Efficiently Estimating Erdos-Renyi Graphs with Node Differential Privacy

TL;DR

This paper presents a simple, efficient, and nearly optimal node-differentially-private algorithm for estimating the parameter p in Erdos-Renyi graphs, with broader applicability to graphs with concentrated degree distributions.

Contribution

It introduces a computationally efficient, private estimation algorithm that nearly matches the best possible accuracy of exponential-time methods for Erdos-Renyi graphs.

Findings

Achieves near-optimal accuracy in estimating p in G(n,p)

Provides an efficient private algorithm for estimating edge-density

Extends to graphs with concentrated degree distributions

Abstract

We give a simple, computationally efficient, and node-differentially-private algorithm for estimating the parameter of an Erdos-Renyi graph---that is, estimating p in a G(n,p)---with near-optimal accuracy. Our algorithm nearly matches the information-theoretically optimal exponential-time algorithm for the same problem due to Borgs et al. (FOCS 2018). More generally, we give an optimal, computationally efficient, private algorithm for estimating the edge-density of any graph whose degree distribution is concentrated on a small interval.

Equations

$${\underset{G}{\mathbb{E}}\mathopen{}\mathclose{{}\left[(p-p_{G})^{2}}\right]}=\frac{p(1-p)}{\binom{n}{2}}.$$

$${\underset{G}{\mathbb{E}}\mathopen{}\mathclose{{}\left[(p-\mathcal{A}_{\textit{na\"{i}ve}}(G))^{2}}\right]}=\Theta\mathopen{}\mathclose{{}\left(\frac{1}{{\varepsilon}^{2}n^{2}}}\right)$$

$${\mathbb{E}\mathopen{}\mathclose{{}\left[(p-\mathcal{A}_{\textit{bcsz}}(G))^{2}}\right]}=\underbrace{\frac{p(1-p)}{\binom{n}{2}}}_{\textrm{non-private error}}+~{}\underbrace{\tilde{O}\mathopen{}\mathclose{{}\left(\frac{p}{{\varepsilon}^{2}n^{3}}}\right)}_{\textrm{overhead due to privacy}}$$

$${\underset{G,A}{\mathbb{E}}\mathopen{}\mathclose{{}\left[(p-\mathcal{A}(G))^{2}}\right]}=\underbrace{\frac{p(1-p)}{\binom{n}{2}}}_{\textrm{non-private error}}+~{}~{}\underbrace{\tilde{O}\mathopen{}\mathclose{{}\left(\frac{p}{{\varepsilon}^{2}n^{3}}+\frac{1}{{\varepsilon}^{4}n^{4}}}\right)}_{\textrm{overhead due to privacy}}$$

$${\underset{\mathcal{A}}{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(p_{G}-\mathcal{A}(G)}\right)^{2}}\right]}=O\mathopen{}\mathclose{{}\left(\frac{k^{2}}{{\varepsilon}^{2}n^{4}}+\frac{1}{{\varepsilon}^{4}n^{4}}}\right)$$

$${\underset{A}{\mathbb{E}}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(p_{G}-\mathcal{A}(G)}\right)^{2}}\right]}=\Omega\mathopen{}\mathclose{{}\left(\frac{k^{2}}{{\varepsilon}^{2}n^{4}}+\frac{1}{{\varepsilon}^{4}n^{4}}}\right)$$

$${\mathbb{P}\mathopen{}\mathclose{{}\left[\mathcal{A}(G)\in R}\right]}\leq e^{{\varepsilon}}\cdot{\mathbb{P}\mathopen{}\mathclose{{}\left[\mathcal{A}(G^{\prime})\in R}\right]}+\delta$$

$$G=G_0\sim G_1\dots G_{c-1}\sim \dots G_c=G^{\prime }$$

$${\mathbb{P}\mathopen{}\mathclose{{}\left[\mathcal{A}(G)\in R}\right]}\leq e^{c{\varepsilon}}{\mathbb{P}\mathopen{}\mathclose{{}\left[\mathcal{A}(G^{\prime})\in R}\right]}+ce^{c{\varepsilon}}\delta$$

$${LS}_f(G)=\underset{G^{\prime }:G^{\prime }\sim G}{\max }|f(G)-f(G^{\prime })|$$

$${GS}_f=\underset{G}{\max }{LS}_f(G)=\underset{G^{\prime }\sim G}{\max }|f(G)-f(G^{\prime })|$$

$$\mathcal{A}(G)=f(G)+\frac{{GS}_f}{\epsilon }\cdot Z,$$

$${\underset{\mathcal{A}}{\mathbb{P}}\mathopen{}\mathclose{{}\left[|\mathcal{A}(G)-f(G)|\geq t\cdot\mathit{GS}_{f}/{\varepsilon}}\right]}\leq\exp(-t).$$

$$\mathcal{A}(G)=f(G)+\frac{S(G)}{\epsilon }\cdot Z,$$

$${\mathsf{wt}}_G(v)=\{\begin{array}{ll}1& \text{if }{t}_v=0\\ 1-\beta t_v& \text{if }{t}_v\in (0,1/\beta ]\\ 0& \text{otherwise}.\end{array}$$

