# Building Topic-Driven Virtual IoTs in a Multiple IoTs Scenario

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## Abstract

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## 1. Introduction

- Differently from SIoT, which introduces a social behavior of objects but still models IoT as one huge network of objects extended worldwide, MIE, and much more MIoT, allow the “breakdown” of the whole huge IoT into multiple networks of smart objects interconnected with each other. This way to proceed is analogous to the evolution of social networking into social internetworking [15]. In particular, MIoT allows the management of situations in which the same object shows different behaviors in different networks it joined. Furthermore, MIoT makes an object to act as a bridge between two objects allowing them to communicate even if they belong to different networks and, therefore, are not directly connected with each other.

## 2. Related Literature

## 3. The MIoT Paradigm

- ${N}_{k}$ is the set of the nodes of ${G}_{k}$; there is a node ${n}_{{j}_{k}}$ for each instance ${\iota}_{{j}_{k}}\in {\mathcal{I}}_{k}$, and vice versa.
- ${A}_{k}$ is the set of the arcs of ${G}_{k}$; there is an arc ${a}_{j{q}_{k}}=({n}_{{j}_{k}},{n}_{{q}_{k}})$ if there exists a physical link from ${n}_{{j}_{k}}$ to ${n}_{{q}_{k}}$.

## 4. Definition of a Thing’s Profile

- $reaso{n}_{j{q}_{{k}_{t}}}$ denotes the reason why ${T}_{j{q}_{{k}_{t}}}$ occurred, chosen among a set of predefined values.
- $sourc{e}_{j{q}_{{k}_{t}}}$ indicates the starting node of the path followed by ${T}_{j{q}_{{k}_{t}}}$.
- $des{t}_{j{q}_{{k}_{t}}}$ represents the final node of the path followed by ${T}_{j{q}_{{k}_{t}}}$.
- $star{t}_{j{q}_{{k}_{t}}}$ denotes the starting timestamp of ${T}_{j{q}_{{k}_{t}}}$.
- $finis{h}_{j{q}_{{k}_{t}}}$ indicates the ending timestamp of ${T}_{j{q}_{{k}_{t}}}$.
- $succes{s}_{j{q}_{{k}_{t}}}$ denotes whether ${T}_{j{q}_{{k}_{t}}}$ was successful or not; it is set to
`true`in the affirmative case, to`false`in the negative one, and to`NULL`if ${T}_{j{q}_{{k}_{t}}}$ is still in progress. - $conten{t}_{j{q}_{{k}_{t}}}$ indicates the content “exchanged” from ${\iota}_{{j}_{k}}$ to ${\iota}_{{q}_{k}}$ during ${T}_{j{q}_{{k}_{t}}}$. In its turn, $conten{t}_{j{q}_{{k}_{t}}}$ presents the following structure:$$conten{t}_{j{q}_{{k}_{t}}}=\langle forma{t}_{j{q}_{{k}_{t}}},fileNam{e}_{j{q}_{{k}_{t}}},siz{e}_{j{q}_{{k}_{t}}},topic{s}_{j{q}_{{k}_{t}}}\rangle $$

- $forma{t}_{j{q}_{{k}_{t}}}$ indicates the format of the content exchanged during ${T}_{j{q}_{{k}_{t}}}$; the possible values are: “audio”, “video”, “image” and “text”.
- $fileNam{e}_{j{q}_{{k}_{t}}}$ denotes the name of the transmitted file.
- $siz{e}_{j{q}_{{k}_{t}}}$ indicates the size in bytes of the content.
- $topic{s}_{j{q}_{{k}_{t}}}$ indicates the set of the content topics; it consists of a set of keywords representing the subjects exchanged during ${T}_{j{q}_{{k}_{t}}}$. It can be formalized as: $topic{s}_{j{q}_{{k}_{t}}}=\{(k{w}_{j{q}_{{k}_{t}}}^{1},nk{w}_{j{q}_{{k}_{t}}}^{1}),(k{w}_{j{q}_{{k}_{t}}}^{2},nk{w}_{j{q}_{{k}_{t}}}^{2}),\dots ,(k{w}_{j{q}_{{k}_{t}}}^{w},nk{w}_{j{q}_{{k}_{t}}}^{w})\}$. In other words, the set of the topics of the ${t}^{th}$ transaction from ${\iota}_{{j}_{k}}$ to ${\iota}_{{q}_{k}}$ consists of w pairs; each pair consists of a keyword and the corresponding number of occurrences.

