2.1. Forman-Ricci Curvature on Networks
- e denotes the edge under consideration that connects the nodes and ;
- denotes the (positive) weight on the edge e;
- denote the (positive) weights associated with the nodes and , respectively;
- denote the set of edges connected to nodes and , respectively.
2.2. Characterizing Large Data Sets with Ricci Curvature
2.3. Ricci-Flow with Forman Curvature
- At each iteration step (i.e., in the process of updating to ), the Forman curvature has to be recomputed for each edge e, since it depends on its respective weight . This clearly increases the computational effort on magnitudes, however, the computation task is less formidable than it might appear at first.
- As already stressed, we consider a discrete time model. Since for smoothing (denoising) a short time flow has to be applied (because, by the theory for the smooth case, a long time flow will produce a limiting state of the network), only a small number of iterations needs be considered. The precise number of necessary iterations is to be determined experimentally. Even though a typical number can be found easily, best results may be obtained for slightly different numbers—depending on the network, and the type and level of the noise, of course.
- Ollivier also devised a continuous flow [11,22]. In the context of the present article, a continuous setting is not required, but for other types of networks, where the evolution is continuous in time, it might be preferable to implement the continuous variant, suitably adapted to the Forman curvature, rather than to Ollivier’s one.
2.4. Characterizing Dynamic Data with Ricci Flow
3.1. Change Detection with Ricci Flow
3.2. Analysis of Gnutella Peer-To-Peer Network
4. Discussion and Future Work
- A task that is almost self evident, is to further experiment with very large data sets (numbers of data points in the order of ten thousand and more);
- Another natural target is the use of our method on different types of networks, with special emphasis on Biological Networks;
- A statistical analysis regarding the Ricci flow, similar to the one presented here and in , should also be performed on various standard types of networks in order to confirm and calibrate the characterization and classifying capabilities of the Ricci curvature and flow.
- The Forman curvature versions of the scalar and Laplace–Beltrami flows. Especially the last one seems to be promising for network denoising, as applications of the analogous flow in image processing showed . Moreover, the Forman-Ricci curvature comes naturally coupled with a fitting version of the so called Bochner Laplacian (and yet with another, intrinsically connected, rough Laplacian). This aspect is subject to ongoing work and will be covered in a forthcoming paper by the authors.
- As for the short time Ricci flow, a statistical analysis should be undertaken to validate the classification potential of the long term Ricci flow. A more ambitious, yet still feasible, future direction would be to explore network stability by considering the long time Forman-Ricci flow (as opposed to the short time one employed for denoising). This approach would exploit, in analogy with the smooth case [19,20], the propensity of the Ricci flow to preserve and quantify the overall, global Geometry (i.e., curvature) and essential topology of the network. This would allow us to study the evolution of a network “under its own pressure” and to detect and examine such catastrophic events as virus attacks and denial of service attempts. Given the basic numerical simplicity of our method, this approach might prove to be an effective alternative to the Persistent Homology method (see, e.g., ) for the 1-dimensional case of networks. Moreover, the Ricci flow does not need to make appeal to higher dimensional structures (namely simplicial complexes) that are necessary for the Persistent Homology based applications, with clear computational advantages (see, e.g., the code described in ), but also theoretically rigor. Furthermore, the here defined Ricci flow can be applied on weighted networks, whereas Persistant Homology requires unweighted complexes.
Conflicts of Interest
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