# Inhomogeneous Long-Range Percolation for Real-Life Network Modeling

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## Abstract

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## 1. Introduction

- Distant particles are typically connected by very few links, i.e., although there are possibly a lot of particles in the network, any two particles are typically connected through only a few other particles. This is called the “small-world effect”. For example, there is the observation that most particles in real-life networks are connected by at most six links, see also [9]. For the Facebook network with 721 million users, where there is a link between two users if they are “friends” on Facebook, the average number of minimal links that connect any two users is around $4.5$, while around $99\%$ of all users are connected by at most six links, see [10]. For the movie actor network, where there is a link between two actors if they appeared in the same film, the average number of minimal links that connect any two actors is also around $4.5$, while the number of actors in the network is over two hundred thousand. See [7] for more examples.
- Linked particles tend to have common friends, which is called the “clustering property”. For instance, if x is friend of both y and z, then it is likely that y and z are also friends. As an example, [10] discovers the following in the Facebook network: given a user with 100 friends, about $14\%$ of the possible friendships among his friends exist.
- The degree distribution, that is, the distribution of the number of links of a given particle, is heavy-tailed, i.e., its survival probability has a power law decay. It is observed that in real-life networks the (power law) tail parameter τ is often between 1 and 2. For instance, for the movie actor network τ is estimated to be around $1.3$. For more explicit examples we refer to [7,8].

**Organization of this article.**In Section 2, we describe the model assumptions and notations. We also state the conditions that are required for a non-trivial phase transition. In Section 3, we state the main results of the article. Namely, we show the continuity of the percolation function in Theorem 5 which is based on a finite box estimate stated in Theorem 6. We also complement the picture about graph distances of [12], see Theorem 8 below. In Section 4, we discuss a concrete financial application of the model, compare the results to homogeneous long-range percolation model results and we discuss open problems. Finally, we provide all proofs of our results in Section 5.

## 2. Model Assumptions and Phase Transition Picture

**Theorem 1**(upper bounds) Fix $d\ge 1$. Assume $min\{\alpha ,\beta \alpha \}>d$.

- (a)
- If $d\ge 2$, then ${\lambda}_{c}<\infty $.
- (b)
- If $d=1$ and $\alpha \in (1,2]$, then ${\lambda}_{c}<\infty $.
- (c)
- If $d=1$ and $min\{\alpha ,\beta \alpha \}>2$, then ${\lambda}_{c}=\infty $.

**Theorem 2**(lower bounds) Fix $d\ge 1$. Assume $min\{\alpha ,\beta \alpha \}>d$.

- (a)
- If $\tau =\beta \alpha /d<2$, then ${\lambda}_{c}=0$.
- (b)
- If $\tau =\beta \alpha /d>2$, then ${\lambda}_{c}>0$.

## 3. Main Results

#### 3.1. Continuity of Percolation Probability

**Theorem 3**Assume $min\{\alpha ,\beta \alpha \}>d$ and $\alpha \in (d,2d)$. Choose $\lambda \in (0,\infty )$ with $\theta (\lambda ,\alpha )>0$. There exist ${\lambda}^{\prime}\in (0,\lambda )$ and ${\alpha}^{\prime}\in (\alpha ,2d)$ such that

**Corollary 4**Assume $\alpha \in (d,2d)$ and $\tau =\beta \alpha /d>2$. There is no infinite cluster $\mathcal{C}$ at criticality ${\lambda}_{c}>0$.

**Theorem 5**For $min\{\alpha ,\beta \alpha \}>d$ and $\alpha \in (d,2d)$, the percolation probability $\lambda \mapsto \theta (\lambda ,\alpha )$ is continuous.

