Federated personalized random forest for human activity recognition

Songfeng Liu; Jinyan Wang; Wenliang Zhang; Songfeng Liu; Jinyan Wang; Wenliang Zhang

doi:10.3934/mbe.2022044

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 953-971. doi: 10.3934/mbe.2022044

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Research article Special Issues

Federated personalized random forest for human activity recognition

1.
Guangxi Key Lab of Multi-source Information Mining and Security, Guangxi Normal University, Guilin, China
2.
College of Computer Science and Engineering, Guangxi Normal University, Guilin, China

Academic Editor: Jun Shen

Received: 30 September 2021 Accepted: 03 November 2021 Published: 22 November 2021

User data usually exists in the organization or own local equipment in the form of data island. It is difficult to collect these data to train better machine learning models because of the General Data Protection Regulation (GDPR) and other laws. The emergence of federated learning enables users to jointly train machine learning models without exposing the original data. Due to the fast training speed and high accuracy of random forest, it has been applied to federated learning among several data institutions. However, for human activity recognition task scenarios, the unified model cannot provide users with personalized services. In this paper, we propose a privacy-protected federated personalized random forest framework, which considers to solve the personalized application of federated random forest in the activity recognition task. According to the characteristics of the activity recognition data, the locality sensitive hashing is used to calculate the similarity of users. Users only train with similar users instead of all users and the model is incrementally selected using the characteristics of ensemble learning, so as to train the model in a personalized way. At the same time, user privacy is protected through differential privacy during the training stage. We conduct experiments on commonly used human activity recognition datasets to analyze the effectiveness of our model.

Keywords:

Citation: Songfeng Liu, Jinyan Wang, Wenliang Zhang. Federated personalized random forest for human activity recognition[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 953-971. doi: 10.3934/mbe.2022044

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Abstract

1. Introduction

In theoretical chemistry, the topological index of a graph, also called molecular structure descriptor, is a real number related to a structural graph of a molecule, and is often used to predict the physico-chemical properties and biological activities of molecules. A large number of molecular structure descriptors have been conceived and several of them have found applications in quantitative structure-activity and structure-property relationships (QSAR/QSPR) studies. In particular, degree-based topological indices and distance-based topological indices are the most important molecular structure descriptors that play an important role in QSAR/QSPR.

Throughout in this paper, $G$ is a simple connected undirected graph with the vertex set $V(G)$ and edge set $E(G)$ . For $u, v \in V(G)$ , $d_v$ is the degree of vertex $v$ in $G$ and $d(u, v)$ is the distance between vertices $u$ and $v$ in $G$ . As a molecular descriptor, the Wiener index, introduced by Wiener ^[1] in 1947, is considered as one of the most used topological indexes with high correlation with many physical and chemical indices of molecular compounds. The Wiener index equals the sum of distances between all pairs of vertices of a graph $G$ , that is,

$W(G) = \sum\limits_{\{u, v \}\subseteq V(G)}d(u, v).$

In 1989, the Schultz index ^[2] of a chemical graph $G$ was put forward as a topological index of alkanes. It is defined as

$S(G) = \sum\limits_{\{u, v \}\subseteq V(G)}(d_u+d_v)d(u, v).$

The proposal of this index has opened up the research on the degree-distance-type index. Plavšić et al. ^[3] showed that the Wiener index and the Schultz index are highly intercorrelated topological indices. For arbitrary catacondensed benzenoid graphs, Dobrynin ^[4] proved that the Schultz index has the same discriminating power with the Wiener index. So, it is both significant and interesting to study the Schultz index for some given class of graphs (or network), no matter whether they are molecular graphs or not.

In 1994, Gutman ^[5] proposed the Schultz index of the second kind, often called the Gutman index, and defined it as

$Gut(G) = \sum\limits_{\{u, v \}\subseteq V(G)}d_ud_vd(u, v).$

Bounds of this index have been extensively studied using mathematical methods; see ^[6]. Moreover, for a tree $T$ on $n$ vertices, the Gutman index and Wiener index are closely related by

$Gut(T) = 4W(T)-(n-1)(2n-1).$

In 2021, from a geometric perspective (degree radius), Gutman ^[7] introduced a novel degree-based topological index called the Sombor index, which is defined as

$SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d^2_u+d^2_v}.$

Note that the Sombor index is the sum of Euclidean distances of the degrees of the two vertices of each edge in the graph. This index is widely studied in mathematics and chemistry; see ^[8].

Inspired by the above research, we propose a new topological index called the Sombor-Wiener (SW) index, and define it as

$SW(G) = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v).$

The new index can be regarded as the sum of the product of degree radius and distance between any two vertices in the graph, which is a novel version of the distance-based topological index.

Naturally, we define a general topological index $DWW(G)$ of a graph $G$ contributed by the degree weights of all vertices as

$DWW(G) = \sum\limits_{\{u, v\}\subseteq V(G)}f(d_u, d_v)d(u, v),$

where $f(d_u, d_v)$ is a real function of $d_u$ and $d_v$ with

$f(d_u, d_v)\geq 0\; \; \text{and}\; \; f(d_u, d_v) = f(d_v, d_u).$

Clearly, the general topological index, called the degree-weighted Wiener index, is the generalization of the Schultz index, the Gutman index, and the SW index.

