Nonparametric statistics of dynamic networks with distinguishable nodes

Abstract

The study of random graphs and networks had an explosive development in the last couple of decades. Meanwhile, techniques for the statistical analysis of sequences of networks were less developed. In this paper, we focus on networks sequences with a fixed number of labeled nodes and study some statistical problems in a nonparametric framework. We introduce natural notions of center and a depth function for networks that evolve in time. We develop several statistical techniques including testing, supervised and unsupervised classification, and some notions of principal component sets in the space of networks. Some examples and asymptotic results are given, as well as two real data examples.

Appendix: Proofs

1.1 Characterization of the central set

Proof of Proposition 1

We have that the expected distance from a network H to a random network ${\mathbf {G}}$ is

$$\begin{aligned} {{\mathbb {E}}}\left( d({{\mathbf {G}}}, H) \right) = \sum _{G \in {\mathscr {G}}} d(G,H)p_G. \end{aligned}$$

(8)

Let A(G) be the adjacency matrix of the network G and ${{\mathbf {A}}}$ the adjacency matrix of the random network ${{\mathbf {G}}}$. Then expression (8) can be written as

$$\begin{aligned} \sum _{G \in {{\mathscr {G}}}} \sum _{i>j} \vert A(G)_{ij} - A(H)_{ij} \vert p_G&= \sum _{i>j} \sum _{G \in {{\mathscr {G}}}} \vert A(G)_{ij} - A(H)_{ij} \vert p_G \\&= \sum _{i>j}{{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij} \ne A(H)_{ij} \right) , \end{aligned}$$

which is minimized by any network H with adjacency matrix $A(H)_{ij}=1$ if and only if ${{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \ge 1/2$. Moreover, if for all i, j

$$\begin{aligned} {{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \ne 1/2, \end{aligned}$$

(9)

there is a unique network S that minimizes expression (8) and the corresponding adjacency matrix satisfies $A(S)_{ij}=1$ if and only if ${{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij} =1 \right) > 1/2$.

On the other hand, if condition (9) does not hold, there are many solutions. The maximal center L is the network whose adjacency matrix fulfills $A(L)_{ij} =1$ when ${{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \ge 1/2$ and the set ${\mathscr {C}}$ contains exactly all subnetworks of L for which S is a subnetwork.

The proof for the empirical version is completely analogous. $\square$

1.2 Depth determines measure

Proof of Proposition 2

Let $\mu = (\mu _1, \ldots , \mu _K)$ and $\nu = (\nu _1, \ldots , \nu _K)$ be two distributions on ${{\mathscr {G}}}$ where $K=2^m$ is the cardinal of the space of networks. For any $H \in {{\mathscr {G}}}$, let $d(H) = (d(H_1,H), \ldots d(H_{K},H))$. Then, the population depth is

$$\begin{aligned} D_\mu (H) = m - d(H)^T \mu . \end{aligned}$$

Therefore, the depth determines the measure if and only if

$$\begin{aligned} d(H)^T \mu - d(H)^T \nu = 0 \text { for all} H\in {{\mathscr {G}}}\text { implies } \mu = \nu . \end{aligned}$$

(10)

Denote by F the matrix with rows given by $d(H_1), \ldots d(H_{K})$. Then expression (10) is equivalent to $F (\mu -\nu ) = 0$, having a unique solution. The result follows from the invertibility of distance matrices. This result was initially proved by Micchelli (1986) and later on by Auer (1995), who provided an elementary proof, that the only uses the triangle inequality. In particular, our metric is just the $L^1$ distance between the adjacency matrices and the result holds. For the sake of completeness, we now state the result in Auer (1995) for distance matrices.

Theorem 4

Let $P_1, \ldots , P_n$ be distinct points in ${\mathbb {R}}^k$, and $d_{i,j} = \Vert P_i - P_j \Vert$. If $F_n$ is the distance matrix with entries $d_{i,j}$, then

• (a) The $\det F_n$ is positive if n is odd and negative if n is even; in particular, $F_n$ is invertible.

• (b) The matrix $F_n$ has one positive and $n-1$ negative eigenvalues.

Applying Theorem A to our setup, we get that the matrix F is invertible. $\square$

1.3 Convergence of empirical depth

Proof of Theorem 2

From the ergodic theorem, we have that ${\hat{D}}_\ell (H) \rightarrow D(H)$ almost surely as $\ell \rightarrow \infty$ for each $H \in {\mathscr {G}}$. Since ${\mathscr {G}}$ is finite, we get uniform convergence. $\square$

Proof of Theorem 3

Recall that a sequence of random elements ${\mathbf {X}} := ({\mathbf {X}}_t, t \ge 1)$ is a strong mixing sequence if it fulfills the following condition. For $1 \le j < \ell \le \infty$, let ${\mathscr {F}}_j^\ell$ denote the $\sigma$-field of events generated by the random elements $X_k,\ j \le k \le \ell \ (k \in \mathbf{N})$ . For any two $\sigma$-fields ${\mathscr {A}}$ and ${\mathscr {B}}$, define

$$\begin{aligned} \alpha ({\mathscr {A}}, {\mathscr {B}}):= \sup _{A \in {\mathscr {A}}, B \in {\mathscr {B}}} \vert {{\mathbb {P}}}\left( A \cap B \right) - {{\mathbb {P}}}\left( A \right) {{\mathbb {P}}}\left( B \right) \vert . \end{aligned}$$

