Communicability Graph and Community Structures in Complex Networks

We use the concept of the network communicability (Phys. Rev. E 77 (2008) 036111) to define communities in a complex network. The communities are defined as the cliques of a communicability graph, which has the same set of nodes as the complex network and links determined by the communicability function. Then, the problem of finding the network communities is transformed to an all-clique problem of the communicability graph. We discuss the efficiency of this algorithm of community detection. In addition, we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network. Finally, we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network. We then analyze the USA airport network, for which we successfully detect two big communities of the eastern airports and of the western/central airports as well as two bridging central communities. In striking contrast, a well-known algorithm groups all but two of the continental airports into one community.Comment: 36 pages, 5 figures, to appear in Applied Mathematics and Computatio

Discussion on "Sparse graphs using exchangeable random measures" by F. Caron and E. B. Fox

Discussion on "Sparse graphs using exchangeable random measures" by F. Caron and E. B. Fox. In this discussion we contribute to the analysis of the GGP model as compared to the Erdos-Renyi (ER) and the preferential attachment (AB) models, using different measures such as number of connected components, global clustering coefficient, assortativity coefficient and share of nodes in the core.Comment: 2 pages, 1 figur

Structural patterns in complex networks through spectral analysis

The study of some structural properties of networks is introduced from a graph spectral perspective. First, subgraph centrality of nodes is defined and used to classify essential proteins in a proteomic map. This index is then used to produce a method that allows the identification of superhomogeneous networks. At the same time this method classify non-homogeneous network into three universal classes of structure. We give examples of these classes from networks in different real-world scenarios. Finally, a communicability function is studied and showed as an alternative for defining communities in complex networks. Using this approach a community is unambiguously defined and an algorithm for its identification is proposed and exemplified in a real-world network

Matrix powers algorithms for trust evaluation in PKI architectures

This paper deals with the evaluation of trust in public-key infrastructures. Different trust models have been proposed to interconnect the various PKI components in order to propagate the trust between them. In this paper we provide a new polynomial algorithm using linear algebra to assess trust relationships in a network using different trust evaluation schemes. The advantages are twofold: first the use of matrix computations instead of graph algorithms provides an optimized computational solution; second, our algorithm can be used for generic graphs, even in the presence of cycles. Our algorithm is designed to evaluate the trust using all existing (finite) trust paths between entities as a preliminary to any exchanges between PKIs. This can give a precise evaluation of trust, and accelerate for instance cross-certificate validation

Community structure detection in the evolution of the United States airport network

This is the post-print version of the Article. Copyright © 2013 World Scientific PublishingThis paper investigates community structure in the US Airport Network as it evolved from 1990 to 2010 by looking at six bi-monthly intervals in 1990, 2000 and 2010, using data obtained from the Bureau of Transportation Statistics of the US Department of Transport. The data contained monthly records of origin-destination pairs of domestic airports and the number of passengers carried. The topological properties and the volume of people traveling are both studied in detail, revealing high heterogeneity in space and time. A recently developed community structure detection method, accounting for the spatial nature of these networks, is applied and reveals a picture of the communities within. The patterns of communities plotted for each bi-monthly interval reveal some interesting seasonal variations of passenger flows and airport clusters that do not occupy a single US region. The long-term evolution of the network between those years is explored and found to have consistently improved its stability. The more recent structure of the network (2010) is compared with migration patterns among the four US macro-regions (West, Midwest, Northeast and South) in order to identify possible relationships and the results highlight a clear overlap between US domestic air travel and migration

Community Detection in Quantum Complex Networks

Determining community structure is a central topic in the study of complex networks, be it technological, social, biological or chemical, in static or interacting systems. In this paper, we extend the concept of community detection from classical to quantum systems---a crucial missing component of a theory of complex networks based on quantum mechanics. We demonstrate that certain quantum mechanical effects cannot be captured using current classical complex network tools and provide new methods that overcome these problems. Our approaches are based on defining closeness measures between nodes, and then maximizing modularity with hierarchical clustering. Our closeness functions are based on quantum transport probability and state fidelity, two important quantities in quantum information theory. To illustrate the effectiveness of our approach in detecting community structure in quantum systems, we provide several examples, including a naturally occurring light-harvesting complex, LHCII. The prediction of our simplest algorithm, semiclassical in nature, mostly agrees with a proposed partitioning for the LHCII found in quantum chemistry literature, whereas our fully quantum treatment of the problem uncovers a new, consistent, and appropriately quantum community structure.Comment: 16 pages, 4 figures, 1 tabl

Space-independent community structure detection in United States air transportation

This article presents an evolution-based model for the US airport network. The topological properties and the volume of people travelling are both studied in detail, revealing high heterogeneity in space and time. A recently developed community structure detection method, accounting for the spatial nature of these networks, reveals a better picture of the communities within. © 2012 IFAC

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

Community structure in the World Trade Network based on communicability distances

In this paper, we investigate the mesoscale structure of the World Trade Network. In this framework, a specific role is assumed by short and long-range interactions, and hence by the distance, between countries. Therefore, we identify clusters through a new procedure that exploits Estrada communicability distance and the vibrational communicability distance, which turn out to be particularly suitable for catching the inner structure of the economic network. The proposed methodology aims at finding the distance threshold that maximizes a specific modularity function defined for general metric spaces. Main advantages regard the computational efficiency of the procedure as well as the possibility to inspect intercluster and intracluster properties of the resulting communities. The numerical analysis highlights peculiar relationships between countries and provides a rich set of information that can hardly be achieved within alternative clustering approaches.Comment: 40 pages, 19 figure