Network Modeling and Inference

Our research group works at the inferface between Computational Statistics, Information Theory, Bayesian Inference, Machine Learning, and Statistical Physics.

We have as main focus the methodological foundations of Network Science and the study of Complex Systems.

Our ambition has been to render obsolete the reliance on ad hoc heuristics in the field of network data analysis, and transition instead to a mature and robust methodological framework that is derived from fundamental principles and is grounded in solid statistical theory.

Furthermore, we seek to incorporate mechanistic modeling, while simultaneously achieving interpretability, algorithmic efficiency, and versatility.

A central concern of ours is the appropriate implementation of inductive reasoning and statistical inference to relational data that come from a variety of complex systems in the real world.

A lot of what we do is framed by the following instrumental questions:

  1. How much complexity can be justified in our explanations of empirical observations?
  2. How can we reconstruct dynamical rules and network structures from indirect information on their behavior?
  3. How do we model the hierarchical and modular structure of complex systems?

This line of work was recognized with the Erdős–Rényi Prize from the Network Science Society.

Most of the methods developed in our group are made available as part of the graph-tool library, which is extensively documented. For a practical introduction to many inference and reconstruction algorithms, please refer to the HOWTO.

Separating structure from statistical noise in complex systems

Networks delineate the constituent interactions of a broad range of large-scale complex systems. They are essential to describe socio-economical relations, the human brain, cell metabolism, ecosystems, epidemic spreading, informational infrastructure, transportation systems, and many more.

The structure of these network systems is typically large and heterogeneous, and the interactions they describe are typically non-linear, and result in nontrivial emergent behavior and self-organization. Network theory offers a wide ranging foundation to untangle such intricate systems, potentially allowing us to predict and control their behavior, as well as to provide scientific explanations.

A significant obstacle for the comprehension of such high-dimensional relational objects lies in discerning between signal and randomness. It is crucial to identify which aspects of these systems arise from random stochastic fluctuations and which convey valuable information about an underlying phenomenon. This is a multifaceted problem that often defies intuition, and lies at the heart of any data-driven analysis.

These three adjacency matrices correspond to the same random graph; the only difference between them is how the nodes are ordered. Each ordering reveals a nontrivial—and seemingly compelling—mixing pattern between the nodes. However, since the graph is completely random, all these patterns are mere statistical illusions, dredged out of pure randomness, and therefore overfit the data. Widespread methods of network data analysis cannot distinguish these spurious patterns from statistically significant ones, posing a significant risk to their application. (Reload this page to see how many different patterns can be found in random graphs!)

In our group, we focus on the development of principled and trustworthy methods to extract scientific understanding from network data, as well as the mathematical modeling of network behavior and evolution.

Our methods are designed to be robust against overfitting, and to be algorithmically efficient. This is achieved by merging analytical tools and concepts from a variety of disciplines, including Information Theory, Bayesian Statistics, and Statistical Mechanics.

We're particularly interested in problems of network inference where meaningful structural and functional patterns cannot be obtained by direct inspection or low-order statistics, and require instead more sophisticated approaches based on large-scale generative models and efficient algorithms derived from them. In more demanding, but nonetheless ubiquitous scenarios, the network data are noisy, incomplete, or even completely hidden, leaving their trace only via an observed dynamical behavior—in which case the network needs to be fully reconstructed from indirect information.


I am an Associate Professor in the Department of Network and Data Science at the Central European University (CEU), Vienna, Austria. I have received my Habilitation in Theoretical Physics at the University of Bremen in 2017. Previously, I have been an Assistant Professor in Applied Mathematics at the University of Bath (2016-2019), External Researcher at the ISI Foundation (2015-2020), and post-doc researcher at the University of Bremen (2011-2016) and Technical University of Darmstadt (2008-2011).

Research highlights

Inferring modular structures in networks
We develop principled methods to infer the hierarchical, modular structure of networks, based on generative models and Bayesian inference. Our approaches are efficient (scaling up to huge networks) and robust. In particular they are able to avoid both overfitting and underfitting the data. See review [B2] for an introduction, and the HOWTO documentation for graph-tool. See also [24,20,33,23,42,43].
Annotated and attributed networks
Network data are often annotated with weights or covariates on the edges, or metadata on the nodes. We develop inference approaches that are able to leverage this formation to uncover latent, statistically meaningful network structures. Our perspective is that such annotations are just more data—not “ground truth”—and hence are also subject to noise, incompleteness, irrelevance, etc. See [37,31].
Dynamical networks
In many instances, networks are dynamical objects and their structure evolves in time. We develop inference methods that are able to characterize how the large-scale structure dynamically changes. Importantly, instead of imposing a priori characteristic time scales, we extract the relevant scales from data by formulating arbitrary-order dynamical models, within a nonparametric Bayesian inference framework. See [B1,36,34,28].
Uncertain network reconstruction
As is unavoidable in any empirical setting, network data are subject to measurement uncertainties and omissions. However, differently from more established empirical traditions, network data often do not contain reported error assessments of any kind. We develop principled methods of error evaluation and network reconstruction that are able to function even in the demanding scenario where only a single network is observed, and the error magnitudes are unknown. See [39].
Reconstruction from dynamics
Certain networks are impossible, or prohibitively expensive, to be measured directly, and we need to infer their structure from an observed dynamics that takes place on them. We develop Bayesian methods that are able to achieve this reconstruction, and demonstrate how the joint inference of modular network structure with the network itself can significantly improve the reconstruction from indirect dynamics as well, specially when coupled with efficient algorithms. This amounts to a unification of network reconstruction with community detection—two central but traditionally isolated problems in network science, statistics and machine learning. See [40].
Disentangling edge formation mechanisms
Networks are often the result of a variety of different and interdependent generative mechanisms that operate on different scales (e.g. global or local). We are able to show that it is possible, in key cases, to decompose the contributions of each mechanism, based only on the traces they leave behind on the network structure. In particular we show how homophily and triadic closure can be disentangled from each other, given only a single network snapshot. This has important consequences to the interpretation of community detection methods, since the effects of both mechanisms are often conflated. See [50].

Open positions

Interested PhD candidates are encouraged to apply for the "PhD Program in Network Science at CEU".