We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Social Network Analysis

The structure and dynamics of multilayer networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others We also survey and discuss existing data sets that can be represented as multilayer networks We review attempts to generalize single-layer-network diagnostics to multilayer networks We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks We conclude with a summary and an outlook

Multilayer Networks

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Cryptocurrencies are distributed systems that allow exchanges of native (and non-) tokens among participants. The availability of the complete historical bookkeeping opens up an unprecedented possibility, i.e. that of understanding the evolution of their network structure while gaining useful insight on the relationships between users' behaviour and cryptocurrency pricing in exchange markets. In this contribution we review some of the most recent results concerning the structural properties of the Bitcoin Transaction Networks, a generic name referring to a set of three different constructs: the Bitcoin Address Network, the Bitcoin User Network and the Bitcoin Lightning Network. The picture that emerges is that of system growing over time, which becomes increasingly sparse and whose mesoscopic structural organization is characterised by the presence of an increasingly significant core-periphery structure. Such a peculiar topology is matched by a highly uneven distribution of bitcoins, a result suggesting that Bitcoin is becoming an increasingly centralized system at different levels.

https://www.zora.uzh.ch/id/eprint/194399/1/fphy-08-00286.pdf

Bitcoin Transaction Networks: an overview of recent results

Bipartite networks provide an insightful representation of many systems, ranging from mutualistic networks of species interactions to investment networks in finance. The analysis of their topological structures has revealed the ubiquitous presence of properties which seem to characterize many - apparently different - systems. Nestedness, for example, has been observed in plants-pollinator as well as in country-product trade networks. This has raised questions about the significance of these patterns, which are often believed to constitute a genuine signature of self-organization. Here, we review several methods that have been developed for the analysis of such evidence. Due to the interdisciplinary character of complex networks, tools developed in one field, for example ecology, can greatly enrich other areas of research, such as economy and finance, and vice versa. With this in mind, we briefly review several entropy-based bipartite null models that have been recently proposed and discuss their application to several real-world systems. The focus on these models is motivated by the fact that they show three very desirable features: analytical character, general applicability and versatility. In this respect, entropy-based methods have been proven to perform satisfactorily both in providing benchmarks for testing evidence-based null hypotheses and in reconstructing unknown network configurations from partial information. On top of that, entropy-based models have been successfully employed to analyze ecological as well as economic systems, thus representing an ideal, interdisciplinary tool to approach the study of bipartite complex systems. [...]

From Ecology to Finance (and Back?): Recent Advancements in the Analysis of Bipartite Networks

Recent events such as the global financial crisis have renewed the interest in the topic of economic networks. One of the main channels of shock propagation among countries is the International Trade Network (ITN). Two important models for the ITN structure, the classical gravity model of trade (more popular among economists) and the fitness model (more popular among networks scientists), are both limited to the characterization of only one representation of the ITN. The gravity model satisfactorily predicts the volume of trade between connected countries, but cannot reproduce the observed missing links (i.e. the topology). On the other hand, the fitness model can successfully replicate the topology of the ITN, but cannot predict the volumes. This paper tries to make an important step forward in the unification of those two frameworks, by proposing a new GDP-driven model which can simultaneously reproduce the binary and the weighted properties of the ITN. Specifically, we adopt a maximum-entropy approach where both the degree and the strength of each node is preserved. We then identify strong nonlinear relationships between the GDP and the parameters of the model. This ultimately results in a weighted generalization of the fitness model of trade, where the GDP plays the role of a `macroeconomic fitness' shaping the binary and the weighted structure of the ITN simultaneously. Our model mathematically highlights an important asymmetry in the role of binary and weighted network properties, namely the fact that binary properties can be inferred without the knowledge of weighted ones, while the opposite is not true.

A GDP-driven model for the binary and weighted structure of the International Trade Network

Link-prediction is an active research field within network theory, aiming at uncovering missing connections or predicting the emergence of future relationships from the observed network structure. This paper represents our contribution to the stream of research concerning missing links prediction. Here, we propose an entropy-based method to predict a given percentage of missing links, by identifying them with the most probable non-observed ones. The probability coefficients are computed by solving opportunely defined null-models over the accessible network structure. Upon comparing our likelihood-based, local method with the most popular algorithms over a set of economic, financial and food networks, we find ours to perform best, as pointed out by a number of statistical indicators (e.g. the precision, the area under the ROC curve, etc.). Moreover, the entropy-based formalism adopted in the present paper allows us to straightforwardly extend the link-prediction exercise to directed networks as well, thus overcoming one of the main limitations of current algorithms. The higher accuracy achievable by employing these methods - together with their larger flexibility - makes them strong competitors of available link-prediction algorithms.

/pdf/entropy-based-approach-to-missing-links-prediction-skaj9rei1p.pdf

Entropy-based approach to missing-links prediction

Social media play a key role in shaping citizens' political opinion. According to the Eurobarometer, the percentage of EU citizens employing online social networks on a daily basis has increased from 18% in 2010 to 48% in 2019. The entwinement between social media and the unfolding of political dynamics has motivated the interest of researchers for the analysis of users online behavior-with particular emphasis on group polarization during debates and echo-chambers formation. In this context, semantic aspects have remained largely under-explored. In this paper, we aim at filling this gap by adopting a two-steps approach. First, we identify the discursive communities animating the political debate in the run up of the 2018 Italian Elections as groups of users with a significantly-similar retweeting behavior. Second, we study the mechanisms that shape their internal discussions by monitoring, on a daily basis, the structural evolution of the semantic networks they induce. Above and beyond specifying the semantic peculiarities of the Italian electoral competition, our approach innovates studies of online political discussions in two main ways. On the one hand, it grounds semantic analysis within users' behaviors by implementing a method, rooted in statistical theory, that guarantees that our inference of socio-semantic structures is not biased by any unsupported assumption about missing information; on the other, it is completely automated as it does not rest upon any manual labelling (either based on the users' features or on their sharing patterns). These elements make our method applicable to any Twitter discussion regardless of the language or the topic addressed.

/pdf/analysing-twitter-semantic-networks-the-case-of-2018-italian-48bp944pgr.pdf

Tiziano Squartini

Papers

Bitcoin Transaction Networks: an overview of recent results

From Ecology to Finance (and Back?): Recent Advancements in the Analysis of Bipartite Networks

A GDP-driven model for the binary and weighted structure of the International Trade Network

Entropy-based approach to missing-links prediction

Analysing Twitter semantic networks: the case of 2018 Italian elections