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Continuum limit of critical inhomogeneous random graphs

Abstract

The last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real-world systems. In this study we analyze the maximal components at criticality of one famous class of such models, the rank-one inhomogeneous random graph model (Norros and Reittu, Adv Appl Probab 38(1):59–75, 2006; Bollobás et al., Random Struct Algorithms 31(1):3–122, 2007, Section 16.4). Viewing these components as measured random metric spaces, under finite moment assumptions for the weight distribution, we show that the components in the critical scaling window with distances scaled by \(n^{-1/3}\) converge in the Gromov–Haussdorf–Prokhorov metric to rescaled versions of the limit objects identified for the Erdős–Rényi random graph components at criticality in Addario-Berry et al. (Probab. Theory Related Fields, 152(3–4):367–406, 2012). A key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called \(\mathbf {p}\)-trees (Camarri and Pitman, Electron. J. Probab 5(2):1–18, 2000; Aldous et al., Probab Theory Related Fields 129(2):182–218, 2004). This is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics (Braunstein et al., Phys Rev Lett 91(16):168701, 2003) for such graph models. By asymptotic equivalence (Janson, Random Struct Algorithms 36(1):26–45, 2010), the same results are true for the Chung–Lu model (Chung and Lu, Proc Natl Acad Sci 99(25):15879–15882, 2002; Chung and Lu, Ann Combin 6(2):125–145, 2002; Chung and Lu, Complex graphs and networks, 2006) and the Britton–Deijfen–Martin–Löf model (Britton et al., J Stat Phys 124(6):1377–1397, 2006). A crucial ingredient of the proof of independent interest are tail bounds for the height of \(\mathbf {p}\)-trees. The techniques developed in this paper form the main technical bedrock for the general program developed in Bhamidi et al. (Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős–Rényi random graph. arXiv preprint, 2014) for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels (Bollobás et al., Random Struct Algorithms 31(1):3–122, 2007).

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Acknowledgments

We thank an anonymous referee for a thorough review of the paper. This significantly improved both the proofs as well as presentation of the paper. We thank Amarjit Budhiraja for many stimulating conversations. We also thank Grégory Miermont for insightful discussions about the results of [8]. SS thanks UNC Chapel Hill for hospitality and support during visits. SB has been partially supported by NSF-DMS Grants 1105581, 1310002, 160683, 161307 and SES Grant 1357622. SS has been supported in part by NSF Grant DMS-1007524 and the Netherlands Organization for Scientific Research (NWO) through the Gravitation Networks Grant 024.002.003. XW has been supported in part by the National Science Foundation (DMS-1004418, DMS-1016441), the Army Research Office (W911NF-0-1-0080, W911NF-10-1-0158) and the US-Israel Binational Science Foundation (2008466).

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Correspondence to Shankar Bhamidi.

Appendix A.

Appendix A.

The aim of this section is to outline a proof of Lemma 7.7.

Proof of Lemma 7.7

By imitating the proof of (10.23), we can show that for any \(\varepsilon >0\),

$$\begin{aligned} {{\mathrm{{\mathbb {P}}}}}\left( q_1-q_1^2-X\geqslant \varepsilon \sigma (\mathbf {p})/2 \right) \leqslant \frac{4K_{10.10}}{\varepsilon ^2}\exp \left( -\frac{\varepsilon ^2}{2^{14}q_1}\right) , \end{aligned}$$

where the meaning of the symbols is as in Lemma 10.10. Now if \(q_1\leqslant K\sigma (\mathbf {p})\) for some fixed \(K>0\), then \(q_1^2\leqslant \varepsilon \sigma (\mathbf {p})/2\) for large n since \(\sigma (\mathbf {p})\rightarrow 0\) under Assumption 7.1. Thus,

