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
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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?
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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.
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4. Comparing Networks
   (a) It’s a Zoo out there!
   (b) https://arxiv.org/abs/2008.02415
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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.
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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
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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.
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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
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9. Related Work
   Network Comparison and the within-ensemble graph distance
   (Harrison Hartle, Brennan Klein, Stefan McCabe, Guillaume St-Onge., et Al.)
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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.
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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.
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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.
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13. Summary/Current Work
    1 Verify and refine, scan larger number of parameter values
    2 Embed Results in 2D Curve
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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
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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
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