- ⊎: it receives a set $\{entitySe{t}_{1},entitySe{t}_{2},\cdots ,entitySe{t}_{t}\}$ of entity sets and performs their union not eliminating the duplicates but reporting the number of their occurrences. Therefore, this operator returns a set of pairs $\{(entit{y}_{1},n{e}_{1}),(entit{y}_{2},n{e}_{2}),\cdots ,(entit{y}_{w},n{e}_{w})\}$ in which the pair $(entit{y}_{r},n{e}_{r})$ indicates the ${r}^{th}$ entity and the number of its occurrences. In counting it, ⊎ takes the presence of synonymies and homonymies into account. These properties can be computed (for terms, images, etc.) by applying the classical approaches proposed in the past literature [57,58].
- $avgFileSize$: it receives a set of files and computes their average size.

- $reasonSe{t}_{j{q}_{k}}={\uplus}_{t=1..v}\left(reaso{n}_{j{q}_{{k}_{t}}}\right)$;
- $sourceSe{t}_{j{q}_{k}}={\uplus}_{t=1..v}\left(sourc{e}_{j{q}_{{k}_{t}}}\right)$;
- $destSe{t}_{j{q}_{k}}={\uplus}_{t=1..v}\left(des{t}_{j{q}_{{k}_{t}}}\right)$;
- $avgSzAudi{o}_{j{q}_{k}}=AvgFileSiz{e}_{t=1..v}\left\{fileNam{e}_{j{q}_{{k}_{t}}}\right|forma{t}_{j{q}_{{k}_{t}}}=\u2018\u2018audio"\}$;
- $avgSzVide{o}_{j{q}_{k}}=AvgFileSiz{e}_{t=1..v}\left\{fileNam{e}_{j{q}_{{k}_{t}}}\right|forma{t}_{j{q}_{{k}_{t}}}=\u2018\u2018video"\}$;
- $avgSzImag{e}_{j{q}_{k}}=AvgFileSiz{e}_{t=1..v}\left\{fileNam{e}_{j{q}_{{k}_{t}}}\right|forma{t}_{j{q}_{{k}_{t}}}=\u2018\u2018image"\}$;
- $avgSzTex{t}_{j{q}_{k}}=AvgFileSiz{e}_{t=1..v}\left\{fileNam{e}_{j{q}_{{k}_{t}}}\right|forma{t}_{j{q}_{{k}_{t}}}=\u2018\u2018text"\}$;
- $successFractio{n}_{j{q}_{k}}=\frac{\left|\right\{{T}_{j{q}_{{k}_{t}}}|{T}_{j{q}_{{k}_{t}}}\in tranSe{t}_{j{q}_{k}},succes{s}_{j{q}_{{k}_{t}}}=true\}|}{v}$;
- $topicSe{t}_{j{q}_{k}}={\uplus}_{t=1..v}\left(topic{s}_{j{q}_{{k}_{t}}}\right)$.

