Tools for the analysis, visualisation and filtering of complex undirected weighted networks.
Complex systems can be represented as networks or graphs in which system components are abstracted as nodes and their interactions as edges connecting them. In addition, the strength or intensity of such interactions can be abstracted by an edge weight. For example, in an air-transportation network, the weight of an edge can be the number of yearly non-stop flights between two airports. In an online social network, on the other hand, the edge weight can be the number of messages shared by two friends.
Considering edge weights in complex networks is very important, as they convey the dynamics of information flowing through the network structure1, as well as the importance or reliability of edges2. As a result, the study of weighted networks is crucial to understand the formation, evolution and dynamics of complex systems. However, this kind of analyses has been hampered by the heterogeneity characterising the degree and weight distributions of weighted networks1,3. This heterogeneity can also be present at the local level, where only a few of the edges touching a node carry most of its strength (i.e. the total weight of the edges in which it takes part)4,5. This feature implies the lack of a characteristic scale and any global filter (i.e. removal of edges with weights below or above a certain cutoff) would overlook the information present above or below the arbitrary cutoff scale4.
Nevertheless, one can take advantage of the above-mentioned heterogeneities to filter out irrelevant edges and preserve a subnetwork comprised of the most salient connections between nodes5. To that end, Serrano et al. proposed a filtering method, the disparity filter, to extract the relevant backbone from a weighted network and, at the same time, retain most of its nodes, total weight, global connectedness, its small-world property, heterogeneous degree distribution and clustering4,5. This filter preserves the edges whose weight represents a statistically significant deviation from a null model. In the null, the normalised weights of the edges incident to a node of degree k are the result of a random assignment from a uniform distribution4.
If the strength of node i is si = ∑jwij, then the normalised weight of the edges linking i with its neighbours is ωij = wij/si and ∑jωij = 1. Note that this normalisation happens at the level of each node, thus ωij can be different from ωji. One can now compare these normalised weights with the ones for node i in the null model, which represent random normalised weights that also sum up to 1. Given this null model, the probability of observing a particular normalised weight x touching a node with degree k is given by a beta distribution with shape parameters 1 and k − 1:
p(x)=(k − 1)(1 − x)k − 2
Salient edges, according to the disparity filter4, are thus those whose normalised weight satisfies the relation:
1 − (k − 1)∫0ωij(1 − x)k − 2dx < α
In other words, relevant edges have weights that are statistically different from what is expected by chance at the significance level α. Since ωij can be different from ωji, weights can be significant for node i but not for j and vice versa. The disparity filter keeps edge ij if it turns out to be relevant to either node. It is also worth noting that the beta distribution requires its shape parameters to be >0, which means that nodes of degree 1 should be treated separately. With this package, the user can choose to always keep them or keep them only if their edge is significant to the node at the other end.
DisparityFilter is an implementation of the above-described identification of salient edges for complex undirected weighted networks, represented as
igraph objects. It also allows for the visual analysis of the topological characteristics of networks filtered at different significance levels. What's more, it provides tools for the computation of node disparities4, defined as Υi(k)=k∑jωij2 for a node i with degree k. Under perfect homogeneity, when all edges incident to i contribute equally to its strength, Υi(k)=1 independently of k, while under perfect heterogeneity, when just one edge carries the whole strength, Υi(k)=k.
- Install the
devtoolspackage from CRAN if you haven't done so:
- Load the
To start using
DisparityFilter, load the package:
Let's now use the US air-transportation network included with
DisparityFilter, analyse its node disparities and apply the disparity filter (for more details about this network type
?air in R). Node disparities can be computed as follows:
disp <- get_node_disparity(net = air)
The relation between node degrees and node disparities can then be visualised with:
plot_degree_vs_disparity(net = air, node_disp = disp)
The above figure shows the disparity measure for individual nodes of the US airport network. Blue dashed lines represent perfect homogeneity and heterogeneity, the area below the red line represents the average plus two standard deviations of disparities given by the null model.
Let's now apply the disparity filter to the network to obtain the p-values associated to each edge:
air_with_pvals <- get_edge_disparity_pvals(net = air)
We can now analyse the topology of the networks resulting from the application of different disparity filters:
analysis <- analyse_disparity_filter(net = air_with_pvals, breaks = 100)
Finally, based on the previous data frame, we can carry out a visual analysis of the resulting network topologies:
p <- plot_disparity_filter_analysis(disp_analysis = analysis)
The following plot shows the remaining fraction of nodes in the extracted backbone of
air (Nbb/Ntot) as a function of the remaining fraction of links (Lbb/Ltot) as different significance levels are applied to the network. In addition, it shows the fractions resulting from the application of a global filter:
In a similar fashion, the following plot shows the remaining fraction of nodes as a function of the remaining fraction of total weight (Wbb/Wtot):
The dashed red lines in the above plots correspond to the significance level that maximises the number of remaining nodes in the network and minimises the average node degree5. Note how the disparity filter manages to retain as many nodes and as much weight as possible, whereas a global filter removes a high fraction of nodes and weight even for the less stringent cutoffs.
How to cite
If you find this package useful, please cite the following papers:
- Serrano M. A., Boguñá, M. & Vespignani, A. Extracting the multiscale backbone of complex weighted networks. PNAS 6(16) 6483-6488 (2009) See paper
- Alanis-Lobato, G. & Andrade-Navarro, M. A reliable and unbiased human protein network with the disparity filter. bioRxiv 10.1101/207761 (2017) See paper
Barrat, A., Barthelemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America 101, 3747–3752 (2004).
Alanis-Lobato, G., Andrade-Navarro, M. A. & Schaefer, M. H. HIPPIE v2.0: Enhancing meaningfulness and reliability of protein–protein interaction networks. Nucleic Acids Research 45, D408–D414 (2017).
Allard, A., Serrano, M. Á., García-Pérez, G. & Boguñá, M. The geometric nature of weights in real complex networks. Nature Communications 8, 14103 (2017).
Serrano, M. Á., Boguñá, M. & Vespignani, A. Extracting the multiscale backbone of complex weighted networks. Proceedings of the National Academy of Sciences 106, 6483–6488 (2009).
García-Pérez, G., Boguñá, M., Allard, A. & Serrano, M. Á. The hidden hyperbolic geometry of international trade: World Trade Atlas 1870–2013. Scientific Reports 6, 33441 (2016).