Characterizing Distances of Networks on the Tensor Manifold

At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a family of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a point on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of geodesics amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems.Comment: This paper is accepted at 8th International Conference on Complex Networks 201

Contrasting Views of Complexity and Their Implications For Network-Centric Infrastructures

There exists a widely recognized need to better understand and manage complex “systems of systems,” ranging from biology, ecology, and medicine to network-centric technologies. This is motivating the search for universal laws of highly evolved systems and driving demand for new mathematics and methods that are consistent, integrative, and predictive. However, the theoretical frameworks available today are not merely fragmented but sometimes contradictory and incompatible. We argue that complexity arises in highly evolved biological and technological systems primarily to provide mechanisms to create robustness. However, this complexity itself can be a source of new fragility, leading to “robust yet fragile” tradeoffs in system design. We focus on the role of robustness and architecture in networked infrastructures, and we highlight recent advances in the theory of distributed control driven by network technologies. This view of complexity in highly organized technological and biological systems is fundamentally different from the dominant perspective in the mainstream sciences, which downplays function, constraints, and tradeoffs, and tends to minimize the role of organization and design

Selection, tinkering and emergence in complex networks: crossing the land of tinkering

Complex biological networks have very different origins than technologic ones. The latter involve extensive design and, as engineered structures, include a high level of optimization. The former involve (in principle) contingency and structural constraints, with new structures being incorporated through tinkering with previously evolved modules or units. However, the observation of the topological features of different biological nets suggests that nature can have a limited repertoire of ”attractors” that essentially optimize communication under some basic constraints of cost and architecture or that allow the biological nets to reach a high degree of homeostasis. Conversely, the topological features exhibited by some technology graphs indicate that tinkering and internal constraints play a key role, in spite of the ”designed” nature of these structures. Previous scenarios suggested to explain the overall trends of evolution are re-analyzed in light of topological patterns.Peer ReviewedPostprint (author's final draft

A simple spatiotemporal evolution model of a transmission power grid

In this paper, we present a model for the spatial and temporal evolution of a particularly large human-made network: the 400-kV French transmission power grid. This is based on 1) an attachment procedure that diminishes the connection probability between two nodes as the network grows and 2) a coupled cost function characterizing the available budget at every time step. Two differentiated and consecutive processes can be distinguished: a first global space-filling process and a secondary local meshing process that increases connectivity at a local level. Results show that even without power system engineering design constraints (i.e., population and energy demand), the evolution of a transmission network can be remarkably explained by means of a simple attachment procedure. Given a distribution of resources and a time span, the model can also be used to generate the probability distribution of cable lengths at every time step, thus facilitating network planning. Implications for network's fragility are suggested as a starting point for new design perspectives in this kind of infrastructures.Peer ReviewedPostprint (author's final draft

Ultrafast Consensus in Small-World Networks

In this paper, we demonstrate a phase transition phenomenon in algebraic connectivity of small-world networks. Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We demonstrate that it is possible to dramatically increase the algebraic connectivity of a regular complex network by 1000 times or more without adding new links or nodes to the network. This implies that a consensus problem can be solved incredibly fast on certain small-world networks giving rise to a network design algorithm for ultra fast information networks. Our study relies on a procedure called "random rewiring" due to Watts & Strogatz (Nature, 1998). Extensive numerical results are provided to support our claims and conjectures. We prove that the mean of the bulk Laplacian spectrum of a complex network remains invariant under random rewiring. The same property only asymptotically holds for scale-free networks. A relationship between increasing the algebraic connectivity of complex networks and robustness to link and node failures is also shown. This is an alternative approach to the use of percolation theory for analysis of network robustness. We also show some connections between our conjectures and certain open problems in the theory of random matrices