For the given random sequence ${\mathbf {X}}$, for any positive integer n, define the dependence coefficient

$$\begin{aligned} \alpha (n) = \alpha ({\mathbf {X}},n) := \sup _{j \ge 1}\; \alpha ({\mathscr {F}}_{1}^j, {\mathscr {F}}_{j + n}^{\infty }). \end{aligned}$$

The random sequence ${\mathbf {X}}$ is said to be “strongly mixing,” or “$\alpha$-mixing,” if $\alpha (n) \rightarrow 0$ as $n \rightarrow \infty$. This condition was introduced by Rosenblatt (1956). By assumption, we have that the sequence of random networks $\{{{\mathbf {G}}}_t: t\ge 1 \}$ is a strongly mixing sequence. In order to prove the theorem, we use the following result (see, for instance, Peligrad 1986).

Theorem 5

Let $\{{\mathbf {X}}_t: t \ge 1\}$ be a strictly stationary centered $\alpha$-mixing sequence, and let ${\mathbf {S}}_\ell = \sum _{t=1}^\ell {\mathbf {X}}_t$. Assume that for some $C>0$

$$\begin{aligned} \vert {\mathbf {X}}_1 \vert< C \ {\textit{almost surely, and}}\; \sum _{n=1}^\infty \alpha (n) < \infty . \end{aligned}$$

Then,

$$\begin{aligned} \sigma ^2 := {{\mathbb {E}}}\left( {\mathbf {X}}_1^2 \right) + 2 \sum _{k=2}^\infty {{\mathbb {E}}}\left( {\mathbf {X}}_1 {\mathbf {X}}_k \right) , \end{aligned}$$

is absolutely summable. If in addition $\sigma ^2 >0$, then ${\mathbf {S}}_\ell /\sqrt{\ell }\sigma$ converges weakly to a standard normal distribution.

First observe that

$$\begin{aligned} \beta ^T {\mathbf {Z}}_{\ell } = \frac{1}{\ell } \sum _{k=1}^{\ell } {\mathbf {W}}_k, \quad \text {with}\quad {\mathbf {W}}_k= \sum _{j=1}^{2^m} \beta _j \left( d(H_j, {{\mathbf {G}}}_k) - {{\mathbb {E}}}\left( d(H_j,{{\mathbf {G}}}_1) \right) \right) , \end{aligned}$$

where $\{{\mathbf {W}}_t: t \ge 1 \}$ is a strictly stationary, bounded, centered $\alpha$-mixing sequence, fulfilling $\sum _{n=1}^\infty \alpha (n) < \infty$. On the other hand, we have that

$$\begin{aligned} {{\mathbb {E}}}\left( {\mathbf {W}}_1^2 \right) = \beta ^T {{\mathbb {E}}}\left( {\mathbf {Y}}_1^T {\mathbf {Y}}_1 \right) \beta \ \text{ and } \ {{\mathbb {E}}}\left( {\mathbf {W}}_1 {\mathbf {W}}_k \right) = \beta ^T {{\mathbb {E}}}\left( {\mathbf {Y}}_1^T {\mathbf {Y}}_k \right) \beta , \end{aligned}$$

and the result follows from Theorem B. $\square$

1.4 Characterization of principal components

Proof of Proposition 3

Note that

$$\begin{aligned}&{\text {var}}\left( \frac{|{{\mathbf {G}}}\wedge Q|}{|Q|} \right) \\&\quad = \frac{1}{\vert Q \vert ^2}\sum _{G \in {\mathscr {G}}} \left( \sum _{i>j} c_{ij} A(G)_{ij} A(Q)_{ij} - \sum _{H \in {\mathscr {G}}} \sum _{i>j} c_{ij} A(H)_{ij} A(Q)_{ij}p_H \right) ^2p_G. \end{aligned}$$

We first consider the case when the network Q has only one link $(k,\ell )$, and find within this family the one that maximizes the objective function. Next we prove that for any other network Q the objective function is bounded by the maximum restricted to the former family. Finally, we show that the principal component space is generated by the one link networks.

