shell}(p)+3\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+3\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]$
$\displaystyle=$ $\displaystyle
3\left(\delta[\delta_{\Lambda}^{(1)}]\right)^{(1+2)}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\frac{17}{7}\left(\delta^{2}[\delta_{\Lambda}^{(1)}]\right)^{(2)}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,,$

where we see that the first two terms contribute to
$\delta[\delta_{\Lambda}^{(1)}]$ (at leading and second order) and the final
term contributes to $\delta^{2}[\delta_{\Lambda}^{(1)}]$. We highlight again
that the contributions to $\left(\delta[\delta_{\Lambda}^{(1)}]\right)^{(1)}$
and $\left(\delta[\delta_{\Lambda}^{(1)}]\right)^{(2)}$ have the same
coefficient.

  * •

$O=\delta^{4}$. In order to understand the expected behaviour of fourth-order
terms, we calculate the shell contribution from $\delta^{4}$:

$\displaystyle\mathcal{S}^{2}_{\delta^{4}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle 6\int_{\bm{p},\bm{p}_{1}}P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{k}-\bm{p}_{1})+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]$
(A.17) $\displaystyle=$ $\displaystyle
6\left(\delta^{2}[\delta_{\Lambda}^{(1)}]\right)^{(2)}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,,$

which contributes only to $\delta^{2}[\delta_{\Lambda}^{(1)}]$. We notice that
the operators of order $n$ can only contribute to operators starting from
order $n-2$.

  * •

$O=\mathcal{G}_{2}$:

$\displaystyle\mathcal{S}^{2}_{\mathcal{G}_{2}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
4\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}\sigma^{2}_{\bm{k}+\bm{p},-\bm{p}}F_{2}(\bm{k},\bm{p})P_{\rm
shell}(p)$ (A.18) $\displaystyle+$ $\displaystyle
4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})$
$\displaystyle\hskip
142.26378pt\times\sigma^{2}_{\bm{p}_{2}-\bm{p},\bm{p}+\bm{p}_{1}}P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle
6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}F_{3}(\bm{p},-\bm{p},\bm{p}_{1})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle
6\int_{\bm{p},\bm{p}_{1},\bm{p}_{1}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{-\bm{p},\bm{k}+\bm{p}}F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\stackrel{{\scriptstyle k\ll p}}{{=}}$
$\displaystyle\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}\,P_{\rm
shell}(p)\left[-\frac{32}{21}\frac{k^{2}}{p^{2}}+\dots\right]+k^{2}[\dots]\,.$

Notice that all terms have a structure like
$\sigma^{2}_{\bm{p}+\dots,-\bm{p}+\dots}$ or
$F_{3}(\bm{p},-\bm{p},\bm{p}_{1})$. Both types of terms scale as $p^{2}$ and
only contribute to higher-derivative terms. That is a consequence of the non-
renormalization theorem for Galileon operators [49, 11].

  * •

$O=\mathcal{G}_{2}\delta$:

$\displaystyle\mathcal{S}^{2}_{\mathcal{G}_{2}\delta}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
2\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}\sigma^{2}_{\bm{p},\bm{k}}P_{\rm
shell}(p)$ (A.19) $\displaystyle+$ $\displaystyle
4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p},\bm{p}_{1}}F_{2}(\bm{p}_{2},-\bm{p})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{k}-\bm{p}_{1})$
$\displaystyle+$ $\displaystyle
4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}+\bm{p}_{1},\bm{p}}F_{2}(\bm{p},\bm{p}_{1})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle
4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}+\bm{p}_{1},\bm{p}_{2}}F_{2}(\bm{p},\bm{p}_{1})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle
2\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{k},\bm{p}}F_{2}(\bm{p}_{1},\bm{p}_{2})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\,.$

The first term

$\displaystyle
2\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}\sigma^{2}_{\bm{k},\bm{p}}P_{\rm
shell}(p)=-\frac{4}{3}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)\,,$ (A.20)

and the last term

$\displaystyle 2\int_{\bm{p}}\sigma^{2}_{\bm{k},\bm{p}}P_{\rm
shell}(p)\int_{\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p}_{2})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
(A.21) $\displaystyle=-\frac{4}{3}\int_{\bm{p}}P_{\rm
shell}(p)\int_{\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})F_{2}(\bm{p}_{1},\bm{p}_{2})$
$\displaystyle\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\,,$

are a correction to $\delta$, where we notice that once more the first and
second order prefactors match. The term in the second line of Eq. (A.19) reads

$\displaystyle 4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p},\bm{p}_{1}}F_{2}(\bm{p}_{2},-\bm{p})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
(A.22) $\displaystyle=\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[-\frac{232}{105}+\frac{16}{105}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})^{2}}{p_{1}^{2}p_{2}^{2}}\right]$
$\displaystyle P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\,,$

and will contribute to $\delta^{2}$ and $\mathcal{G}_{2}$. The term in the
fourth line of Eq. (A.19) leads to

$\displaystyle 4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}+\bm{p}_{1},\bm{p}_{2}}F_{2}(\bm{p},\bm{p}_{1})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
(A.23) $\displaystyle=\quad\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[-\frac{52}{21}+\frac{20}{21}\frac{(\bm{p}_{1}\cdot\bm{p}_{2})^{2}}{p_{1}^{2}p_{2}^{2}}\right]P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\,,$

Finally

$\displaystyle\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}+\bm{p}_{1},\bm{p}}F_{2}(\bm{p},\bm{p}_{1})P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
(A.24)

sources higher-derivatives due to $\sigma^{2}_{\bm{p}+\bm{p}_{1},\bm{p}}$.
Summing everything up, we find

$\displaystyle\mathcal{S}^{2}_{\mathcal{G}_{2}\delta}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$
$\displaystyle\left[-\frac{4}{3}\delta^{(1+2)}(\bm{k})-\frac{376}{105}\delta^{2,(2)}(\bm{k})+\frac{116}{105}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}(p)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\mbox{h.d.}+\mathcal{O}\left[\left(\delta^{(1)}_{\Lambda}\right)^{3}\right]\,.$

  * •

$O=\mathcal{G}_{3}$: Following the non-renormalization properties of Galileons
[49], this term will only contribute to higher-derivatives as well [11].

  * •

$O=\Gamma_{3}$. Similar to the Galileon operator, all terms have a structure
like $\sigma^{2}_{\bm{p}+\dots,-\bm{p}+\dots}$ or
$F_{3}(\bm{p},-\bm{p},\bm{p}_{1})$:

$\displaystyle\mathcal{S}^{2}_{\Gamma_{3}}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=$ $\displaystyle
4\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}\sigma^{2}_{\bm{k}+\bm{p},-\bm{p}}\left[F_{2}(\bm{k},\bm{p})-G_{2}(\bm{k},\bm{p})\right]P_{\rm
shell}(p)$ $\displaystyle+$ $\displaystyle
4\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\left[F_{2}(\bm{p},\bm{p}_{1})F_{2}(-\bm{p},\bm{p}_{2})-G_{2}(\bm{p},\bm{p}_{1})G_{2}(-\bm{p},\bm{p}_{2})\right]\sigma^{2}_{\bm{p}_{2}-\bm{p},\bm{p}+\bm{p}_{1}}$
$\displaystyle\hskip 227.62204pt\times P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle
6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}\left[F_{3}(\bm{p},-\bm{p},\bm{p}_{1})-G_{3}(\bm{p},-\bm{p},\bm{p}_{1})\right]$
$\displaystyle\hskip 227.62204pt\times P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle+$ $\displaystyle
6\int_{\bm{p},\bm{p}_{1},\bm{p}_{2}}\delta_{\rm
D}(\bm{k}-\bm{p}_{12})\sigma^{2}_{-\bm{p},\bm{k}+\bm{p}}\left[F_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})-G_{3}(\bm{p},\bm{p}_{1},\bm{p}_{2})\right]$
$\displaystyle\hskip 227.62204pt\times P_{\rm
shell}(p)\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\,,$

which only contributes to higher-derivative terms.

