Abstract: Identification of central elements in networks is an ill-defined problem. Hence, a large number of centrality measures have been proposed in the literature. We present a survey of existing axioms, which characterize certain properties of centralities. We also perform a perturbation analysis of centrality measures in real and artificial networks.

Understanding of real systems relies on the identification of its central elements. Over the years, a large number of centrality measures have been proposed to assess the importance of nodes in complex networks. However, most real networks are incomplete and contain incorrect data, resulting in a high sensitivity of centrality indices. In this paper, we examine the robustness of centrality to the presence of errors in the network structure. Our experiments are performed on weighted and unweighted real-world networks ranging from the criminal network to the trade food network. As a result, we discuss a sensitivity of centrality measures to different data imputation techniques.

In recent decades, a large number of centrality measures have been proposed to assess the importance of nodes in complex networks. The choice of the most appropriate centrality index for specific applications is one of the biggest challenges. This paper performs the perturbation analysis of 8 centrality measures. Since most real networks are incomplete and prone to bias, we compare centrality measures in order to evaluate their sensitivity to small changes in a graph structure. Our experiments are performed on 8 classical graph structures ranging from a simple path graph to a Watts-Strogatz graph model. As a result, we provide a sensitivity of centrality measures on different graph structures.

Over the last number of years there has been a growing interest in the analysis of complex networks which describe a wide range of real-world systems in nature and society. Identification of the central elements in such networks is one of the key research areas. Solutions to this problem are important for making strategic decisions and studying the behavior of dynamic processes, e.g. epidemic spread. The importance of nodes has been studied using various centrality measures. Generally, it should be considered that most real systems are not homogeneous: nodes may have individual attributes and influence each other in groups while connections between nodes may describe different types of relations. Thus, critical nodes detection is not a straightforward process. New Centrality Measures in Networks presents a class of new centrality measures which take into account individual attributes of nodes, the possibility of group influence and long-range interactions and discusses all their new features. The book provides a wide range of applications of network analysis in several fields - financial networks, international migration, global trade, global food network, arms transfers, networks of terrorist groups, and networks of international journals in economics. Real-world studies of networks indicate that the proposed centrality measures can identify important nodes in different applications. Starting from the basic ideas, the development of the indices and their advantages compared to existing centrality measures are presented.

We consider an application of long-range interaction centrality (LRIC) to the problem of the influence assessment in the global retail food network. Firstly, we reconstruct an initial graph into the graph of directed intensities based on individual node’s characteristics and possibility of the group influence. Secondly, we apply different models of the indirect influence estimation based on simple paths and random walks. This approach can help us to estimate node-to-node influence in networks. Finally, we aggregate node-to-node influence into the influence index. The model is applied to the food trade network based on the World International Trade Solution database. The results obtained for the global trade by different product commodities are compared with classical centrality measures.

Power evaluation in networks has been studied by many scientists. The problem has been usually considered from different areas as the problem of decision making by a group of agents, the problem of epidemic spreading, etc. Based on the intuition that a node becomes affected if its individual threshold is exceeded, we present a model how to measure the influence. The model considers chain reactions in a network and takes into account individual characteristics and the group influence of nodes as well as indirect interactions between them. Our main focus is to evaluate a pairwise influence between nodes and reveal the most powerful elements.

Our study employs the network approach to the problem of international migration. During the last years, migration has attracted a lot of attention and has been examined from many points of view. However, very few studies considered it from the network perspective. The international migration can be represented as a network (or weighted directed graph) where the nodes correspond to countries and the edges correspond to migration flows. The main focus of our study is to reveal a set of critical or central elements in the network. To do it, we calculated different existing and new centrality measures. In our research the United Nations International Migration Flows Database (version 2015) was used. As a result, we obtained information on critical elements for the migration process in 2013.

We consider an application of power indices, which take into account preferences of agents for coalition formation proposed for an analysis of power distribution in elected bodies to reveal most powerful (central) nodes in networks. These indices take into account the parameters of the nodes in networks, a possibility of group influence from the subset of nodes to single nodes, and intensity of short and long interactions among the nodes.

The problem of quick detection of central nodes in large networks is studied. There are many measures that allow to evaluate a topological importance of nodes of the network. Unfortunately, most of them cannot be applied to large networks due to their high computational complexity. However, if we narrow the initial network and apply these centrality measures to the sparse network, it is possible that the obtained set of central nodes will be similar to the set of central nodes in large networks. If these sets are similar, the centrality measures with a high computational complexity can be used for central nodes detection in large networks. To check the idea, several random networks were generated and different techniques of network reduction were considered. We also adapted some rules from social choice theory for the key nodes detection. As a result, we show how the initial network should be narrowed in order to apply centrality measures with a high computational complexity and maintain the set of key nodes of a large network.