Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.

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Graph Curvature for Differentiating Cancer Networks.

Quantifying the systemic risk and fragility of financial systems is of vital importance in analyzing market efficiency, deciding on portfolio allocation, and containing financial contagions. At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow (for example, debt, stock returns, and shareholder ownership). Such a representation often turns out to provide keen insights. We show that fragility is a system-level characteristic of "business-as-usual" market behavior and that financial crashes are invariably preceded by system-level changes in robustness. This was done by leveraging previous work, which suggests that Ricci curvature, a key geometric feature of a given network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily returns from a set of stocks comprising the Standard and Poor's 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature constitute a financial "crash hallmark." This work lays the foundation of understanding how to design (banking) systems and policy regulations in a manner that can combat financial instabilities exposed during the 2007-2008 crisis.

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Ricci curvature: An economic indicator for market fragility and systemic risk

Many complex networks in the real world have community structures – groups of well-connected nodes with important functional roles. It has been well recognized that the identification of communities bears numerous practical applications. While existing approaches mainly apply statistical or graph theoretical/combinatorial methods for community detection, in this paper, we present a novel geometric approach which  enables us to borrow powerful classical geometric methods and properties. By considering networks as geometric objects and communities in a network as a geometric decomposition, we apply curvature and discrete Ricci flow, which have been used to decompose smooth manifolds with astonishing successes in mathematics, to break down communities in networks. We  tested our method on networks with ground-truth community structures, and experimentally confirmed the effectiveness of this geometric approach.

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Community Detection on Networks with Ricci Flow.

Identification of community structures in complex network is of crucial importance for understanding the system’s function, organization, robustness and security. Here, we present a novel Ollivier-Ricci curvature (ORC) inspired approach to community identification in complex networks. We demonstrate that the intrinsic geometric underpinning of the ORC offers a natural approach to discover inherent community structures within a network based on interaction among entities. We develop an ORC-based community identification algorithm based on the idea of sequential removal of negatively curved edges symptomatic of high interactions (e.g., traffic, attraction). To illustrate and compare the performance with other community identification methods, we examine the ORC-based algorithm with stochastic block model artificial networks and real-world examples ranging from social to drug-drug interaction networks. The ORC-based algorithm is able to identify communities with either better or comparable performance accuracy and to discover finer hierarchical structures of the network. This opens new geometric avenues for analysis of complex networks dynamics.

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Ollivier-Ricci Curvature-Based Method to Community Detection in Complex Networks.

This paper assesses the performance of the D-Wave 2X (DW) quantum annealer for finding a maximum clique in a graph, one of the most fundamental and important NP-hard problems Because the size of the largest graphs DW can directly solve is quite small (usually around 45 vertices), we also consider decomposition algorithms intended for larger graphs and analyze their performance For smaller graphs that fit DW, we provide formulations of the maximum clique problem as a quadratic unconstrained binary optimization (QUBO) problem, which is one of the two input types (together with the Ising model) acceptable by the machine, and compare several quantum implementations to current classical algorithms such as simulated annealing, Gurobi, and third-party clique finding heuristics We further estimate the contributions of the quantum phase of the quantum annealer and the classical post-processing phase typically used to enhance each solution returned by DW We demonstrate that on random graphs that fit DW, no quantum speedup can be observed compared with the classical algorithms On the other hand, for instances specifically designed to fit well the DW qubit interconnection network, we observe substantial speed-ups in computing time over classical approaches

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Finding Maximum Cliques on the D-Wave Quantum Annealer

Negative curvature in the relatively new Ollivier-Ricci sense of a wireless network graph is shown to be the culprit behind large queue occupancy, large routing energy, and restricted capacity region under any throughput optimal protocol. This is the wireless counterpart of the congestion phenomenon occurring in a Gromov negatively curved wired network under least cost path routing. Significantly different protocols call for significantly different curvature concepts to explain the “congestion” phenomenon. The rationale for the Ollivier-Ricci curvature is that it is defined in terms of a transportation cost-formalized by the Wasserstein distance-under a diffusion process. The Heat-Diffusion protocol used in this paper is driven by the queue differential, so that interpreting packets as calories, it lends itself to a genuine heat diffusion, yet retaining the 3-stage process of weighting-scheduling-forwarding of well-known Back-Pressure protocol. The main result is that the transportation definition of the Ollivier-Ricci curvature allows for the direct connection-without resorting to the Laplacian operator of heat calculus-between curvature and queue occupancy, routing energy, and capacity region.