$${\mathsf{wt}}_G(e)=\min ({\mathsf{wt}}_G(u),{\mathsf{wt}}_G(v))$$

$${\mathsf{val}}_G(e)={\mathsf{wt}}_G(e)\cdot x_e+(1-{\mathsf{wt}}_G(e))\cdot p_G$$

$$f(G)=\sum _{u,v\in V}{\mathsf{val}}_G(\{u,v\}),$$

$$f_u(G):=\sum _{v\ne u}{\mathsf{val}}_G(\{u,v\})=\sum _{v\ne u}{\mathsf{wt}}_G(\{u,v\})\cdot x_{\{u,v\}}+\sum _{v\ne u}(1-{\mathsf{wt}}_G(\{u,v\}))p_G.$$

$$\sum _{v\ne u}{\mathsf{wt}}_G(u)x_{\{u,v\}}\pm k_G{\mathsf{wt}}_G(u)={\mathrm{deg}}_G(u){\mathsf{wt}}_G(u)\pm k_G{\mathsf{wt}}_G(u)$$

$$p_{G}\cdot\mathopen{}\mathclose{{}\left((n-1)-\sum_{v\neq u}\mathsf{wt}_{G}(\{u,v\})}\right)=\bar{d}_{G}-\bar{d}_{G}\mathsf{wt}_{G}(u)\pm p_{G}k_{G}\mathsf{wt}_{G}(u).$$

$$f_u(G)={\bar{d}}_G+({\mathrm{deg}}_G(u)-{\bar{d}}_G){\mathsf{wt}}_G(u)\pm 2k_G{\mathsf{wt}}_G(u).$$

$$|f_u(G)-f_u(G^{\prime })|\le 7+3\beta (k^{\prime \prime }+3)+9\beta +6\beta k_G=O(1+\beta (k_G+k^{\ast })).$$

$$f_{v^{\ast }}(G)=\sum _{v\ne v^{\ast }}{\mathsf{val}}_G(\{v^{\ast },v\})=(n-1)p_G={\bar{d}}_G.$$

$$f_{v^{\ast }}(G)$$

$$={\mathrm{deg}}_G(v^{\ast })\pm k_G$$

$$={\bar{d}}_G\pm k^{\prime }\pm k_G.$$

$$f_{v^{\ast }}(G)$$

$$={\bar{d}}_G+({\mathrm{deg}}_G(v^{\ast })-{\bar{d}}_G){\mathsf{wt}}_G(v^{\ast })\pm k_G{\mathsf{wt}}_G(v^{\ast })$$

$$={\bar{d}}_G\pm ({\mathrm{deg}}_G(v^{\ast })-{\bar{d}}_G)\pm k_G,$$

$$g(k_G,k^{\ast },\beta )=C((k_G+k^{\ast })(1+\beta k_G)+\frac{1}{\beta })$$

$$S(G)=\underset{\ell \ge 0}{\max }{e}^{-\ell \beta }g(k_G+\ell ,k^{\ast },\beta ).$$

Full text

1 Introduction

Network data modeling individuals and relationships between individuals are increasingly central in data science. However, while there is a highly successful literature on differentially private statistical estimation for traditional iid data, the literature on estimating network models is far less well developed.

Early work on private network data focused on edge-differential-privacy, in which the algorithm is required to “hide” the presence or absence of a single edge in the graph (see, e.g. [NRS07, HLMJ09, KRSY14, GRU12, BBDS12, XCT14, KS16], and many others). A more desirable notion of privacy is node-differential privacy (node-DP), which requires the algorithm to hide the presence or absence of an arbitrary set of edges incident on a single node. Although node-DP is difficult to achieve without compromising accuracy, the beautiful works of Blocki et al. [BBDS13] and Kasiviswanathan et al. [KNRS13] showed how to design accurate node-DP estimators for many interesting graph statistics via Lipschitz extensions. However, many of the known constructions of Lipschitz extensions require exponential running time, and constructions of computationally efficient Lipschitz extensions [RS16, CD18, CKM*+*19] lag behind. As a result, even for estimating very simple graph models, there are large gaps in accuracy between the best known computationally efficient algorithms and the information theoretically optimal algorithms.

In this work we focus on what is arguably the simplest model of network data, the Erdős-Rényi graph. In this model, denoted $G(n,p)$, we are given a number of nodes $n$ and a parameter $p\in[0,1]$, and we sample an $n$-node graph $G$ by independently including each edge $(i,j)$ for $1\leq i<j\leq n$ with probability $p$. The goal is to design a node-DP algorithm that takes as input a graph $G\sim G(n,p)$ and outputs an estimate $\hat{p}$ of the edge density parameter $p$.

Surprisingly, until an elegant recent work of Borgs et al. [BCSZ18], the optimal accuracy for estimating the parameter $p$ in a $G(n,p)$ via node-DP algorithms was unknown. Although that work essentially resolved the optimal accuracy of node-DP algorithms, their construction is again based on generic Lipschitz extensions, and thus results in an exponential-time algorithm, and, in our opinion, gives little insight for how to construct an efficient estimator with similar accuracy.

The main contribution of this work is to give a simple, polynomial-time estimator for Erdős-Rényi graphs whose error very nearly matches that of Borgs et al.’s estimator, and indeed matches it in a wide range of parameters. We achieve this by giving a more general result, showing how to optimally estimate the edge-density of any graph whose degree distribution is concentrated in a small interval.

