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 

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 

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 

Comparing Networks (a) It’s a Zoo out there! (b) https://arxiv.org/abs/2008.02415 LSD Presentation 

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 

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 

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 

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 

Related Work Network Comparison and the within-ensemble graph distance (Harrison Hartle, Brennan Klein, Stefan McCabe, Guillaume St-Onge., et Al.) LSD Presentation 

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 

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 

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 

Summary/Current Work 1 Verify and refine, scan larger number of parameter values 2 Embed Results in 2D Curve LSD Presentation 

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 

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