Constructing a compact metric space of network ensembles

Network comparison has immediate applications to problems in image recognition, network reconstruction, analysis of time-dependent networks, and fitting of network models. Many of these applications deal with comparison of network ensembles --- both because real networks are rarely uniquely defined and because network models produce sets of possible networks --- such that simple pairwise comparison of networks is often insufficient.

Here, we extend the problem of network comparison to the comparison of network ensembles. We leverage tools from the theory of probabilistic metric spaces (PMS) to explore the surfaces defined by both ensembles of real networks as well as synthetic generative models. In particular, our PMS approach uses a Modified Lévy Metric (MLM) to build surfaces in a compact and complete space for network ensembles characterized by any underlying network comparison approach whether they themselves define a metric or not.

Our framework has many advantages over simple pairwise comparison of networks. Mainly, it captures key features of a network ensemble that are not contained in single instances of the ensemble, such as the diversity of networks therein. In doing so, it naturally allows us to fit generative models, which generate network ensembles by definition, to sets of real networks (e.g. a set of food webs or power grids) or even to interpolate between different ensembles. Moreover, it provides a natural approach to comparing network models which can, for example, identify important structural decisions in different parameterizations of the same model or simply explore the space of possible networks models.

Transcript

1. Constructing a compact metric space of network ensembles A. Daniels1 L.Hebert-Dufresne2 1PhD Candidate University of Vermont 2Professor University of Vermont September 21, 2020 LSD Presentation
2. Motivational Questions Can ensembles of networks be cast into objects unto a topology? What can we find at the intersection of topology/geometry and network science? Is there a natural measure of distance (metric, pseudo-metric, divergence, ...) between ensembles of graphs, given certain features we wish to compare? How can the features of an ensemble be parameterized or tuned? What structures lie between two ensembles? LSD Presentation
3. Network Structure and Function (Example: The Resilient Internet) Figure: Carmi, Shai, et al. ”A model of Internet topology using k-shell decomposition.” Proceedings of the National Academy of Sciences 104.27 (2007): 11150-11154. LSD Presentation
4. Comparing Networks (a) It’s a Zoo out there! (b) https://arxiv.org/abs/2008.02415 LSD Presentation
5. Comparing Groups of Networks Figure: Herculano-Houzel, Suzana. ”The remarkable, yet not extraordinary, human brain as a scaled-up primate brain and its associated cost.” Proceedings of the National Academy of Sciences 109.Supplement 1 (2012): 10661-10668. LSD Presentation
6. Comparing Ensembles of Networks: Methodology Figure: Apples Vs. Oranges Method 1: Intra-ensemble Comparison Step 1: Characterize the group of oranges/apples separately Step 2: Create a summary or archetype Step 3: Compare summaries or archetypes Method 2: Inter-ensemble Comparison Step 1: Compare features between Apple-Orange Pairs Step 2: Combine comparisons of pairs Step 3: Compare combined comparisons LSD Presentation
7. The Notion of a Space A space is a structure, a set with relational properties - Operations, invariants, symmetries, and a notion of ’closeness’ between points are examples of some features of a space. A topological space has a clearly defined notion of ’closeness’ between points. A topology is a set of sets (including empty and entire) closed under set union and finite intersection [Collection of hierarchical groupings]. Figure: Netsimile - Berlingerio et al. LSD Presentation
8. Recipe 1 for Network Ensemble Comparison 1 Choose how individual graphs will be described (description). 2 Choose the discrepancy measure used to calculate ’distance’ between graphs. 3 Build ensembles 4 Sample pairs of graphs from within each ensemble to compute a distribution of distances. 5 Compare averages of distances between ensembles LSD Presentation
9. Related Work Network Comparison and the within-ensemble graph distance (Harrison Hartle, Brennan Klein, Stefan McCabe, Guillaume St-Onge., et Al.) LSD Presentation
10. Recipe 2 for Network Ensemble Comparison 1 Choose how individual graphs will be described (description). 2 Choose the discrepancy measure used to calculate ’distance’ between graphs. 3 Build ensembles 4 Sample pairs of graphs from within each ensemble to compute a distribution of distances. 5 Compute the corresponding cumulative probability distributions (CPD) for each ensemble. 6 Compute the Modified L´ evy Distance between CPDs. LSD Presentation
11. Some Results Modified Levy Metric Distance for Intraensemble G(N,P) Comparison Onion Decomposition 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Netsimile Distance 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Modified Levy Distance Modified Levy Distance Figure: Definitions given in (RIGHT) Berlingerio, Michele, et al. ”Netsimile: A scalable approach to size-independent network similarity.” arXiv preprint arXiv:1209.2684 (2012). (LEFT) H´ ebert-Dufresne, Laurent, Joshua A. Grochow, and Antoine Allard. ”Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition.” Scientific reports 6 (2016): 31708. LSD Presentation
12. Building a Metric Space 1 Consider a metric which takes pairs of (cumulative probability) distribution functions as inputs. 2 Implement for discrete distributions. 3 Find the greatest lower bound which satisfies the conditions using a Bisection Method like approach. The (Modified) L´ evy Metric Let F, G be CPDFS: dL(F, G) = inf{h|[F, G; h] & [G, F; h] } [F, G; h] ≡ G(x) ≤ F(x + h) + h, x ∈ 0, 1 h This metric allows us to build a topology (induces a topology), where the points are ensembles of graphs. LSD Presentation
13. Summary/Current Work 1 Verify and refine, scan larger number of parameter values 2 Embed Results in 2D Curve LSD Presentation
14. References and Related Reading I chweizer, Berthold and Abe Sklar.(2011) Probabilistic metric spaces Courier Corporation 2011. Amari, Shun-ichi, and Andrzej Cichocki. (2010) Information geometry of divergence functions. Bulletin of the polish academy of sciences. Technical sciences 58.1 (2010): 183-195. tevanovic, Dragan. (2014) Metrics for graph comparison: A practitioner’s guide. Plos one 15.2 (2020) e0228728. H´ ebert-Dufresne, Laurent, Joshua A. Grochow, and Antoine Allard. (2018) Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition. Scientific reports Scientific reports 6, 31708 Bagrow, James P., and Erik M. Bollt. (2018) An information-theoretic, all-scales approach to comparing networks. arXiv preprint arXiv:1804.03665 LSD Presentation
15. References and Related Reading II Apolloni, Neural Nets WIRN10 B. (2011) An introduction to spectral distances in networks. Neural Nets WIRN10: Proceedings of the 20th Italian Workshop on Neural Nets. Vol. 226. IOS Press, 2011. Carr´ e, Bernard. (1979) Graphs and Networks Kraetzl, M., and W. D. Wallis. (2006) Modality distance between graphs., Utilitas Mathematica Utilitas Mathematica 69: 97-102. Stevanovic, Dragan. (2014) Spectral radius of graphs., Academic Press Academic Press LSD Presentation