In this paper, we study the basic properties of the SW index, and the linear regression analysis of the SW index, with respect to acentric factor of octane isomers. In addition, we give the calculation formula of degree-weighted Wiener index for generalized Bethe trees. Our results generalize some known formulae on the Schultz index and Gutman index.

2. Basic properties of the SW index

Theorem 2.1. Let $G$ be a connected graph with $n$ vertices.

(i) If $G = P_n$ , then

$SW(G) = (n-1)\left(\frac{\sqrt{2}(n^2-5n+9)}{3}+\sqrt{5}(n-2)\right).$

(ii) If $G$ is $r$ -regular, then

$SW(G) = \sqrt{2}rW(G).$

Moreover, if $G = K_n$ , then

$SW(G) = \frac{\sqrt{2}n(n-1)^2}{2}.$

If $G = C_n$ , then

$SW(G) =\left\{ \begin{array}{l}\frac{\sqrt{2}n^3}{4} , \qquad \qquad \mathit{\text{if n is even;}}\\ \frac{\sqrt{2}n(n^2-1)}{4}, \ \quad \mathit{\text{if n is odd.}} \end{array} \right.$

(iii) If $G = K_{n_1, \, n_2}$ , then

$SW(G) = n_1n_2\left[\sqrt{n_1^2+n_2^2}+\sqrt{2}(n_1+n_2)-2\sqrt{2}\right].$

In particular, if $G = K_{1, \, n-1}$ , then

$SW(G) = (n-1)(\sqrt{n^2-2n+2}+\sqrt{2}n-2\sqrt{2}).$

Proof. (i) If $G = P_n$ , then

$\begin{align*} SW(G) = & \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ = & \sqrt{5}(1+2+\cdots+n-2)+\sqrt{2}(n-1)\\ & +2\sqrt{2}(1+2+\cdots+n-3)\\ & +\sqrt{5}(n-2)+\cdots+2\sqrt{2}+2\sqrt{5}+\sqrt{5}\\ = & \frac{\sqrt{5}(n-1)(n-2)}{2}+\sqrt{2}(n-1)\\ & +2\sqrt{2}\left(1+3+\cdots+\frac{(n-3)(n-2)}{2}\right)\\ & +\sqrt{5}(1+2+\cdots+n-2)\\ = & \sqrt{2}(n-1)+\sqrt{5}(n-1)(n-2)\\ & +\frac{2\sqrt{2}(n-1)(n-2)(n-3)}{6}\\ = & (n-1)\left(\frac{\sqrt{2}(n^2-5n+9)}{3}+\sqrt{5}(n-2)\right). \end{align*}$

(ii) If $G$ is $r$ -regular, then

$SW(G) = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{r^2+r^2}d(u, v) = \sqrt{2}rW(G).$

In particular, if $G = K_n$ , then

$\begin{align*} SW(G) & = \sqrt{2}(n-1)W(G)\\& = \sqrt{2}(n-1)\frac{n(n-1)}{2}\\ & = \frac{\sqrt{2}n(n-1)^2}{2}. \end{align*}$

If $G$ is a cycle $C_n$ , from ^[9], we have

$SW(G) = \left\{ \begin{array}{l}\frac{\sqrt{2}n^3}{4} , \qquad \qquad \text{if } ~~n ~~\text{ is even;}\\ \frac{\sqrt{2}n(n^2-1)}{4}, \ \quad \text{if } ~~n ~~ \text{is odd.} \end{array} \right.$

(iii) If $G = K_{n_1, \, n_2}$ , then

$\begin{align*} SW(G) = & \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ = & n_1n_2\sqrt{n_1^2+n_2^2}+2\binom{n_1}{2}\sqrt{n_2^2+n_2^2}\\ & +2\binom{n_2}{2}\sqrt{n_1^2+n_1^2}\\ = & n_1n_2\left[\sqrt{n_1^2+n_2^2}+\sqrt{2}(n_1+n_2)-2\sqrt{2}\right]. \end{align*}$

Let $n_1 = 1$ and $n_2 = n-1$ , then

$SW(K_{1, \, n-1}) = (n-1)(\sqrt{n^2-2n+2}+\sqrt{2}n-2\sqrt{2}).$

This completes the proof. □

Theorem 2.2. Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$ , then

$\sqrt{2}\delta W(G)\leq SW(G)\leq \sqrt{2}\Delta W(G)$

with equality if, and only if, $G$ is regular.

Proof. By definition of $SW(G)$ , we have the proof. □

Corollary 2.3. Let $G$ be a connected graph with $n$ vertices, then

$\sqrt{2} W(G)\leq SW(G)\leq \sqrt{2}(n-1) W(G).$

Theorem 2.4. Let $G$ be a connected graph with the minimum degree $\delta$ , then

$\begin{align} \frac{1}{\sqrt{2}}S(G)\leq SW(G) \leq S(G)-(2-\sqrt{2})\delta W(G) \end{align}$

(2.1)

with equality (left and right) if, and only if, $G$ is regular.

