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

2018 ESC/ESH Guidelines for the management of arterial hypertension.

The four Visegrad states — Poland, the Czech Republic, Slovakia (until 1993 Czechoslovakia) and Hungary — form a compact area between Germany and Austria in the west and the states of the former USSR in the east. They are bounded by the Baltic in the north and the Danube river in the south. They are cut by the Sudeten and Carpathian mountain ranges, which divide Poland off from the other states. Poland is an extension of the North European plain and like the latter is drained by rivers that flow from south to north west — the Oder, the Vlatava and the Elbe, the Vistula and the Bug. The Danube is the great exception, flowing from its source eastward, turning through two 90-degree turns to end up in the Black Sea, forming the barrier and often the political frontier between central Europe and the Balkans. Hungary to the east of the Danube is also an open plain. The region is historically and culturally part of western Europe, but its eastern Marches now represents a vital strategic zone between Germany and the core of the European Union to the west and the Russian zone to the east.

Economy and Society

https://toreopsahl.files.wordpress.com/2010/04/node_centrality_in_weighted_networks1.pdf

Node centrality in weighted networks: Generalizing degree and shortest paths.

The structure and dynamics of multilayer networks

Clustering in weighted networks

Complex systems are often characterized by large-scale hierarchical organizations. Whether the prominent elements, at the top of the hierarchy, share and control resources or avoid one another lies at the heart of a system's global organization and functioning. Inspired by network perspectives, we propose a new general framework for studying the tendency of prominent elements to form clubs with exclusive control over the majority of a system's resources. We explore associations between prominence and control in the fields of transportation, scientific collaboration, and online communication.

https://www.epicx-lab.com/uploads/9/6/9/4/9694133/physrevlett_101_168702.pdf

Prominence and Control: The Weighted Rich-Club Effect

This research draws on longitudinal network data from an online community to examine patterns of users' behavior and social interaction, and infer the processes underpinning dynamics of system use. The online community represents a prototypical example of a complex evolving social network in which connections between users are established over time by online messages. We study the evolution of a variety of properties since the inception of the system, including how users create, reciprocate, and deepen relationships with one another, variations in users' gregariousness and popularity, reachability and typical distances among users, and the degree of local redundancy in the system. Results indicate that the system is a “small world” characterized by the emergence, in its early stages, of a hub-dominated structure with heterogeneity in users' behavior. We investigate whether hubs are responsible for holding the system together and facilitating information flow, examine first-mover advantages underpinning users' ability to rise to system prominence, and uncover gender differences in users' gregariousness, popularity, and local redundancy. We discuss the implications of the results for research on system use and evolving social networks, and for a host of applications, including information diffusion, communities of practice, and the security and robustness of information systems. © 2009 Wiley Periodicals, Inc.

Patterns and dynamics of users' behavior and interaction: Network analysis of an online community

Many complex systems can be described as multiplex networks in which the same nodes can interact with one another in different layers, thus forming a set of interacting and co-evolving networks Examples of such multiplex systems are social networks where people are involved in different types of relationships and interact through various forms of communication media The ranking of nodes in multiplex networks is one of the most pressing and challenging tasks that research on complex networks is currently facing When pairs of nodes can be connected through multiple links and in multiple layers, the ranking of nodes should necessarily reflect the importance of nodes in one layer as well as their importance in other interdependent layers In this paper, we draw on the idea of biased random walks to define the Multiplex PageRank centrality measure in which the effects of the interplay between networks on the centrality of nodes are directly taken into account In particular, depending on the intensity of the interaction between layers, we define the Additive, Multiplicative, Combined, and Neutral versions of Multiplex PageRank, and show how each version reflects the extent to which the importance of a node in one layer affects the importance the node can gain in another layer We discuss these measures and apply them to an online multiplex social network Findings indicate that taking the multiplex nature of the network into account helps uncover the emergence of rankings of nodes that differ from the rankings obtained from one single layer Results provide support in favor of the salience of multiplex centrality measures, like Multiplex PageRank, for assessing the prominence of nodes embedded in multiple interacting networks, and for shedding a new light on structural properties that would otherwise remain undetected if each of the interacting networks were analyzed in isolation

Multiplex PageRank

One of the most important challenges in network science is to quantify the information encoded in complex network structures. Disentangling randomness from organizational principles is even more demanding when networks have a multiplex nature. Multiplex networks are multilayer systems of nodes that can be linked in multiple interacting and co-evolving layers. In these networks, relevant information might not be captured if the single layers were analyzed separately. Here we demonstrate that such partial analysis of layers fails to capture significant correlations between weights and topology of complex multiplex networks. To this end, we study two weighted multiplex co-authorship and citation networks involving the authors included in the American Physical Society. We show that in these networks weights are strongly correlated with multiplex structure, and provide empirical evidence in favor of the advantage of studying weighted measures of multiplex networks, such as multistrength and the inverse multiparticipation ratio. Finally, we introduce a theoretical framework based on the entropy of multiplex ensembles to quantify the information stored in multiplex networks that would remain undetected if the single layers were analyzed in isolation.

/pdf/weighted-multiplex-networks-2374ei48a5.pdf

Pietro Panzarasa

Papers

Clustering in weighted networks

Prominence and Control: The Weighted Rich-Club Effect

Patterns and dynamics of users' behavior and interaction: Network analysis of an online community

Multiplex PageRank

Weighted Multiplex Networks