1.1 Background: Node-Private Algorithms for Erdős-Rényi Graphs

Without privacy, the optimal estimator is simply to output the edge-density $p_{G}=|E|/\binom{n}{2}$ of the realized graph $G\sim G(n,p)$, which guarantees

[TABLE]

The simplest way to achieve ${\varepsilon}$-node-DP is to add zero-mean noise to the edge-density with standard-deviation calibrated to its global-sensitivity, which is the amount that changing the neighborhood of a single node in a graph can change its edge-density. The global sensitivity of $p_{G}$ is $\Theta(1/n)$, and thus the resulting private algorithm $\mathcal{A}_{\textit{na\"{i}ve}}$ satisfies

[TABLE]

Note that this error is at least on the same order as the non-private error, and can asymptotically dominate the non-private error.

Borgs et al. [BCSZ18] gave an improved ${\varepsilon}$-node-DP algorithm such that, when both $p$ and ${\varepsilon}$ are $\gtrsim\log(n)/n$,

[TABLE]

What is remarkable about their algorithm is that, unless ${\varepsilon}$ is quite small (roughly ${\varepsilon}\lesssim n^{-1/2}$), the first term dominates the error, in which case privacy comes essentially for free. That is, the error of the private algorithm is only larger than that of the optimal non-private algorithm by a $1+o(1)$ factor. However, as we discussed above, this algorithm is not computationally efficient.

The only computationally efficient node-DP algorithms for computing the edge-density apply to graphs with small maximum degree [BBDS13, KNRS13, RS16], and thus do not give optimal estimators for Erdős-Rényi graphs unless $p$ is very small.

1.2 Our Results

Our main result is a computationally efficient estimator for Erdős-Rényi graphs.

Theorem 1.1** (Erdős-Rényi Graphs, Informal).**

There is an $O(n^{2})$-time ${\varepsilon}$-node-DP algorithm $\mathcal{A}$ such that for every $n$ and every $p\gtrsim 1/n$ if $G\sim G(n,p)$ then

[TABLE]

The error of Theorem 1.1 matches that of the exponential-time estimator of Borgs et al. [BCSZ18] up to the additive $\tilde{O}(1/{\varepsilon}^{4}n^{4})$ term, which is often not the dominant term in the overall error. In particular, the error of our estimator is still within a 1+o(1) factor of the optimal non-private error unless ${\varepsilon}$ or $p$ is quite small—for example, when $p$ is a constant and ${\varepsilon}\gtrsim n^{-1/2}$.

Our estimator actually approximates the edge density for a significantly more general class of graphs than merely Erdős-Rényi graphs. Specifically, Theorem 1.1 follows from a more general result for the family of concentrated-degree graphs. For $k\in\mathbb{N}$, define $\mathcal{G}_{n,k}$ to be the set of $n$-node graphs such that the degree of every node is between $\bar{d}-k$ and $\bar{d}+k$, where $\bar{d}=2|E|/n$ is the average degree of the graph.

Theorem 1.2** (Concentrated-Degree Graphs, Informal).**

For every $k\in\mathbb{N}$, there is an $O(n^{2})$-time ${\varepsilon}$-node-DP algorithm $\mathcal{A}$ such that for every $n$ and every $G\in\mathcal{G}_{n,k}$,

[TABLE]

where $p_{G}=|E|/\binom{n}{2}$ is the empirical edge density of $G$.

Theorem 1.1 follows from Theorem 1.2 by using the fact that for an Erdős-Rényi graph, with overwhelming probability the degree of every node lies in an interval of width $\tilde{O}(\sqrt{pn})$ around the average degree.

The main technical ingredient in Theorem 1.2 is to construct a low sensitivity estimator $f(G)$ for the number of edges. The first property we need is that when $G$ satisfies the concentrated degrees property, $f(G)$ equals the number of edges in $G$. The second property of the estimator we construct is that its smooth sensitivity [NRS07] is low on these graphs $G$. At a high level, the smooth sensitivity of $f$ at a graph $G$ is the most that changing the neighborhood of a small number of nodes in $G$ can change the value of $f(G)$. Once we have this property, it is sufficient to add noise to $f(G)$ calibrated to its smooth sensitivity. We construct $f$ by carefully reweighting edges that are incident on nodes that do not satisfy the concentrated-degree condition.

Finally, we are able to show that Theorem 1.2 is optimal for concentrated-degree graphs. In additional to being a natural class of graphs in its own right, this lower bound demonstrates that in order to improve Theorem 1.1 we will need techniques that are more specialized to Erdős-Rényi graphs.

Theorem 1.3** (Lower Bound, Informal).**

For every $n$ and $k$, and every ${\varepsilon}$-node-DP algorithm $A$, there is some $G\in\mathcal{G}_{n,k}$ such that

[TABLE]

The same bound applies to $({\varepsilon},\delta)$-node-DP algorithms with sufficiently small $\delta\lesssim{\varepsilon}$.

