Rubén J. Sánchez-García

Hierarchical spectral clustering of power grids

A power transmission system can be represented by a network with nodes and links representing buses and electrical transmission lines, respectively. Each line can be given a weight, representing some electrical property of the line, such as line admittance or average power flow at a given time. We use a hierarchical spectral clustering methodology to reveal the internal connectivity structure of such a network. Spectral clustering uses the eigenvalues and eigenvectors of a matrix associated to the network, it is computationally very efficient, and it works for any choice of weights. When using line admittances, it reveals the static internal connectivity structure of the underlying network, while using power flows highlights islands with minimal power flow disruption, and thus it naturally relates to controlled islanding. Our methodology goes beyond the standard k-means algorithm by instead representing the complete network substructure as a dendrogram. We provide a thorough theoretical justification of the use of spectral clustering in power systems, and we include the results of our methodology for several test systems of small, medium and large size, including a model of the Great Britain transmission network.

Dimensionality reduction and spectral properties of multilayer networks.

Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered systems include multiple subsystems and layers of connectivity. This new paradigm has attracted a great deal of attention and one fundamental challenge is to characterize multilayer networks both structurally and dynamically. One way to address this question is to study the spectral properties of such networks. Here we apply the framework of graph quotients, which occurs naturally in this context, and the associated eigenvalue interlacing results to the adjacency and Laplacian matrices of undirected multilayer networks. Specifically, we describe relationships between the eigenvalue spectra of multilayer networks and their two most natural quotients, the network of layers and the aggregate network, and show the dynamical implications of working with either of the two simplified representations. Our work thus contributes in particular to the study of dynamical processes whose critical properties are determined by the spectral properties of the underlying network.

Spectral characteristics of network redundancy

Many real-world complex networks contain a significant amount of structural redundancy, in which multiple vertices play identical topological roles. Such redundancy arises naturally from the simple growth processes which form and shape many real-world systems. Since structurally redundant elements may be permuted without altering network structure, redundancy may be formally investigated by examining network automorphism (symmetry) groups. Here, we use a group-theoretic approach to give a complete description of spectral signatures of redundancy in undirected networks. In particular, we describe how a networks automorphism group may be used to directly associate specific eigenvalues and eigenvectors with specific network motifs.

International scientific collaboration in HIV and HPV: a network analysis.

Research endeavours require the collaborative effort of an increasing number of individuals. International scientific collaborations are particularly important for HIV and HPV co-infection studies, since the burden of disease is rising in developing countries, but most experts and research funds are found in developed countries, where the prevalence of HIV is low. The objective of our study was to investigate patterns of international scientific collaboration in HIV and HPV research using social network analysis. Through a systematic review of the literature, we obtained epidemiological data, as well as data on countries and authors involved in co-infection studies. The collaboration network was analysed in respect to the following: centrality, density, modularity, connected components, distance, clustering and spectral clustering. We observed that for many low- and middle-income countries there were no epidemiological estimates of HPV infection of the cervix among HIV-infected individuals. Most studies found only involved researchers from the same country (64%). Studies derived from international collaborations including high-income countries and either low- or middle-income countries had on average three times larger sample sizes than those including only high-income countries or low-income countries. The high global clustering coefficient (0.9) coupled with a short average distance between researchers (4.34) suggests a “small-world phenomenon.” Researchers from high-income countries seem to have higher degree centrality and tend to cluster together in densely connected communities. We found a large well-connected community, which encompasses 70% of researchers, and 49 other small isolated communities. Our findings suggest that in the field of HIV and HPV, there seems to be both room and incentives for researchers to engage in collaborations between countries of different income-level. Through international collaboration resources available to researchers in high-income countries can be efficiently used to enroll more participants in low- and middle-income countries.

Microdynamics and Criticality of Adaptive Regulatory Networks

We present a model of adaptive regulatory networks consisting of a simple biologically motivated rewiring procedure coupled to an elementary stability criterion. The resulting networks exhibit a characteristic stationary heavy-tailed degree distribution, show complex structural microdynamics, and self-organize to a dynamically critical state. We show analytically that the observed criticality results from the formation and breaking of transient feedback loops during the adaptive process.

Bredon homology and equivariant K-homology of SL (3,Z)

Abstract We obtain the equivariant K -homology of the classifying space for proper actions E ¯ S L ( 3 , Z ) from the computation of its Bredon homology with respect to finite subgroups and coefficients in the representation ring. We also obtain the corresponding results for G L ( 3 , Z ) . Our calculations give therefore the topological side of the Baum–Connes conjecture for these groups.

Equivariant K-homology for some Coxeter groups

We obtain the equivariant K-homology of the classifying space \underline{E}W for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of \underline{E}W in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C^*-algebra of W, via the Baum-Connes assembly map.

Topics in algebraic and topological K-theory

K-theory for group C*-algebras.- Universal Coefficient Theorems and assembly maps in KK-theory.- Algebraic v. topological K-theory: a friendly match.- Higher algebraic K-theory (after Quillen, Thomason and others).- Lectures on DG-categories

Noise-processing by signaling networks

Signaling networks mediate environmental information to the cell nucleus. To perform this task effectively they must be able to integrate multiple stimuli and distinguish persistent signals from transient environmental fluctuations. However, the ways in which signaling networks process environmental noise are not well understood. Here we outline a mathematical framework that relates a network’s structure to its capacity to process noise, and use this framework to dissect the noise-processing ability of signaling networks. We find that complex networks that are dense in directed paths are poor noise processors, while those that are sparse and strongly directional process noise well. These results suggest that while cross-talk between signaling pathways may increase the ability of signaling networks to integrate multiple stimuli, too much cross-talk may compromise the ability of the network to distinguish signal from noise. To illustrate these general results we consider the structure of the signalling network that maintains pluripotency in mouse embryonic stem cells, and find an incoherent feedforward loop structure involving Stat3, Tfcp2l1, Esrrb, Klf2 and Klf4 is particularly important for noise-processing. Taken together these results suggest that noise-processing is an important function of signaling networks and they may be structured in part to optimize this task.

Equivariant K-homology for hyperbolic reflection groups

We compute the equivariant K-homology of the classifying space for proper actions, for cocompact 3-dimensional hyperbolic reflection groups. This coincides with the topological K-theory of the reduced C∗-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated K-theory groups are torsion-free. As a result we can promote previous rational computations to integral computations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.