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.
A related line of research studies the so-called stochastic actor-oriented model @cite_40 @cite_27 . This model combines multiple formation dynamics in a multinomial logit functional form, and develops connections between network formation and Markov chains in the space of graphs. However, they are impractical to estimate, especially for larger data sets. should increase distancing here? Need to return to the distancing here. Why is it impractical for large datasets?
@cite_27 @cite_40
2099815494 2104725117 2508909698 2001667451
Abstract Stochastic actor-based models are models for network dynamics that can represent a wide variety of influences on network change, and allow to estimate parameters expressing such influences, and test corresponding hypotheses. The nodes in the network represent social actors, and the collection of ties represents a social relation. The assumptions posit that the network evolves as a stochastic process ‘driven by the actors’, i.e., the model lends itself especially for representing theories about how actors change their outgoing ties. The probabilities of tie changes are in part endogenously determined, i.e., as a function of the current network structure itself, and in part exogenously, as a function of characteristics of the nodes (‘actor covariates’) and of characteristics of pairs of nodes (‘dyadic covariates’). In an extended form, stochastic actor-based models can be used to analyze longitudinal data on social networks jointly with changing attributes of the actors: dynamics of networks and behavior. This paper gives an introduction to stochastic actor-based models for dynamics of directed networks, using only a minimum of mathematics. The focus is on understanding the basic principles of the model, understanding the results, and on sensible rules for model selection. A class of statistical models is proposed for longitudinal network data. The dependent variable is the changing (or evolving) relation network, represented by two or more observations of a directed graph with a fixed set of actors. The network evolution is modeled as the consequence of the actors making new choices, or withdrawing existing choices, on the basis of functions, with fixed and random components, that the actors try to maximize. Individual and dyadic exogenous variables can be used as covariates. The change in the network is modeled as the stochastic result of network effects (reciprocity, transitivity, etc.) and these covariates. The existing network structure is a dynamic constraint for the evolution of the structure itself. The models are continuous-time Markov chain models that can be implemented as simulation models. The model parameters are estimated from observed data. For estimating and testing these models, statistical procedures are proposed that are based on the method of moments. The statistical procedures are implemented using a stochastic approximation algorithm based on computer simulations of the network evolution process. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. Determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas. Knowledge of such a structure can aid in identifying cause and effect relationships, clustering of similar entities, identification of representative elements and model reduction. The majority of existing results are based on correlation based indices which effectively assume a static relationship between the time series data and are not suitable for detecting interrelatedness when the time series are dynamically related or when the time series involve loops. In this paper, a methodology for identifying the interrelatedness structure of dynamically related time series data is presented that also allows for the presence of loops in the connectivity structure. A linear dynamic graph model is presented where it is assumed that each time series data is the sum of an independent stochastic noise source and a dynamically weighted sum of other time series data. A link is assumed to be present between two time series if the weight of a time series, which is a linear time-invariant filter, is nonzero in the formation of the other. Reconstruction of the link connectivity structure under various scenarios is considered. It is shown that when the linear dynamic graph is allowed to admit non-causal weights, then the links structure can be recovered with the possibility of identifying spurious connections. However, it is shown that the spurious links remain local, where, a spurious link is restricted to be within one hop of a true link. Furthermore, strategies for exact reconstruction of the link structure when the weights are restricted to be causal are developed. The main tools for determining the network topology are based on variations of Wiener filtering. A significant insight provided by the article is that, in the class of network models considered in the paper, the Wiener filter estimating a stochastic process based on other processes remains local in the sense that the Wiener filter utilizes only measurements local to the node being estimated.