Proof. First, we prove the left-hand side of (2.1). By Cauchy-Schwarz's inequality, we have

$\begin{align*} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \geq \sum\limits_{\{u, v\}\subseteq V(G)}\frac{1}{\sqrt{2}}(d_u+d_v)d(u, v)\\ & = \frac{1}{\sqrt{2}}S(G) \end{align*}$

with equality if, and only if, $d_u = d_v$ for $u, v\in V(G)$ , that is, $G$ is regular.

Second, we prove the righthand side of (2.1). For any $u, v \in V(G)$ ( $d_u\geq d_v$ ), we have

$\sqrt{d^2_u+d^2_v}\leq d_u+(\sqrt{2}-1)d_v$

with equality if, and only if, $d_u = d_v$ . Thus,

$\begin{align*} \begin{aligned} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \leq \sum\limits_{\{u, v\}\subseteq V(G)} [d_u+(\sqrt{2}-1)d_v]d(u, v)\\ & \leq \sum\limits_{\{u, v\}\subseteq V(G)} (d_u+d_v)d(u, v)-\sum\limits_{\{u, v\}\subseteq V(G)} (2-\sqrt{2})\delta d(u, v)\\ & \leq S(G)-(2-\sqrt{2})\delta W(G)\end{aligned} \end{align*}$

with equality if, and only if, $G$ is regular.

This completes the proof. □

Theorem 2.5. Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$ , then

$\frac{\sqrt{2}}{\Delta}Gut(G)\leq SW(G) \leq \frac{\sqrt{2}}{\delta}Gut(G)$

with equality (left and right) if, and only if, $G$ is regular.

Proof. Note that

$\sqrt{d^2_u+d^2_v} = d_ud_v\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}},$

then we have

$\begin{align*} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \leq \sum\limits_{\{u, v\}\subseteq V(G)} d_ud_v\sqrt{\frac{1}{\delta^2}+\frac{1}{\delta^2}}d(u, v)\\ & = \frac{\sqrt{2}}{\delta}\sum\limits_{\{u, v\}\subseteq V(G)} d_ud_vd(u, v)\\ & = \frac{\sqrt{2}}{\delta}Gut(G) \end{align*}$

and

$\begin{align*} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \geq \sum\limits_{\{u, v\}\subseteq V(G)} d_ud_v\sqrt{\frac{1}{\Delta^2}+\frac{1}{\Delta^2}}d(u, v)\\ & = \frac{\sqrt{2}}{\Delta}\sum\limits_{\{u, v\}\subseteq V(G)} d_ud_vd(u, v)\\ & = \frac{\sqrt{2}}{\Delta}Gut(G). \end{align*}$

This completes the proof. □

3. Degree-weighted Wiener index of generalized Bethe trees

The generalized Bethe tree is an important graph structure that has wide applications in many fields. The investigation on topological indices of generalized Bethe trees and dendrimer trees frequently appeared in various journals. A Bethe tree $B_{k, d}$ is a rooted tree at $k$ levels whose root is on level $1$ and has degree equal to $d$ , the vertices of levels from $2$ to $k-1$ have degrees equal to $d+1$ , and the vertices on the level $k$ have degree equal to $1$ ; see ^[10]. In 2007, Rojo ^[11] generalized the notion of a Bethe tree as follows: A generalized Bethe tree $B_{k}$ is a rooted tree whose vertices at the same level have equal degrees. Moreover, a regular dendrimer tree $T_{k, d}$ is a generalized Bethe tree of $k+1$ levels with each non-pendent vertex having degree $d$ .

Theorem 3.1. Let $B_{k+1}$ be a generalized Bethe tree of $k+1$ levels. If $d_1$ denotes the degree of rooted vertex and $d_i+1$ denotes the degree of vertices on the $i$ -th level of $B_{k+1}$ for $i < 1\leq k$ , then

$DWW(B_{k+1}) = \sum\limits_{l = 1}^{k+1}A_l,$

where $n_j$ is the number of vertices on the $j$ -th level of $B_{k+1}$ , and

$n_1 = 1\; \; \mathit{\text{and}}\; \; n_{j+1} = d_1d_2\cdots d_j$

for $1\leq j\leq k$ , and

$\begin{align*} \begin{aligned} A_1 = & \sum\limits_{j = 2}^{k}n_j(j-1)f(d_1, d_j+1 )+kn_{k+1}f(d_1, 1), \\ A_l = & \Big[2\binom{d_{l-1}}{2}+4\binom{d_{l-1}}{1}\binom{d_{l-1}d_{l-2}-d_{l-1}}{1}+\cdots\\ & +2(l-1)\binom{d_{l-1}\cdots d_{2}}{1}\binom{n_l-d_{l-1}\cdots d_{2}}{1}\Big]f(d_l+1, d_l+1)\\ & +\sum\limits_{j = l+1}^{k}n_j(j-l)f(d_l+1, d_j+1)\\&+(k-l+1)n_{k+1}f(d_l+1, 1)\\ & +(d_{l-1}-1)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+2)f(d_l+1, d_j+1)\\ & +(k-l+3)n_{k+1}f(d_l+1, 1)\Big]\\ & +(d_{l-1}d_{l-2}-d_{l-1})\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+4)f(d_l+1, d_j+1)\\ & +(k-l+5)n_{k+1}f(d_l+1, 1)\Big]+\cdots\\ & +(n_l-d_{l-1}\cdots d_2)\Big[\sum\limits_{j = l+1}^{k}n_j(j+l-2)f(d_l+1, d_j+1)\\ & +(k+l-1)n_{k+1}f(d_l+1, 1)\Big], \\ A_{k+1} = & f(1, 1)\Big[2\binom{d_{k}}{2}+4\binom{d_{k}}{1}\binom{d_{k}d_{k-1}-d_{k}}{1}+\cdots\\ & +2k\binom{d_{k}\cdots d_{2}}{1}\binom{n_{k+1}-d_{k}\cdots d_{2}}{1}\Big].\end{aligned} \end{align*}$