##  Appendix B Solution when including proxy for higher-order operators

We display here the solution for the ODE system Eqs. (2.34)-(2.36) when taking
the ansatz Eq. (2.40) for the evolution of higher-order bias parameters:

$\displaystyle b_{\delta}(\sigma^{2})$ $\displaystyle=$ $\displaystyle
b_{\delta}^{\ast}+\frac{b^{*(\delta)}_{n=3}\left(e^{-c^{(\delta)}\sigma^{2}}-e^{-c^{(\delta)}\text{$\sigma^{2}_{\ast}$}}\right)}{c^{(\delta)}}$
$\displaystyle+$
$\displaystyle\frac{e^{\text{$\sigma^{2}_{\ast}$}c_{\delta^{2},\delta^{2}}}c_{\delta,\delta^{2}}\left[c^{(\delta^{2})}\left(b_{\delta^{2}}^{\ast}c^{(\delta^{2})}-b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\text{$\sigma^{2}_{\ast}$}}\right)\left(e^{-\sigma^{2}c_{\delta^{2},\delta^{2}}}-e^{-\text{$\sigma_{\ast}$}c_{\delta^{2},\delta^{2}}}\right)\right]}{c^{(\delta^{2})}c_{\delta^{2},\delta^{2}}\left(c^{(\delta^{2})}-c_{\delta^{2},\delta^{2}}\right)}$
$\displaystyle+$
$\displaystyle\frac{c_{\delta,\delta^{2}}\left[c_{\delta^{2},\delta^{2}}\left(b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\sigma^{2}}-b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\text{$\sigma^{2}_{\ast}$}}-b_{\delta^{2}}^{\ast}c^{(\delta^{2})}+b_{\delta^{2}}^{\ast}c^{(\delta^{2})}e^{(\text{$\sigma^{2}_{\ast}$}-\text{$\sigma^{2}$})c_{\delta^{2},\delta^{2}}}\right)\right]}{c^{(\delta^{2})}c_{\delta^{2},\delta^{2}}\left(c^{(\delta^{2})}-c_{\delta^{2},\delta^{2}}\right)}\,,$
$\displaystyle b_{\delta^{2}}(\sigma^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{-(\sigma^{2}-\text{$\sigma^{2}_{\ast}$})c_{\delta^{2},\delta^{2}}}\left(b_{\delta^{2}}^{\ast}c_{\delta^{2},\delta^{2}}+b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\text{$\sigma^{2}_{\ast}$}}-b_{\delta^{2}}^{\ast}c^{(\delta^{2})}\right)-b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\sigma^{2}}}{c_{\delta^{2},\delta^{2}}-c^{(\delta^{2})}}\,,$
(B.2) $\displaystyle b_{\mathcal{G}_{2}}(\sigma^{2})$ $\displaystyle=$
$\displaystyle
b_{\mathcal{G}_{2}}^{\ast}+\frac{b^{*(\mathcal{G}_{2})}_{n=3+4}\left(e^{-c^{(\mathcal{G}_{2})}\sigma^{2}}-e^{-c^{(\mathcal{G}_{2})}\text{$\sigma^{2}_{\ast}$}}\right)}{c^{(\mathcal{G}_{2})}}$
$\displaystyle+$
$\displaystyle\frac{e^{\text{$\sigma^{2}_{\ast}$}c_{\delta^{2},\delta^{2}}}c_{\mathcal{G}_{2},\delta^{2}}\left[c^{(\delta^{2})}\left(b_{\delta^{2}}^{\ast}c^{(\delta^{2})}-b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\text{$\sigma^{2}_{\ast}$}}\right)\left(e^{-\sigma^{2}c_{\delta^{2},\delta^{2}}}-e^{-\text{$\sigma_{\ast}$}c_{\delta^{2},\delta^{2}}}\right)\right]}{c^{(\delta^{2})}c_{\delta^{2},\delta^{2}}\left(c^{(\delta^{2})}-c_{\delta^{2},\delta^{2}}\right)}$
$\displaystyle+$
$\displaystyle\frac{c_{\mathcal{G}_{2},\delta^{2}}\left[c_{\delta^{2},\delta^{2}}\left(b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\sigma^{2}}-b^{*(\delta^{2})}_{n=3+4}e^{-c^{(\delta^{2})}\text{$\sigma^{2}_{\ast}$}}-b_{\delta^{2}}^{\ast}c^{(\delta^{2})}+b_{\delta^{2}}^{\ast}c^{(\delta^{2})}e^{(\text{$\sigma^{2}_{\ast}$}-\text{$\sigma^{2}$})c_{\delta^{2},\delta^{2}}}\right)\right]}{c^{(\delta^{2})}c_{\delta^{2},\delta^{2}}\left(c^{(\delta^{2})}-c_{\delta^{2},\delta^{2}}\right)}\,.$

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# Cooperation and Federation in Distributed Radar Point Cloud Processing
††thanks: Funded by the European Union. Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union or European Innovation Council and SMEs Executive Agency
(EISMEA). Neither the European Union nor the granting authority can be held
responsible for them. Grant Agreement No: 101099491

S. Savazzi, V. Rampa, S. Kianoush  Consiglio Nazionale delle Ricerche (CNR)
IEIIT institute, Milano
Email<EMAIL_ADDRESS>A. Minora, L. Costa  CogniMade S.r.l,
https://www.cognimade.com
Via C. Colombo 10/A, I-20066 Melzo, Italy
Email<EMAIL_ADDRESS>
###### Abstract

The paper considers the problem of human-scale RF sensing utilizing a network
of resource-constrained MIMO radars with low range-azimuth resolution. The
radars operate in the mmWave band and obtain time-varying 3D point cloud (PC)
information that is sensitive to body movements. They also observe the same
scene from different views and cooperate while sensing the environment using a
sidelink communication channel. Conventional _cooperation_ setups allow the
radars to mutually exchange raw PC information to improve ego sensing. The
paper proposes a _federation_ mechanism where the radars exchange the
parameters of a Bayesian posterior measure of the observed PCs, rather than
raw data. The radars act as distributed parameter servers to reconstruct a
global posterior (i.e., federated posterior) using Bayesian tools. The paper
quantifies and compares the benefits of radar federation with respect to
cooperation mechanisms. Both approaches are validated by experiments with a
real-time demonstration platform. Federation makes minimal use of the sidelink
communication channel ($20\div 25$ times lower bandwidth use) and is less
sensitive to unresolved targets. On the other hand, cooperation reduces the
mean absolute target estimation error of about $20\%$.

###### Index Terms:

Distributed and federated radar networks, Point Cloud processing,
localization, RF sensing, Bayesian estimation.

##  I Introduction

The emerging paradigm of integrated radar sensing and communication represents
a promising enabler for pervasive smart environments. These systems target
both network performance improvements [1] and enhanced spatial discovery
functions over which applications/verticals can be optimally implemented [2].
Targeting indoor environments, recent advances in hardware and digitalization
technology enabled the deployment of low-cost, low-power COTS (Commercial Off-
The-Shelf) radar devices of small dimension and suitable for smart home and
industrial applications [3]. In this framework, a crucial challenge is related
to the development of accurate modelling tools to optimize the interactions
between radar devices and environment as well as among the radars themselves.

Radar technology has recently evolved to support a multitude of functions,
ranging from detection, classification, multi-subject localization [3], human-
robot cooperation [4], human gait, gesture, and activity recognition [5, 6].
Indoor classification and identification of human subjects is based on the
processing of point cloud (PC) information extracted from millimeter-wave
radars with relative low angular resolution and number of antennas [7, 8]. PCs
or raw data returned by the radar can be used as input to machine learning [7,
9] to extract the features of the target. Merging of PC data from multiple
radar sensors have been reported in [10] for better detection. A deep learning
approach to fuse the features of both PCs and range–Doppler for classifying
activities has been proposed in [9]. A Bayesian approach for fusing/grouping
of radar data [11] is shown to help in angular resolution and provide added
information about the object which can be used for subsequent target/object
detection. On the other hand, the real-time exchange of radar data can be
infeasible in typical localization and tracking applications found in smart
home environments.

Figure 1:  Model of the proposed federated radar system and its point cloud
representation. Figure 2: Cooperation v. Federation PC processing
architectures: examples from real data. From left to right: (a) scene
$\mathbb{S}_{t}$ consist of 2 targets co-present and moving in opposite
directions with two radars monitoring the same scene; (b) _cooperation mode_ :
the radars exchange pre-processed PCs $\mathbb{\widehat{Q}}_{t,k}$ and obtain
a target probability measure via local fusion of the received PCs; (c)
_federation mode_ : the radars exchange the parameters $\mathbb{P}_{k}$
characterizing the local posterior measure approximated as Gaussian mixture
(number of Gaussian components $M_{k}$, weights $\beta_{k,m}$, mean
$\mathbf{\boldsymbol{\mu}}_{k,m}$, and covariance $\boldsymbol{\Gamma}_{k,m}$
of each component $m$).