2 Preliminaries

Let $\mathcal{G}_{n}$ be the set of $n$-node graphs. We say that two graphs $G,G^{\prime}\in\mathcal{G}_{n}$ are node-adjacent, denoted $G\sim G^{\prime}$, if $G^{\prime}$ can be obtained by $G$ modifying the neighborhood of a single node $i$. That is, there exists a single node $i$ such that for every edge $e$ in the symmetric difference of $G$ and $G^{\prime}$, $e$ is incident on $i$. As is standard in the literature on differential privacy, we treat $n$ as a fixed quantity and define adjacency only for graphs with the same number of nodes. We could easily extend our definition of adjacency to include adding or deleting a single node itself.

Definition 2.1** (Differential Privacy [DMNS06]).**

A randomized algorithm $\mathcal{A}\colon\mathcal{G}_{n}\to\mathcal{R}$ is $({\varepsilon},\delta)$-node-differentially private if for every $G\sim G^{\prime}\in\mathcal{G}_{n}$ and every $R\subseteq\mathcal{R}$,

[TABLE]

If $\delta=0$ we will simply say that $\mathcal{A}$ is ${\varepsilon}$-node-differentially private. As we only consider node differential privacy in this work, we will frequently simply say that $\mathcal{A}$ satisfies differential privacy.

The next lemma is the basic composition property of differential privacy.

Lemma 2.2** (Composition [DMNS06]).**

If $\mathcal{A}_{1},\mathcal{A}_{2}\colon\mathcal{G}_{n}\to\mathcal{R}$ are each $({\varepsilon},\delta)$-node-differentially private algorithms, then the mechanism $\mathcal{A}(G)=(\mathcal{A}_{1}(G),\mathcal{A}_{2}(G))$ satisfies $(2{\varepsilon},2\delta)$-node-differential privacy. The same holds if $\mathcal{A}_{2}$ may depend on the output of $\mathcal{A}_{1}$.

We say that two graphs $G,G^{\prime}$ are at node distance $c$ if there exists a sequence of graphs

[TABLE]

The standard group privacy property of differential privacy yields the following guarantees for graphs at node distance $c>1$.

Lemma 2.3** (Group Privacy [DMNS06]).**

If $\mathcal{A}\colon\mathcal{G}_{n}\to\mathcal{R}$ is $({\varepsilon},\delta)$-node-differentially-private and $G,G^{\prime}$ are at node-distance $c$ then for every $R\subseteq\mathcal{R}$,

[TABLE]

Sensitivity and Basic DP Mechanisms. The main differentially private primitive we will use is smooth sensitivity [NRS07]. Let $f\colon\mathcal{G}_{n}\to\mathbb{R}$ be a real-valued function. For a graph $G\in\mathcal{G}_{n}$, we can define the local sensitivity of $f$ at $G$ to be

[TABLE]

and the global sensitivity of $f$ to be

[TABLE]

A basic result in differential privacy says that we can achieve privacy for any real-valued function $f$ by adding noise calibrated to the global sensitivity of $f$.

Theorem 2.4** (DP via Global Sensitivity [DMNS06]).**

Let $f:\mathcal{G}_{n}\to\mathbb{R}$ be any function. Then the algorithm

[TABLE]

where $Z$ is sampled from a standard Laplace distribution, satisfies $({\varepsilon},0)$-differential privacy.111The standard Laplace distribution $Z$ has {\mathbb{E}\mathopen{}\mathclose{{}\left[Z}\right]}=0,{\mathbb{E}\mathopen{}\mathclose{{}\left[Z^{2}}\right]}=2, and density $\mu(z)\propto e^{-|z|}$. Moreover, this mechanism satisfies {\underset{\mathcal{A}}{\mathbb{E}}\mathopen{}\mathclose{{}\left[(\mathcal{A}(G)-f(G))^{2}}\right]}=O(\mathit{GS}_{f}/{\varepsilon}), and for all $t>0$ we have that

[TABLE]

In many cases the global sensitivity of $f$ is too high, and we want to use a more refined mechanism that adds instance-dependent noise that is more comparable to the local sensitivity. This can be achieved via the smooth sensitivity framework of Nissim et al. [NRS07].

Definition 2.5** (Smooth Upper Bound [NRS07]).**

Let $f:\mathcal{G}_{n}\to\mathbb{R}$ be a real-valued function and $\beta>0$ be a parameter. A function $S:\mathcal{G}_{n}\to\mathbb{R}$ is a $\beta$-smooth upper bound on $\mathit{LS}_{f}$ if

for all $G\in\mathcal{G}_{n}$, $S(G)\geq LS_{f}(G)$, and 2. 2.

for all neighboring $G\sim G^{\prime}\in\mathcal{G}_{n}$, $S(G)\leq e^{\beta}\cdot S(G^{\prime})$.

The key result in smooth sensitivity is that we can achieve differential privacy by adding noise to $f(G)$ proportional to any smooth upper bound $S(G)$.

