The relationship between the dynamics of a process and the underlying network topology is crucial to understand complex systems. On one hand, there is significant interest in understanding how the structure of the network impacts the dynamics of processes. On the other hand, there is also interest in the inverse problem: given the dynamics of a particular process, to what extent can the underlying network structure be inferred? This problem is crucial in various fields, including biology, recommendation systems, and communication networks, where the initial structure of the network is unknown while understanding the underlying graph structure can provide valuable insights and predictions.

In this talk, we infer the initial topology of the graph G from the contact data of random walkers. More precisely, we consider M independent random walkers that traverse an unknown underlying graph G with N nodes and L links with respect to the NxN probability transition matrix P. The nodes in the graph G represent different physical locations (e.g. workplaces, homes, hospitals, schools or public transport stations) and links are physical paths between locations. What can we infer about the graph G only from the interactions of random walkers? Although it is tempting to conclude that network reconstruction is impossible from the K-length random contact sequence, which represents only one possible realization of the Markov process, we show the opposite. We demonstrate that if the NxN probability transition matrix P of the random walkers admits a steady-state distribution (the process is ergodic) and the K-length contacts sequence is sufficiently long, we can infer

- the number of nodes N in the underlying graph G,

- the number of links L in the underlying graph G,

- the Markov chain with matrix P and the reversed Markov chain with matrix P',

- the underlying topology of graph G.

We would like to point out that the formulation of the network reconstruction problem is motivated by the analysis of empirical datasets that provide contact information between people but lack details about the underlying graph topology and the location of contacts. We also believe that the obtained results represent a significant step forward in gaining a deeper understanding of network evolutionary processes.

The biased net framework, originally introduced by Solomonoff and Rapoport in the 1950s, has undergone many iterations over the decades. Recent work (building on work by Skvoretz and colleagues) has proposed an approach based on a Markovian specification in which observed networks arise from a latent dynamic process driven by "activation" events (which result in tie formation or persistence) and "inhibition" events (which prevent or remove ties). Although the likelihood arising from this process is incomputable, approximate Bayesian inference can be performed using an approximate prevision strategy, in which a non-parametric least-squares learner trained on synthetic data is used to infer posterior expectations from observed graphs. While proof-of-principle work with this approach has made use of model terms based on traditional biased net concepts (e.g., parent, sibling, and double-role effects), it is also possible to specify models using terms more closely related to more modern exponential-family random graph and stochastic actor-oriented frameworks. These allow for more natural specification of covariate and dependence effects, including both activation and inhibition variants. Here, we discuss some of these new specifications, and illustrate their application to empirical cases.

This presentation focuses on the analysis of multilevel networks over time. Multilevel structures are complex systems that combine different types of ties in various domains where the aim of the modelling is to reduce the structural complexity of networks and provide a meaningful interpretation. We will demonstrate a graphical representation of multilevel structures over time and conduct a positional analysis of multilevel networks with different types of ties. Algebraic methods will be used to represent the role algebra for the constructed positional system, and the presentation will also discuss challenges and future directions in analysing multilevel structures and dynamics in complex networks.

Recently, network methods have become increasingly popular for analysing various types of mobility. In these methods, the rows and columns of mobility tables (e.g., occupations) are treated as network nodes connected by mobile individuals, creating a directed, weighted mobility network. Originally, descriptive network methods like community detection have been used alongside statistical methods such as exponential random graph models (ERGMs) developed for binary networks. Recent extensions to log-linear models have further integrated network approaches to the study of mobility. These models challenge the dominant covariate-based approach to mobility by explaining mobility through endogenous network structures, such as reciprocation or clustering, that represent emergent social mechanisms. In this study, we empirically assess the necessity of modelling endogenous structures across five types of mobility: intergenerational occupational mobility, marriage mobility, residential mobility, international student mobility, and “money mobility” derived from a behavioural experiment. For each type, we compare the fit of (i) a log-linear model with one endogenous (network) parameter, (ii) a log-linear model with one measured covariate, and (iii) such a model with one latent covariate estimated to maximize the model fit. Model fit is assessed by the extent to which individuals from the same origin select the same destination, i.e., bandwagon effects in mobility. We find that the network model consistently fits the data best. Our results suggest that researchers interested in such bandwagon effects should consider using network methods. Furthermore, our study offers a method to compare different models using flexible fit criteria that relate to researchers’ modelling goals.