$$\begin{aligned} {{\mathrm{{\mathbb {P}}}}}\left( q_1-X\geqslant \varepsilon \sigma (\mathbf {p}) \right) \leqslant \frac{4K_{10.10}}{\varepsilon ^2}\exp \left( -\frac{\varepsilon ^2}{2^{14}K\sigma (\mathbf {p})}\right) \end{aligned}$$
(A.1)

for large n provided \(q_1\leqslant K\sigma (\mathbf {p})\). Similarly, by imitating the proof of (10.24), we can show that for any \(\varepsilon >0\) and integer \(r\geqslant 2\),

$$\begin{aligned} {{\mathrm{{\mathbb {P}}}}}\left( X \geqslant q_1+\varepsilon \sigma (\mathbf {p}) \right) \leqslant K'_{10.10}\left( \frac{K^r}{\varepsilon ^{2r}}\right) \sigma (\mathbf {p})^{r-1} \end{aligned}$$
(A.2)

if \(q_1\leqslant K\sigma (\mathbf {p})\) for some \(K>0\). Now, with notation as in the proof of Proposition 10.9,

$$\begin{aligned}&{{\mathrm{{\mathbb {P}}}}}\left( \left| \frac{{\mathscr {G}}(v)}{2}-F^{{{\mathrm{exc}}}, \mathbf {p}}(e(v)) \right| \geqslant \varepsilon \sigma (\mathbf {p}),\ {\mathscr {G}}(v)\leqslant K\sigma (\mathbf {p})\right) \nonumber \\&\quad \leqslant {{\mathrm{{\mathbb {P}}}}}\left( \left| \frac{{\mathscr {G}}(v)}{2}-\sum \nolimits _1 p_j\mathbbm {1}\left\{ i\leadsto j,\ U_{ij}\leqslant V_i\right\} \right| \geqslant \frac{\varepsilon \sigma (\mathbf {p})}{2},\ {\mathscr {G}}(v)\leqslant K\sigma (\mathbf {p}) \right) \nonumber \\&\qquad +{{\mathrm{{\mathbb {P}}}}}\left( \left| F^{{{\mathrm{exc}}}, \mathbf {p}}(e(v))-\sum \nolimits _1 p_j\mathbbm {1}\left\{ i\leadsto j,\ U_{ij}\leqslant V_i\right\} \right| \geqslant \frac{\varepsilon \sigma (\mathbf {p})}{2}\right) \nonumber \\&\quad =: Z_1+Z_2. \end{aligned}$$
(A.3)

Under Assumption 7.1, \(p_{\max }^{3/4}\leqslant \sigma (\mathbf {p})^{9/8}\leqslant \varepsilon \sigma (\mathbf {p})/2\) for large n. In view of (10.37) and (10.40),

$$\begin{aligned} Z_2\leqslant mK_{10.10}\exp \left( -p_{\max }^{-1/4}\right) . \end{aligned}$$
(A.4)

As in the proof of Proposition 10.9, we use the following bound for \(Z_1\):

$$\begin{aligned} Z_1&\leqslant {{\mathrm{{\mathbb {P}}}}}\left( \left| {\mathscr {G}}(v)-\sum \nolimits _1 p_j\mathbbm {1}\left\{ i\leadsto j\right\} \right| \geqslant \frac{\varepsilon \sigma (\mathbf {p})}{2},\ {\mathscr {G}}(v)\leqslant K\sigma (\mathbf {p}) \right) \nonumber \\&\quad +{{\mathrm{{\mathbb {P}}}}}\left( \left| \sum \nolimits _1 \frac{p_j}{2}\mathbbm {1}\left\{ i\leadsto j\right\} -\sum \nolimits _1 p_j\mathbbm {1}\left\{ i\leadsto j,\ U_{ij}\leqslant V_i\right\} \right| \geqslant \frac{\varepsilon \sigma (\mathbf {p})}{4},\ {\mathscr {G}}(v)\leqslant K\sigma (\mathbf {p}),\right. \nonumber \\&\quad \times \left. \sum \nolimits _1 p_j\mathbbm {1}\left\{ i\leadsto j\right\} \leqslant {\mathscr {G}}(v)+\frac{\varepsilon \sigma (\mathbf {p})}{2},\ \max _{i\in \mathbb {B}(v)}\sum _j p_j\mathbbm {1}\left\{ i\leadsto j\right\} \leqslant p_{\max }^{3/4} \right) \nonumber \\&\quad +{{\mathrm{{\mathbb {P}}}}}\left( \max _{i\in \mathbb {B}(v)}\sum _j p_j\mathbbm {1}\left\{ i\leadsto j\right\} \geqslant p_{\max }^{3/4} \right) =:Z_{11}+Z_{12}+Z_{13}. \end{aligned}$$
(A.5)

