## Section: New Results

### Modeling of Dynamics of Complex Networks

Participants : Christophe Crespelle, Éric Fleury, Márton Karsai, Yannick Leo, Matteo Morini.

**Non-Altering Time Scales for Aggregation of Dynamic Networks into Series of Graphs**[29]-
Many dynamic networks coming from real-world contexts are

*link streams*, i.e. a finite collection of triplets $(u,v,t)$ where $u$ and $v$ are two nodes having a link between them at time $t$. A great number of studies on these objects start by aggregating the data on disjoint time windows of length $\Delta $ in order to obtain a series of graphs on which are made all subsequent analyses. Here we are concerned with the impact of the chosen $\Delta $ on the obtained graph series. We address the fundamental question of knowing whether a series of graphs formed using a given $\Delta $ faithfully describes the original link stream. We answer the question by showing that such dynamic networks exhibit a threshold for $\Delta $, which we call the*saturation scale*, beyond which the properties of propagation of the link stream are altered, while they are mostly preserved before. We design an automatic method to determine the saturation scale of any link stream, which we apply and validate on several real-world datasets. **Termination of the Iterated Strong-Factor Operator on Multipartite Graphs**[10]-
The clean-factor operator is a multipartite graph operator that has been introduced in the context of complex network modelling. Here, we consider a less constrained variation of the clean-factor operator, named strong-factor operator, and we prove that, as for the clean-factor operator, the iteration of the strong-factor operator always terminates, independently of the graph given as input. Obtaining termination for all graphs using minimal constraints on the definition of the operator is crucial for the modelling purposes for which the clean-factor operator has been introduced. Moreover we show that the relaxation of constraints we operate not only preserves termination but also preserves the termination time, in the sense that the strong-factor series always terminates before the clean-factor series.

**On the Termination of Some Biclique Operators on Multipartite Graphs**[9]-
We define a new graph operator, called the

*weak-factor graph*, which comes from the context of complex network modelling. The weak-factor operator is close to the well-known clique-graph operator but it rather operates in terms of bicliques in a multipartite graph. We address the problem of the termination of the series of graphs obtained by iteratively applying the weak-factor operator starting from a given input graph. As for the clique-graph operator, it turns out that some graphs give rise to series that do not terminate. Therefore, we design a slight variation of the weak-factor operator, called*clean-factor*, and prove that its associated series terminates for all input graphs. In addition, we show that the multipartite graph on which the series terminates has a very nice combinatorial structure: we exhibit a bijection between its vertices and the chains of the inclusion order on the intersections of the maximal cliques of the input graph. **Directed Cartesian-Product Decomposition**[11] .-
In this paper, we design an algorithm that, given a directed graph $G$ and the Cartesian-product decomposition of its underlying undirected graph $\tilde{G}$, produces the directed Cartesian-product decomposition of $G$ in linear time. This is the first time that the linear complexity is achieved for this problem, which has two major consequences. Firstly, it shows that the directed and undirected versions of the Cartesian-product decomposition of graphs are linear-time equivalent problems. And secondly, as there already exists a linear-time algorithm for solving the undirected version of the problem, combined together, it provides the first linear-time algorithm for computing the directed Cartesian-product decomposition of a directed graph.

**An $O\left({n}^{2}\right)$ time Algorithm for the Minimal Permutation Completion Problem**[28]-
We provide an $O\left({n}^{2}\right)$ time algorithm computing a minimal permutation completion of an arbitrary graph $G=(V,E)$, i.e., a permutation graph $H=(V,F)$ on the same vertex set, such that $E\subseteq F$ and $F$ is inclusion-minimal among all possibilities.

**Linearity is Strictly More Powerful than Contiguity for Encoding Graphs**[27]-
Linearity and contiguity are two parameters devoted to graph encoding. Linearity is a generalisation of contiguity in the sense that every encoding achieving contiguity $k$ induces an encoding achieving linearity $k$, both encoding having size $\Theta (k.n)$, where $n$ is the number of vertices of $G$. In this paper, we prove that linearity is a strictly more powerful encoding than contiguity, i.e. there exists some graph family such that the linearity is asymptotically negligible in front of the contiguity. We prove this by answering an open question asking for the worst case linearity of a cograph on $n$ vertices: we provide an $O(logn/loglogn)$ upper bound which matches the previously known lower bound.

