## Section: New Results

### Modeling of Dynamics of Complex Networks

Participants : Christophe Crespelle, Éric Fleury, Márton Karsai, Yannick Léo, Philippe Nain, Matteo Morini.

#### Data Driven studies on socioeconomic data and communication networks

The study of correlations between the social network and economic status of individuals is difficult due to the lack of large-scale multimodal data disclosing both the social ties and economic indicators of the same population. Thanks to our collaboration with GranData, we close this gap through the analysis of coupled datasets recording the mobile phone communications and bank transaction history of one million anonymised individuals living in a Latin American country. From this large scale data set based on a representative, society-large population we empirically demonstrate some long-lasting hypotheses on socioeconomic correlations, which potentially lay behind social segregation, and induce differences in human mobility. More precisely, in [12] we show that wealth and debt are unevenly distributed among people in agreement with the Pareto principle; the observed social structure is strongly stratified, with people being better connected to others of their own socioeconomic class rather than to others of different classes; the social network appears to have assortative socioeconomic correlations and tightly connected ârich clubsâ; and that individuals from the same class live closer to each other but commute further if they are wealthier. In [41], we show that typical consumption patterns are strongly correlated with identified socioeconomic classes leading to patterns of stratification in the social structure. In addition we measure correlations between merchant categories and introduce a correlation network, which emerges with a meaningful community structure. We detect multivariate relations between merchant categories and show correlations in purchasing habits of individuals. Our work provides novel and detailed insight into the relations between social and consuming behaviour with potential applications in recommendation system design. In [36] we provide insight about the effects of marking events on the structure and the dynamics of egocentric networks. More precisely, we study the impact of university admission on the composition and evolution of the egocentric networks of freshmen. In other words, we study whether university helps to build connections between egos from different socioeconomic classes, or new social ties emerge via homophilic effects between students of similar economic status. Finally, in [44],

#### Generalisation of multilayer and temporal graphs

In [16] we introduce the concept of MultiAspect Graph (MAG) as a graph generalisation that we prove to be isomorphic to a directed graph, and also capable of representing all previous generalisations of multilayer and temporal networks. In our proposal, the set of vertices, layers, time instants, or any other independent features are considered as an aspect of the MAG. For instance, a MAG is able to represent multilayer or time-varying networks, while both concepts can also be combined to represent a multilayer time-varying network and even other higher-order networks. Since the MAG structure admits an arbitrary (finite) number of aspects, it hence introduces a powerful modelling abstraction for networked complex systems. In [17] we develop the algebraic representation and basic algorithms for MultiAspect Graphs (MAGs). In particular, we show that, as a consequence of the properties associated with the MAG structure, a MAG can be represented in matrix form. Moreover, we also show that any possible MAG function (algorithm) can be obtained from this matrix-based representation. This is an important theoretical result since it paves the way for adapting well-known graph algorithms for application in MAGs. We present a set of basic MAG algorithms, constructed from well-known graph algorithms, such as degree computing, Breadth First Search (BFS), and Depth First Search (DFS).

Multilayer networks arise in scenarios when a common set of nodes form multiple networks via different co-existing, and sometimes interdependent means of connectivity. In [6] we studied the threshold on the occupation density in the individual network layers for long-range connectivity to emerge in a large multilayer network. For a multilayer network formed via merging $M$ random instances of a graph $G$ with site-occupation probability $q$ in each layer, we showed that when $q$ exceeds a threshold ${q}_{c}\left(M\right)$, a giant connected component appears in the $M$-layer network. We showed that ${q}_{c}\left(M\right)\lesssim \sqrt{-ln(1-{p}_{c})}\phantom{\rule{0.166667em}{0ex}}/\sqrt{M}$, where ${p}_{c}$ is the bond percolation threshold of $G$, and ${q}_{c}\left(1\right)\equiv {q}_{c}$ is by definition the site percolation threshold of $G$. We found ${q}_{c}\left(M\right)$ exactly for when $G$ is a large random graph with any given node-degree distribution. We calculated ${q}_{c}\left(M\right)$ numerically for various regular lattices, and obtained an exact lower bound for the kagome lattice. Finally, we established an intriguing close connection between the aforesaid multilayer percolation model and the well-studied problem of site-bond (or, mixed) percolation, in the sense that both models provide a bridge between the traditional independent site and independent bond percolation models. Using this connection, and leveraging some analytical approximations to the site-bond critical region developed in the 1990s, we derived an excellent general approximation to the multilayer threshold ${q}_{c}\left(M\right)$ for regular lattices, which are not only functions solely of the ${p}_{c}$ and ${q}_{c}$ of the respective lattices, but also closely match the true values of ${q}_{c}\left(M\right)$ for a large class of lattices, even for small (single-digit) vales of $M$.

#### User-based representation of dynamical multimodal public transportation networks

In this project published as an invited paper [9], we provide a novel user-based representation of public transportation systems, which combines representations, accounting for the presence of multiple lines and reducing the effect of spatial embeddedness, while considering the total travel time, its variability across the schedule, and taking into account the number of transfers necessary. After the adjustment of earlier techniques to the novel representation framework, we analyse the public transportation systems of several French municipal areas and identify hidden patterns of privileged connections. Furthermore, we study their efficiency as compared to the commuting flow. The proposed representation could help to enhance resilience of local transportation systems to provide better design policies for future developments.

#### Local cascades induced global contagion

In this paper [8] we analyse and model product adoption dynamics in the world's largest voice over internet service, the social network of Skype. We provide empirical evidence about the heterogeneous distribution of fractional behavioural thresholds, which appears to be independent of the degree of adopting egos. We show that the structure of real-world adoption clusters is radically different from previous theoretical expectations, since vulnerable adoptions induced by a single adopting neighbour appear to be important only locally, while spontaneous adopters arriving at a constant rate and the involvement of unconcerned individuals govern the global emergence of social spreading.

#### Asymptotic theory of time-varying social networks with heterogeneous activity and tie allocation

In this work [15] we empirically characterise social activity and memory in seven real networks describing temporal human interactions in three different settings: scientific collaborations, Twitter mentions, and mobile phone calls. We find that the individuals' social activity and their strategy in choosing ties where to allocate their social interactions can be quantitatively described and encoded in a simple stochastic network modelling framework. The Master Equation of the model can be solved in the asymptotic limit. The analytical solutions provide an explicit description of both the system dynamic and the dynamical scaling laws characterising crucial aspects about the evolution of the networks. The analytical predictions match with accuracy the empirical observations, thus validating the theoretical approach. Our results provide a rigorous dynamical system framework that can be extended to include other processes shaping social dynamics and to generate data driven predictions for the asymptotic behaviour of social networks.

#### Link prediction in the Twitter mention network

In this project [35] we analyse a large Twitter data corpus and quantify similarities between people by considering the set of their common friends and the set of their commonly shared hashtags in order to predict mention links among them. We show that these similarity measures are correlated among connected people and that the combination of contextual and local structural features provides better predictions as compared to cases where they are considered separately.