#### 3.2. Percolation on Finite Boxes

**Theorem 6**Assume $min\{\alpha ,\beta \alpha \}>d$ and $\alpha \in (d,2d)$. Choose $\lambda \in (0,\infty )$ with $\theta (\lambda ,\alpha )>0$. For each ${\alpha}^{\prime}\in (\alpha ,2d)$ there exist $\rho >0$ and ${N}_{0}<\infty $ such that for all $n\ge {N}_{0}$ we have

**Corollary 7**Under the assumptions of Theorem 6 we have the following.

- (i)
- There exists $\rho >0$ such that for any $x\in {\mathbb{Z}}^{d}$$$\underset{n\to \infty}{lim}\mathbb{P}\left[|{\mathcal{C}}_{n}\left(x\right)|\ge \rho |{\Lambda}_{n}\left(x\right)\left|\right|x\in \mathcal{C}\right]=1$$
- (ii)
- For any ${\alpha}^{\prime}\in (\alpha ,2d)$ there exist $\rho >0$ and ${\ell}_{0}$ such that for any ℓ and n with ${\ell}_{0}\le \ell \le n/{\ell}_{0}$$$\mathbb{P}\left[|{\mathcal{D}}_{n}^{(\rho ,\ell )}|\ge \rho |{\Lambda}_{n}|\right]\ge 1-{e}^{-\rho {n}^{2d-{\alpha}^{\prime}}}$$

#### 3.3. Graph Distances

**Theorem 8**Assume $min\{\alpha ,\beta \alpha \}>d$.

- (a)
- (infinite variance of degree distribution). Assume $\tau =\beta \alpha /d<2$. For any $\lambda >{\lambda}_{c}=0$ there exists ${\eta}_{1}>0$ such that for every $\u03f5>0$$$\underset{\left|x\right|\to \infty}{lim}\mathbb{P}\left[\left.{\eta}_{1}\le \frac{d(0,x)}{loglog\left|x\right|}\le (1+\u03f5)\frac{2}{|log(\beta \alpha /d-1\left)\right|}\right|0,x\in \mathcal{C}\right]=1$$
- (b1)
- (finite variance of degree distribution case 1). Assume $\tau =\beta \alpha /d>2$ and $\alpha \in (d,2d)$. For any $\lambda >{\lambda}_{c}$ and any $\u03f5>0$$$\underset{\left|x\right|\to \infty}{lim}\mathbb{P}\left[\left.1-\epsilon \le \frac{logd(0,x)}{loglog\left|x\right|}\le (1+\u03f5)\frac{log2}{log(2d/\alpha )}\right|0,x\in \mathcal{C}\right]=1$$
- (b2)
- (finite variance of degree distribution case 2). Assume $min\{\alpha ,\beta \alpha \}>2d$. There exists ${\eta}_{2}>0$ such that$$\underset{\left|x\right|\to \infty}{lim}\mathbb{P}\left[{\eta}_{2}<\frac{d(0,x)}{\left|x\right|}\right]=1$$

## 4. Example and Discussion

## 5. Proofs

#### 5.1. Bounds on Percolation on Finite Boxes

**Lemma 9**Assume $min\{\alpha ,\beta \alpha \}>d$ and $\alpha \in (d,2d)$. Choose $\lambda \in (0,\infty )$ with $\theta (\lambda ,\alpha )>0$ and let ${\alpha}^{\prime}\in [\alpha ,2d)$. For every $\epsilon \in (0,1)$ and $\rho >0$ there exists ${N}_{0}\ge 1$ such that for all $m\ge {N}_{0}$

**Sketch of proof of Lemma 9.**This lemma corresponds to Lemma 2.3 of [22] in our model. Its proof is based on renormalization arguments which only depend on the fact that $\alpha \in (d,2d)$ and that the edge probabilities are bounded from below by $1-exp(-\lambda |x-y{|}^{-\alpha})$ for any $x,y\in {\mathbb{Z}}^{d}$. Using that ${W}_{x}\ge 1$ for all $x\in {\mathbb{Z}}^{d}$, a.s., we see by stochastic dominance that the renormalization holds also true for our model. Renormalization shows that for m sufficiently large, the probability of {${B}_{m}$ contains at least a positive fraction of ${m}^{d}$ vertices that are connected within a fixed enlargement of ${B}_{m}$} is bounded by a multiple of the probability of the same event but on a much smaller scale. To bound the latter probability we then use the fact that the model is percolating, and from this we can conclude Lemma 9. We skip the details of the proof of Lemma 9 and refer to the proof of Lemma 2.3 of [27] for the details, in particular, the bound on ${\psi}_{n}$ in our homogeneous percolation model (see proof of Lemma 2.3 in [27]) also applies to the inhomogeneous percolation model.

