'Conjectural' links in complex networks
📝 Abstract
This paper introduces the concept of Conjectural Link for Complex Networks, in particular, social networks. Conjectural Link we understand as an implicit link, not available in the network, but supposed to be present, based on the characteristics of its topology. It is possible, for example, when in the formal description of the network some connections are skipped due to errors, deliberately hidden or withdrawn (e.g. in the case of partial destruction of the network). Introduced a parameter that allows ranking the Conjectural Link. The more this parameter - the more likely that this connection should be present in the network. This paper presents a method of recovery of partially destroyed Complex Networks using Conjectural Links finding. Presented two methods of finding the node pairs that are not linked directly to one another, but have a great possibility of Conjectural Link communication among themselves: a method based on the determination of the resistance between two nodes, and method based on the computation of the lengths of routes between two nodes. Several examples of real networks are reviewed and performed a comparison to know network links prediction methods, not intended to find the missing links in already formed networks.
💡 Analysis
This paper introduces the concept of Conjectural Link for Complex Networks, in particular, social networks. Conjectural Link we understand as an implicit link, not available in the network, but supposed to be present, based on the characteristics of its topology. It is possible, for example, when in the formal description of the network some connections are skipped due to errors, deliberately hidden or withdrawn (e.g. in the case of partial destruction of the network). Introduced a parameter that allows ranking the Conjectural Link. The more this parameter - the more likely that this connection should be present in the network. This paper presents a method of recovery of partially destroyed Complex Networks using Conjectural Links finding. Presented two methods of finding the node pairs that are not linked directly to one another, but have a great possibility of Conjectural Link communication among themselves: a method based on the determination of the resistance between two nodes, and method based on the computation of the lengths of routes between two nodes. Several examples of real networks are reviewed and performed a comparison to know network links prediction methods, not intended to find the missing links in already formed networks.
📄 Content
1 Corresponding author at: Kyiv Polytechnic Institute, Kyiv, Ukraine. D.I. Zorinets. deniszorinets@gmail.com “Conjectural” links in complex networks A.A. Snarskii a,b, D.I. Zorinets a*, D.V. Lande a,b
a NTUU “ Kyiv Polytechnic Institute ” Kyiv, Ukraine b Institute for Information Recording NAS of Ukraine
PACS numbers: 89.75.Fb, 89.75.Hc, 89.65.-s
Abstract
This paper introduces the concept of Conjectural Link for Complex Networks, in particular, social
networks. Conjectural Link we understand as an implicit link, not available in the network, but supposed to be present, based on the characteristics of its topology. It is possible, for example, when in the formal description of the network some connections are skipped due to errors, deliberately hidden or withdrawn (e.g. in the case of partial destruction of the network). Introduced a parameter that allows ranking the Conjectural Link. The more this parameter - the more likely that this connection should be present in the network. This paper presents a method of recovery of partially destroyed Complex Networks using Conjectural Links finding. Presented two methods of finding the node pairs that are not linked directly to one another, but have a great possibility of Conjectural Link communication among themselves: a method based on the determination of the resistance between two nodes, and method based on the computation of the lengths of routes between two nodes. Several examples of real networks are reviewed and performed a comparison to know network links prediction methods, not intended to find the missing links in already formed networks.
Keywords: complex networks, centrality, conjectural links, link ranking, network restoration
Introduction
Simulation of complex networks (CNs) and the research of their parameters [1, 2, 3, 4] comprise a challenging problem. There are a lot of network parameters. Some of them describe networks in whole, e.g., the average node degree and the clustering coefficient. Some other network characteristics, e.g., the node degree distribution function, provide more detailed information. There are parameters that rank the nodes or links in accordance with a certain algorithm. They are known under the common name “centrality” and include, e.g., the degree centrality, the Katz centrality, the Page Rank, and others. There are also CN parameters which characterize nodes in pairs. For example, in the problem of missing link prediction [5, 6, 7, 8] (sometimes called “Link Prediction” [5, 6, 7, 8]), such parameters are considered which enable one to estimate the probability for a link to emerge between two nodes when the network grows. In this work, we are interested in such a characteristic for a pair of nodes, which would evaluate how strongly those nodes are connected with each other. We will assume that the larger is the number of paths connecting two nodes and the shorter are those paths, the stronger the nodes are linked together. The simplest variant of this characteristic is the shortest path length between the nodes. When 2
the nodes are linked directly, the path length is equal to unity. It is easy to see that such a simple variant of characteristic as the shortest path length between the nodes is not suitable for the description of, e.g., two nodes that are not linked directly, but have many common neighbors. We believe that the introduced characteristic has to be large both for a pair of directly linked nodes and in the case when the nodes are not linked directly, but have plenty of common links with various lengths. In real networks, e.g., in social ones, the majority of nodes are known not to be linked directly, with the number of existing links being much less than the possible maximum. Therefore, first of all, we are interested in how strongly the nodes that are not linked directly are linked with each other. The absent direct link will be called a “conjectural” link (CL). Accordingly, all CLs can be ranked using a certain numerical parameter. It is important to notice that CLs with large values of this parameter may really exist, but they were overlooked while describing the network, e.g., the social one. It occurred, besides other reasons, because those links were artificially hidden. An example of networks containing “hidden” links can be the so-called “terrorist networks”, i.e. networks containing links between terrorists [9]. As CL parameters, we cam use characteristics introduced when studying the Link Prediction problem [5, 6, 7, 8]. The problem of revealing CLs with large parameter values (the top-most ones in the ranking list) is interesting per se and allows one, for instance, to find unexpected (conjectural) links between the literary characters or the members of real social networks. However, not less interesting and important is the fact that the link ranking enables one to address another problem: the restoratio
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