Proof. Let $A_i$ be the value of degree-weighted Wiener index of vertices on the $i$ -th level of $B_{k+1}$ , then

$DWW(B_{k+1}) = \sum\limits_{i = 1}^{k+1}A_i.$

By definition of $B_{k+1}$ , we have

$\begin{align*} A_1 = & n_2f(d_1, d_2+1)+2n_3f(d_1, d_3+1)+\cdots\\ & +(k-1)n_kf(d_1, d_k+1)+kn_{k+1}f(d_1, 1)\\ = & \sum\limits_{j = 2}^{k}n_j(j-1)f(d_1, d_j+1)+kn_{k+1}f(d_1, 1), \\ A_2 = & 2 \binom{d_1}{2}f(d_2+1, d_2+1)+n_2\Big[d_2f(d_2+1, d_3+1)\\ & +2d_2d_3f(d_2+1, d_4+1) + \cdots\\ &+ d_2d_3\cdots d_{k-1}(k-2)f(d_2+1, d_k+1)\\ & +d_2d_3\cdots d_{k}(k-1)f(d_2+1, 1)\\ & +(n_2-1)(3d_2f(d_2+1, d_3+1)\\ & +4d_2d_3f(d_2+1, d_4+1)+\cdots\\ & +d_2d_3\cdots d_{k-1}kf(d_2+1, d_k+1)\\ &+d_2d_3\cdots d_{k}(k+1)f(d_2+1, 1))\Big]\\ = & d_1(d_1-1)f(d_2+1, d_2+1)\\ & +n_2\Big[n_3/d_1f(d_2+1, d_3+1) \\ & +2n_4/d_1f(d_2+1, d_4+1) + \cdots\\ &+ n_k/d_1(k-2)f(d_2+1, d_k+1)\\ & +n_{k+1}/d_1(k-1)f(d_2+1, 1)\\ & +(n_2-1)(3n_3/d_1f(d_2+1, d_3+1)\\ & +4n_4/d_1f(d_2+1, d_4+1)+\cdots\\ & +kn_k/d_1f(d_2+1, d_k+1)\\ & +n_{k+1}/d_1(k+1)f(d_2+1, 1))\Big]\\ = & d_1(d_1-1)f(d_2+1, d_2+1)\\ &+ n_2/d_1\Big[\sum\limits_{j = 3}^{k}(j-2)n_jf(d_2+1, d_j+1)\\ & +(k-1)n_{k+1}f(d_2+1, 1)\\ & +(n_2-1)(\sum\limits_{j = 3}^{k}jn_jf(d_2+1, d_j+1)\\ & +(k+1)n_{k+1}f(d_2+1, 1))\Big]\\ = &d_1(d_1-1)f(d_2+1, d_2+1)\\ & + \sum\limits_{j = 3}^{k}(j-2)n_jf(d_2+1, d_j+1)\\ & +(k-1)n_{k+1}f(d_2+1, 1)\\ & +(n_2-1)\Big[\sum\limits_{j = 3}^{k}jn_jf(d_2+1, d_j+1)\\ & +(k+1)n_{k+1}f(d_2+1, 1)\Big], \\ A_3 = & \Big[2\binom{d_2}{2}+4\binom{d_2}{1}\binom{n_3-d_2}{1}\Big]f(d_3+1, d_3+1)\\ & +n_3\Big[d_3f(d_3+1, d_4+1)\\ & +2d_3d_4f(d_3+1, d_5+1)+\cdots\\ & +(k-3)d_3d_4\cdots d_{k-1}f(d_3+1, d_{k}+1)\\ & +(k-2)d_3d_4\cdots d_k f(d_3+1, 1)\\ & +(d_2-1)(3d_3f(d_3+1, d_4+1)\\ & +4d_3d_4 f(d_3+1, d_5+1)+\cdots\\ & +(k-1)d_3d_4\cdots d_{k-1}f(d_3+1, d_{k}+1)\\ & +kd_3d_4\cdots d_k f(d_3+1, 1))\\ & +(n_3-d_2)(5d_3f(d_3+1, d_4+1)\\ & +6d_3d_4f(d_3+1, d_5+1)+\cdots\\ & +(k+1)d_3d_4\cdots d_{k-1}f(d_3+1, d_{k}+1)\\ & +(k+2)d_3d_4\cdots d_k f(d_3+1, 1))\Big]\\ = & \Big[2\binom{d_2}{2}+4\binom{d_2}{1}\binom{n_3-d_2}{1}\Big])f(d_3+1, d_3+1)\\ & +\sum\limits_{j = 4}^{k}n_j(j-3)f(d_3+1, d_j+1)\\ & +(k-2)n_{k+1}f(d_3+1, 1)\\ & +(d_2-1)\Big[\sum\limits_{j = 4}^{k}n_j(j-1)f(d_3+1, d_j+1)\\ & +kn_{k+1}f(d_3+1, 1)\Big]\\&+(n_3-d_2)\Big[\sum\limits_{j = 4}^{k}n_j(j+1)f(d_3+1, d_j+1)\\ & +n_{k+1}(k+2)f(d_3+1, 1)\Big]. \end{align*}$