Wireless network capacity versus Ollivier-Ricci curvature under Heat-Diffusion (HD) protocol

Quantum annealing (QA) serves as a specialized optimizer that is able to solve many NP-hard problems and that is believed to have a theoretical advantage over simulated annealing (SA) via quantum tunneling. With the introduction of the D-Wave programmable quantum annealer, a considerable amount of effort has been devoted to detect and quantify quantum speedup. While the debate over speedup remains inconclusive as of now, instead of attempting to show general quantum advantage, here, we focus on a novel real-world application of D-Wave in wireless networking—more specifically, the scheduling of the activation of the air-links for maximum throughput subject to interference avoidance near network nodes. In addition, D-Wave implementation is made error insensitive by a novel Hamiltonian extra penalty weight adjustment that enlarges the gap and substantially reduces the occurrence of interference violations resulting from inevitable spin bias and coupling errors. The major result of this paper is that quantum annealing benefits more than simulated annealing from this gap expansion process, both in terms of ST99 speedup and network queue occupancy. It is the hope that this could become a real-word application niche where potential benefits of quantum annealing could be objectively assessed.

Quantum versus simulated annealing in wireless interference network optimization.

This paper proceeds from the premise that the topology of interference constrained wireless networks heavily impacts their node-to-node delay, routing energy, and capacity region. We quantitatively analyze how the discrete Ollivier-Ricci curvature of a network affects the performance metrics of several routing protocols. Since different protocols are optimal relative to different metrics under different topologies, an adaptive control system is proposed that identifies the topology curvature and selects the best protocol under current circumstances subject to user needs. Also, we analyze how sensitive the four routing protocols (Heat Diffusion, Dirichlet, Back Pressure and Shortest Path Routing) under examination are to varying topological environment, as it would commonly be encountered in wireless networks.

Interference constrained network control based on curvature

The D-Wave quantum computer is designed to solve a specific class of problems - The Quadratic Unconstrained Binary Optimization (QUBO) problem. One of the key processes in this pathway to the solution consists in embedding the problem graph into a hardware graph. It is a nontrivial task to determine whether a problem graph is minor embeddable in a hardware graph. One method that singles out cases where minor embeddability fails requires calculation of the tree-width of both the problem and the hardware graphs. The latter computation is known to be NP-Complete. In this paper, we propose a novel, fast approximation to tree-width based on the differential geometric concept of Ollivier-Ricci curvature. This latter runs in linear time and thus could significantly reduce the overall complexity of determining whether a QUBO problem is solvable on the D-Wave architecture.

Ollivier-Ricci curvature and fast approximation to tree-width in embeddability of QUBO problems

The D-Wave adiabatic quantum computing platform is designed to solve a particular class of problems--the Quadratic Unconstrained Binary Optimization (QUBO) problems. Due to the particular "Chimera" physical architecture of the D-Wave chip, the logical problem graph at hand needs an extra process called minor embedding in order to be solvable on the D-Wave architecture. The latter problem is itself NP-hard. In this paper, we propose a novel polynomial-time approximation to the closely related treewidth based on the differential geometric concept of Ollivier---Ricci curvature. The latter runs in polynomial time and thus could significantly reduce the overall complexity of determining whether a QUBO problem is minor embeddable, and thus solvable on the D-Wave architecture.

Chi Wang

Papers

Wireless network capacity versus Ollivier-Ricci curvature under Heat-Diffusion (HD) protocol

Quantum versus simulated annealing in wireless interference network optimization.

Interference constrained network control based on curvature

Ollivier-Ricci curvature and fast approximation to tree-width in embeddability of QUBO problems

Differential geometric treewidth estimation in adiabatic quantum computation