The Physics of Communicability in Complex Networks

A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and applied to a wide variety of real-world networks in recent years. Several such communicability functions are reviewed in this paper. It is emphasized that communication and correlation in networks can take place through many more routes than the shortest paths, a fact that may not have been sufficiently appreciated in previously proposed correlation measures. In contrast to these, the communicability measures reviewed in this paper are defined by taking into account all possible routes between two nodes, assigning smaller weights to longer ones. This point of view naturally leads to the definition of communicability in terms of matrix functions, such as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either the adjacency matrix or the graph Laplacian associated with the network. Considerable insight on communicability can be gained by modeling a network as a system of oscillators and deriving physical interpretations, both classical and quantum-mechanical, of various communicability functions. Applications of communicability measures to the analysis of complex systems are illustrated on a variety of biological, physical and social networks. The last part of the paper is devoted to a review of the notion of locality in complex networks and to computational aspects that by exploiting sparsity can greatly reduce the computational efforts for the calculation of communicability functions for large networks.Comment: Review Article. 90 pages, 14 figures. Contents: Introduction; Communicability in Networks; Physical Analogies; Comparing Communicability Functions; Communicability and the Analysis of Networks; Communicability and Localization in Complex Networks; Computability of Communicability Functions; Conclusions and Prespective

Updating and downdating techniques for optimizing network communicability

The total communicability of a network (or graph) is defined as the sum of the entries in the exponential of the adjacency matrix of the network, possibly normalized by the number of nodes. This quantity offers a good measure of how easily information spreads across the network, and can be useful in the design of networks having certain desirable properties. The total communicability can be computed quickly even for large networks using techniques based on the Lanczos algorithm. In this work we introduce some heuristics that can be used to add, delete, or rewire a limited number of edges in a given sparse network so that the modified network has a large total communicability. To this end, we introduce new edge centrality measures which can be used to guide in the selection of edges to be added or removed. Moreover, we show experimentally that the total communicability provides an effective and easily computable measure of how "well-connected" a sparse network is.Comment: 20 pages, 9 pages Supplementary Materia

On the limiting behavior of parameter-dependent network centrality measures

We consider a broad class of walk-based, parameterized node centrality measures for network analysis. These measures are expressed in terms of functions of the adjacency matrix and generalize various well-known centrality indices, including Katz and subgraph centrality. We show that the parameter can be "tuned" to interpolate between degree and eigenvector centrality, which appear as limiting cases. Our analysis helps explain certain correlations often observed between the rankings obtained using different centrality measures, and provides some guidance for the tuning of parameters. We also highlight the roles played by the spectral gap of the adjacency matrix and by the number of triangles in the network. Our analysis covers both undirected and directed networks, including weighted ones. A brief discussion of PageRank is also given.Comment: First 22 pages are the paper, pages 22-38 are the supplementary material

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Two betweenness centrality measures based on Randomized Shortest Paths

This paper introduces two new closely related betweenness centrality measures based on the Randomized Shortest Paths (RSP) framework, which fill a gap between traditional network centrality measures based on shortest paths and more recent methods considering random walks or current flows. The framework defines Boltzmann probability distributions over paths of the network which focus on the shortest paths, but also take into account longer paths depending on an inverse temperature parameter. RSP's have previously proven to be useful in defining distance measures on networks. In this work we study their utility in quantifying the importance of the nodes of a network. The proposed RSP betweenness centralities combine, in an optimal way, the ideas of using the shortest and purely random paths for analysing the roles of network nodes, avoiding issues involving these two paradigms. We present the derivations of these measures and how they can be computed in an efficient way. In addition, we show with real world examples the potential of the RSP betweenness centralities in identifying interesting nodes of a network that more traditional methods might fail to notice.Comment: Minor updates; published in Scientific Report

Polarization and multiscale structural balance in signed networks

Polarization is a common feature of social systems. Structural Balance Theory studies polarization of positive in-group and negative out-group ties in terms of semicycles within signed networks. However, enumerating semicycles is computationally expensive, so approximations are often needed to assess balance. Here we introduce Multiscale Semiwalk Balance (MSB) approach for quantifying the degree of balance (DoB) in (un)directed, (un)weighted signed networks by approximating semicycles with closed semiwalks. MSB allows principled selection of a range of cycle lengths appropriate for assessing DoB and interpretable under the Locality Principle (which posits that patterns in shorter cycles are crucial for balance). This flexibility overcomes several limitations affecting walk-based approximations and enables efficient, interpretable methods for measuring DoB and clustering signed networks. We demonstrate the effectiveness of our approach by applying it to real-world social systems. For instance, our methods capture increasing polarization in the U.S. Congress, which may go undetected with other methods.Comment: 29 pages; 7 figures; preprint before peer revie

Communicability distance reveals hidden patterns of alzheimer’s disease

The communicability distance between pairs of regions in human brain is used as a quantitative proxy for studying Alzheimer’s disease. Using this distance, we obtain the shortest communicability path lengths between different regions of brain networks from patients with Alzheimer’s disease (AD) and healthy cohorts (HC). We show that the shortest communicability path length is significantly better than the shortest topological path length in distinguishing AD patients from HC. Based on this approach, we identify 399 pairs of brain regions for which there are very significant changes in the shortest communicability path length after AD appears. We find that 42% of these regions interconnect both brain hemispheres, 28% connect regions inside the left hemisphere only, and 20% affect vermis connection with brain hemispheres. These findings clearly agree with the disconnection syndrome hypothesis of AD. Finally, we show that in 76.9% of damaged brain regions the shortest communicability path length drops in AD in relation to HC. This counterintuitive finding indicates that AD transforms the brain network into a more efficient system from the perspective of the transmission of the disease, because it drops the circulability of the disease factor around the brain regions in relation to its transmissibility to other regions

A walk-based measure of balance in signed networks. Detecting lack of balance in social networks

There is a longstanding belief that in social networks with simultaneous friendly and hostile interactions (signed networks) there is a general tendency to a global balance. Balance represents a state of the network with a lack of contentious situations. Here we introduce a method to quantify the degree of balance of any signed (social) network. It accounts for the contribution of all signed cycles in the network and gives, in agreement with empirical evidence, more weight to the shorter cycles than to the longer ones. We found that, contrary to what is generally believed, many signed social networks, in particular very large directed online social networks, are in general very poorly balanced. We also show that unbalanced states can be changed by tuning the weights of the social interactions among the agents in the network

Graphs and networks theory

This chapter discusses graphs and networks theory