By calculating similarly to the above, for any $2\leq l \leq k$ , we have

$\begin{align*} A_l = & \Big[2\binom{d_{l-1}}{2}+4\binom{d_{l-1}}{1}\binom{d_{l-1}d_{l-2}-d_{l-1}}{1}+\cdots\\ & +2(l-1)\binom{d_{l-1}\cdots d_{2}}{1}\binom{n_l-d_{l-1}\cdots d_{2}}{1}\Big]f(d_l+1, d_l+1)\\ & +\sum\limits_{j = l+1}^{k}n_j(j-l)f(d_l+1, d_j+1)\\&+(k-l+1)n_{k+1}f(d_l+1, 1)\\ & +(d_{l-1}-1)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+2)f(d_l+1, d_j+1)\\ & +(k-l+3)n_{k+1}f(d_l+1, 1)\Big]\\ & +(d_{l-1}d_{l-2}-d_{l-1})\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+4)f(d_l+1, d_j+1)\\ & +(k-l+5)n_{k+1}f(d_l+1, 1)\Big]+\cdots\\ & +(n_l-d_{l-1}\cdots d_2)\Big[\sum\limits_{j = l+1}^{k}n_j(j+l-2)f(d_l+1, d_j+1)\\ & +(k+l-1)n_{k+1}f(d_l+1, 1)\Big]. \end{align*}$

In particular, we have

$\begin{align*} A_{k+1} = & f(1, 1)\Big[2\binom{d_{k}}{2}+4\binom{d_{k}}{1}\binom{d_{k}d_{k-1}-d_{k}}{1}+\cdots\\ & +2k\binom{d_{k}\cdots d_{2}}{1}\binom{n_{k+1}-d_{k}\cdots d_{2}}{1}\Big]. \end{align*}$

This completes the proof. □

Corollary 3.2. The degree-weighted Wiener index of a Bethe tree $B_{k, d}$ is

$DWW(B_{k, d}) = \sum\limits_{l = 1}^{k}A_l,$

where

$\begin{align*} A_1 = & \sum\limits_{j = 2}^{k-1}d(d+1)^{j-2}(j-1)f(d, d+1)+(k-1)d(d+1)^{k-2}f(d, 1), \\ A_l = & \Big[2\binom{d+1}{2}+4\binom{d+1}{1}\binom{(d+1)^2-(d+1)}{1}+\cdots\\ & +2(l-1)\binom{d^{l-2}}{1}\binom{n_l-d^{l-2}}{1}\Big]f(d+1, d+1)\\ & +\sum\limits_{j = l+1}^{k-1}n_j(j-l)f(d+1, d+1)+(k-l)n_{k}f(d+1, 1)\\ & +(d_{l-1}-1)\Big[\sum\limits_{j = l+1}^{k-1}f(d+1, d+1)n_j(j-l+2)\\ & +(k-l+2)n_{k}f(d+1, 1)\Big]\\ & +(d_{l-1}d_{l-2}-d_{l-1})\Big[\sum\limits_{j = l+1}^{k-1}n_j(j-l+4)f(d+1, d+1)\\ & +(k-l+4)n_{k}f(d+1, 1)\Big] +\cdots\\ & +(n_l-d_{l-1}\cdots d_2)\Big[\sum\limits_{j = l+1}^{k-1}n_j(j+l-2)f(d+1, d+1)\\ & +(k+l-2)n_{k}f(d+1, 1)\Big], \\ A_{k} = & f(1, 1)\Big[2\binom{d}{2}+4\binom{d}{1}\binom{d(d-1)}{1}+\cdots\\ & +2(k-1)\binom{d^{k-2}}{1}\binom{(d-1)d^{k-2}}{1}\Big]. \end{align*}$

Corollary 3.3. The degree-weighted Wiener index of a regular dendrimer tree $T_{k, d}$ is