Theorem 2.6** (DP via Smooth Sensitivity [NRS07, BS19]).**

Let $f:\mathcal{G}_{n}\to\mathbb{R}$ be any function and $S$ be a $\beta$-smooth upper bound on the local sensitivity of $f$ for any $\beta\leq{\varepsilon}$. Then the algorithm

[TABLE]

*where $Z$ is sampled from a Student’s $t$-distribution with $3$ degrees of freedom, satisfies $(O({\varepsilon}),0)$-differential privacy.222The Student’s $t$-distribution with 3 degrees of freedom can be efficiently sampled by choosing $X,Y_{1},Y_{2},Y_{3}\sim\mathcal{N}(0,1)$ independently from a standard normal and returning $Z=X/\sqrt{Y_{1}^{2}+Y_{2}^{2}+Y_{3}^{2}}$. This distribution has {\mathbb{E}\mathopen{}\mathclose{{}\left[Z}\right]}=0 and {\mathbb{E}\mathopen{}\mathclose{{}\left[Z^{2}}\right]}=3, and its density is $\mu(z)\propto 1/(1+z^{2})^{2}$. Moreover, for any $G\in\mathcal{G}_{n}$, this algorithm satisfies {\underset{\mathcal{A}}{\mathbb{E}}\mathopen{}\mathclose{{}\left[(\mathcal{A}(G)-f(G))^{2}}\right]}=O(S(G)^{2}/{\varepsilon}^{2}). *

3 An Estimator for Concentrated-Degree Graphs

In this section we describe and analyze a node-differentially-private estimator for the edge density of a concentrated-degree graph.

3.1 The Estimator

In order to describe the estimator we introduce some key notation. The input to the estimator is a graph $G=(V,E)$ and a parameter $k^{*}$. Intuitively, $k^{*}$ should be an upper bound on the concentration parameter of the graph, although we obtain more general results when $k^{*}$ is not an upper bound, in case the user does not have an a priori upper bound on this quantity.

For a graph $G=(V,E)$, let $p_{G}=|E|/\binom{n}{2}$ be the empirical edge density of $G$, and let $\bar{d}_{G}=(n-1)p_{G}$ be the empirical average degree of $G$. Let $k_{G}$ be the smallest positive integer value such that at most $k_{G}$ vertices of $G$ have degrees differing from $\bar{d}_{G}$ by more than $k^{\prime}:=k^{*}+3k_{G}$. Define $I_{G}=[\bar{d}_{G}-k^{\prime},\bar{d}_{G}+k^{\prime}]$. For each vertex $v\in V$, let $t_{v}=\min\{|t|:\deg_{G}(v)\pm t\in I_{G}\}$ be the distance between $\deg_{G}(v)$ and the interval $I_{G}$, and define the weight $\mathsf{wt}_{G}(v)$ of $v$ as follows. For a parameter $\beta>0$ to be specified later, let

[TABLE]

That is, $\mathsf{wt}_{G}(v)=\max(0,1-\beta t_{v})$. For each pair of vertices $e=\{u,v\}$, define the weight $\mathsf{wt}_{G}(e)$ and value $\mathsf{val}_{G}(e)$ as follows. Let

[TABLE]

and let

[TABLE]

where $x_{e}$ denotes the indicator variable on whether $e\in E$. As above, define the function $f$ to be the total value of all pairs of vertices in the graph,

[TABLE]

where the sum is over unordered pairs of distinct vertices.

Once we construct this function $f$, we add noise to $f$ proportional to a $\beta$-smooth upper bound on the sensitivity of $f$, which we derive in this section. Pseudocode for our estimator is given in Algorithm 1.

3.2 Analysis using Smooth Sensitivity

We begin by bounding the local sensitivity $LS_{f}(G)$ of the function $f$ defined above.

Lemma 3.1**.**

$LS_{f}(G)=O((k_{G}+k^{*})(1+\beta k_{G})+\frac{1}{\beta})$.

Proof.

Consider any pair of graphs $G,G^{\prime}$ differing in only a single vertex $v^{*}$, and note that the empirical edge densities $p_{G}$ and $p_{G^{\prime}}$ can differ by at most $\frac{2}{n}<\frac{2}{n-1}$, so $\bar{d}_{G}$ and $\bar{d}_{G^{\prime}}$ can differ by at most $2$. Moreover, for any vertex $v\neq v^{*}$, the degree of $v$ can differ by at most $1$ between $G$ and $G^{\prime}$. Consequently, by the Triangle Inequality, for any $v\neq v^{*}$, $|\bar{d}_{G}-\deg_{G}(v)|$ can differ from $|\bar{d}_{G^{\prime}}-\deg_{G^{\prime}}(v)|$ by at most $3$ and $\mathsf{wt}_{G}(v)$ can differ from $\mathsf{wt}_{G^{\prime}}(v)$ by at most $3\beta$. It follows from the former statement that $k_{G}$ and $k_{G^{\prime}}$ differ by at most $1$.

Let $\mathsf{Far}_{G}$ denote the set of at most $k_{G}$ vertices whose degree differs from $\bar{d}_{G}$ by more than $k^{\prime}=k^{*}+3k_{G}$. For any vertices $u,v\notin\mathsf{Far}_{G}\cup\mathsf{Far}_{G^{\prime}}\cup\{v^{*}\}$, we have that $\mathsf{wt}_{G}(\{u,v\})=\mathsf{wt}_{G^{\prime}}(\{u,v\})=1$, and so $\mathsf{val}_{G}(\{u,v\})=\mathsf{val}_{G^{\prime}}(\{u,v\})$, since the edge $\{u,v\}$ is present in $G$ if and only if it is present in $G^{\prime}$.

