1.5 SIoT network characterization
1.5.2 Numerical results
In the following I show and analyze the numerical results obtained as explained in the previ- ous section. More specifically, I study the characteristics of the random variable X(A). This is
defined as the random variable representing the distance between two nodes that are tied by a social relationship of typeA (in my caseA ∈ {POR, C-LOR, OOR, SOR, C-WOR}). I am interested in the probability density function ofX(A).
POR and C-LOR
Parental Object Relationships (POR) are independent (in the scales of interest) from the specific positions of nodes. In fact, in most cases objects tied by POR are distributed uniformly in the area of interest. Accordingly, I do not focus on the distribution of X(P OR). Here, I only stress
that POR can be utilized to build long links in the SIoT.
The distribution ofX(C−LOR)is obvious as well. In fact, in this case, a link exists between
two objects only if their distance is very small. Accordingly,
fXC−LOR(x)≈δ(x) (1.2)
Observe that relationships of C-LOR type can be utilized by the applications to explore the environment surrounding a given object and, therefore, are extremely important in the context of smart environment applications.
OOR
In Figure1.4I represent the probability density function in case of Ownership Object Relation- ship, that is, I show fX(OOR)(x) versus the value of the distance x. In the same figure, I also
show the probability density function of the exponential and Gamma distributions that have av- erage value and variance equal to those ofX(OOR). In the figure it is evident that the exponential
distribution does not provide an accurate approximation ofX(OOR).
To better assess the accuracy of the approximation provided by the Gamma distribution, I need to clean the measured fX(OOR)(x) and to this purpose I filter it. Specifically, I define
the operator Φ(f) which can be applied to any sequencef of values and that returns another sequence{Φ(f)}such that itsi-th value is the sum of the firstivalues in the sequencef, that is
{Φ(f)}i =
i
X
j=1
{f}j (1.3)
1.5. SIoT network characterization 26
Figure 1.4: Probability density function of the variableX(OOR). In the figure I show the pdf of
the exponential and Gamma distributions with the same average value and variance.
Figure 1.5: Values of{Φ(fX(OOR)(x)}and filtered pdf of the Gamma distributions with the same
1.5. SIoT network characterization 27
Figure 1.6: Probability density functions,fX(SOR)(x), obtained for different values ofTC.
distribution that approximatesfX(OOR)(x). In Figure1.5 it is evident that the Gamma distribu-
tion does not provide an accurate approximation of fX(OOR)(x). In the same figure it is also
evident that {Φ(fX(OOR)(x))} has a linear behavior when represented in log-log scale, which
means that fX(OOR)(x) is power-law. To demonstrate this, in Figure1.5 I show the line which
approximates{Φ(fX(OOR)(x))}in the log-log scale. In other words it is possible to approximate
fX(OOR)(x)∝xβOOR (1.4)
For example, in the case discussed above, I have thatβOORis equal to -0.827.
SOR
In Figure1.6I represent the probability density functions,fX(SOR)(x), of the distance between
nodes connected by Social Object Relationships for different values of the parameterTC. I have assumed that a relationship of the SOR type is established between objects if their owners meet at leastNC times, if successive meetings occur at intervals of duration longer thanTI, and if each of the meetings lasts longer thanTC. More specifically, in Figure1.6I assume thatNC = 2 andTI = 8hours. In order to “clean” the figure, I represent the values of{Φ(fX(SOR)(x))} in
Figure 1.7. In the same figure I show the filtered pdf of the exponential distribution and the power law distribution which approximatefX(SOR)(x). By observing the figure, one notices that
1.5. SIoT network characterization 28
Figure 1.7: “Filtered” probability density functions,fX(SOR)(x), obtained for different values of TC, and filtered exponential distribution that approximates them.
TC . Additionally, it arises that the exponential distribution provides an accurate approximation of X(SOR) for large values of x, while the power law distribution is more accurate for small
values ofx. Accordingly,fX(SOR)(x)can be approximated as follows: fX(SOR)(x)∝ ( xβSOR ifx < x thresh. e−γSOR·x ifx > x thresh. (1.5)
In my case, for example,βSOR= 0.12,γSOR = 3.87, andxThresh = 1.