$DWW(T_{k, d}) = \sum\limits_{l = 1}^{k+1}A_l,$

where

$\begin{align*} A_1 = & \sum\limits_{j = 2}^{k}n_j(j-1)f(d, d)+kd(d-1)^{k-1}f(d, 1), \\ A_l = & \Big[2\binom{d-1}{2}+4\binom{d-1}{1}\binom{(d-1)(d-2)}{1}+\cdots\\ & +2(l-1)\binom{(d-1)^{l-2}}{1}\binom{n_l-(d-1)^{l-2}}{1}\Big]f(d, d)\\ & +\sum\limits_{j = l+1}^{k}n_j(j-l)f(d, d)+(k-l+1)n_{k+1}f(d, 1)\\ & +(d-2)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+2)f(d, d)\\ & +(k-l+3)n_{k+1}f(d, 1)\Big]\\ & +(d-1)(d-2)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+4)f(d, d)\\ & +(k-l+5)n_{k+1}f(d, 1)\Big] +\cdots\\ & +(n_l-(d-1)^{l-2})\Big[\sum\limits_{j = l+1}^{k}n_j(j+l-2)f(d, d)\\ & +(k+l-1)n_{k+1}f(d, 1)\Big], \\ A_{k+1} = & f(1, 1)\Big[2\binom{d-1}{2}+4\binom{d-1}{1}\binom{(d-1)(d-2)}{1}+\cdots\\ & +2k\binom{(d-1)^{k-1}}{1}\binom{(d-1)^{k}}{1}\Big]. \end{align*}$

4. Applications of SW indices to the acentric factor of octane isomers

In this section, the chemical applicability of the SW index is investigated. The acentric factor (AcenFac) is a measure of the non-sphericity of molecules. We consider the correlation between acentric factors of octane isomers and the respective SW indices. The experimental values of acentric factors of octane isomers were taken from http://www.moleculardescriptors.eu/dataset/dataset.htm.

Using the data from Table 1, we find the correlation of AcenFac with the value of SW index for octane isomers; see Figure 1. The following equations give the regression models for the SW index:

$AcenFac = 0.00198\times SW+0.008141.$

Table 1. Experimental values of AcenFac and SW index for octane isomers.

Molecule	AcenFac	SW
Octane	0.397898	202.8093
2-methyl-heptane	0.377916	191.2453
3-methyl-heptane	0.371002	182.2057
4-methyl-heptane	0.371504	179.1925
3-ethyl-hexane	0.362472	170.5225
2, 2-dimethyl-hexane	0.339426	170.4970
2, 3-dimethyl-hexane	0.348247	166.8181
2, 4-dimethyl-hexane	0.344223	169.9447
2, 5-dimethyl-hexane	0.35683	179.0977
3, 3-dimethyl-hexane	0.322596	158.2653
3, 4-dimethyl-hexane	0.340345	160.7917
2-methyl-3-ethyl-pentane	0.332433	157.9633
3-methyl-3-ethyl-pentane	0.306899	149.3210
2, 2, 3-trimethyl-pentane	0.300816	148.2544
2, 2, 4-trimethyl-pentane	0.30537	157.5862
2, 3, 3-trimethyl-pentane	0.293177	145.1517
2, 3, 4-trimethyl-pentane	0.317422	153.9314
2, 2, 3, 3-tetramethylbutane	0.255294	135.0271

| Show Table

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Figure 1. Scatter plot between acentric factor of Octane isomers and their SW index.

DownLoad: Full-Size Img PowerPoint

Thus, the SW index can also help to predict the properties of octane isomers.

5. Conclusions

In this paper, we propose the SW index, and establish some mathematical relations between the Harary-Sombor index and other classic topological indices. Morover, we obtain the calculation formula of degree-weighted Wiener index for generalized Bethe trees. In addition, some numerical results are discussed. We calculate the SW index of octane isomers. The regression models show that the AcenFac and SW index of octane isomers are highly correlated.

In 1993, Klein and Randić ^[12] introduced the notion of resistance distance. Naturally, from the perspective of distance, we similarly propose the degree-weighted resistance-distance index of a graph $G$ and define it as

$DWR(G) = \sum\limits_{\{u, v \}\subseteq V(G)}f(d_u, d_v)r(u, v),$

where $r(u, v)$ is the resistance distance between $u$ and $v$ . It would be interesting to explore chemical and mathematical properties and possible predictive potential of this index.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are thankful to the anonymous referees for their helpful comments that improved the quality of the manuscript. This work was funded by the National Natural Science Foundation of China under Grant No. 12261074.

Conflict of interest

The authors declare no conflicts of interest to this work.

