Information Recovery from Pairwise Measurements

A variety of information processing tasks in practice involve recovering $n$ objects from single-shot graph-based measurements, particularly those taken over the edges of some measurement graph $\mathcal{G}$. This paper concerns the situation where each object takes value over a group of $M$ different values, and where one is interested to recover all these values based on observations of certain pairwise relations over $\mathcal{G}$. The imperfection of measurements presents two major challenges for information recovery: 1) $\textit{inaccuracy}$: a (dominant) portion $1-p$ of measurements are corrupted; 2) $\textit{incompleteness}$: a significant fraction of pairs are unobservable, i.e. $\mathcal{G}$ can be highly sparse. Under a natural random outlier model, we characterize the $\textit{minimax recovery rate}$, that is, the critical threshold of non-corruption rate $p$ below which exact information recovery is infeasible. This accommodates a very general class of pairwise relations. For various homogeneous random graph models (e.g. Erdos Renyi random graphs, random geometric graphs, small world graphs), the minimax recovery rate depends almost exclusively on the edge sparsity of the measurement graph $\mathcal{G}$ irrespective of other graphical metrics. This fundamental limit decays with the group size $M$ at a square root rate before entering a connectivity-limited regime. Under the Erdos Renyi random graph, a tractable combinatorial algorithm is proposed to approach the limit for large $M$ ($M=n^{\Omega(1)}$), while order-optimal recovery is enabled by semidefinite programs in the small $M$ regime. The extended (and most updated) version of this work can be found at (http://arxiv.org/abs/1504.01369).Comment: This version is no longer updated -- please find the latest version at (arXiv:1504.01369

When is a Network a Network? Multi-Order Graphical Model Selection in Pathways and Temporal Networks

We introduce a framework for the modeling of sequential data capturing pathways of varying lengths observed in a network. Such data are important, e.g., when studying click streams in information networks, travel patterns in transportation systems, information cascades in social networks, biological pathways or time-stamped social interactions. While it is common to apply graph analytics and network analysis to such data, recent works have shown that temporal correlations can invalidate the results of such methods. This raises a fundamental question: when is a network abstraction of sequential data justified? Addressing this open question, we propose a framework which combines Markov chains of multiple, higher orders into a multi-layer graphical model that captures temporal correlations in pathways at multiple length scales simultaneously. We develop a model selection technique to infer the optimal number of layers of such a model and show that it outperforms previously used Markov order detection techniques. An application to eight real-world data sets on pathways and temporal networks shows that it allows to infer graphical models which capture both topological and temporal characteristics of such data. Our work highlights fallacies of network abstractions and provides a principled answer to the open question when they are justified. Generalizing network representations to multi-order graphical models, it opens perspectives for new data mining and knowledge discovery algorithms.Comment: 10 pages, 4 figures, 1 table, companion python package pathpy available on gitHu

von Neumann-Morgenstern and Savage Theorems for Causal Decision Making

Causal thinking and decision making under uncertainty are fundamental aspects of intelligent reasoning. Decision making under uncertainty has been well studied when information is considered at the associative (probabilistic) level. The classical Theorems of von Neumann-Morgenstern and Savage provide a formal criterion for rational choice using purely associative information. Causal inference often yields uncertainty about the exact causal structure, so we consider what kinds of decisions are possible in those conditions. In this work, we consider decision problems in which available actions and consequences are causally connected. After recalling a previous causal decision making result, which relies on a known causal model, we consider the case in which the causal mechanism that controls some environment is unknown to a rational decision maker. In this setting we state and prove a causal version of Savage's Theorem, which we then use to develop a notion of causal games with its respective causal Nash equilibrium. These results highlight the importance of causal models in decision making and the variety of potential applications.Comment: Submitted to Journal of Causal Inferenc