Now consider edges $\{u,v\}$ such that $u,v\neq v^{*}$ but $u\in\mathsf{Far}_{G}\cup\mathsf{Far}_{G^{\prime}}$ (and $v$ may or may not be as well). If $\deg_{G}(u)\notin[\bar{d}_{G}-k^{\prime\prime},\bar{d}_{G}+k^{\prime\prime}]$ for $k^{\prime\prime}=k^{\prime}+1/\beta+3$, then $\mathsf{wt}_{G}(u)=\mathsf{wt}_{G^{\prime}}(u)=0$ and so $|\mathsf{val}_{G}(\{u,v\})-\mathsf{val}_{G^{\prime}}(\{u,v\})|=|p_{G}-p_{G^{\prime}}|\leq 2/n$. Otherwise, $\deg_{G}(u)\in[\bar{d}_{G}-k^{\prime\prime},\bar{d}_{G}+k^{\prime\prime}]$. We can break up the sum

[TABLE]

Since at most $k_{G}$ other vertices can have weight less than the weight of $u$, we can bound the first term by

[TABLE]

and the second term by

[TABLE]

so the total sum is bounded by

[TABLE]

Since $|\mathsf{wt}_{G}(u)-\mathsf{wt}_{G^{\prime}}(u)|\leq 3\beta$, it follows that

[TABLE]

Since there are at most $k_{G}+k_{G}^{\prime}\leq 2k_{G}+1$ vertices in $u\in\mathsf{Far}_{G}\cup\mathsf{Far}_{G^{\prime}}\setminus\{v^{*}\}$, the total difference in the terms of $f(G)$ and $f(G^{\prime})$ corresponding to such vertices is at most $O(k_{G}+\beta k_{G}(k_{G}+k^{*})).$ However, we are double-counting any edges between two vertices in $u\in\mathsf{Far}_{G}\cup\mathsf{Far}_{G^{\prime}}$; the number of such edges is $O(k_{G}^{2})$, and for any such edge $e$, $|\mathsf{val}_{G}(e)-\mathsf{val}_{G^{\prime}}(e)|\pm O(\beta)$. Consequently the error induced by this double-counting is at most $O(\beta k_{G}^{2})$, so the total difference between the terms of $f(G)$ and $f(G^{\prime})$ corresponding to such vertices is still $O(k_{G}+\beta k_{G}(k_{G}+k^{*})).$

Finally, consider the edges $\{u,v^{*}\}$ involving vertex $v^{*}$. If $\mathsf{wt}_{G}(v^{*})=0$ then

[TABLE]

If $\mathsf{wt}_{G}(v^{*})=1$ then $\deg_{G}(v^{*})\in[\bar{d}_{G}-k^{\prime},\bar{d}_{G}+k^{\prime}]$, so

[TABLE]

Otherwise, $\deg_{G}(v^{*})\in[\bar{d}_{G}-k^{\prime}-1/\beta,\bar{d}_{G}+k^{\prime}+1/\beta]$. Then we have that

[TABLE]

so in either case we have that $f_{v^{*}}(G)\in[\bar{d}_{G}-O(k_{G}+k^{*}+1/\beta),\bar{d}_{G}+O(k_{G}+k^{*}+1/\beta)]$. Consequently $|f_{v^{*}}(G)-f_{v^{*}}(G^{\prime})|\leq O(k_{G}+k^{*}+1/\beta)$.

Putting everything together, we have that $LS_{f}(G)=O((k_{G}+k^{*})(1+\beta k_{G})+1/\beta)$. ∎

We now compute a smooth upper bound on $LS_{f}(G)$. From the proof of Lemma 3.1, we have that there exists some constant $C>0$ such that $LS_{f}(G)\leq C((k_{G}+k^{*})(1+\beta k_{G})+\frac{1}{\beta})$. Let

[TABLE]

be this upper bound on $LS_{f}(G)$, and let

[TABLE]

Lemma 3.2**.**

$S(G)$* is a $\beta$-smooth upper bound on the local sensitivity of $f$. Moreover,*

[TABLE]

Proof.

For neighboring graphs $G,G^{\prime}$, we have that

[TABLE]

Moreover, for fixed $k_{G},k^{*},\beta$, consider the function $h(\ell)=e^{-\ell\beta}g(k_{G}+\ell,k^{*}\beta)$, and consider the derivative $h^{\prime}(\ell)$. We have that

[TABLE]

Consequently the only possible local maximum for $\ell>0$ would occur for $\ell=1/\beta-k_{G}-k^{*}$; note that the function $h$ decreases as $\ell\to\infty$. Consequently the maximum value of $h$ occurs for some $\ell\leq 1/\beta$, and so

[TABLE]

as desired. ∎

Theorem 3.3**.**

Algorithm 1 is $(O(\varepsilon),0)$-differentially private. Moreover, for any $k$-concentrated $n$-vertex graph $G=(V,E)\in\mathcal{G}_{n,k}$ with $k\geq 1$, we have that Algorithm 1 satisfies

[TABLE]

Proof.