## 5. Topic-Guided Virtual IoTs in a MIoT and Approaches to Constructing Them

#### 5.1. Supervised Approach

- It starts when a user specifies a query Q consisting of r keywords:$$Q=\{k{w}_{1},k{w}_{2},\cdots ,k{w}_{r}\}$$It searches for all the instances of the MIoT having at least one topic whose keyword is identical to, or synonymous of, at least one keyword specified in Q. These instances, as a whole, represent the set of candidate instances to be included in the new thematic view. We call this set $\mathcal{CI}$ (Candidate Instances).
- However, the fact that an instance $\iota \in \mathcal{CI}$ has a keyword in common with Q is necessary but not sufficient for it to be chosen. In fact, it is advisable that $\iota $ has more keywords in common with Q and, possibly, that the common keywords are among the ones of $\iota $ with the highest number of occurrences. This condition can be guaranteed by the usage of the operator ${J}^{*}$.In particular, our approach first constructs ${Q}^{\prime}=\left\{(kw,1)\right|kw\in Q\}$ in such a way as to make the application of ${J}^{*}$ on the keywords specified by the user possible. Then, it constructs the set $\mathcal{RI}$ (Real Instances) of those instances of $\mathcal{CI}$ whose topics have a significant similarity with the keywords of Q:$$\mathcal{RI}=\{\iota \in \mathcal{CI}|{J}^{*}(topicSe{t}_{\iota},{Q}^{\prime})>t{h}_{J}\}$$Here, $t{h}_{J}$ is a suitable tuning threshold.
- Now, our approach can start to construct the thematic view ${\mathcal{V}}_{Q}$ corresponding to Q.
- -
- It first creates a node ${n}_{\iota}$ in ${\mathcal{V}}_{Q}$ for each instance $\iota $ of $\mathcal{RI}$. Let ${n}_{{\iota}_{1}}$ and ${n}_{{\iota}_{2}}$ be the nodes corresponding to two instances ${\iota}_{1}$ and ${\iota}_{2}$ belonging to $\mathcal{RI}$.
- *
- If an i-arc exists between the nodes corresponding to ${\iota}_{1}$ and ${\iota}_{2}$ in the MIoT $\mathcal{M}$, then an i-arc is also created between the nodes ${n}_{{\iota}_{1}}$ and ${n}_{{\iota}_{2}}$ in ${\mathcal{V}}_{Q}$.
- *
- Instead, if a c-arc exists between the nodes corresponding to ${\iota}_{1}$ and ${\iota}_{2}$ in $\mathcal{M}$, then ${n}_{{\iota}_{1}}$ and ${n}_{{\iota}_{2}}$ are merged in a unique node ${n}_{{\iota}_{12}}$ in ${\mathcal{V}}_{Q}$. This task is motivated by the fact that ${n}_{{\iota}_{1}}$ and ${n}_{{\iota}_{2}}$ represent different instances of the same object in different real IoTs, but they represent the same instance in the same virtual IoT; as a consequence, they must be merged and no cross arc can exist between them. The profile $\overline{{\mathcal{P}}_{12}}$ of ${n}_{{\iota}_{12}}$ is obtained by applying the operator ⨆ on the profiles $\overline{{\mathcal{P}}_{1}}$ of ${\iota}_{1}$ and $\overline{{\mathcal{P}}_{2}}$ of ${\iota}_{2}$.

- Finally, our approach adds a disconnected node in ${\mathcal{V}}_{Q}$ for each keyword in Q such that there is no MIoT instance having at least one topic whose keyword is identical to, or synonymous of, it (The rationale underlying this step will be clearer in the following.).
- At this point, two cases may occur. In particular:
- -
- It could happen that ${\mathcal{V}}_{Q}$ is connected. In this case, it is returned as the answer to the query Q submitted by the user.
- -
- If ${\mathcal{V}}_{Q}$ is not connected and if the number of its connected components is less than a certain threshold, our approach adds the minimum number of “fictitious” i-arcs necessary to make ${\mathcal{V}}_{Q}$ connected.
- -
- Otherwise, if the number of connected components of ${\mathcal{V}}_{Q}$ is higher than a certain threshold, our approach concludes that a unique thematic virtual IoT corresponding to the keywords specified by the user does not exist and returns the thematic views related to the connected components of ${\mathcal{V}}_{Q}$. At this point, the user can decide whether to accept these thematic views or to modify the query in such a way as to construct a unique thematic view by re-applying all the above mentioned steps starting from the new query.

#### 5.2. Unsupervised Approach

- For each node ${n}_{{\iota}_{k}}$ of $\mathcal{M}$, a node $\overline{{n}_{{\iota}_{k}}}$ is added in $\mathcal{N}$.
- For each i-arc $({n}_{{\iota}_{{j}_{k}}},{n}_{{\iota}_{{q}_{k}}})$ in $\mathcal{M}$, an (unoriented) arc $(\overline{{n}_{{\iota}_{{j}_{k}}}},\overline{{n}_{{\iota}_{{q}_{k}}}})$ is added in $\mathcal{N}$. The arcs of $\mathcal{N}$ are weighted. The weight of the arc $(\overline{{n}_{{\iota}_{{j}_{k}}}},\overline{{n}_{{\iota}_{{q}_{k}}}})$ is obtained by applying the operator ${J}^{*}$ on the topic sets $topicSe{t}_{{j}_{k}}$ and $topicSe{t}_{{q}_{k}}$ of ${\iota}_{{j}_{k}}$ and ${\iota}_{{q}_{k}}$, respectively. Therefore, the weight of an arc in $\mathcal{N}$ belongs to the real interval $[0,1]$; the higher this weight the higher the semantic similarity between the topics of the profiles $\overline{{\mathcal{P}}_{{j}_{k}}}$ and $\overline{{\mathcal{P}}_{{q}_{k}}}$ of ${\iota}_{{j}_{k}}$ and ${\iota}_{{q}_{k}}$, respectively.
- For each c-arc in $\mathcal{M}$, which relates two instances ${n}_{{\iota}_{{j}_{k}}}$ and ${n}_{{\iota}_{{j}_{q}}}$ of the same object ${o}_{j}$ in two different IoTs ${\mathcal{I}}_{k}$ and ${\mathcal{I}}_{q}$, the two nodes $\overline{{n}_{{\iota}_{{j}_{k}}}}$ and $\overline{{n}_{{\iota}_{{j}_{q}}}}$ in $\mathcal{N}$, corresponding to the nodes ${n}_{{\iota}_{{j}_{k}}}$ and ${n}_{{\iota}_{{j}_{q}}}$ in $\mathcal{M}$, are merged into a unique node $\overline{{n}_{{\iota}_{j}}}$. This node inherits all the arcs of $\overline{{n}_{{\iota}_{{j}_{k}}}}$ and $\overline{{n}_{{\iota}_{{j}_{q}}}}$.

