A large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength. In recent years, the study of an increasing number of large scale networks has highlighted the statistical heterogeneity of their interaction pattern, with degree and weight distributions which vary over many orders of magnitude. These features, along with the large number of elements and links, make the extraction of the truly relevant connections forming the network's backbone a very challenging problem. More specifically, coarse-graining approaches and filtering techniques are at struggle with the multiscale nature of large scale systems. Here we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks, preserving the edges that represent statistical significant deviations with respect to a null model for the local assignment of weights to edges. An important aspect of the method is that it does not belittle smallscale interactions and operates at all scales defined by the weight distribution. We apply our method to real world network instances and compare the obtained results with alternative backbone extraction techniques.  n recent years, a huge amount of data on large scale social, biological, and communication networks, meticulously collected and catalogued, has become available for scientific analysis and study. Examples can be found in all domains; from technological to social systems and transportation networks on a local and global scale, and down to the microscopic scale of biochemical networks [1, 2, 3]. Common traits of these networks can be found in the statistical properties characterized by large scale heterogeneity with statistical observables such as nodes' degree and traffic varying over a wide range of scales [4]. The sheer size and multiscale nature of these networks make very difficult the extraction of the relevant information that would allow a reduced representation while preserving the key features we want to highlight. A typical example is faced in the visualization of networks. While it is generally possible to create wonderful images of large scale heterogeneous networks, the amount of valuable information gathered is in most cases very little because of the redundant intricacy generated by the overwhelming number of connections. Problems such as the extraction of the relevant backbone or the isolation of the statistically relevant structures/signal that would allow reduced but meaningful representations of the system are indeed major challenges in the analysis of large-scale networks. In complex weighted networks, the discrimination of the right trade-off between the level of network reduction and the amount of relevant information preserved in the new representation faces us with additional problems. In many cases, the probability distribution P (ω) that any given link is carrying a weight ω is broadly distributed, spanning several orders of magnitude. This feature implies the lack of a characteristic scale and any method based on thresholding would simply overlook the information present above or below the arbitrary cut-off scale. While this issue would not be a major drawback in networks where the intensities of all the edges are independently and identically distributed, the cut off of the P (ω) tail would destroy the multiscale nature of more realistic networks where weights are locally correlated on edges incident to the same node and non-trivially coupled to topology [5]. Thus, the presence of multiscale fluctuations calls for reduction techniques that consistently highlight the relevant structures and hierarchies without favoring any particular resolution scale. Furthermore, it also demands to change the focus towards a local perspective rather than a global one, where the relevance of the connections could be decided at the level of nodes in relative terms. In this work, we concentrate on a particular technique that operates at all the scales defined by the weighted network structure. This method, based on the local identification of the statistically relevant weight heterogeneities, is able to filter out the backbone of dominant connections in weighted networks with strong disorder, preserving structural properties and hierarchies at all scales. We discuss our multiscale filter in relation to the appropriate null model that provides the basis for the statistical significance of the heterogeneity measurements. We apply the technique to two real world networks, the U.S. airport network and the Florida Bay food web, and compare the results to those obtained by the application of thresholding methods. In Statistical Mathematics, as in other areas, filtering techniques aimed at uncovering the relevant information in data sets are popular and successful. One could cite, for instance, the Principal Components Analysis to identify hidden patterns by reducing the effective dimension of multivariate data [6]. In the following, we will refer to the network reduction as the construction of a network that contains far less data (in our case links) and allows the discrimination and computational tractability of the relevant features of the original networks; for instance, the traffic backbone of a large scale transportation infrastructure. Reduction schemes can be divided into two main categories: coarse-graining and filtering/pruning. In the first case, nodes sharing a common attribute could be gathered together in the same class -group, community, etc.-and then substituted by a single new unit which represents the whole class in a new network representation of the system [7, 8, 9, 10] . This coarse-graining is indeed zooming out the system so that it can be observed at different scales. Something completely different is done when a filter is applied. In this case, the observation scale is fixed and the representation that the network symbolizes is not changed. Instead, those elements -nodes and edges-that carry relevant information about the network structure are kept while the rest are discarded. An example of a well-known hierarchical topological filter, although usually not referred as such, is the k-core decomposition of a network [11] , with a filtering rule that acts on the connectivity of the nodes.
The Disparity Filter algorithm is a network reduction technique developed by Serrano et al. REF to identify the 'backbone' structure of a weighted network.
Extracting the multiscale backbone of complex weighted networks
Proc. Natl. Acad. Sci. USA 106, 6483-6488 (2009)
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