Let $Q_1$ such that $A(Q_1)_{k_1\ell _1}=1$, and 0 otherwise. Also, let ${\mathscr {G}}_{Q_1}^+ = \{G\in {\mathscr {G}}: A(G)_{k_1\ell _1} = 1\}$ and ${\mathscr {G}}_{Q_1}^- = \{G\in {\mathscr {G}}: A(G)_{k_1\ell _1} = 0 \}$. When we search within the one link networks, the objective function reduces to

$$\begin{aligned} {\text {var}}\left( \frac{|{{\mathbf {G}}}\wedge Q|}{|Q|} \right)&= \frac{1}{c_{k_1\ell _1}^2}\sum _{G \in {\mathscr {G}}} \left( c_{k_1\ell _1} A(G)_{k_1\ell _1} - \sum _{H \in {\mathscr {G}}} c_{k_1\ell _1} A(H)_{k_1\ell _1} p_H \right) ^2 p_G \\&=\sum _{G \in {\mathscr {G}}} \left( A(G)_{k_1\ell _1} -{{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) \right) ^2 p_G \\&=\sum _{G \in {\mathscr {G}}^+} \left( 1- {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) \right) ^2 p_G + \sum _{G \in {\mathscr {G}}^-} {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) ^2 p_G \\&= ( 1- {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) )^2 {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) + {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) ^2 {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=0 \right) \\&= {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) (1- {{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right) ), \end{aligned}$$

and the solution is the one link graph for which ${{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right)$ is closest to 1 / 2.

In the general case, we want to find Q that maximizes

$$\begin{aligned} {\text {var}}\left( \frac{|{{\mathbf {G}}}\wedge Q|}{|Q|} \right)&= \frac{1}{\vert Q \vert ^2} \sum _{G \in {\mathscr {G}}} \left( \sum _{(i,j)\in Q} c_{ij} A(G)_{ij} - \sum _{H \in {\mathscr {G}}} \sum _{(i,j)\in Q} c_{ij} A(H)_{ij} p_H \right) ^2 p_G \\&= \sum _{G \in {\mathscr {G}}} \ \sum _{(i,j)\in Q} w_{ij} \left( A(G)_{ij} - {{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \right) ^2 p_G, \end{aligned}$$

where $w_{ij}= c_{ij} / \sum _{(p,q)\in Q} c_{pq}$. Now, since the weights $w_{ij}$ add to one, we have

$$\begin{aligned} {\text {var}}\left( \frac{|{{\mathbf {G}}}\wedge Q|}{|Q|} \right)&\le \sum _{(i,j)\in Q} w_{ij}\; \max _{(i,j)}\; {{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \left( 1 - {{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \right) \\&= \max _{(i,j)}\; {{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \left( 1-{{\mathbb {P}}}\left( {{\mathbf {A}}}_{ij}=1 \right) \right) , \end{aligned}$$

which corresponds to the one link optimum.

If there exists a unique one link graph ($Q_1$) that verifies ${{\mathbb {P}}}\left( {{\mathbf {A}}}_{k_1\ell _1}=1 \right)$ is closest to 1 / 2, then the principal component space is generated just by $Q_1$, i.e., ${{\mathscr {S}}}_1 ={\mathscr {G}}_{Q_1}^+$. If there exist multiple one link graphs, $Q_1, Q_2, \ldots , Q_p$, that minimize $|{{\mathbb {P}}}\left( {{\mathbf {A}}}_{k\ell }=1 \right) -1/2|$, then the principal component space is ${{\mathscr {S}}}_1 = \cup _{i=1}^p {\mathscr {G}}_{Q_i}^+$. The second principal component space verifies the same. In this case, the maximization of the variance is over $\{G\in {\mathscr {G}}: G \notin {{\mathscr {S}}}_1 \}$. Analogous for the rest of the components, for example, for finding the $k-esima$ principal component space just maximizes the variance over $\{G\in {\mathscr {G}}: G \notin {{\mathscr {S}}}_1, G \notin {{\mathscr {S}}}_2, G \notin {{\mathscr {S}}}_{k-1} \}.$ $\square$

1.5 Consistency of principal components

Proof of Proposition 4

For each $Q \in {\mathscr {G}}$ from the ergodic theorem, we have the following.

• (a) $\displaystyle \varLambda _{\ell } (Q) = \frac{1}{\ell } \sum _{k=1}^{\ell } \vert G_k \wedge Q \vert \rightarrow {{\mathbb {E}}}\left( \vert {{\mathbf {G}}}\wedge Q\vert \right)$, almost surely as $\ell \rightarrow \infty$.

• (b) which converges almost surely to

  $$\begin{aligned} {{\mathbb {E}}}\left( \frac{\vert {{\mathbf {G}}}\wedge Q \vert ^2}{\vert Q \vert ^2} \right) + {{\mathbb {E}}}\left( \frac{\vert {{\mathbf {G}}}\wedge Q \vert }{\vert Q \vert ^2} \right) ^2 - 2 {{\mathbb {E}}}\left( \frac{\vert {{\mathbf {G}}}\wedge Q \vert }{\vert Q \vert ^2} \right) ^2 = {\text {var}}\left( \frac{\vert {{\mathbf {G}}}\wedge Q \vert }{\vert Q \vert }\right) . \end{aligned}$$

  (11)

• (c) Since the space ${\mathscr {G}}$ is finite, expression (11) entails that $\hat{{\mathscr {Q}}}_1 \rightarrow {\mathscr {Q}}_1$ almost surely, i.e., $\hat{{\mathscr {Q}}}_1 = {\mathscr {Q}}_1$ for $\ell$ large enough almost surely, which entails that the principal components converge because the geodesics coincide eventually.

For the next principal component, the proof is analogous. $\square$