From (A.1) and (A.2), for every \(r\geqslant 2\), there exists \(C_r>0\) such that

$$\begin{aligned} Z_{11}&\leqslant \max _{\begin{array}{c} A\subset [m]{\setminus }\left\{ v\right\} :\\ p(A)\leqslant K\sigma (\mathbf {p}) \end{array}} {{\mathrm{{\mathbb {P}}}}}\left( \left| {\mathscr {G}}(v)-\sum \nolimits _1 p_j\mathbbm {1}\left\{ i\leadsto j\right\} \right| \geqslant \frac{\varepsilon \sigma (\mathbf {p})}{2}\ \bigg |\ \mathbb {A}(v)=A\right) \nonumber \\&\leqslant C_r\left( \frac{K^r}{\varepsilon ^{2r}}\right) \sigma (\mathbf {p})^{r-1} \end{aligned}$$
(A.6)

for sufficiently large n. Similarly, conditioning on \(\mathbb {A}(v)\) and the children of \(\mathbb {B}(v)\) and using Lemma 10.11,

$$\begin{aligned} Z_{12}&\leqslant K_{10.11}\left( \frac{4}{\varepsilon \sigma (\mathbf {p})}\right) ^{2r}p_{\max }^{3r/4} \left[ \left( K+\frac{\varepsilon }{2}\right) \sigma (\mathbf {p})\right] ^r \leqslant K_{10.11}\left[ \frac{16}{\varepsilon ^2}\left( K+\frac{\varepsilon }{2}\right) \right] ^{r}\sigma (\mathbf {p})^{r/8}, \end{aligned}$$
(A.7)

where the last step uses the bound \(p_{\max }\leqslant \sigma (\mathbf {p})^{3/2}\) for large n. Finally, from (10.40),

$$\begin{aligned} Z_{13}\leqslant mK_{10.12}\exp \left( -p_{\max }^{-1/4}\right) . \end{aligned}$$
(A.8)

Combining (A.3)–(A.8) and using a union bound, we conclude that under Assumption 7.1,

$$\begin{aligned} {{\mathrm{{\mathbb {P}}}}}\left( \left| \frac{{\mathscr {G}}(v)}{2}-F^{{{\mathrm{exc}}}, \mathbf {p}}(e(v)) \right| \geqslant \varepsilon \sigma (\mathbf {p})\hbox { for some }v\hbox { with }{\mathscr {G}}(v)\leqslant K\sigma (\mathbf {p})\right) =o(1). \end{aligned}$$

This shows that [8, Equation (46)] remains valid under Assumption 7.1. Now the claim follows from the arguments given after Equation (46) in [8]. \(\square \)

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Bhamidi, S., Sen, S. & Wang, X. Continuum limit of critical inhomogeneous random graphs. Probab. Theory Relat. Fields 169, 565–641 (2017). https://doi.org/10.1007/s00440-016-0737-x

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Keywords

  • Multiplicative coalescent
  • Continuum random tree
  • Critical random graphs
  • Branching processes
  • \(\mathbf {p}\)-trees
  • Scaling limits

Mathematics Subject Classification

  • Primary 60C05
  • 05C80