Max-linear models on infinte graphs generated by Bernoulli bond percolation

We extend previous work of max-linear models on finite directed acyclic graphs to infinite graphs, and investigate their relations to classical percolation theory. We formulate results for the oriented square lattice graph $\mathbb{Z}^2$ and nearest neighbor bond percolation. Focus is on the dependence introduced by this graph into the max-linear model. As a natural application we consider communication networks, in particular, the distribution of extreme opinions in social networks.Comment: 18 page

Batch kernel SOM and related Laplacian methods for social network analysis

Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts

Exponential-family Random Network Models

Random graphs, where the connections between nodes are considered random variables, have wide applicability in the social sciences. Exponential-family Random Graph Models (ERGM) have shown themselves to be a useful class of models for representing com- plex social phenomena. We generalize ERGM by also modeling nodal attributes as random variates, thus creating a random model of the full network, which we call Exponential-family Random Network Models (ERNM). We demonstrate how this framework allows a new formu- lation for logistic regression in network data. We develop likelihood-based inference for the model and an MCMC algorithm to implement it. This new model formulation is used to analyze a peer social network from the National Lon- gitudinal Study of Adolescent Health. We model the relationship between substance use and friendship relations, and show how the results differ from the standard use of logistic regression on network data

Multi-GCN: Graph Convolutional Networks for Multi-View Networks, with Applications to Global Poverty

With the rapid expansion of mobile phone networks in developing countries, large-scale graph machine learning has gained sudden relevance in the study of global poverty. Recent applications range from humanitarian response and poverty estimation to urban planning and epidemic containment. Yet the vast majority of computational tools and algorithms used in these applications do not account for the multi-view nature of social networks: people are related in myriad ways, but most graph learning models treat relations as binary. In this paper, we develop a graph-based convolutional network for learning on multi-view networks. We show that this method outperforms state-of-the-art semi-supervised learning algorithms on three different prediction tasks using mobile phone datasets from three different developing countries. We also show that, while designed specifically for use in poverty research, the algorithm also outperforms existing benchmarks on a broader set of learning tasks on multi-view networks, including node labelling in citation networks

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

Modeling social networks from sampled data

Network models are widely used to represent relational information among interacting units and the structural implications of these relations. Recently, social network studies have focused a great deal of attention on random graph models of networks whose nodes represent individual social actors and whose edges represent a specified relationship between the actors. Most inference for social network models assumes that the presence or absence of all possible links is observed, that the information is completely reliable, and that there are no measurement (e.g., recording) errors. This is clearly not true in practice, as much network data is collected though sample surveys. In addition even if a census of a population is attempted, individuals and links between individuals are missed (i.e., do not appear in the recorded data). In this paper we develop the conceptual and computational theory for inference based on sampled network information. We first review forms of network sampling designs used in practice. We consider inference from the likelihood framework, and develop a typology of network data that reflects their treatment within this frame. We then develop inference for social network models based on information from adaptive network designs. We motivate and illustrate these ideas by analyzing the effect of link-tracing sampling designs on a collaboration network.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS221 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org