#### 5.3. Discussion

## 6. Experiments

#### 6.1. Adopted Dataset

- Python, powered with the NetworkX library, as programming language;
- Neo4J (Version 3.4.5) as underlying DBMS; we also exploited some plugins of Neo4J to perform community detection and to compute clustering coefficients.

#### 6.2. Cohesion of the Obtained Topic-Guided Virtual IoTs

#### 6.2.1. Supervised Approach

#### 6.2.2. Unsupervised Approach

#### 6.3. Average Fraction of Merged C-Nodes and Analysis of Node Distribution in Virtual IoTs

#### 6.4. Computation Time

#### 6.5. Our Approaches’ Capability of Improving the Efficiency of Information Dissemination

#### 6.6. Number and Size of Returned Virtual IoTs

- increases when the MIoT size ranges from 176 to 946;
- slightly increases when the MIoT size ranges from 946 to 2028;
- remains essentially constant when the MIoT size is higher than 2028.

- slightly increases when the MIoT size ranges from 176 to 946;
- increases when the MIoT size ranges from 946 to 2028;
- highly increases when the MIoT size is higher than 2028.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Computation time (in seconds) against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach)—first part.

**Figure 2.**Computation time (in seconds) against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach)—second part

MIoT (Size) | Number of Arcs | Mean In-Degree | Mean Out-Degree | Number of i-arcs | Number of c-arcs |
---|---|---|---|---|---|

${\mathcal{M}}_{1}$ (176) | 1176 | 6.29 | 6.61 | 980 | 126 |

${\mathcal{M}}_{2}$ (301) | 2050 | 7.76 | 7.74 | 1709 | 341 |

${\mathcal{M}}_{3}$ (485) | 3756 | 8.80 | 8.54 | 3130 | 626 |

${\mathcal{M}}_{4}$ (778) | 5866 | 8.89 | 9.11 | 4895 | 971 |

${\mathcal{M}}_{5}$ (946) | 7624 | 8.64 | 8.84 | 6422 | 1202 |

${\mathcal{M}}_{6}$ (1256) | 9860 | 7.87 | 7.98 | 7917 | 1943 |

${\mathcal{M}}_{7}$ (1725) | 12,263 | 7.94 | 8.18 | 9964 | 2299 |

${\mathcal{M}}_{8}$ (2028) | 15,568 | 8.22 | 8.38 | 12,857 | 2711 |

${\mathcal{M}}_{9}$ (3544) | 26,428 | 8.36 | 8.42 | 22,718 | 3710 |

${\mathcal{M}}_{10}$ (5024) | 38,642 | 8.44 | 8.54 | 33,724 | 4918 |

**Table 2.**Values of the clustering coefficient for real and virtual IoTs against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Avg. Clustering Coeff. (Real IoTs) | Avg. Clustering Coeff. (Virtual IoTs) | |||||
---|---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | ||

${\mathcal{M}}_{1}$ (176) | 0.230 | 0.318 | 0.368 | 0.389 | 0.394 | 0.401 | 0.408 |

${\mathcal{M}}_{2}$ (301) | 0.272 | 0.343 | 0.388 | 0.419 | 0.424 | 0.434 | 0.446 |

${\mathcal{M}}_{3}$ (485) | 0.293 | 0.396 | 0.437 | 0.477 | 0.482 | 0.488 | 0.497 |

${\mathcal{M}}_{4}$ (778) | 0.353 | 0.447 | 0.478 | 0.503 | 0.508 | 0.511 | 0.517 |