**Lemma 10**

- (a)
- Assume $min\{\alpha ,\beta \alpha \}>d$ and $\alpha \in (d,2d)$. Choose $\lambda \in (0,\infty )$ such that $\theta (\lambda ,\alpha )>0$. For each $\xi <\infty $ and $r\in (0,1)$ there exist $m<\infty $ and an integer $\delta >0$ such that$$\begin{array}{ccc}& & \mathbb{P}\left[|{C}_{m}\left(x\right)|<\delta |{B}_{m}\left(x\right)|\right]\le 1-r,\hfill \\ & & \mathbb{P}\left.[{B}_{m}\left(x\right)\iff {B}_{m}\left(y\right)\left|\right|{C}_{m}\left(x\right)|\ge \delta |{B}_{m}\left(x\right)|,\phantom{\rule{3.33333pt}{0ex}}|{C}_{m}\left(y\right)|\ge \delta |{B}_{m}\left(y\right)|\right]\ge 1-{e}^{-\xi {\left(\frac{|x-y|}{m}\right)}^{-\alpha}},\hfill \end{array}$$
- (b)
- [Lemma 3.6, [11]] Let $d\ge 1$ and consider the site-bond percolation model on ${\mathbb{Z}}^{d}$ with sites being alive with probability $r\in [0,1]$ and sites $x,y\in {\mathbb{Z}}^{d}$ are attached with probability ${\tilde{p}}_{x,y}={1-exp(-\xi |x-y|}^{-\alpha})$ where $\alpha \in (d,2d)$ and $\xi \ge 0$. Let $|{\tilde{\mathcal{C}}}_{N}|$ be the size of the largest attached cluster ${\tilde{\mathcal{C}}}_{N}$ of living sites in box ${B}_{N}$. For each ${\alpha}^{\prime}\in (\alpha ,2d)$ there exist ${N}_{0}\ge 1$, $\nu >0$ and ${\xi}_{0}<\infty $ such that$${\mathbb{P}}_{\xi ,r}\left[|{\tilde{\mathcal{C}}}_{N}|\ge \nu |{B}_{N}|\right]\ge 1-{e}^{-\nu \xi {N}^{2d-{\alpha}^{\prime}}}$$

**Proof of Lemma 10 (a).**We adapt the proof of Lemma 3.5 of [11] to our model. Fix $r\in (0,1)$ and $\xi <\infty $. Choose $\rho >0$ such that

