric estimates of the importance of degree show a highly linear trend, and we expose the importance of looking carefully at nodes with degree zero. Examining the formation of a large citation graph, we find evidence for an increased role of degree when accounting for age.
  @cite_67: 
  @cite_7: A key ingredient of many current models proposed to capture the topological evolution of complex networks is the hypothesis that highly connected nodes increase their connectivity faster than their less connected peers, a phenomenon called preferential attachment. Measurements on four networks, namely the science citation network, Internet, actor collaboration and science coauthorship network indicate that the rate at which nodes acquire links depends on the node's degree, offering direct quantitative support for the presence of preferential attachment. We find that for the first two systems the attachment rate depends linearly on the node degree, while for the last two the dependence follows a sublinear power law.
  @cite_41: We study empirically the time evolution of scientific collaboration networks in physics and biology. In these networks, two scientists are considered connected if they have coauthored one or more papers together. We show that the probability of scientists collaborating increases with the number of other collaborators they have in common, and that the probability of a particular scientist acquiring new collaborators increases with the number of his or her past collaborators. These results provide experimental evidence in favor of previously conjectured mechanisms for clustering and power-law degree distributions in networks.
  @cite_52: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
  @cite_19: 
  @cite_73: Publicly available data reveal long-term systematic features about citation statistics and how papers are referenced. The data also tell fascinating citation histories of individual articles.
  @cite_17: A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites that link to earlier sites with a probability A(k) which depends on the number of preexisting links k to that site. For homogeneous connection kernels, A(k) approximately k(gamma), different behaviors arise for gamma 1, and gamma = 1. For gamma 1, a single site connects to nearly all other sites. In the borderline case A(k) approximately k, the power law N(k) approximately k(-nu) is found, where the exponent nu can be tuned to any value in the range 2<nu<infinity.
summary: There is a long history of parameter estimation in network formation processes @cite_67 @cite_19 . Probably the most studied formation model is preferential attachment @cite_52 , which relates the likelihood of new nodes connecting to a node @math to @math 's degree. More references on PA, since we claim it is the most studied? The original justification of the model was based on the degree distribution of the resulting graph qualitatively matching those of several early large-scale empirical networks. Shortly after, a number of authors proposed non-parametric methods to measure preferential attachment by counting what the degrees of the nodes getting edges are and normalizing them by the relative likelihood of those degrees @cite_41 @cite_7 @cite_73 . There are several generalizations of preferential attachment, where node @math is chosen proportional to some function @math of the degree @math of node @math (here, @math is standard preferential attachment). TODO: Need references if claiming there are several generalizations Much of this research has focused on the case when @math , i.e., attachment is preferential to some (latent) power @math of the degree @cite_17 .