**Socioeconomic correlations in communication networks**[37] , [38]-
In this work we study the socioeconomic structure of a communication network by combining mobile communication records and bank credit informations of a large number of individuals living in Mexico. We provide empirical evidences about present economic unbalances suggesting not only the distribution of wealth but also the distribution of debts to follow the Pareto principle. Further we study the internal and interconnected structure of socioeconomic groups. Through a weighted core analysis we signal assortative correlations between people regarding their economic capacities, and show the existence of "rich-clubs" indicating present social stratification in the social structure. This project is ongoing with final results expected in 2016.

**Detecting global bridges in networks**[15]-
The identification of nodes occupying important positions in a network structure is crucial for the understanding of the associated real-world system. Usually, betweenness centrality is used to evaluate a node capacity to connect different graph regions. However, we argue here that this measure is not adapted for that task, as it gives equal weight to "local" centers (i.e. nodes of high degree central to a single region) and to "global" bridges, which connect different communities. This distinction is important as the roles of such nodes are different in terms of the local and global organisation of the network structure. In this paper we propose a decomposition of betweenness centrality into two terms, one highlighting the local contributions and the other the global ones. We call the latter bridgeness centrality and show that it is capable to specifically spot out global bridges. In addition, we introduce an effective algorithmic implementation of this measure and demonstrate its capability to identify global bridges in air transportation and scientific collaboration networks.

**Collective attention in the age of (mis)information**[17]-
We study, on a sample of 2.3 million individuals, how Facebook users consumed different information at the edge of political discussion and news during the last Italian electoral competition. Pages are categorized, according to their topics and the communities of interests they pertain to, in a) alternative information sources (diffusing topics that are neglected by science and main stream media); b) online political activism; and c) main stream media. We show that attention patterns are similar despite the different qualitative nature of the information, meaning that unsubstantiated claims (mainly conspiracy theories) reverberate for as long as other information. Finally, we categorize users according to their interaction patterns among the different topics and measure how a sample of this social ecosystem (1279 users) responded to the injection of 2788 false information posts. Our analysis reveals that users which are prominently interacting with alternative information sources (i.e. more exposed to unsubstantiated claims) are more prone to interact with false claims.

**The Scaling of Human Contacts in Reaction-Diffusion Processes**[22]-
We present new empirical evidence, based on millions of interactions on Twitter, confirming that human contacts scale with population sizes. We integrate such observations into a reaction-diffusion metapopulation framework providing an analytical expression for the global invasion threshold of a contagion process. Remarkably, the scaling of human contacts is found to facilitate the spreading dynamics. Our results show that the scaling properties of human interactions can significantly affect dynamical processes mediated by human contacts such as the spread of diseases, and ideas.

**From calls to communities: a model for time varying social networks**[16]-
Social interactions vary in time and appear to be driven by intrinsic mechanisms, which in turn shape the emerging structure of the social network. Large-scale empirical observations of social interaction structure have become possible only recently, and modelling their dynamics is an actual challenge. Here we propose a temporal network model which builds on the framework of activity-driven time-varying networks with memory. The model also integrates key mechanisms that drive the formation of social ties - social reinforcement, focal closure and cyclic closure, which have been shown to give rise to community structure and the global connectedness of the network. We compare the proposed model with a real-world time-varying network of mobile phone communication and show that they share several characteristics from heterogeneous degrees and weights to rich community structure. Further, the strong and weak ties that emerge from the model follow similar weight-topology correlations as real-world social networks, including the role of weak ties.

**Kinetics of Social Contagion**[21]-
Diffusion of information, behavioural patterns or innovations follows diverse pathways depending on a number of conditions, including the structure of the underlying social network, the sensitivity to peer pressure and the influence of media. Here we study analytically and by simulations a general model that incorporates threshold mechanism capturing sensitivity to peer pressure, the effect of `immune' nodes who never adopt, and a perpetual flow of external information. While any constant, non-zero rate of dynamically-introduced innovators leads to global spreading, the kinetics by which the asymptotic state is approached show rich behaviour. In particular we find that, as a function of the density of immune nodes, there is a transition from fast to slow spreading governed by entirely different mechanisms. This transition happens below the percolation threshold of fragmentation of the network, and has its origin in the competition between cascading behaviour induced by innovators and blocking of adoption due to immune nodes. This change is accompanied by a percolation transition of the induced clusters.