The paper investigates radar cooperation models where a sidelink communication
channel of limited bandwidth is used to coordinate the sensing task. According
to the scenario in Fig. 1, a network of radars is deployed to monitor the same
scene $\mathbb{S}_{t}$ at time $t$. Each radar is equipped with multiple
antennas, operates in frequency modulated continuous wave (FMCW) mode [5], and
obtains time-varying 2D/3D Point Cloud (PC) information, through
local/independent Digital Signal Processing (DSP) stages. Usage of multiple
radars in the same environment can mitigate shadowing effects of the sensed
PCs as well as improve resolution that would be obtained from ego behavior. On
the other hand, effective denoising algorithms are needed to limit
parallel/cross chirp interference sources, Signal to Noise Ratio (SNR)
degradation, blind spots, and ghost object effects. The joint analysis of the
spatio-temporal structure of the radar PCs is here explored with the goal of
improving both accuracy and resolution of the individual radars.

Contributions. Unlike classical distributed radar processing that merge/fuse
raw data and/or PC information, the paper proposes a _federation model_ that
does not rely on data intensive fusion of the radars raw samples on a server.
The model is rooted at federation tools [12], and exploits sidelink
communication channels of limited bandwidths. The radars keep the raw data on
the devices, thus protecting data ownership as in as in classical federation
approaches. On the other hand, rather than exchanging the machine learning
model parameters, we propose an architecture where the radars share the
parameters of a Bayesian local posterior probability measure that describes
the (local) “beliefs” on the observed scene $\mathbb{S}_{t}$. Local posteriors
follow a Gaussian mixture model that is chosen for point clouds (PC)
representation. Proposed federation model is evaluated experimentally
targeting a building automation scenario. It is compared with classical
cooperation, where the radars exchange raw or pre-processed PC info. The
benefits of federation vs. cooperation are quantified considering accuracy,
resolution, and communication requirements as well.

The paper is organized as follows. Sect. II introduces the system model and
the PC processing tools. Cooperation and Bayesian federation settings are
analyzed in Sect. III, while Sect. IV describes the developed prototype and
presents the results of the on-field experiments. Finally, future activities
are discussed in Sect. V.

##  II Point cloud model and processing

As depicted in Fig. 1, the focus is on an indoor scenario where $K$ radio
devices operate as MIMO radars to sense and track the same time-varying scene
$\mathbb{S}_{t}$ from different points of view. The scene might be indicative
of specific body trajectories followed by one or more users (i.e., the
targets) in the monitored space, or specific actions performed by a
user/subject in a specific location(s). The observed scene is represented in
terms of (true) PCs $\mathbf{q}_{i}\in\mathbb{S}_{t}$ relative to the moving
targets in a 2D or 3D space: these points can be generally described through
an empirical probability mass function on a 2D or 3D pixel/voxel grid. At any
time $t>0,$ the scene $\mathbb{S}_{t}$ follows a stochastic random process
with probability function $p(\cdot)$ where
$p(\mathbf{q}_{i}\in\mathbb{S}_{t+1}|\mathbb{S}_{t})$ is assumed to be
memoryless (i.e., Markovian): therefore, the PC distribution of the scene at
time $t+1$ depends only on the scene at present time $t$.

Each $k$-th device is equipped with a radar and measures time-varying (3D)
noisy PC information $\mathbf{\widehat{q}}_{i,k}\in\mathbb{Q}_{t,k}$. In
particular, we adopt the following PC formation model

$\mathbb{Q}_{t,k}=\left\\{\mathbf{\widehat{q}}_{i,k}\right\\}=\left\\{\mathbf{q}_{i}+\mathbf{n}_{i,k}\right\\}_{\mathbf{q}_{i}\in\mathbb{S}_{t}}\cup\left\\{\mathbf{o}_{j}\right\\}_{\mathbf{o}_{j}\in\mathbb{O}_{t,k}},$
(1)

where $\mathbf{n}_{i,k}$ is the additive noise produced by $k$-th radar PC
processing hardware, and $\mathbf{o}_{j}\in\mathbb{O}_{t,k}$ is a set of
outlier points, typically caused by chirp interference, and SNR degradation.
In what follows, the PCs are obtained locally using the Cell Averaging (CA)
Constant False Alarm Rate (CA-CFAR) method. Other approaches may be
implemented as in [13].

The radar devices operate autonomously and independently to process and
extract local point clouds $\mathbb{Q}_{t,k}$. On the other hand, a sidelink
communication channel is used by the radars to share selected information with
the goal of improving ego knowledge. The topology of the radar network is
modelled as a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{\xi})$ with
the set of nodes $\mathcal{V}=\left\\{1,...,K\right\\}$ and edges, or links,
$\mathcal{\xi}=\left\\{1,...,L\right\\}$. The neighbor set of the $k$-th
device is denoted as $\mathcal{N}_{k}=\left\\{h\in\mathcal{V},h\neq
k:(h,k)\in\mathcal{\xi}\right\\}$, with cardinality
$\left|\mathcal{N}_{k}\right|$.

The goal of the $k$-th radar at time $t$ is to output a higher quality
reconstructed scene $\mathbb{\widehat{S}}_{t,k}$ by using side information
obtained from the radar neighborhood. In the following, we define the ensemble
of PCs in the neighbor set $\mathcal{N}_{k}$, as
$\mathbb{Q}_{t,k}\cup\left\\{\mathbb{Q}_{t+\bigtriangleup
t_{h},h}\right\\}_{h\in\mathcal{N}_{k}},$ where $\mathbb{Q}_{t+\bigtriangleup
t_{h},h}$ collects the PCs obtained from neighbor $h\in\mathcal{N}_{k}$ and
$\bigtriangleup t_{h}$ accounts for the relative time offset w.r.t. the $k$-th
radar. Compared to the observed points $\mathbb{Q}_{t,k}$, the estimated scene
$\mathbb{\widehat{S}}_{t,k}$ should be closer to the true one
$\mathbb{S}_{t}$, and suitable for further machine learning based processing,
e.g. based on PointNet methods [14, 15].

##  III Distributed radar models and settings

In the following, we compare two distributed settings as summarized in the
Fig. 2. The example refers to $2$ radars but can be easily generalized. For
the cooperation case (Fig. 2(b)), the radars mutually share a subset
$\mathbb{\widehat{Q}}_{t,k}\subseteq\mathbb{Q}_{t,k}$ of the local PCs (after
displacement correction and outlier removal). Scene
$\mathbb{\widehat{S}}_{t,k}$ reconstruction at radar $k$ is obtained from the
global Bayesian _global posterior probability_ measure
$\pi_{t}^{(\mathrm{C})}$. Considering the federation case (Fig. 2(c)), scene
$\mathbb{\widehat{S}}_{t,k}$ reconstruction is solved by a loose cooperation
(federation) of radars that share the parameters of a _local_ posterior
probability __ measure, rather than exchanging the raw points
$\mathbb{\widehat{Q}}_{t,k}$. The local posterior measures are approximated by
a Gaussian mixture function: therefore, the radars exchange the parameters
$\mathbb{P}_{k}$ characterizing the mixture model (number of Gaussian
components $M_{k}$, weights $\beta_{k,m}$, mean
$\mathbf{\boldsymbol{\mu}}_{k,m}$, and covariance
$\boldsymbol{\boldsymbol{\Gamma}}_{k,m}$ of each component $m$). Scene
$\mathbb{\widehat{S}}_{t,k}$ reconstruction is now obtained from the
_federated posterior probability_ $\pi_{t}^{(\mathrm{F})}$. Notice that the
proposed federation model has roots in Bayesian federated learning [16, 17]
and decentralized setups [12].