References

[1]	O. D. Lara, M. A. Labrador, A survey on human activity recognition using wearable sensors, IEEE Commun. Surv. Tutorials, 15 (2013), 1192–1209. doi: 10.1109/SURV.2012.110112.00192. doi: 10.1109/SURV.2012.110112.00192
[2]	J. P. Queralta, T. N. Gia, H. Tenhunen, T. Westerlund, Edge-AI in LoRa-based health monitoring: fall detection system with fog computing and LSTM cecurrent neural networks, in Proceedings of the 42nd International Conference on Telecommunications, Signal Processing (TSP), Budapest, (2019), 601–604. doi: 10.1109/TSP.2019.8768883.
[3]	E. Shirin, Z. Ahmed, M. Andreas, S. Severin, A. S. Thomas, E. Tarek, et al., Health management and pattern analysis of daily living activities of people with dementia using in-home sensors and machine learning techniques, PLoS ONE, 13 (2018), e0195605. doi: 10.1371/journal.pone.0195605. doi: 10.1371/journal.pone.0195605
[4]	J. A. Ward, G. Pirkl, P. Hevesi, P. Lukowicz, Towards recognising collaborative activities using multiple on-body sensors, in Proceedings of the 2016 ACM International Joint Conference on Pervasive and Ubiquitous Computing (UbiComp Adjunct), (2016), 221–224. doi: 10.1145/2968219.2971429.
[5]	J. Wang, Y. Chen, S. Hao, X. Peng, L. Hu, Deep learning for sensor-based activity recognition: a survey, Pattern Recognit. Lett., 119 (2019), 3–11. doi: 10.1016/j.patrec.2018.02.010. doi: 10.1016/j.patrec.2018.02.010
[6]	P. Voigt, A. Bussche, The eu general data protection regulation (gdpr), A Practical Guide, 1st Ed., Cham: Springer International Publishing, 10 (2017), 3152676.
[7]	B. Mcmahan, E. Moore, D. Ramage, S. Hampson, B. A. Arcas, Communication-efficient learning of deep networks from decentralized data, in Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS), 54 (2017), 1273–1282.
[8]	Y. Zhao, M. Li, L. Lai, N. Suda, D. Civin, V. Chandra, Federated learning with Non-IID data, preprint, arXiv: 1806.00582.
[9]	Y. Liu, Y. Liu, Z. Liu, J. Zhang, C. Meng, Y. Zheng, Federated forest, IEEE Trans. Big Data, 2020. doi: 10.1109/TBDATA.2020.2992755. doi: 10.1109/TBDATA.2020.2992755
[10]	K. Cheng, T. Fan, Y. Jin, Y. Liu, T. Chen, Q. Yang, Secureboost: A lossless federated learning framework, IEEE Intell. Syst., 2021. doi: 10.1109/MIS.2021.3082561. doi: 10.1109/MIS.2021.3082561
[11]	Q. Li, Z. Wen, B. He, Practical federated gradient boosting decision trees, in Proceedings of the AAAI Conference on Artificial Intelligence, 34 (2020), 4642–4649. doi: 10.1609/aaai.v34i04.5895
[12]	Y. Liu, M. Chen, W. Zhang, J. Zhang, Y. Zheng, Federated extra-trees with privacy preserving, preprint, arXiv: 2002.07323.
[13]	L. T. Phong, Y. Aono, T. Hayashi, L. Wang, S. Moriai, Privacy-preserving deep learning via additively homomorphic encryption, IEEE Trans. Inf. Forensics Secur., 13 (2017), 1333–1345. doi: 10.1109/TIFS.2017.2787987. doi: 10.1109/TIFS.2017.2787987
[14]	C. Dwork, Differential privacy, in Proceedinds of the 33rd International Colloquium on Automata, Languages and Programming (ICALP), 4052 (2006), 1–12. doi: 10.1007/11787006_1.
[15]	C. Gentry, A Fully Homomorphic Encryption Scheme, Ph.D thesis, Stanford University, 2009.
[16]	F. Mo, H. Haddadi, K. Katevas, E. Marin, D. Perino, N. Kourtellis, PPFL: privacy-preserving federated learning with trusted execution environments, in Proceedings of the 19th Annual International Conference on Mobile Systems, Applications and Services (Mobisys), (2021), 94–108. doi: 10.1145/3458864.3466628.
[17]	D. Polap, G. Srivastava, K. Yu, Agent architecture of an intelligent medical system based on federated learning and blockchain technology, J. Inf. Secur. Appl., 58 (2021), 102748. doi: 10.1016/j.jisa.2021.102748. doi: 10.1016/j.jisa.2021.102748
[18]	K. Sozinov, V. Vlassov, S. Girdzijauskas, Human activity recognition using federated learning, in 2018 IEEE International Conference on Big Data and Cloud Computing (BDCloud), (2018), 1103–1111. doi: 10.1109/BDCloud.2018.00164.
[19]	A. Gionis, P. Indyk, R. Motwani, Similarity search in high dimensions via hashing, in Proceedings of 25th International Conference on Very Large Data Bases, (1999), 518–529.
[20]	K. Wang, R. Mathews, C. Kiddon, H. Eichner, F. Beaufays, D. Ramage, Federated evaluation of on-device personalization, preprint, arXiv: 1910.10252.
[21]	Y. Chen, X. Qin, J. Wang, C. Yu, W. Gao, Fedhealth: A federated transfer learning framework for wearable healthcare, IEEE Intell. Syst., 35 (2020), 83–93. doi: 10.1109/MIS.2020.2988604. doi: 10.1109/MIS.2020.2988604
[22]	R. Caruana, Multitask learning, Mach. learn., 28 (1997), 41–75. doi: 10.1023/A:1007379606734. doi: 10.1023/A:1007379606734
[23]	V. Smith, C. Chiang, M. Sanjabi, A. Talwalkar, Federated multi-task learning, in Annual Conference on Neural Information Processing Systems (NIPS), (2017), 4424–4434.
[24]	T. Yu, E. Bagdasaryan, V. Shmatikov, Salvaging federated learning by local adaptation, preprint, arXiv: 2002.04758.