**Proof of Theorem 6.**The proof follows as in Theorem 3.2 of [11], we briefly sketch the main argument. Choose the constants ${N}_{0}\ge 1$, $\nu >0$, $\xi >{\xi}_{0}$, $r\ge 1-{e}^{-\nu \xi}$ and $\delta >0$ as in Lemma 10, and note that it is sufficient to prove the theorem for $L=mN$, where $N\ge {N}_{0}$ and m is chosen (fixed) as in Lemma 10 (a). In this set up ${B}_{L}$ can be viewed as a disjoint union of ${B}_{m}\left(x\right)$ for $x\in (m{\mathbb{Z}}^{d}\cap {B}_{L})$. There are ${N}^{d}$ such disjoint boxes. We call ${B}_{m}\left(x\right)$ alive if $|{C}_{m}\left(x\right)|\ge \delta |{B}_{m}|$ and we say that disjoint ${B}_{m}\left(x\right)$ and ${B}_{m}\left(y\right)$ are pairwise attached if their largest connected components ${C}_{m}\left(x\right)$ and ${C}_{m}\left(y\right)$ share an occupied edge. Part (a) of Lemma 10 provides that ${B}_{m}\left(x\right)$ is alive with probability exceeding r and ${B}_{m}\left(x\right)$ and ${B}_{m}\left(y\right)$ are pairwise attached with probability exceeding ${\tilde{p}}_{x,y}$ for living boxes ${B}_{m}\left(x\right)$ and ${B}_{m}\left(y\right)$ with $x,y\in m{\mathbb{Z}}^{d}$ (note that in this site-bond percolation model the attachedness property is only considered between living vertices because these form the clusters). For any $N\ge {N}_{0}$, let ${A}_{N,m}$ be the event that box ${B}_{L}$ contains a connected component formed by attaching at least $\nu |{B}_{N}|$ of the living boxes. On event ${A}_{N,m}$ we have for the largest connected component in ${B}_{L}$

#### 5.2. Proof of Continuity of the Percolation Probability

**Proof of Theorem 3.**Note that $min\{\alpha ,\beta \alpha \}>d$ and $\alpha \in (d,2d)$ imply that ${\lambda}_{c}<\infty $. Therefore, there exists $\lambda \in ({\lambda}_{c},\infty )$ with $\theta =\theta (\lambda ,\alpha )>0$. For these choices of $\lambda >0$ we have a unique infinite cluster $\mathcal{C}$, a.s., and we can apply Lemma 9.

**Proof of Theorem 5.**We need to modify Proposition 1.3 of [23] because in our model, edges are not occupied independently induced by the random choices of weights ${\left({W}_{x}\right)}_{x\in {\mathbb{Z}}^{d}}$. (i) From Theorem 3 it follows that $\theta (\lambda ,\alpha )=0$ for all $\lambda \in (0,{\lambda}_{c}]$, which proves continuity of $\lambda \mapsto \theta (\lambda ,\alpha )$ on $(0,{\lambda}_{c}]$.

#### 5.3. Proofs of the Graph Distances

**Proposition 11**Let $\alpha \in (d,2d)$ and $\tau =\beta \alpha /d>2$ and $\lambda >{\lambda}_{c}$. For each ${\Delta}^{\prime}>\Delta =\Delta (\alpha ,2d)=log2/log(2d/\alpha )$ and each $\epsilon >0$, there exists ${N}_{0}<\infty $ such that

**Sketch of proof of Proposition 11.**We only sketch the proof because it is almost identical to the one in [11]. Definition 1 and Figure 1 of [11] define for $x,y\in {\mathbb{Z}}^{d}$ a hierarchy of depth $m\in \mathbb{N}$ connecting x and y as the following collection of vertices:

- (1)
- ${z}_{0}=x$ and ${z}_{1}=y$,
- (2)
- ${z}_{\sigma 00}={z}_{\sigma 0}$ and ${z}_{\sigma 11}={z}_{\sigma 1}$ for all $k=0,\dots ,m-2$ and $\sigma \in {\{0,1\}}^{k}$,
- (3)
- for all $k=0,\dots ,m-2$ and and $\sigma \in {\{0,1\}}^{k}$ such that ${z}_{\sigma 01}\ne {z}_{\sigma 10}$ the edge between ${z}_{\sigma 01}$ and ${z}_{\sigma 10}$ is occupied,
- (4)
- each bond $({z}_{\sigma 01},{z}_{\sigma 10})$ specified in (3) appears only once in ${\mathcal{H}}_{k}(x,y)$.