Algorithm 1 computes function $f$ and releases it with noise proportional to a $\beta$-smooth upper bound on the local sensitivity for $\beta\leq\varepsilon$. Consequently $(O(\varepsilon),0)$-differential privacy follows immediately from Theorem 2.6.

We now analyze its accuracy on $k$-concentrated graphs $G$. If $G$ is $k$-concentrated and $k^{*}\geq k$, then $\mathsf{wt}_{G}(v)=1$ for all vertices $v\in V$ and $\mathsf{val}_{G}(\{u,v\})=x_{\{u,v\}}$ for all $u,v\in V$, and so $f(G)=|E|$. Consequently Algorithm 1 computes the edge density of a $k$-concentrated graph with noise distributed according to the Student’s $t$-distribution scaled by a factor of $S(G)/(\varepsilon\binom{n}{2})$.

Since $G$ is $k$-concentrated, we also have that $k_{G}=1$, and so $S(G)=O(k+\beta(k+1)+1/\beta)\leq O(k+1/\varepsilon)$ by Lemma 3.2. The variance of the Student’s $t$-distribution with three degrees of freedom is $O(1)$, so the expected squared error of the algorithm is

[TABLE]

as desired. ∎

4 Application to Erdős-Rényi Graphs

In this section we show how to apply Algorithm 1 to estimate the parameter of an Erdős-Rényi graph. Pseudocode is given in Algorithm 2.

It is straightforward to prove that this mechanism satisfies differential privacy.

Theorem 4.1**.**

Algorithm 2 satisfies $(O({\varepsilon}),0)$-node-differential privacy.

Proof.

The first line computes the empirical edge density of the graph $G$, which is a function with global sensitivity $(n-1)/\binom{n}{2}=2/n$. Therefore by Theorem 2.4 this step satisfies $({\varepsilon},0)$-differential privacy. The third line runs an algorithm that satisfies $(O({\varepsilon}),0)$-differential privacy for every fixed parameter $\tilde{k}$. By Lemma 2.2, the composition satisfies $(O({\varepsilon}),0)$-differential privacy. ∎

Next, we argue that this algorithm satisfies the desired accuracy guarantee.

Theorem 4.2**.**

For every $n\in\mathbb{N}$ and $\frac{1}{2}\geq p\geq 0$, and an appropriate parameter $\alpha>0$, Algorithm 2 satisfies

[TABLE]

Proof.

We will prove the result in the case where $p\geq\frac{\log n}{n}$. The case where $p$ is smaller will follow immediately by using $\frac{\log n}{n}$ as an upper bound on $p$. The first term in the bound is simply the variance of the empirical edge-density $\bar{p}$. For the remainder of the proof we will focus on bounding {\mathbb{E}\mathopen{}\mathclose{{}\left[(\bar{p}-\hat{p})^{2}}\right]}.

A basic fact about $G(n,p)$ for $p\geq\frac{\log n}{n}$ is that with probability at least $1-2\alpha$: (1) $|\bar{p}-p|\leq 2\log(1/\alpha)/n$, and (2) the degree of every node $i$ lies in the interval $[\bar{d}\pm\sqrt{pn\log(n/\alpha)}]$ where $\bar{d}$ is the average degree of $G$. We will assume for the remainder that these events hold.

Using Theorem 2.4, we also have that with probability at least $1-\alpha$, the estimate $\tilde{p}^{\prime}$ satisfies $|\bar{p}-\tilde{p}^{\prime}|\leq 4\log(1/\alpha)/{\varepsilon}n$. We will also assume for the remainder that this latter event holds. Therefore, we have $p\leq\tilde{p}$ and $p\geq\tilde{p}-8\log(1/\alpha)/{\varepsilon}n.$

Assuming this condition holds, the graph will have $\tilde{k}$-concentrated degrees for $\tilde{k}$ as specified on line 2 of the algorithm. Since this assumption holds, we have

[TABLE]

To complete the proof, we can plug in a suitably small $\alpha=1/\mathrm{poly}(n)$ so that the $O(\alpha)$ probability of failure will not affect the overall mean-squared error in a significant way. ∎

5 Lower Bounds for Concentrated-Degree Graphs

In this section we prove a lower bound for estimating the number of edges in concentrated-degree graphs. Theorem 1.3, which lower bounds the mean squared error follows by applying Jensen’s Inequality.

Theorem 5.1**.**

For every $n,k\in\mathbb{N}$, every ${\varepsilon}\in[\frac{2}{n},\frac{1}{4}]$ and $\delta\leq\frac{{\varepsilon}}{32}$, and every $({\varepsilon},\delta)$-node-DP algorithm $A$, there exists $G\in\mathcal{G}_{n,k}$ such that {\underset{A}{\mathbb{E}}\mathopen{}\mathclose{{}\left[|p_{G}-A(G)|}\right]}=\Omega\mathopen{}\mathclose{{}\left(\frac{k}{{\varepsilon}n^{2}}+\frac{1}{{\varepsilon}^{2}n^{2}}}\right).

The proof relies on the following standard fact about differentially private algorithms. Since we are not aware of a formal treatment in the literature, we include a proof for completeness.