###  III-A Cooperation: point cloud data exchange

Radars engaged in device cooperation share the PCs obtained from the local DSP
processor. Before transmission, the local points $\mathbb{Q}_{t,k}$ in (1) are
pre-processed

$\mathbb{\widehat{Q}}_{t,k}=\left\\{\mathbf{\widehat{q}}_{i,k}+\widehat{\mathbf{d}}_{i,k}\right\\}_{\mathbf{\widehat{q}}_{i,k}\in\mathbb{Q}_{t,k}\setminus\mathbb{\widehat{O}}_{t,k}}$
(2)

to correct the displacement errors $\widehat{\mathbf{d}}_{i,k}$ (i.e.,
depending on the relative position of the radar) and to remove the outliers
$\mathbb{\widehat{O}}_{t,k}$ detected in $\mathbb{Q}_{t,k}$. The dBscan method
[18, 19] is adopted for outlier estimation: considering the neighborhood
$\mathcal{N}_{k}$, the PC ensemble $\mathbb{Q}_{t,\mathcal{N}_{k}}$ received
by the $k$-th radar is

$\mathbb{Q}_{t,\mathcal{N}_{k}}=\mathbb{\widehat{Q}}_{t,k}\cup\left\\{\mathbb{\widehat{Q}}_{t+\bigtriangleup
t_{h},h}\right\\}_{h\in\mathcal{N}_{k}}.$ (3)

Scene reconstruction is based on the _global_ measure $\pi_{t}^{(\mathrm{C})}$
of the posterior probability, namely the posterior obtained from the PCs
observed in the radar neighborhood:

$\pi_{t}^{(\mathrm{C})}=p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{Q}_{t,\mathcal{N}_{k}}).$
(4)

Finally, the PCs representing the true scene are here identified as those with
largest posterior probability measure:

$\mathbb{\widehat{S}}_{t,k}=\left\\{\mathbf{q}_{i}:\pi_{t}^{(\mathrm{C})}>\tau\right\\},$
(5)

with threshold $\tau$ defined at calibration time (see Sect. IV). As analyzed
in Sect. IV, the posterior measure (5) is used to track the positions of the
targets ($\widehat{x}_{t},\widehat{y}_{t}$) in the space: these correspond to
the local maxima of the posterior, according to maximum a-posteriori (MAP)
estimation.

According to Bayesian filtering tools, the posterior $\pi_{t}^{(\mathrm{C})}$
is updated at time $t$ by an iterative approach that uses the PCs representing
the previous reconstructed scene $\widehat{\mathbb{S}}_{t-1}$, as

$\pi_{t}^{(\mathrm{C})}\propto
p(\mathbf{\widehat{q}}_{i}\in\mathbb{Q}_{t,\mathcal{N}_{k}}|\mathbb{S}_{t})\cdot
p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k}),$ (6)

with the conditional likelihood
$p(\mathbf{\widehat{q}}_{i}\in\mathbb{Q}_{t,\mathcal{N}_{k}}|\mathbb{S}_{t})$
and the prior measure
$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k})$. Prior and
conditional likelihood models suitable for tracking of human subjects are
discussed in the following sections. The interested reader may also refer to
[20, 22]. Finally, notice that
$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k})$ approximates
the a-priori probability term
$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{Q}_{t-1,\mathcal{N}_{k}})=p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{S}_{t-1})\cdot\pi_{t-1}^{(\mathrm{C})}$.

###  III-B Federation: local posterior exchange

Rather than exchanging the PC set $\mathbb{\widehat{Q}}_{t,k}$, in the
proposed federation model the radars share the parameters of the _local
posterior_ probability __ measure
$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\widehat{\mathbb{Q}}_{t,k})$. The PC
samples are kept on the individual radars and not transferred: therefore, the
global posterior $\pi_{t}^{(\mathrm{C})}$ in (4) is now approximated by a
mixture of the local posteriors to obtain $\pi_{t}^{(\mathrm{F})}$ as

$\pi_{t}^{(\mathrm{F})}=\alpha_{k}\cdot
p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{Q}}_{t,k})+\sum_{h\in\mathcal{N}_{k}}\alpha_{h}\cdot
p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{Q}}_{t,h}),$ (7)

where $\alpha_{k}+\sum_{h\in\mathcal{N}_{k}}\alpha_{h}=1$ and
$\alpha_{k}=\frac{Q_{k}}{Q_{k}+\mathop{\sum_{h}Q_{h}}}$. $Q_{k}$ is the number
of points measured by radar $k$. The federated posterior
$\pi_{t}^{(\mathrm{F})}$ can be evaluated either centrally by a server
collecting the local posterior parameters, namely the Parameter Server, (PS)
or distributedly. Scene reconstruction is represented similarly as in (5), now
with
$\mathbb{\widehat{S}}_{t,k}=\left\\{\mathbf{q}_{i}:\pi_{t}^{(\mathrm{F})}>\tau\right\\}$.
Notice that, the estimated scene $\mathbb{\widehat{S}}_{t,k}$ from the
federated posterior is used to update the _local_ posterior on the next step
$t+1$, thus augmenting the $k$-th radar knowledge about the monitored space.
Likewise equation (6), the local posterior can be also evaluated iteratively
as $p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\widehat{\mathbb{Q}}_{t,k})\propto
p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k})\cdot
p(\mathbf{\widehat{q}}_{i}\in\widehat{\mathbb{Q}}_{t,k}|\mathbb{S}_{t})$ using
the prior $p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k})$
and the (local) conditional likelihood
$p(\mathbf{\widehat{q}}_{i}\in\widehat{\mathbb{Q}}_{t,k}|\mathbb{S}_{t})$,
respectively.

Cooperation (4) versus federation (7) tradeoffs are discussed in Sect. IV
targeting the problem of multi body tracking. In what follows, we adopt the
multivariate Gaussian mixture distributions to represent both the local, the
global posterior probability measures, and the priors.

Figure 3: Proposed architecture and system model. Radar $k$ monitoring the
sidelink (S) and receiving information from radar $h$. The radar MIMO HW
supports range-azimuth and elevation tracking and it is equipped with a DSP to
extract time-varying 2D/3D point cloud information. The radar CFAR
configuration parameters can be controlled via UART communication with the
MCU.  Figure 4: Scene example featuring 2 subjects moving from position L to H
and from M to F. Posteriors observed over three consecutive time samples
($t=1,2,3$): a) no-cooperation case with $k=1,2,3$; and b) federation and
cooperation cases for $K=3$ radars with deployment described above.  Figure 5:
Kullback-Leibler (KL) $\mathrm{D_{KL}}[\cdot]$ divergence sample probability
functions $\mathrm{Pr}[\mathrm{D_{KL}}]$: solid line compares global and
federated posteriors
$(\mathrm{D_{KL}}[\pi_{t}^{(\mathrm{C})}\|\pi_{t}^{(\mathrm{F})}]$), dashed
lines compare global and local posteriors for the three deployed radars.
Probability measure is obtained by collecting $900$ consecutive sample
divergences ($9$ sec. with $10$ms per localization update) corresponding to
$2$ subjects moving from position L to H and from M to F (same scenario as in
Figure 4).

###  III-C Gaussian mixture approximations

Local/global posteriors follow a multivariate Gaussian mixture model. The
local posterior is therefore approximated as

$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\widehat{\mathbb{Q}}_{t,k})\simeq\sum_{m=1}^{M_{k}}\beta_{k,m}\cdot\mathcal{N}(\mathbf{\mu}_{k,m},\mathbf{\Sigma}_{k,m}),$
(8)

with parameters
$\mathbb{P}_{k}=\left\\{\beta_{k,m},\mathbf{\boldsymbol{\mu}}_{k,m},\mathbf{\boldsymbol{\Sigma}}_{k,m}\right\\}_{m=1}^{M_{k}}$
for each $k$-th radar. $M_{k}$ is the number of components observed by radar
$k$, while
($\mathbf{\boldsymbol{\mu}}_{k,m},\mathbf{\boldsymbol{\Sigma}}_{k,m}$) are the
mean and covariance of the $m$-th component111for 3D PCs,
$\mathbf{\boldsymbol{\mu}}_{k,m}$ has dimension $3\times 1$ and covariance
$\mathbf{\boldsymbol{\Sigma}}_{k,m}$ is $3\times 3$. and
$\sum_{m=1}^{M_{k}}\beta_{k,m}=1$. Number of components $M_{k}$ depends on the
post-processed PCs $\widehat{\mathbb{Q}}_{t,k}$ after outlier removal [19] and
will be assessed in the following experimental analysis. The federation model
(7) requires the radars share the parameters $\mathbb{P}_{k}$ of the Gaussian
mixture (8). Federated posterior measure $\pi_{t}^{(\mathrm{F})}$ is thus
obtained by substituting (8) in (7). For the cooperation model, the global
posterior $\pi_{t}^{(\mathrm{C})}$ is

$\pi_{t}^{(\mathrm{C})}\negthinspace=\negthinspace
p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{Q}_{t,\mathcal{N}_{k}})\negthinspace\simeq\negthinspace\sum_{m=1}^{N}\beta_{\mathrm{C},m}\cdot\mathcal{N}(\mathbf{\mu}_{\mathrm{C},m},\mathbf{\Sigma}_{\mathrm{C},m}),$
(9)

with $\sum_{m=1}^{N}\beta_{C,m}=1$, and typically $N\gg M_{k}$. Global and
federated posteriors are compared in the following section in terms of the
Kullback-Leibler (KL) divergence.