${\mathcal{M}}_{5}$ (946) | 0.371 | 0.452 | 0.492 | 0.512 | 0.522 | 0.524 | 0.526 |

${\mathcal{M}}_{6}$ (1256) | 0.385 | 0.486 | 0.511 | 0.529 | 0.530 | 0.532 | 0.535 |

${\mathcal{M}}_{7}$ (1725) | 0.386 | 0.501 | 0.524 | 0.536 | 0.537 | 0.538 | 0.539 |

${\mathcal{M}}_{8}$ (2028) | 0.388 | 0.519 | 0.536 | 0.541 | 0.541 | 0.542 | 0.543 |

${\mathcal{M}}_{9}$ (3544) | 0.392 | 0.522 | 0.540 | 0.544 | 0.544 | 0.545 | 0.546 |

${\mathcal{M}}_{10}$ (5024) | 0.395 | 0.534 | 0.546 | 0.546 | 0.546 | 0.547 | 0.548 |

**Table 3.**Values of the density for real and virtual IoTs against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Average Density (Real IoTs) | Average Density (Virtual IoTs) | |||||
---|---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | ||

${\mathcal{M}}_{1}$ (176) | 0.348 | 0.260 | 0.264 | 0.280 | 0.289 | 0.296 | 0.301 |

${\mathcal{M}}_{2}$ (301) | 0.262 | 0.292 | 0.303 | 0.309 | 0.315 | 0.320 | 0.324 |

${\mathcal{M}}_{3}$ (485) | 0.274 | 0.390 | 0.395 | 0.400 | 0.402 | 0.405 | 0.408 |

${\mathcal{M}}_{4}$ (778) | 0.269 | 0.476 | 0.483 | 0.490 | 0.501 | 0.509 | 0.514 |

${\mathcal{M}}_{5}$ (946) | 0.276 | 0.492 | 0.509 | 0.521 | 0.536 | 0.534 | 0.556 |

${\mathcal{M}}_{6}$ (1256) | 0.284 | 0.547 | 0.556 | 0.567 | 0.572 | 0.576 | 0.581 |

${\mathcal{M}}_{7}$ (1725) | 0.278 | 0.582 | 0.582 | 0.594 | 0.598 | 0.598 | 0.601 |

${\mathcal{M}}_{8}$ (2028) | 0.273 | 0.609 | 0.610 | 0.620 | 0.626 | 0.630 | 0.639 |

${\mathcal{M}}_{9}$ (3544) | 0.269 | 0.626 | 0.628 | 0.630 | 0.634 | 0.636 | 0.637 |

${\mathcal{M}}_{10}$ (5024) | 0.262 | 0.636 | 0.636 | 0.638 | 0.638 | 0.640 | 0.642 |

**Table 4.**Values of both clustering coefficient and density of real and virtual IoTs against the size of MIoTs (unsupervised approach).

MIoT (Size) | Average Clustering Coefficient | Average Density | ||
---|---|---|---|---|

Real IoTs | Virtual IoTs | Real IoTs | Virtual IoTs | |

${\mathcal{M}}_{1}$ (176) | 0.230 | 0.473 | 0.348 | 0.315 |

${\mathcal{M}}_{2}$ (301) | 0.272 | 0.499 | 0.262 | 0.350 |

${\mathcal{M}}_{3}$ (485) | 0.293 | 0.500 | 0.274 | 0.375 |

${\mathcal{M}}_{4}$ (778) | 0.353 | 0.511 | 0.269 | 0.318 |

${\mathcal{M}}_{5}$ (946) | 0.372 | 0.509 | 0.276 | 0.316 |

${\mathcal{M}}_{6}$ (1256) | 0.385 | 0.506 | 0.284 | 0.314 |

${\mathcal{M}}_{7}$ (1725) | 0.386 | 0.522 | 0.280 | 0.328 |

${\mathcal{M}}_{8}$ (2028) | 0.388 | 0.535 | 0.273 | 0.360 |

${\mathcal{M}}_{9}$ (3544) | 0.394 | 0.547 | 0.271 | 0.364 |

${\mathcal{M}}_{10}$ (5024) | 0.398 | 0.562 | 0.269 | 0.368 |

**Table 5.**Average fraction of merged c-nodes against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Average Fraction of Merged C-Nodes | |||||
---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | |

${\mathcal{M}}_{1}$ (176) | 0.304 | 0.455 | 0.517 | 0.532 | 0.554 | 0.572 |

${\mathcal{M}}_{2}$ (301) | 0.380 | 0.515 | 0.608 | 0.627 | 0.652 | 0.679 |

${\mathcal{M}}_{3}$ (485) | 0.539 | 0.661 | 0.782 | 0.798 | 0.813 | 0.823 |

${\mathcal{M}}_{4}$ (778) | 0.690 | 0.786 | 0.860 | 0.874 | 0.883 | 0.892 |

${\mathcal{M}}_{5}$ (946) | 0.724 | 0.812 | 0.884 | 0.898 | 0.916 | 0.924 |

${\mathcal{M}}_{6}$ (1256) | 0.808 | 0.883 | 0.939 | 0.943 | 0.946 | 0.948 |

${\mathcal{M}}_{7}$ (1725) | 0.862 | 0.908 | 0.952 | 0.961 | 0.961 | 0.963 |

${\mathcal{M}}_{8}$ (2028) | 0.908 | 0.959 | 0.974 | 0.975 | 0.976 | 0.977 |

${\mathcal{M}}_{9}$ (3544) | 0.928 | 0.963 | 0.976 | 0.977 | 0.977 | 0.978 |

${\mathcal{M}}_{10}$ (5024) | 0.936 | 0.968 | 0.978 | 0.979 | 0.980 | 0.981 |

**Table 6.**Average fraction of real IoTs involved in a virtual IoT against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Average Fraction of Involved Real IoTs | |||||
---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | |

${\mathcal{M}}_{1}$ (176) | 0.373 | 0.467 | 0.488 | 0.476 | 0.452 | 0.448 |

${\mathcal{M}}_{2}$ (301) | 0.365 | 0.469 | 0.525 | 0.501 | 0.488 | 0.480 |

${\mathcal{M}}_{3}$ (485) | 0.482 | 0.477 | 0.448 | 0.442 | 0.435 | 0.432 |

${\mathcal{M}}_{4}$ (778) | 0.457 | 0.432 | 0.418 | 0.415 | 0.413 | 0.411 |

${\mathcal{M}}_{5}$ (946) | 0.455 | 0.482 | 0.624 | 0.628 | 0.647 | 0.644 |

${\mathcal{M}}_{6}$ (1256) | 0.453 | 0.514 | 0.805 | 0.864 | 0.917 | 0.924 |

${\mathcal{M}}_{7}$ (1725) | 0.482 | 0.577 | 0.815 | 0.872 | 0.917 | 0.924 |

${\mathcal{M}}_{8}$ (2028) | 0.514 | 0.672 | 0.833 | 0.898 | 0.917 | 0.924 |

${\mathcal{M}}_{9}$ (3544) | 0.584 | 0.704 | 0.844 | 0.905 | 0.924 | 0.926 |

${\mathcal{M}}_{10}$ (5024) | 0.624 | 0.727 | 0.888 | 0.911 | 0.928 | 0.934 |

**Table 7.**Average fraction of merged c-nodes and average fraction of real IoTs involved in a virtual IoT against the size of MIoTs (unsupervised approach).

MIoT (Size) | Average Fraction of Merged C-Nodes | Average Fraction of Involved Real IoTs |
---|---|---|

${\mathcal{M}}_{1}$ (176) | 0.227 | 0.361 |

${\mathcal{M}}_{2}$ (301) | 0.306 | 0.353 |

${\mathcal{M}}_{3}$ (485) | 0.309 | 0.357 |

${\mathcal{M}}_{4}$ (778) | 0.342 | 0.356 |

${\mathcal{M}}_{5}$ (946) | 0.334 | 0.359 |

${\mathcal{M}}_{6}$ (1256) | 0.326 | 0.361 |

${\mathcal{M}}_{7}$ (778) | 0.332 | 0.360 |

${\mathcal{M}}_{8}$ (2028) | 0.335 | 0.358 |

${\mathcal{M}}_{9}$ (3544) | 0.341 | 0.371 |

${\mathcal{M}}_{10}$ (5024) | 0.344 | 0.378 |

**Table 8.**Average Herfindahl Index of virtual IoTs against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Average Herfindhal Index | |||||
---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | |

${\mathcal{M}}_{1}$ (176) | 0.207 | 0.186 | 0.177 | 0.175 | 0.173 | 0.172 |

${\mathcal{M}}_{2}$ (301) | 0.204 | 0.183 | 0.174 | 0.173 | 0.172 | 0.171 |

${\mathcal{M}}_{3}$ (485) | 0.178 | 0.173 | 0.170 | 0.170 | 0.169 | 0.168 |

${\mathcal{M}}_{4}$ (778) | 0.172 | 0.172 | 0.170 | 0.170 | 0.169 | 0.168 |

${\mathcal{M}}_{5}$ (946) | 0.172 | 0.170 | 0.169 | 0.169 | 0.169 | 0.168 |

${\mathcal{M}}_{6}$ (1256) | 0.173 | 0.168 | 0.167 | 0.169 | 0.168 | 0.167 |

${\mathcal{M}}_{7}$ (1725) | 0.170 | 0.168 | 0.167 | 0.169 | 0.168 | 0.167 |

${\mathcal{M}}_{8}$ (2028) | 0.168 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 |

${\mathcal{M}}_{9}$ (3544) | 0.168 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 |

${\mathcal{M}}_{10}$ (5024) | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 |

**Table 9.**Average Herfindahl Index of virtual IoTs against the size of MIoTs (unsupervised approach).

MIoT (Size) | Average Herfindahl Index |
---|---|

${\mathcal{M}}_{1}$ (176) | 0.658 |

${\mathcal{M}}_{2}$ (301) | 0.543 |

${\mathcal{M}}_{3}$ (485) | 0.658 |

${\mathcal{M}}_{4}$ (778) | 0.636 |

${\mathcal{M}}_{5}$ (946) | 0.654 |

${\mathcal{M}}_{6}$ (1256) | 0.694 |

${\mathcal{M}}_{7}$ (1725) | 0.656 |

${\mathcal{M}}_{8}$ (2028) | 0.635 |

${\mathcal{M}}_{9}$ (3544) | 0.664 |

${\mathcal{M}}_{10}$ (5024) | 0.686 |

**Table 10.**Average values of ${f}_{st}$ against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Average ${\mathit{f}}_{\mathit{st}}$ | |||||
---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | |

${\mathcal{M}}_{1}$ (176) | 0.144 | 0.220 | 0.290 | 0.304 | 0.336 | 0.347 |

${\mathcal{M}}_{2}$ (301) | 0.126 | 0.170 | 0.177 | 0.175 | 0.178 | 0.179 |

${\mathcal{M}}_{3}$ (485) | 0.104 | 0.112 | 0.074 | 0.052 | 0.041 | 0.037 |

${\mathcal{M}}_{4}$ (778) | 0.057 | 0.051 | 0.028 | 0.038 | 0.047 | 0.049 |

${\mathcal{M}}_{5}$ (946) | 0.048 | 0.034 | 0.022 | 0.028 | 0.032 | 0.024 |

${\mathcal{M}}_{6}$ (1256) | 0.031 | 0.015 | 0.017 | 0.011 | 0.007 | 0.007 |

${\mathcal{M}}_{7}$ (1725) | 0.026 | 0.014 | 0.011 | 0.010 | 0.008 | 0.008 |

${\mathcal{M}}_{8}$ (2028) | 0.016 | 0.010 | 0.009 | 0.009 | 0.009 | 0.009 |

${\mathcal{M}}_{9}$ (3544) | 0.012 | 0.009 | 0.009 | 0.009 | 0.009 | 0.009 |

${\mathcal{M}}_{10}$ (5024) | 0.011 | 0.008 | 0.007 | 0.007 | 0.007 | 0.007 |

MIoT (Size) | Average ${\mathit{f}}_{\mathit{st}}$ |
---|---|

${\mathcal{M}}_{1}$(176) | 0.904 |

${\mathcal{M}}_{2}$(301) | 0.722 |

${\mathcal{M}}_{3}$(485) | 0.635 |

${\mathcal{M}}_{4}$(778) | 0.584 |

${\mathcal{M}}_{5}$(946) | 0.580 |

${\mathcal{M}}_{6}$(1256) | 0.576 |

${\mathcal{M}}_{7}$(1725) | 0.516 |

${\mathcal{M}}_{8}$(2028) | 0.477 |

${\mathcal{M}}_{9}$(3544) | 0.452 |

${\mathcal{M}}_{10}$(5024) | 0.426 |

**Table 12.**Average values of ${g}_{st}$ against the size of MIoTs and queries used to generate the virtual IoTs (supervised approach).

