1,364 research outputs found
A New Methodology for Generalizing Unweighted Network Measures
Several important complex network measures that helped discovering common
patterns across real-world networks ignore edge weights, an important
information in real-world networks. We propose a new methodology for
generalizing measures of unweighted networks through a generalization of the
cardinality concept of a set of weights. The key observation here is that many
measures of unweighted networks use the cardinality (the size) of some subset
of edges in their computation. For example, the node degree is the number of
edges incident to a node. We define the effective cardinality, a new metric
that quantifies how many edges are effectively being used, assuming that an
edge's weight reflects the amount of interaction across that edge. We prove
that a generalized measure, using our method, reduces to the original
unweighted measure if there is no disparity between weights, which ensures that
the laws that govern the original unweighted measure will also govern the
generalized measure when the weights are equal. We also prove that our
generalization ensures a partial ordering (among sets of weighted edges) that
is consistent with the original unweighted measure, unlike previously developed
generalizations. We illustrate the applicability of our method by generalizing
four unweighted network measures. As a case study, we analyze four real-world
weighted networks using our generalized degree and clustering coefficient. The
analysis shows that the generalized degree distribution is consistent with the
power-law hypothesis but with steeper decline and that there is a common
pattern governing the ratio between the generalized degree and the traditional
degree. The analysis also shows that nodes with more uniform weights tend to
cluster with nodes that also have more uniform weights among themselves.Comment: 23 pages, 10 figure
Economic Small-World Behavior in Weighted Networks
The small-world phenomenon has been already the subject of a huge variety of
papers, showing its appeareance in a variety of systems. However, some big
holes still remain to be filled, as the commonly adopted mathematical
formulation suffers from a variety of limitations, that make it unsuitable to
provide a general tool of analysis for real networks, and not just for
mathematical (topological) abstractions. In this paper we show where the major
problems arise, and how there is therefore the need for a new reformulation of
the small-world concept. Together with an analysis of the variables involved,
we then propose a new theory of small-world networks based on two leading
concepts: efficiency and cost. Efficiency measures how well information
propagates over the network, and cost measures how expensive it is to build a
network. The combination of these factors leads us to introduce the concept of
{\em economic small worlds}, that formalizes the idea of networks that are
"cheap" to build, and nevertheless efficient in propagating information, both
at global and local scale. This new concept is shown to overcome all the
limitations proper of the so-far commonly adopted formulation, and to provide
an adequate tool to quantitatively analyze the behaviour of complex networks in
the real world. Various complex systems are analyzed, ranging from the realm of
neural networks, to social sciences, to communication and transportation
networks. In each case, economic small worlds are found. Moreover, using the
economic small-world framework, the construction principles of these networks
can be quantitatively analyzed and compared, giving good insights on how
efficiency and economy principles combine up to shape all these systems.Comment: 17 pages, 10 figures, 4 table
Generalized Erdos Numbers for network analysis
In this paper we consider the concept of `closeness' between nodes in a
weighted network that can be defined topologically even in the absence of a
metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable
properties as a measure of topological closeness when nodes share a finite
resource between nodes as they are real-valued and non-local, and can be used
to create an asymmetric matrix of connectivities. We show that they can be used
to define a personalized measure of the importance of nodes in a network with a
natural interpretation that leads to a new global measure of centrality and is
highly correlated with Page Rank. The relative asymmetry of the GENs (due to
their non-metric definition) is linked also to the asymmetry in the mean first
passage time between nodes in a random walk, and we use a linearized form of
the GENs to develop a continuum model for `closeness' in spatial networks. As
an example of their practicality, we deploy them to characterize the structure
of static networks and show how it relates to dynamics on networks in such
situations as the spread of an epidemic
Local dependency in networks
Many real world data and processes have a network structure and can usefully be represented as graphs. Network analysis focuses on the relations among the nodes exploring the properties of each network. We introduce a method for measuring the strength of the relationship between two nodes of a network and for their ranking. This method is applicable to all kinds of networks, including directed and weighted networks. The approach extracts dependency relations among the network's nodes from the structure in local surroundings of individual nodes. For the tasks we deal with in this article, the key technical parameter is locality. Since only the surroundings of the examined nodes are used in computations, there is no need to analyze the entire network. This allows the application of our approach in the area of large-scale networks. We present several experiments using small networks as well as large-scale artificial and real world networks. The results of the experiments show high effectiveness due to the locality of our approach and also high quality node ranking comparable to PageRank.Web of Science25229328
- …