Network Models

Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce "network models" to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to $\mathbf{Cat}$, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.Comment: 46 page

Network Transitivity and Matrix Models

This paper is a step towards a systematic theory of the transitivity (clustering) phenomenon in random networks. A static framework is used, with adjacency matrix playing the role of the dynamical variable. Hence, our model is a matrix model, where matrices are random, but their elements take values 0 and 1 only. Confusion present in some papers where earlier attempts to incorporate transitivity in a similar framework have been made is hopefully dissipated. Inspired by more conventional matrix models, new analytic techniques to develop a static model with non-trivial clustering are introduced. Computer simulations complete the analytic discussion.Comment: 11 pages, 7 eps figures, 2-column revtex format, print bug correcte

Network Reconstruction with Realistic Models

We extend a recently proposed gradient-matching method for inferring interactions in complex systems described by differential equations in various respects: improved gradient inference, evaluation of the influence of the prior on kinetic parameters, comparative evaluation of two model selection paradigms: marginal likelihood versus DIC (divergence information criterion), comparative evaluation of different numerical procedures for computing the marginal likelihood, extension of the methodology from protein phosphorylation to transcriptional regulation, based on a realistic simulation of the underlying molecular processes with Markov jump processes

Model Checking Social Network Models

A social network service is a platform to build social relations among people sharing similar interests and activities. The underlying structure of a social networks service is the social graph, where nodes represent users and the arcs represent the users' social links and other kind of connections. One important concern in social networks is privacy: what others are (not) allowed to know about us. The "logic of knowledge" (epistemic logic) is thus a good formalism to define, and reason about, privacy policies. In this paper we consider the problem of verifying knowledge properties over social network models (SNMs), that is social graphs enriched with knowledge bases containing the information that the users know. More concretely, our contributions are: i) We prove that the model checking problem for epistemic properties over SNMs is decidable; ii) We prove that a number of properties of knowledge that are sound w.r.t. Kripke models are also sound w.r.t. SNMs; iii) We give a satisfaction-preserving encoding of SNMs into canonical Kripke models, and we also characterise which Kripke models may be translated into SNMs; iv) We show that, for SNMs, the model checking problem is cheaper than the one based on standard Kripke models. Finally, we have developed a proof-of-concept implementation of the model-checking algorithm for SNMs.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Generalized Network Psychometrics: Combining Network and Latent Variable Models

We introduce the network model as a formal psychometric model, conceptualizing the covariance between psychometric indicators as resulting from pairwise interactions between observable variables in a network structure. This contrasts with standard psychometric models, in which the covariance between test items arises from the influence of one or more common latent variables. Here, we present two generalizations of the network model that encompass latent variable structures, establishing network modeling as parts of the more general framework of Structural Equation Modeling (SEM). In the first generalization, we model the covariance structure of latent variables as a network. We term this framework Latent Network Modeling (LNM) and show that, with LNM, a unique structure of conditional independence relationships between latent variables can be obtained in an explorative manner. In the second generalization, the residual variance-covariance structure of indicators is modeled as a network. We term this generalization Residual Network Modeling (RNM) and show that, within this framework, identifiable models can be obtained in which local independence is structurally violated. These generalizations allow for a general modeling framework that can be used to fit, and compare, SEM models, network models, and the RNM and LNM generalizations. This methodology has been implemented in the free-to-use software package lvnet, which contains confirmatory model testing as well as two exploratory search algorithms: stepwise search algorithms for low-dimensional datasets and penalized maximum likelihood estimation for larger datasets. We show in simulation studies that these search algorithms performs adequately in identifying the structure of the relevant residual or latent networks. We further demonstrate the utility of these generalizations in an empirical example on a personality inventory dataset.Comment: Published in Psychometrik