MIoT (Size) | Average ${\mathit{g}}_{\mathit{st}}$ | |||||
---|---|---|---|---|---|---|

$\left|\mathit{Q}\right|=\mathbf{1}$ | $\left|\mathit{Q}\right|=\mathbf{2}$ | $\left|\mathit{Q}\right|=\mathbf{4}$ | $\left|\mathit{Q}\right|=\mathbf{6}$ | $\left|\mathit{Q}\right|=\mathbf{8}$ | $\left|\mathit{Q}\right|=\mathbf{10}$ | |

${\mathcal{M}}_{1}$ (176) | 4.018 | 2.792 | 2.223 | 1.918 | 1.331 | 1.321 |

${\mathcal{M}}_{2}$ (301) | 3.563 | 2.619 | 2.445 | 2.009 | 1.683 | 1.664 |

${\mathcal{M}}_{3}$ (485) | 3.269 | 2.370 | 1.426 | 1.528 | 1.626 | 1.674 |

${\mathcal{M}}_{4}$ (778) | 3.130 | 2.168 | 2.367 | 1.916 | 1.494 | 1.325 |

${\mathcal{M}}_{5}$ (946) | 3.232 | 2.102 | 1.864 | 1.712 | 1.461 | 1.391 |

${\mathcal{M}}_{6}$ (1256) | 3.467 | 1.979 | 1.378 | 1.412 | 1.438 | 1.452 |

${\mathcal{M}}_{7}$ (1725) | 3.476 | 2.224 | 1.414 | 1.444 | 1.494 | 1.492 |

${\mathcal{M}}_{8}$ (2028) | 3.496 | 2.669 | 1.489 | 1.491 | 1.521 | 1.545 |

${\mathcal{M}}_{9}$ (3544) | 3.507 | 2.712 | 1.612 | 1.624 | 1.631 | 1.632 |

${\mathcal{M}}_{10}$ (5024) | 3.517 | 2.926 | 1.783 | 1.841 | 1.864 | 1.874 |

MIoT (Size) | Average ${\mathit{g}}_{\mathit{st}}$ |
---|---|

${\mathcal{M}}_{1}$ (176) | 1.341 |

${\mathcal{M}}_{2}$ (301) | 1.269 |

${\mathcal{M}}_{3}$ (485) | 1.211 |

${\mathcal{M}}_{4}$ (778) | 1.177 |

${\mathcal{M}}_{5}$ (946) | 1.173 |

${\mathcal{M}}_{6}$ (1256) | 1.171 |

${\mathcal{M}}_{7}$ (1725) | 1.194 |

${\mathcal{M}}_{8}$ (2028) | 1.273 |

${\mathcal{M}}_{9}$ (3544) | 1.281 |

${\mathcal{M}}_{10}$ (5024) | 1.301 |

**Table 14.**Average size and number of virtual IoTs against the increase of the MIoT size (unsupervised approach).

MIoT (Size) | Average Size of Virtual IoTs | Number of Virtual IoTs |
---|---|---|

${\mathcal{M}}_{1}$ (176) | 22.44 | 10 |

${\mathcal{M}}_{2}$ (301) | 28.21 | 13 |

${\mathcal{M}}_{3}$ (485) | 36.64 | 16 |

${\mathcal{M}}_{4}$ (778) | 40.82 | 22 |

${\mathcal{M}}_{5}$ (946) | 44.66 | 24 |

${\mathcal{M}}_{6}$ (1256) | 46.74 | 30 |

${\mathcal{M}}_{7}$ (1725) | 48.12 | 39 |

${\mathcal{M}}_{8}$ (2028) | 50.24 | 45 |

${\mathcal{M}}_{9}$ (3544) | 50.46 | 78 |

${\mathcal{M}}_{10}$ (5024) | 50.64 | 105 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lo Giudice, P.; Nocera, A.; Ursino, D.; Virgili, L.
Building Topic-Driven Virtual IoTs in a Multiple IoTs Scenario. *Sensors* **2019**, *19*, 2956.
https://doi.org/10.3390/s19132956

**AMA Style**

Lo Giudice P, Nocera A, Ursino D, Virgili L.
Building Topic-Driven Virtual IoTs in a Multiple IoTs Scenario. *Sensors*. 2019; 19(13):2956.
https://doi.org/10.3390/s19132956

**Chicago/Turabian Style**

Lo Giudice, Paolo, Antonino Nocera, Domenico Ursino, and Luca Virgili.
2019. "Building Topic-Driven Virtual IoTs in a Multiple IoTs Scenario" *Sensors* 19, no. 13: 2956.
https://doi.org/10.3390/s19132956