This dichotomy in the behavior ofX(SOR)– that is, it is power-law for low values ofxand
exponential for high values ofx– is in line with what has been recently demonstrated in [42]. In Figure1.8I show the number of SOR relationships established versus the value ofTC. As expected, the number of relationships decreases as the value ofTC increases.
Same discussions can be done by observing Figure1.9where I show the probability density function fX(SOR)(x) for different values ofTI. In this case, I have assumed thatNC = 2 and
TC = 30 min. Also in this case the number of relationships established decreases as the value ofTI increases.
Finally, similar observations can be done by considering Figures1.10and1.11which are analogous to1.6and1.7, respectively, but have been obtained by using different values ofNC.
1.5. SIoT network characterization 29
Figure 1.8: Number of SOR relationships established versus the value ofTC, whenNC = 2and
TI = 8hours.
1.5. SIoT network characterization 30
Figure 1.10: Probability density functions,fX(SOR)(x), obtained for different values ofNC.
Figure 1.11: “Filtered” probability density function,fX(SOR)(x), obtained for different values of NC, and filtered exponential distribution which approximates them.
1.5. SIoT network characterization 31
Figure 1.12: Probability density functions, fX(C−W OR)(x), obtained for different values of TC, and Gamma distribution which approximates them.
CWOR
In Figure 1.12 I show the pdf of X(C−W OR) obtained when I impose that a co-work social
relationship is established only when the objects “meet” in a certain set of locations (offices, fabrics, laboratories, etc.) and that such meetings last for longer thanTC. More specifically, in the figure I represent the results obtained by considering different values ofTC; furthermore I show the Gamma distribution that approximates the above pdfs. By observing Figure 1.12, I notice that the value ofTC does not have a significant impact on the probability distributions
fX(C−W OR)(x) and that the Gamma distribution provides an accurate approximation of such
pdfs. Indeed, I have
fX(C−W OR)(x) =x
k−1 e−x/θ
Γ(k)θk (1.6)
whereΓ(k)is defined as follows:
Γ(k) =
Z ∞
0
tk−1e−tdt (1.7)
andθ= 15.93whereask= 2.11.
In Figure 1.13 I show analogous curves when there are no predetermined locations in which co-work object relationships can be established.
1.5. SIoT network characterization 32
Figure 1.13: Probability density functions, fX(C−W OR)(x), obtained for different values of TC,
Chapter 2
Network Navigability
In the previous Chapter, I presented the new paradigm known as Social Internet of Things, which proposes the integration of social networking concepts into the Internet of Things. The underneath idea is that every object can look for the desired service using its friendships, in a distributed manner.
A SIoT network is based on the idea that every object can look for the desired service by using its relationships, querying its friends, the friends of its friends and so on in a distributed manner, in order to guarantee an efficient and scalable discovery of objects and services follow- ing the same principles that characterize the social networks between humans. The assumption that a SIoT network will be navigable is based on the principle of the sociologist Stanley Mil- gram about the small-world phenomenon. This paradigm refers to the existence of short chains of acquaintances among individual in societies [43].
According to this paradigm, each object has to store and manage the information related to the friendships, implement the search functions, and eventually employ additional tools such as the trustworthiness relationship module to evaluate the reliability of each friend. Clearly, the number of relationships affects the memory consumption, the use of computational power and battery, and the efficacy of the service search operations. It results that the selection of the friendships is key for a successful deployment of the SIoT. In this Chapter, I intend to address this issue by analyzing possible strategies for selection of appropriate links for the benefit of overall network navigability. I first propose five heuristics which are based on local network properties and that are expected to have an impact on the overall network structures. I then perform extensive experiments, which are intended to analyze the performance in terms of giant components, average degree of connections, local clustering and average path length.
The Chapter is organized as follows. In Section2.1I present the scenario of the social IoT and provide a quick survey of the solutions for the search of services in the IoT. In Section2.2