Finite-size analysis of the detectability limit of the stochastic block model

Jean-Gabriel Young, Patrick Desrosiers, Laurent Hébert-Dufresne, Edward Laurence, and Louis J. Dubé

Phys. Rev. E 95, 062304 – Published 19 June 2017

Abstract

It has been shown in recent years that the stochastic block model is sometimes undetectable in the sparse limit, i.e., that no algorithm can identify a partition correlated with the partition used to generate an instance, if the instance is sparse enough and infinitely large. In this contribution, we treat the finite case explicitly, using arguments drawn from information theory and statistics. We give a necessary condition for finite-size detectability in the general SBM. We then distinguish the concept of average detectability from the concept of instance-by-instance detectability and give explicit formulas for both definitions. Using these formulas, we prove that there exist large equivalence classes of parameters, where widely different network ensembles are equally detectable with respect to our definitions of detectability. In an extensive case study, we investigate the finite-size detectability of a simplified variant of the SBM, which encompasses a number of important models as special cases. These models include the symmetric SBM, the planted coloring model, and more exotic SBMs not previously studied. We conclude with three appendices, where we study the interplay of noise and detectability, establish a connection between our information-theoretic approach and random matrix theory, and provide proofs of some of the more technical results.

Received 31 December 2016
Revised 1 May 2017

DOI:https://doi.org/10.1103/PhysRevE.95.062304

Physics Subject Headings (PhySH)

Community structure Continuous phase transition

Erdos-Renyi graph Stochastic networks Uncorrelated network

Block models Metropolis algorithm

Networks

Authors & Affiliations

Jean-Gabriel Young^1,*, Patrick Desrosiers^1,2, Laurent Hébert-Dufresne³, Edward Laurence¹, and Louis J. Dubé^1,†

¹Département de Physique, de Génie Physique, et d'Optique, Université Laval, Québec (Québec), Canada G1V 0A6
²Centre de recherche de l'Institut universitaire en santé mentale de Québec, Québec (Québec), Canada G1J 2G3
³Santa Fe Institute, Santa Fe, New Mexico 87501, USA

^*jean-gabriel.young.1@ulaval.ca
^†ljd@phy.ulaval.ca

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Vol. 95, Iss. 6 — June 2017

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Images

Figure 1
Stochastic block model with (a, c, e) nonuniform density matrix and (b, d, f) nearly uniform density matrix. (a, b) Density matrix of the two ensembles. Notice the difference in scale. (c, d) One instance of each ensemble, with $n = [50, 50, 50, 100, 200, 200]$ . Each color denotes a block [46]. (e, f) Empirical distribution of the normalized log-likelihood obtained from $100 000$ samples of $L$ . The bins in which the instances (c, d) fall are colored in red. Notice that a negative log-likelihood ratio is associated with some instances in (f).
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Figure 2
Accuracy of the CLT approximation for the (a) nonuniform and (b) nearly uniform SBM of Fig. 1. Both histograms aggregate $100 000$ samples of $L$ . The prediction of the CLT is shown in red [see Eqs. (23b)–(23e)]. We plot for comparison the Gaussian kernel density estimate (KDE) of $ρ (λ)$ (dashed black line, hidden by the CLT curve). Equation (25) predicts $η_{(a)} = 1$ and $η_{(b)} = 0.981 (2)$ ; for the sample shown, numerical estimates yield ${\hat{η}}_{(a)} = 1$ and ${\hat{η}}_{(b)} = 0.980 (7)$ .
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Figure 3
Detectability in the general modular graph model. All figures use the same indicator matrix $W$ [panel (a)] and the size vector $n = [10, 30, 20, 20, 20]$ ( $n = 100$ nodes). (a) Example of density matrix $P$ allowed in the GMGM. Dark squares indicate block pairs of the inner type and light squares indicate pairs of the outer type. (b) Average detectability in the density space of the GMGM. Both the numerical solution of $〈 L 〉 = λ$ (solid black line) and the prediction of Eq. (34) (dashed white line) are shown, for $λ = 0.05$ and 0.3. (c) $η (ρ, Δ; β)$ in the density space of the GMGM; notice the change of $Δ$ axis. Solid white lines are curves where $η (ρ, Δ; β) = η^{*}$ , with $η^{*} = 0.7$ (central curve) and $η^{*} = 0.99$ (outer curve). Equation (25) is used to compute both $η$ and $η^{*}$ .
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Figure 4
Inference of the GMGM. All figures show results for the special case $q = 2$ , $W = I_{2}$ , and $n = [n / 2, n / 2]$ , corresponding to the $q = 2$ SSBM [23]. All empirical results are averaged over $10^{4}$ independent instances of the SBM. (a) Average rNMI of the planted and the inferred partition in the density space of the model of size $n = 100$ . Solid red lines mark the boundaries of the 1-detectability region, with tolerance threshold $T = 4 \sqrt{2}$ ; see Eq. (28). Dotted black lines show the two solutions of $Δ^{*} (ρ; λ = 1 / 2 n, β)$ ; see Eq. (34). White lines show the finite-size KS bound before it is adjusted for the symmetries of the problem. (b, c) Phase transition at constant $ρ = 0.25$ for networks of $n = 100$ nodes (b) and $n = 500$ nodes (c). Circles indicate the fraction of instances for which a correlated partition could be identified, while diamonds show the average of the rNMI (lines are added to guide the eye). Blue solid curves show $η (Δ; ρ, β, n)$ ; see Eq. (25). The shaded region lies below the finite-size Kesten-Stigum bound $Δ = \pm q \sqrt{ρ / n}$ (here with $q = 2$ ). The dotted lines show the two solutions of $Δ^{*} (ρ; λ = 1 / 2 n, β = 1 / 2)$ .
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