[25]	B. Wang, S. Yu, W. Lou, Y. T. Hou, Privacy-preserving multi-keyword fuzzy search over encrypted data in the cloud, in Proceedings of the 2014 IEEE Conference on Computer Communications (INFOCOM), (2014), 2112–2120. doi: 10.1109/INFOCOM.2014.6848153.
[26]	L. Qi, X. Zhang, W. Dou, Q. Ni, A distributed locality-sensitive hashing-based approach for cloud service recommendation from multi-source data, IEEE J. Sel. Areas Commun., 35 (2017), 2616–2624. doi: 10.1109/JSAC.2017.2760458. doi: 10.1109/JSAC.2017.2760458
[27]	M. Datar, N. Immorlica, P. Indyk, V. Mirrokni, Locality-sensitive hashing scheme based on p-stable distributions, in Proceedings of the 20th Annual Symposium on Computational Geometry, 34 (2004), 253–262. doi: 10.1145/997817.997857.
[28]	C. Dwork, F. Mcsherry, K. Nissim, A. Smith, Calibrating noise to sensitivity in private data analysis, in Proceedings of the Third conference on Theory of Cryptography, 3876 (2006), 265–284. doi: 10.1007/11681878_14.
[29]	F. Mcsherry, K. Talwar, Mechanism design via differential privacy, in 48th Annual IEEE Symposium on Foundations of Computer Science, (2007), 94–103. doi: 10.1109/FOCS.2007.41.
[30]	P. Xiong, T. Q. Zhu, X. F. Wang, A survey on differential privacy and applications, Chin. J. Comput., 37 (2014), 101–122. doi: 10.3724/SP.J.1016.2014.00101. doi: 10.3724/SP.J.1016.2014.00101
[31]	F. Mcsherry, Privacy integrated queries: an extensible platform for privacy-preserving data analysis, in Proceedings of the 2009 ACM SIGMOD International Conference on Management of data, (2009), 19–30. doi: 10.1145/1559845.1559850.
[32]	P. Patarasuk, X. Yuan, Bandwidth optimal all-reduce algorithms for clusters of workstations, J. Parallel Distrib. Comput., 69 (2009), 117–124. doi: 10.1016/j.jpdc.2008.09.002. doi: 10.1016/j.jpdc.2008.09.002
[33]	L. Breiman, J. Friedman, C. J. Stone, R. A. Olshen, Classification and regression trees, The Wadsworth and Brooks-Cole statistics-probability series, 1984.
[34]	Z. H. Zhou, J. Wu, W. Tang, Ensembling neural networks: many could be better than all, Artif. Intell., 137 (2002), 239–263. doi: 10.1016/S0004-3702(02)00190-X. doi: 10.1016/S0004-3702(02)00190-X
[35]	H. Chen, P. Tiňo, X. Yao, Predictive ensemble pruning by expectation propagation, IEEE Trans. Knowl. Data Eng., 21 (2009), 999–1013. doi: 10.1109/TKDE.2009.62. doi: 10.1109/TKDE.2009.62
[36]	A. Friedman, A. Schuster, Data mining with differential privacy, in Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining, (2010), 493–502. doi: 10.1145/1835804.1835868.
[37]	S. Fletcher, M. Z. Islam, Decision tree classification with differential privacy: a survey, ACM Comput. Surv. (CSUR), 52 (2019), 1–33. doi: 10.1145/3337064. doi: 10.1145/3337064
[38]	A. Patil, S. Singh, Differential private random forest, in 2014 International Conference on Advances in Computing, Communications and Informatics (ICACCI), (2014), 2623–2630. doi: 10.1109/ICACCI.2014.6968348.
[39]	D. Anguita, A. Ghio, L. Oneto, X. Parra, J. L. Reyes-Ortiz, Human activity recognition on smartphones using a multiclass hardware-friendly support vector machine, Int. workshop ambient assisted living, 7657 (2012), 216–223. doi: 10.1007/978-3-642-35395-6_30. doi: 10.1007/978-3-642-35395-6_30
[40]	R. K. Jennifer, M. W. Gary, M. Samuel, Activity recognition using cell phone accelerometers, ACM SigKDD Explor. Newsl., 12 (2010), 74–82. doi: 10.1145/1964897.1964918. doi: 10.1145/1964897.1964918

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Mathematical Biosciences and Engineering

Federated personalized random forest for human activity recognition

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Abstract

1. Introduction

2. Basic properties of the SW index

3. Degree-weighted Wiener index of generalized Bethe trees

4. Applications of SW indices to the acentric factor of octane isomers

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

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Abstract

1. Introduction

2. Basic properties of the SW index

3. Degree-weighted Wiener index of generalized Bethe trees

4. Applications of SW indices to the acentric factor of octane isomers

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Mathematical Biosciences and Engineering

Federated personalized random forest for human activity recognition

Related Papers:

Abstract

1. Introduction

2. Basic properties of the SW index

3. Degree-weighted Wiener index of generalized Bethe trees

4. Applications of SW indices to the acentric factor of octane isomers

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

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Abstract

1. Introduction

2. Basic properties of the SW index

3. Degree-weighted Wiener index of generalized Bethe trees

4. Applications of SW indices to the acentric factor of octane isomers

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References