**Definition 12**(good n-stage boxes) Choose $n\in {\mathbb{N}}_{0}$ and $x\in {\mathbb{Z}}^{d}$ fixed.

- 0-stage box ${B}_{{m}_{0}}\left(x\right)$ is good under a given edge configuration if there is no occupied edge in ${B}_{{m}_{0}}\left(x\right)$ with size larger than ${m}_{0}/100$.
- n-stage box ${B}_{{m}_{n}}\left(x\right)$, $n\ge 1$, is good under a given edge configuration if for all $j\in {\{-1,0,1\}}^{d}$
- (a)
- there is no occupied edge in ${B}_{{m}_{n}}\left(x+j\frac{{m}_{n-1}}{2}\right)$ with size larger than ${m}_{n-1}/100$; and
- (b)
- among the children of ${B}_{{m}_{n}}\left(x+j\frac{{m}_{n-1}}{2}\right)$ there are at most ${3}^{d}$ that are not good.

**Lemma 13**Assume $min\{\alpha ,\beta \alpha \}>d$. For all $\delta \in (0,\alpha (\beta \wedge 1)-d)$ there exist ${t}_{0}\ge 1$ and a constant ${c}_{1}>0$ such that for all $t\ge {t}_{0}$ and all $s\ge 1$,

**Proof of Lemma 13.**Let ${W}_{1}$ and ${W}_{2}$ be two independent random variables each having a Pareto distribution with parameters $\theta =1$ and $\beta >0$. For $u\ge 1$ we have, using integration by parts in the first step,

**Lemma 14**Assume $min\{\alpha ,\beta \alpha \}>2d$. For ${a}_{n}={n}^{2}$, $n\ge 1$, and ${a}_{0}$ sufficiently large we have

**Proof of Lemma 14.**We prove by induction that ${\psi}_{n}=\mathbb{P}\left[{B}_{{m}_{n}}\left(0\right)\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{not}\phantom{\rule{4.pt}{0ex}}\text{good}\right]$ is summable. Choose $\delta \in (0,\alpha (\beta \wedge 1)-2d)$ and set $\gamma =min\{\alpha ,\beta \alpha \}-2d-\delta >0$. For ${m}_{0}$ sufficiently large we obtain by Lemma 13,

**Lemma 15**(Proposition 3 of [13]) Choose ${a}_{n}={n}^{2}$ for $n\ge 1$. There exists a constant ${c}_{3}>0$ such that for every n sufficiently large, if for every $j\in {\{-1,0,1\}}^{d}$ the n-stage box ${B}_{{m}_{n}}\left(0+j\frac{{m}_{n}}{2}\right)$ is good and for every $l>n$ the l-stage boxes ${\widehat{B}}_{{m}_{l}}$ centered at ${B}_{{m}_{n}}\left(0\right)$ are good, then if $x,y\in {B}_{{m}_{n}}\left(0\right)$ satisfy $|x-y|>{m}_{n}/8$ then $d(x,y)\ge {c}_{3}|x-y|$.

**Proof of Theorem 8 (b2).**Lemma 14 says that, a.s., the l-stage boxes ${\widehat{B}}_{{m}_{l}}$ are eventually good for all $l\ge n$. Moreover, from Lemma 15 we obtain the linearity in the distance for these good boxes which says that, a.s., for n sufficiently large and $\left|x\right|>{m}_{n}/8$ we have $d(0,x)\ge {c}_{3}\left|x\right|$.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Deprez, P.; Hazra, R.S.; Wüthrich, M.V.
Inhomogeneous Long-Range Percolation for Real-Life Network Modeling. *Risks* **2015**, *3*, 1-23.
https://doi.org/10.3390/risks3010001

**AMA Style**

Deprez P, Hazra RS, Wüthrich MV.
Inhomogeneous Long-Range Percolation for Real-Life Network Modeling. *Risks*. 2015; 3(1):1-23.
https://doi.org/10.3390/risks3010001

**Chicago/Turabian Style**

Deprez, Philippe, Rajat Subhra Hazra, and Mario V. Wüthrich.
2015. "Inhomogeneous Long-Range Percolation for Real-Life Network Modeling" *Risks* 3, no. 1: 1-23.
https://doi.org/10.3390/risks3010001