Lemma 5.2**.**

Suppose there are two graphs $G_{0},G_{1}\in\mathcal{G}_{n,k}$ at node distance at most $\frac{1}{{\varepsilon}}$ from one another. Then for every $({\varepsilon},\frac{{\varepsilon}}{32})$-node-DP algorithm $A$, there exists $b\in\{0,1\}$ such that

[TABLE]

Proof.

Let $A$ be any ${\varepsilon}$-node-DP algorithm. Since $G_{0},G_{1}$ have node distance at most $\frac{1}{{\varepsilon}}$, by group privacy (Lemma 2.3), for every set $S$ and every $b\in\{0,1\}$

[TABLE]

Now, let S_{b}=\mathopen{}\mathclose{{}\left\{y:\mathopen{}\mathclose{{}\left|y-p_{G_{b}}}\right|<\frac{1}{2}\mathopen{}\mathclose{{}\left|p_{G_{0}}-p_{G_{1}}}\right|}\right\} and note that $S_{0}$ and $S_{1}$ are disjoint by construction. Let \rho=\min\{{\mathbb{P}\mathopen{}\mathclose{{}\left[A(G_{0})\in S_{0}}\right]},{\mathbb{P}\mathopen{}\mathclose{{}\left[A(G_{1})\in S_{1}}\right]}\}. Then we have

[TABLE]

from which we can deduce $\rho\leq\frac{4}{5}$. Therefore, for some $b\in\{0,1\}$, we have

[TABLE]

from which the lemma follows. ∎

We will construct two simple pairs of graphs to which we can apply Lemma 5.2.

Lemma 5.3** (Lower bound for large $k$).**

For every $n,k\in\mathbb{N}$ and ${\varepsilon}\geq 2/n$, there is a pair of graphs $G_{0},G_{1}\in\mathcal{G}_{n,k}$ at node distance $1/{\varepsilon}$ such that $|p_{G_{0}}-p_{G_{1}}|=\Omega(\frac{k}{{\varepsilon}n^{2}})$.

Proof.

Let $G_{0}$ be the empty graph on $n$ nodes. Note that $p_{G_{0}}=0$, $\bar{d}_{G_{0}}=0$, and $G_{0}$ is in $\mathcal{G}_{n,k}$.

We construct $G_{1}$ as follows. Start with the empty bipartite graph with $\frac{1}{{\varepsilon}}$ nodes on the left and $n-\frac{1}{{\varepsilon}}$ nodes on the right. We connect the first node on the left to each of the first $k$ nodes on the right, then the second node on the left to each of the next $k$ nodes on the right and so on, wrapping around to the first node on the right when we run out of nodes. By construction, $p_{G_{1}}=k/{\varepsilon}\binom{n}{2}$, $\bar{d}_{G_{1}}=2k/{\varepsilon}n$. Moreover, each of the first $\frac{1}{{\varepsilon}}$ nodes has degree exactly $k$ and each of the nodes on the right has degree

[TABLE]

Thus, for $n$ larger than some absolute constant, every degree lies in the interval $[\bar{d}_{G_{1}}\pm k]$ so we have $G_{1}\in\mathcal{G}_{n,k}$. ∎

Lemma 5.4** (Lower bound for small $k$).**

For every $n\geq 4$ and ${\varepsilon}\in[2/n,1/4]$, there is a pair of graphs $G_{0},G_{1}\in\mathcal{G}_{n,1}$ at node distance $1/{\varepsilon}$ such that $|p_{G_{0}}-p_{G_{1}}|=\Omega(\frac{1}{{\varepsilon}^{2}n^{2}})$.

Proof.

Let $i=\lceil n\varepsilon\rceil$, and let $G_{0}$ be the graph consisting of $i$ disjoint cliques each of size $\lfloor n/i\rfloor$ or $\lceil n/i\rceil$. Let $G_{1}$ be the graph consisting of $i+1$ disjoint cliques each of size $\lfloor n/(i+1)\rfloor$ or $\lceil n/(i+1)\rceil$. We can obtain $G_{0}$ from $G_{1}$ by taking one of the cliques and redistributing its vertices among the $i$ remaining cliques, so $G_{0}$ and $G_{1}$ have node distance $\ell:=\lfloor n/(i+1)\rfloor\leq 1/\varepsilon$. For $1/4\geq\varepsilon\geq 2/n$ we have that $\ell\geq\lfloor 1/2\varepsilon\rfloor>1/4\varepsilon$. Transforming $G_{1}$ into $G_{0}$ involves removing a clique of size $\ell$, containing $\binom{\ell}{2}$ edges, and then inserting these $\ell$ vertices into cliques that already have size $\ell$, adding at least $\ell^{2}$ new edges. Consequently $G_{0}$ contains at least $\ell^{2}-\ell(\ell-1)/2=\ell(\ell+1)/2$ more edges than $G_{1}$, so

[TABLE]

as desired. ∎

Theorem 5.1 now follows by combining Lemmas 5.2, 5.3, and 5.4.

Acknowledgments

Part of this work was done while the authors were visiting the Simons Institute for the Theory of Computing. AS is supported by NSF MACS CNS-1413920, DARPA/NJIT Palisade 491512803, Sloan/NJIT 996698, and MIT/IBM W1771646. JU is supported by NSF grants CCF-1718088, CCF-1750640, and CNS-1816028. The authors are grateful to Adam Smith for helpful discussions.