As far as the prior
$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k})$ is
concerned, we assume that the objects/targets in the scene consist of smooth
and well separated patches moving at maximum speed $v$. For $S$ points in the
scene $\mathbb{\widehat{S}}_{t-1,k}$, the prior measure

$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{S}}_{t-1,k})\simeq\frac{1}{S}\sum_{\mathbf{q}_{i}\in\mathbb{\widehat{S}}_{t-1,k}}\mathcal{N}\left[\mathbf{q}_{i},\sigma^{2}(v)\times\mathbf{I}\right]$
(10)

is obtained as the superposition of Gaussian distributions each modelling the
motion of individual particle $\mathbf{q}_{i}$ with $\sigma^{2}(v)$ obtained
from a random walk model [23]. Other priors can be also exploited to quantify
the smoothness and sharpness of the targets as proposed in [20, 21] for
imaging applications.

##  IV Experimental data and validation

This section discusses cooperation and federation models based on a prototype
implementation and experimental data. Proposed architecture is summarized in
Fig. 3: in the example, the radar $k$ is monitoring the sidelink (S) and
receiving information from radar $h$. The radars work in the $77$ GHz band and
support MIMO-FMCW radio featuring an array of $3$ TX and $4$ uniformly spaced
RX antennas. Each radar has an azimuth Field Of View (FOV) of -/+ $60$° angle
and range resolution of $25$° and $4.2$cm, 3.9 GHz band (sweep time
$36$$\mu$s). It integrates a C674x DSP and an ARM R4F MCU with TX output power
set to $12$ dBm. PC samples are obtained from the radar DSP using the CA-CFAR
method configured with range and Doppler detection thresholds of $15$dB. Then,
PCs are further processed using a ARM-Cortex-A57 SoC (Jetson Nano device
model). Notice that the radar CFAR configuration parameters can be controlled
via UART communication with the MCU.

Three wall-mounted radars are deployed as detailed in Fig. 4 (positions are
relative to radar $k=1$). The scene $\mathbb{S}_{t}$, detailed in Fig. 4 (on
top), consists of $2$ co-present human subjects moving inside an indoor lab
environment. Trajectories follow $6$ landmark points (A-M), with corresponding
relative ground-truth locations detailed in the same figure. Radars exchange
either raw PCs (cooperation) or local model parameters $\mathbb{P}_{k}$
(federation) via WiFi links while synchronization is implemented using NTP
(Network Time Protocol). Impact of inaccurate clock source on performance [24]
is here assessed experimentally: however, a dedicated analysis is out of
scope.

###  IV-A Global, federated and local posterior analysis

Figure 4 compares the local posterior distributions
$p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{Q}}_{t,k})$ for three
(isolated) radars $k=1,2,3$ (Fig. 4.a) with the global
$\pi_{t}^{(\mathrm{C})}$ and federated posterior $\pi_{t}^{(\mathrm{F})}$
(Fig. 4.b). Posteriors are observed at $3$ consecutive time instants
$t=1,2,3$. The radar $k=1$ correctly identifies the target T2 following the
path M$\rightarrow$G$\rightarrow$F, but it is less effective for T1, moving in
the opposite direction, from L$\rightarrow$H, due to low SNR. On the other
hand, the radar $k=2$ identifies T1, while T2 is now shadowed. Cooperation and
federation cases refer to a scenario where the radar $k=1$ obtains side
information from neighbors $k=2,3$: in both cases, the radar is able to
effectively resolve both targets.

Figure 5 now assess in more detail the capability of the federated posterior
$\pi_{t}^{(\mathrm{F})}$ in (7) to effectively reproduce the global posterior
$\pi_{t}^{(\mathrm{C})}$ as obtained by fusion of the PCs on the radar. We
thus analyze and compare the Kullback-Leibler (KL) distance $\mathrm{D_{KL}}$
[25] between the global, the federated and the local posterior measures.
Notice that the global posterior in (4) approximates the posterior that would
be obtained by letting the radar exchange the (full) raw 3D radar data.
However, exchanging radar data over sidelinks is not considered in this paper
as infeasible due to bandwidth constraints introduced by the real-time data
sharing process (Fig. 3). Considering that the posterior is computed on each
localization update, we collect consecutive KL distances corresponding to $2$
subjects moving from position L to H and from M to F (see Figure 4) and obtain
a sample probability $\mathrm{Pr}[\mathrm{D_{KL}}]$. Solid line compares
global and federated posteriors
$(\mathrm{D_{KL}}[\pi_{t}^{(\mathrm{C})}\|\pi_{t}^{(\mathrm{F})}]$), while
dashed lines compare global and local posteriors
($\mathrm{D_{KL}}[\pi_{t}^{(\mathrm{C})}\|p(\mathbf{q}_{i}\in\mathbb{S}_{t}|\mathbb{\widehat{Q}}_{t,k})],$
$k=1,2,3$) for the three radars. Proposed federation setup confirms the
property of being a loose form of cooperation that makes the federated
posterior closer to the global one, compared with the local measures, but with
no exchange of the raw PCs.

Figure 6:  Probability distribution of estimated positions $\widehat{x}_{t}$ for targets T1, T2 located in C, I (red bars) moving to G, E (blue bars). $\mathrm{P}_{u}$ is the probability of unresolved targets. Cooperation, federation and isolated ($k=1$) cases are compared. TABLE I: MAE $\sigma(x),\sigma(y)$ [m], probability of unresolved targets ($\mathrm{P}_{u}$) and sidelink overhead $B$ [MB] considering T1 and T2 in positions (C, I) and (G, E). | $k=1$ | Cooperation | Federation
---|---|---|---
$\sigma$ (acc.) | I:$\underset{\sigma(x)}{0.08},\underset{\sigma(y)}{0.15}$ | I:$\underset{\sigma(x)}{0.08},\underset{\sigma(y)}{0.13}$ | I:$\underset{\sigma(x)}{0.07},\underset{\sigma(y)}{0.15}$
C:$0.09,0.16$ | C: $0.09$$,0.07$ | C:$0.08,0.09$
G:$0.23,0.14$ | G:$0.07,0.09$ | G:$0.11,0.09$
E:$0.12,0.15$ | E:$0.08,0.08$ | E:$0.1,0.12$
$\mathrm{P}_{u}$ | (C, I):$10\%$ | (C, I):$1\%$ | (C, I):$1\%$
(G, E):$59.3\%$ | (G, E):$40.9\%$ | (G, E):$19.8\%$
$B$ [MB] | (C, I):$0.0$ | (C, I):$12$ Mbit/s | (C, I):$0.28$ Mbit/s
|  $62.500$ PCs/s |  $4.400$ $\mathbb{P}_{k}$/s
(G, E):$0.0$ | (G, E):$15.6$ MB/s | (G, E):$0.28$ Mbit/s
|  $81.500$ PCs/s |  $4.400$ $\mathbb{P}_{k}$/s

###  IV-B MAE, unresolved targets and communication overhead

Fig. 6 and Tab. I provide a quantitative evaluation of the localization
accuracy and communication overhead. T1 and T2 are now located in positions
C,I and move towards the corresponding landmarks E,G, by following the
trajectory C$\rightarrow$D$\rightarrow$E and I$\rightarrow$M$\rightarrow$G,
respectively. The estimated positions of the subjects
($\widehat{x}_{t},\widehat{y}_{t}$) in the space correspond to the local
maxima of the posteriors ($\pi_{t}^{(\mathrm{C})}$,$\pi_{t}^{(\mathrm{F})}$),
according to MAP estimation. Notice that before obtaining the local maxima, we
apply a thresholding (5) of the posteriors, with $\tau=0.45$ to limit the
search time of the local maxima to the PCs that belong to the reconstructed
scene $\mathbb{\widehat{S}}_{t,k}$. Fig. 6 shows the probability distribution
of estimated positions $\widehat{x}_{t}$ obtained by radar $k=1$ alone, via
cooperation and federation, respectively. Low azimuth resolution of the radar
$k=1$ causes targets in positions G,E be unresolved with large probability
($\mathrm{P}_{u}$). Cooperation limits these impairments, while federation
shows superior performance being less sensitive to the number of observed PCs.
For the same scene, Tab. I shows the localization mean absolute error (MAE),
namely $\sigma(x)=\mathbb{E}_{t}\left[\left|\widehat{x}_{t}-x\right|\right]$,
$\sigma(y)=\mathbb{E}_{t}\left[\left|\widehat{y}_{t}-y\right|\right]$, the
probability of unresolved targets $\mathrm{P}_{u}$, and the communication
overhead $B$ [Mbit/s], quantified here as the sum of bits exchanged over the
sidelink per second (localization updates are issued on every $10$ ms). The
number of exchanged PCs (cooperation) and the mixture model parameters
$\mathbb{P}_{k}$ (federation) per second are also reported.

Cooperation generally improves the average accuracy ($\sigma(x)=0.09$m)
compared with federation ($\sigma(x)=0.11$m). On the other hand, federation is
more effective in minimizing unresolved targets $(\mathrm{P}_{u}$) as less
sensitive to outliers: this effect can be appreciated for co-located subjects
G-E. Federation also keeps the required communication overhead about $20\div
25$ times lower than cooperation. In cooperation mode, the number of PCs
exchanged by each radar depend on the subject position. Subjects in positions
C and I make each radar send $625$ PCs per localization update on average
($62.500$ PC/s), or $12$ Mbit/s, considering that 3D PCs are $64$-bit
quantized and formatted in JSON [4]. Subjects in positions G and E are
represented on average by $815$ PCs per update with required sidelink
throughput of $15.6$ MB/s. In federation mode, each radar exchanges (on
average) $44$ local posterior parameters ($\mathbb{P}_{k}$) per localization
update, namely $281.6$ Kbit/s. Parameters exchanged contain: the points
$Q_{k}$ measured by radar $k$ and used to represent the federated posterior as
in (7), $M_{k}=3$ of components (on average), and, for each component, $14$
parameters namely, $\beta_{k,m}$, $\mathbf{\boldsymbol{\mu}}_{k,m}$ (3 terms),
$\mathbf{\boldsymbol{\Sigma}}_{k,m}$ (9 terms) and the number of represented
PCs $Q_{k,m}$, being $\sum_{m=1}^{M_{k}}Q_{k,m}=Q_{k}$.

##  V Conclusion and future activities

The paper proposed and compared cooperation and federation models suitable for
distributed radar point cloud processing. The considered radars are
characterized by limited range-angular resolution when operating in isolation;
however, they exchange side information using a communication channel of
limited bandwidth. The paper discusses a prototype and related experiments
that quantify the benefits of federation vs. cooperation targeting the
localization and tracking of two people in an indoor environment. Federation
and cooperation benefits are quantified in terms of latency/sidelink bandwidth
use, resolution, probability of unresolved targets and mean absolute
localization error performance.

Cooperation generally provides improved average mean absolute localization
error ($0.09$m) than federation ($0.11$m). Federation forces the radar to
exchange model parameters of observed PCs probability (local posterior
measure) and it is thus more effective in minimizing unresolved targets and
outliers. Federation is a promising solution for scaling up the system to
support the deployment of more radars as it keeps the required communication
overhead about $20\div 25$ times lower than cooperation. Required sidelink
throughput is also independent from the number of observed PCs. Future
activities will consider the impact of synchronization, radar network
topologies and heterogeneity in terms of angular resolution and field of view.

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# Importance of intrinsic and non-network contribution in PageRank centrality
and its effect on PageRank localization

Krishanu Deyasi<EMAIL_ADDRESS>Department of Mathematics and
Statistics,
Indian Institute of Science Education and Research Kolkata,
Mohanpur - 741 246, WB, India

###### Abstract

PageRank centrality is used by Google for ranking web-pages to present search
result for a user query. Here, we have shown that PageRank value of a vertex
also depends on its intrinsic, non-network contribution. If the intrinsic,
non-network contributions of the vertices are proportional to their degrees or
zeros, then their PageRank centralities become proportion to their degrees.
Some simulations and empirical data are used to support our study. In
addition, we have shown that localization of PageRank centrality depends upon
the same intrinsic, non-network contribution.

##  I Introduction

Complex network tools have been used in different fields from social,
technological and biological networks Albert02 ; Newmanbook , to analyze the
structure and dynamics of the network. Different centrality measures have been
used to find the important vertices in a network e.g., degree centrality,
eigenvector centrality, Katz centrality, vertex betweenness centrality, edge
betweenness centrality, closeness centralityNewmanbook , etc. Among them, the
simplest measure is the degree centrality. Degree of a vertex is the number of
edges connected to it.

PageRank is another important measure and is used in network tasks like,
information retrieval, link prediction and community detectionGleich15 ;
chung09 . In 1998, Brin and Page BP98  developed the PageRank algorithm to
rank websites in their search engine. PageRank score, $x_{i}$, of a vertex $i$
in an undirected and unweighted network, is defined as
$x_{i}=\alpha\sum_{j}A_{ij}\frac{x_{j}}{k_{j}}+\beta_{i}$, where $0<\alpha<1$
is the damping factor, also known as teleportation factor, $A$ is the
adjacency matrix corresponding to the network, $k_{j}$ is the degree of the
vertex $j$ and $\beta_{i}$ is the intrinsic, non-network contribution. Many
researchers have studied the significance of $\alpha$ in PageRank Langville04
; Langville06 ; Pretto02 ; Fortunato07 ; Ding09 . However, till now, no study
has been taken place for the same for $\beta_{i}$. In many cases the value of
$\beta_{i}$ has been taken as $1$ Langville04 ; Pretto02 ; Fortunato07 .

Localization of eigenvector is a common phenomenon in most of the networks and
it is frequently suggested that the main cause of localization is due to the
presence of high degree vertices Goltsev12 ; Martin14 . Localization means
that most of the weight of the centrality accumulates in a few number of
vertices. Recently, empirical studies in PageRank suggest that PageRank does
not show localization Ermann13 .

In this paper, we show that depending upon intrinsic and non-network
contribution, PageRank value changes. This article has been organized in the
following way: in Chapter II, we show that if the intrinsic and non-network
contribution of the vertices are the same or proportional to their degrees,
then the PageRanks remain same or proportional to the degree of the vertices.
We also show that if the intrinsic and non-network contribution of the
vertices is zero then PageRank values become proportional to the degrees. In
Chapter III, our study on simulated and empirical networks support our
findings. On the contrary to the common belief we show that, for some
networks, PageRank centrality can be localized and the value of $\beta_{i}$
can effect in the localization of the PageRank centrality. Chapter IV
concludes with some discussion.

##  II PageRank centrality: Intrinsic and non-network contribution
proportional to degree

Consider a simple undirected connected network $G$ of size $n$, with adjacency
matrix $A$. The PageRank centrality $x_{i}$ of a vertex $i$ in the network $G$
is defined as:

$x_{i}=\alpha\sum_{j=1}^{n}A_{ij}\frac{x_{j}}{k_{j}}+\beta_{i},$ (1)

where $0<\alpha<1$ is the damping factor (Google search engine uses
$\alpha=0.85$ Langville06 ), $k_{j}$ denotes the degree of vertex $j$, and
$\beta_{i}$ is the intrinsic and non-network contribution to the vertex $i$.

In matrix form, the Equation (1) can be written as:

$\mathbf{x}=\alpha\mathbf{AD}^{-1}\mathbf{x}+\mathbf{\beta},$ (2)

where $\mathbf{D}$ is the diagonal matrix with elements $D_{ii}=k_{i}$ and
$\beta=(\beta_{1},\beta_{2},\dots,\beta_{n})^{T}$. Equation (2) can be written
as:

$\mathbf{x}=(\mathbf{I}-\alpha\mathbf{AD}^{-1})^{-1}\mathbf{\beta},$ (3)

Since $||\alpha\mathbf{AD}^{-1}||_{1}\leq\alpha<1,$ Equation (3) can be
written as the sum of an infinite series:

$\mathbf{x}=\sum_{i=0}^{\infty}\alpha^{i}\left(\mathbf{A}\mathbf{D}^{-1}\right)^{i}\mathbf{\beta}.$
(4)

Now $\mathbf{AD^{-1}}=\mathbf{P}$, is a column stochastic matrix. Since $G$ is
undirected and connected $\mathbf{P}$ is diagonalizable. One can write
$\mathbf{\beta}$ as a linear combination of the eigenvectors $\mathbf{w_{i}}$
of $\mathbf{P}$, as

$\mathbf{\beta}=\sum_{j=1}^{n}\mathbf{c}_{j}\mathbf{w}_{j}.$ (5)

Then the Equation (4) becomes

$\mathbf{x}=\sum_{i=0}^{\infty}\alpha^{i}\mathbf{P}^{i}\left(\sum_{j=1}^{n}\mathbf{c}_{j}\mathbf{w}_{j}\right).$
(6)

Let $1=\mu_{1}\geq\mu_{2}\geq\dots\geq\mu_{n}\geq 0$ be the eigenvalues of
$\mathbf{P}$ corresponding to the eigenvectors
$\mathbf{w}_{1},\mathbf{w}_{2},\dots,\mathbf{w}_{n}$, respectively. Then

$\mathbf{x}=\sum_{i=0}^{\infty}\alpha^{i}\left(\sum_{j=1}^{n}\mathbf{c}_{j}\mu_{j}^{i}\mathbf{w}_{j}\right).$
(7)

After some simple calculation, we get

$\mathbf{x}=\sum_{j=1}^{n}c_{j}\left(\sum_{i=0}^{\infty}\left(\alpha\mu_{j}\right)^{i}\right)\mathbf{w}_{j}.$
(8)

Since, $\alpha\mu_{i}<1,\forall i=1,\dots,n$, we get

$\mathbf{x}=\sum_{j=1}^{n}c_{j}\frac{1}{1-\alpha\mu_{j}}\mathbf{w}_{j}.$ (9)

Therefore, the PageRank vector $\mathbf{x}$ of a network is the linear sum of
eigenvectors of $\mathbf{P}$.

######  Remark 1.

If the value of $\mathbf{\beta}$ is proportional to degrees, then Equation (5)
becomes:

$\mathbf{\beta}=c_{1}\mathbf{w}_{1},$ (10)

where $\mathbf{w}_{1}$ is the leading eigenvector of $\mathbf{P}$. Note that
$\mathbf{w}_{1}$ is proportional to the degree of the network. Therefore,
Equation (9) reduces to

$\mathbf{x}=c_{1}\frac{1}{1-\alpha}\mathbf{w}_{1}.$ (11)

Hence, PageRank centrality is proportional to the degree of the network.

######  Remark 2.

PageRank can also be quantified successively with an initial estimation of
$\mathbf{x}$ from Equation (1) or (equivalently Equation (2)). Equation (2)
will be,

$\mathbf{x}_{1}=\alpha\mathbf{A}\mathbf{D}^{-1}\mathbf{x}_{0}+\beta,$ (12)

where $\mathbf{x}_{0}$ is any initial value of $\mathbf{x}$ usually taken
$\mathbf{x}=0$ Newmanbook . If the intrinsic and non-network contribution is
zero, then from Equation (12), we get

$\mathbf{x}_{1}=\alpha\mathbf{A}\mathbf{D}^{-1}\mathbf{x}_{0}.$ (13)

In this case, we take $\mathbf{x}_{0}$ any non-zero vector as $\mathbf{x}=0$
will not contribute anything. After $t$ iterative steps we have

$\mathbf{x}_{t+1}=\left(\alpha\mathbf{P}\right)^{t}\mathbf{x}_{0}.$ (14)

Now, writing $\mathbf{x}_{0}$ as a linear combination of the eigenvectors
$\mathbf{w}_{j}$ of $\mathbf{P}$, we get

$\mathbf{x}_{0}=\sum_{i=1}^{n}c_{i}^{\prime}\mathbf{w}_{i},$ (15)

for some appropriate choice of $c_{i}^{\prime}$. Then

$\mathbf{x}_{t+1}=\alpha^{t}\sum_{i=1}^{n}c_{i}^{\prime}\mu_{i}^{t}\mathbf{w}_{i}.$
(16)

Since $|\mu_{i}|<1$ for all $i\neq 1$, we get $\mathbf{x}_{t+1}\to
c_{1}^{\prime}\alpha^{t}\mathbf{w}_{1}$ in the limit $t\to\infty$. As
$c^{\prime}_{1}$ and $\alpha$ are constants and $\mathbf{w}_{1}$ is
proportional to degree, it follows that PageRank centrality is proportional to
the degree centrality.

##  III Results

We explore four different values of intrinsic and non-network contribution,
$\beta$.

  1. 1.

$\beta_{i}$ is inversely proportional to the degree of vertex $i$.

  2. 2.

All $\beta_{i}$ are equal to one.

  3. 3.

$\beta_{i}$ is proportional to the degree of vertex $i$.

  4. 4.

$\beta_{i}$ is proportional to the square of the degree of vertex $i$.

In all the above cases, we have taken $\alpha=0.85$.

Our numerical simulation on Erdös-Rényi random network Bollobas01 , we have
observed that if the value of $\beta_{i}$ is proportional to the inverse of
the degree of vertex $i$ then PageRank score increases for a few number of low
degree vertices and for other, PageRank is stable (see Figure 1(a)). Here
‘stable’ means, whether the PageRank scores are positively correlated with the
degree. If $\beta$ is equal or proportional to the degrees of the vertices
then PageRank centrality is proportional to the degree of the vertices (see
Figure 1(c)). PageRank values are stable when $\beta$ is either equal to one
or proportional to the square of the degree of vertex (see Figure 1(b) and
Figure 1(d)). In comparison with random network, small-world and scale-free
network are stable with any value of $\beta$ that we have considered (see
Figure 2 and Figure 3).

However, in real-world networks, the results are not consistent for the above
mentioned cases. In Blog network, we have observed that PageRank score
fluctuates when $\beta$ is inversely proportional to the degree of vertex (see
Figure 4(a)). When all $\beta_{i}$ are equal to one, PageRank value fluctuates
but less than that in the previous case (see Figure 4(b)). When $\beta_{i}$ is
proportional to the degree of vertex $i$, $\beta$ and PageRank score correlate
positively (see Figure 4(c)). PageRank values are stable when $\beta_{i}$ is
proportional to the square of the degree of vertex $i$ (see Figure 4(d)).

In P2P (peer-to-peer) network, we have seen high fluctuation of PageRank value
for some high degree vertices when $\beta$ is inversely proportional to the
degree of a vertex (see Figure 5(a)). Similar observation is also found when
all $\beta_{i}$ are equal to one (see Figure 5(b)). When $\beta_{i}$ is
proportional to the degree of vertex $i$, PageRank score is also proportional
to the degree of vertex $i$ (see Figure 5(c)). PageRank score is stable when
$\beta_{i}$ is proportional to the square of the degree of vertex $i$ (see
Figure 5(d)).

For Email network, we have observed strong fluctuation of PageRank value when
$\beta_{i}$ is inversely proportional to the degree of vertex $i$ (see Figure
6(a)). When all $\beta_{i}$ are equal to one, we have seen less fluctuation of
PageRank value than that in the first case (see Figure 6(b)). PageRank value
also becomes proportional to the vertex degree when $\beta_{i}$ is
proportional to the same (see Figure 6(c)). When $\beta_{i}$ is proportional
to the square of the degree of vertex $i$, higher degree vertices hold higher
PageRank values (see Figure 6(d)).

In PGP (Pretty Good Privacy) network, some high degree vertices fluctuate in
PageRank values when $\beta_{i}$ is inversely proportional to the degree of a
vertex $i$ (see Figure 7(a)). Similar result has been found when all
$\beta_{i}$ are equal to one (see Figure 7(b)). When $\beta_{i}$ is equal or
proportional to the degree of vertex $i$, PageRank value is proportional to
the same (see Figure 7(c)). PageRank value increases with the increment of
degree of vertices when $\beta_{i}$ is proportional to the square of the
degree of vertex $i$ (see Figure 7).

Internet network is comparatively stable than other networks when $\beta_{i}$
is inversely proportional to the degree of vertex $i$ and when all $\beta_{i}$
are equal to one (see Figure 8(a) and Figure 8(b)). When $\beta_{i}$ is equal
or proportional to the degree of vertex $i$, PageRank value is also
proportional to the same (see Figure 8(c)). When $\beta_{i}$ is proportional
to the square of the degree of vertex $i$, high degree vertices take the high
PageRank score (see Figure 8(d)).

Localization of PageRank centrality: The extent of localization can be
measured by calculating inverse participation ratio (IPR) Kawamoto16 ;
Martin14 . The IPR of a PageRank vector $\mathbf{x}$ is calculated as:

$\text{IPR}=\frac{\sum_{i=1}^{n}x_{i}^{4}}{(\sum_{i=1}^{n}x_{i}^{2})^{2}}.$
(17)

When IPR is of order $O(1)$ as $n\to\infty$, the PageRank vector $\mathbf{x}$
is localized. If IPR $\to 0$, $\mathbf{x}$ is delocalized Goltsev12 ;
Kawamoto16 .

We observe that random and small-world network do not show localization of
PageRank centrality. In these two networks, the lowest IPRs observe when
$\beta$ is inversely proportional to the degree of vertex and the highest IPRs
observe when $\beta$ is the square of the degree of the vertices (see Figure 9
(A) and 9 (B)). However, in scale-free network the PageRank is localized and
the highest IPR is observed when $\beta$ is proportional to the square of the
degree of a vertex and the lowest IPR is observed when all $\beta_{i}$ are
equal to one (see Figure 9 (C)).

In Blogs network, the PageRank is localized and the highest IPR is observed
when the value of $\beta$ is inversely proportional to the degree of the
vertices. The lowest value of IPR is seen when $\beta$ is proportional to the
degree of the vertex (see Figure 9 (D)).

The PageRank is not localized in P2P and Email networks. In both of these
networks, the highest value of IPRs are observed when $\beta$ is proportional
to the square of the degree of vertex and the lowest IPR is observed when all
$\beta_{i}$ are equal to one (see Figure 9 (E) and 9 (F)).

The PageRank is localized in PGP network and the highest value IPR is observed
when $\beta$ is proportional to the square of the degree and the lowest IPR is
seen when $\beta$ is proportional to the degree of the vertex (see Figure 9
(G)).

In Internet network, PageRank centrality is localized. The highest value of
IPR is observed when $\beta$ is the square of the degree of the network and
the lowest IPR is observed when $\beta$ is proportional to the degree of the
vertex (see Figure 9 (H)).

##  IV Discussion and conclusion

In this work, we have shown that PageRank centrality depends upon intrinsic,
non-network contribution of the vertices. If the intrinsic and non-network
contribution of the vertices is proportional to the degree then PageRank score
becomes proportional to the degree of the vertices. We have also shown that if
the intrinsic and non-network contribution of PageRank centrality is zero,
then it also becomes proportional to the degree of the network. Our numerical
simulation of three network models (Erdös-Rényi random network, Watts-Strogatz
small-world network and Barabási-Albert scale-free network) shows that the
PageRank scores are more resilient in the intrinsic and non-network
contribution in small-world and scale-free networks than in the random
network. Among all real-world networks studied here, Blog, P2P, E-mail and
PGP, PageRank score fluctuates heavily when intrinsic and non-network
contribution are either inversely proportional to the degree or equal to one.
When the value of $\beta$ is proportional to degree, PageRank keeps the same
proportional value. High degree vertices possess high PageRank value when the
$\beta$ is proportional to the square of the degree. Internet network shows
very small fluctuation of PageRank score when $\beta$ is either inversely
proportional or equal to one and for other two values of $\beta$ show similar
result as other real networks.

On the contrary to the common belief we show that PageRank centrality can show
localization for some networks and the extent of localization depends upon the
intrinsic and non-network contribution of the PageRank centrality. In
synthetic networks, we have seen that the value of IPR increases with the
increase in the value of $\beta$. However, in the real-world networks studied
here, no such correlation is found. In Blog network, IPR attains the highest
value when $\beta$ is inversely proportional to the degree and for other
networks the highest value of IPR was observed when $\beta$ is proportion to
the square of the degree. In a nutshell, we have observed that the intrinsic
and non-network contribution plays an important role in PageRank centrality
calculation.

Here, we have considered the underlying undirected structure of all the
networks. The giant component (i.e., the largest connected sub-network) has
been considered when the network is not connected.

## Network construction and source :

Three simulated networks: Erdös-Rényi random network is constructed with 1000
vertices and two vertices are connected with probability $p=0.01$ Bollobas01 .
In Watts-Strogatz small-world network, the number of vertices is $1000$,
average degree of initial regular graph is $10$ and the rewiring probability
is $0.4$ WS98 . Barabási-Albert scale-free network is generated with $1000$
vertices and size of the seed network $m_{0}=5$. A new vertex is added with
the existing $m=5$ vertices BA99 .

Blogs network: Here vertices are the blogs and an edge represents a hyperlink
between two blogs. The number of vertices and edges are 1222 and 16714,
respectively. The data were downloaded from KONECT konect  and used in
Adamic05

Gnutella peer-to-peer (P2P) network: Gnutella is a system where individual can
exchange files over the Internet directly without going through a Web site.
Here, peer-to-peer means persons to persons. The vertices are hosts in the
Gnutella network topology and the edges represent connection between them. The
number of vertices are 22663 and number of edges are 54693. P2P network data
were downloaded from SNAPSNAP  and used in Leskovec07 ; Ripeanu02 .

E-mail network: E-mail network was constructed based on the exchange of mails
between the members of the University of Rovira i Virgili (Tarragone). Here
the vertices are the users, and two users are connected by an edge if one has
sent or received email from other. The number of vertices and edges are 1133
and 5451, respectively. The data were downloaded from alex  and used in
Emaildata

Pretty Good Privacy network: Here the vertices are the users of Pretty Good
Privacy (PGP) algorithm and edges are the interactions between them. The
number of vertices and edges are 10680 and 24316, respectively. The data were
downloaded from alex  and used in PGPdata .

Internet network: The vertices in this network are autonomous systems
(collection of computers and routersNewmanbook ) and the edges are the routes
taken by the data traveling between them. The number of vertices and edges are
22963 and 48436, respectively. Internet data were downloaded from
Newmanwebsite  and used in Newman02 .

###### Acknowledgements.

The author thanks Anirban Banerjee and Abhijit Chakraborty for critical
reading and suggestion. Author also thanks Prof. Asok Kumar Nanda for helping
to prepare the manuscript. Author gratefully acknowledges the financial
support from CSIR, India through a doctoral fellowship.

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(a) (b)
(c) (d)

Figure 1: PageRank on random network for different values $\mathbf{\beta}$:
(a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional
to the vertex degree, (d) square of the vertex degree. Random network model
proposed by Erdös-Réyni Bollobas01 . Here random network generated with $1000$
vertices and the probability of connecting two vertices is $0.01$. Simulated
result is average over 100 iterations.

(a) (b)
(c) (d)

Figure 2: PageRank on small-world network for different values of
$\mathbf{\beta}$: (a) inverse of the vertex degree, (b) equal for all
vertices, (c) proportional to the vertex degree, (d) square of the vertex
degree. Small-world network proposed by Watts-Strogatz WS98 . Here small-world
network generated by rewiring regular ring lattice of size $1000$ and average
degree $10$ with rewiring probability $0.4$. Simulated result is average over
100 iterations.

(a) (b)
(c) (d)

Figure 3: PageRank stability on scale-free network for different values of
$\mathbf{\beta}$: (a) inverse of the vertex degree, (b) equal for all
vertices, (c) proportional to the vertex degree, (d) square of the vertex
degree. Scale-free network proposed by Barabási-Albert BA99 . Here scale-free
network generated of size $1000$ and size of the seed network is $m_{0}=5$ and
a new vertex added with existing $m=5$ vertices. Simulated results is average
over 100 iterations.

(a) (b)
(c) (d)

Figure 4: PageRank on Blogs network for different values of $\mathbf{\beta}$:
(a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional
to the vertex degree, (d) square of the vertex degree. In Blog networkAdamic05
, vertices are the blogs and an edge represents a hyperlink between two blogs.

(a) (b)
(c) (d)

Figure 5: PageRank on Gnutella peer-to-peer network for different values of
$\mathbf{\beta}$: (a) inverse of the vertex degree, (b) equal for all
vertices, (c) proportional to the vertex degree, (d) square of the vertex
degree. In Gnutella peer-to-peer network Leskovec07 , the vertices are hosts
in the Gnutella network topology and edges are the routes taken by data
traveling between them.

(a) (b)
(c) (d)

Figure 6: PageRank on Email network for different values of $\mathbf{\beta}$:
(a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional
to the vertex degree, (d) square of the vertex degree. In Email
networkEmaildata , the vertices are the users and two users are connected by
an edge if one sent or received email from other.

(a) (b)
(c) (d)

Figure 7: PageRank on PGP network for different values of $\mathbf{\beta}$:
(a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional
to the vertex degree, (d) square of the vertex degree. In PGP network PGPdata
, vertices are the user of Pretty Good Privacy (PGP) algorithm and edges are
the interactions between them.

(a) (b)
(c) (d)

Figure 8: PageRank on Internet network for different values of
$\mathbf{\beta}$: (a) inverse of the vertex degree, (b) equal for all
vertices, (c) proportional to the vertex degree, (d) square of the vertex
degree. In Internet network Newman02 , vertices are the autonomous systems
(collection of computers and routers) and the edges show the route taken by
the data traveling between them.

(A) (B) (C) (D)
(E) (F) (G) (H)

Figure 9: Inverse participation ratio of various networks for different values
of $\mathbf{\beta}$: (a) inverse of the vertex degree, (b) equal for all
vertices, (c) proportional to the vertex degree, (d) square of the vertex
degree. Networks are (A) Random network, (B) Small-world network, (C) Scale-
free network, (D) Blog network, (E) P2P network, (F) Email network, (G) PGP
network and (H) Internet network.