Egocentric network focused community aware multicast routing for DTNs

Abstract

Multicasting for delay-tolerant networks (DTNs) in sparse social network scenarios is a challenge due to the deficiency of end-to-end paths. In social network scenarios, the behaviors of their nodes are controlled by human beings, and node mobility is the same as that of humans. To design the multicasting algorithms for DTNs, therefore, it would be promising to capture the intrinsic characteristics of relationships among these nodes. In this paper, multicasting in DTNs is regarded as a message dissemination issue in social networks, and an egocentric network focused community aware multicast routing algorithm (ENCAR) is proposed. As distinct from some social-based routing algorithms which only focus on centrality analysis, ENCAR is an utility based and hierarchical routing algorithm, its utility function is constructed on the basis of centrality analysis and destination-oriented contact probability. We take notice of clustering phenomenon in social networks, and present the community aware forwarding schemes. In addition, to simulate the mobility of individuals in social networks, a novel community based random way point mobility model is also presented. In this paper, the performance of ENCAR is theoretically analyzed and further evaluated on simulator ONE. Simulation results show that ENCAR outperforms most of the existing multicast routing algorithms in routing overhead, on condition that delivery ratio is relatively high, with other significant parameters guaranteed to perform well.

Appendix

Theorem 1

For the proposed mobility model CB_RWP, the encounter process between a pair of nodes is Possion process, and the inter-contact time follows exponential distribution.

Proof

A continuous-time counting process $\{N(t), t>0\}$ can be classified into Possion process if it possesses the following properties:

• 1. $N(0)=0$;

• 2. The numbers of occurrences counted in disjoint intervals are independent from each other;

• 3. The probability distribution for the number of occurrences counted in any interval only depends on the length of the interval and follows Possion distribution.

In DTNs, an occurrence represents the contact between a pair of nodes. If $t=0$, any two nodes do not contact with each other, of course, $N(0)=0$. In CB_RWP, the speed of each node is randomly chosen between [3 m/s, 10 m/s], the moving direction of each node is randomly chosen between $[0,2\pi ]$, and the next waypoint is also randomly chosen within the simulation area. Apparently, every node stays in or departs from its community independently. Thus, the motions of nodes will not influence each other. As we know, the contacts of node pairs only occur when two nodes encounter each other. With any pair of nodes, the numbers of contacts in disjoint intervals are independent. Hence, the encounter process N(t) surely has the properties (1) and (2).

To prove the third property, we make the following notations firstly:

• 1. Let $v_{i}$ and $v_{j}$ be a pair of nodes in CB_RWP, and $\lambda _{i,j}$ denote the contact ratio between them. $\lambda _{i,j}$ represents the number of contacts in a unit interval and can be estimated as $\lambda = \frac{n}{\sum \nolimits _{i=1}^{n}T_{i}}$, where $T_{1}, T_{2}, \ldots , T_{n}$ are inter-contact time samples;

• 2. Let $\triangle t$ be the unit interval. $\triangle t$ represents a small duration, and a period of time T can be divided into $n=\frac{T}{\triangle t}$ units;

• 3. Let $P_{0}$ be the probability that $v_{i}$ encounters $v_{j}$ only once during $\triangle t$.

The number of contacts during T is denoted as X. Obviously, X follows the Bernoulli distribution. The probability that $v_{i}$ encounters $v_{j}$ for k times during T can be formulated as:

$$\begin{aligned} P(X=k)= C^k_n\times {P_0}^k\times (1-P_0)^{n-k} \end{aligned}$$

(13)

$$\begin{aligned} E(X)= n\times P_0 \end{aligned}$$

(14)

The average number of encounters in T can also be represented as $\lambda _{i,j}\cdot T$, and we have $\lambda _{i,j}\cdot T=n\cdot P_0$, $P_0=\frac{\lambda _{i,j}\cdot T}{n}$. Thus, Eq. (13) can be transformed into:

$$\begin{aligned} \begin{aligned}&P\{[N(t+T)-N(t)]=k\}\\&=\frac{n\cdot (n-1)\cdots (n-k+1)}{k!}\times \left( \frac{\lambda _{i,j}\cdot T}{n}\right) ^k\times \left( 1-\frac{\lambda _{i,j}\cdot T}{n}\right) ^{n-k}\\&=\frac{n\cdot (n-1)\cdots (n-k+1)}{n^k}\times \frac{(\lambda _{i,j}\cdot T)^k}{k!}\times \left( 1-\frac{\lambda _{i,j}\cdot T}{n}\right) ^{n-k} \end{aligned} \end{aligned}$$

(15)

When $\triangle t \rightarrow 0$, $n\rightarrow \infty$

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{n\cdot (n-1)\cdot \cdot \cdot (n-k+1)}{n^k}=1 \end{aligned}$$

(16)

$$\begin{aligned}&\lim _{n\rightarrow \infty }\left( 1-\frac{\lambda _{i,j}\cdot T}{n}\right) ^{n-k}=e^{\lim \limits _{n\rightarrow \infty }\ln \left( 1-\frac{\lambda _{i,j}\cdot T}{n}\right) ^{n-k}}\nonumber \\&=e^{\lim \limits _{n\rightarrow \infty } \left( (n-k)\cdot \left( -\frac{\lambda _{i,j}\cdot T}{n}\right) \right) }\nonumber \\&=e^{-\lambda _{i,j}\cdot T} \end{aligned}$$

(17)

Finally,

$$\begin{aligned} \begin{aligned}&P\{[N(t+T)-N(t)]=k\}=\frac{(\lambda _{i,j}\cdot T)^k\times e^{-\lambda _{i,j}\cdot T}}{k!} \end{aligned} \end{aligned}$$

(18)

Consequently, in CB_RWP, the probability distribution for the number of contacts counted in any time interval only depends on the length of the interval and follows Possion distribution. Thus, the third property is proven.

To sum up, it can be asserted that the encounter process $\{N(t), t>0\}$ in CB_RWP is Possion process. From Eq. (18), the probability that $v_{i}$ does not encounter $v_{j}$ during T is

$$\begin{aligned} P\{[N(t+T)-N(t)]=0\}=e^{-\lambda _{i,j}\cdot T} \end{aligned}$$

(19)

As a result, we can derive that the probability that $v_{i}$ encounters $v_{j}$ during T is

$$\begin{aligned} F(T)=1 - e^{-\lambda _{i,j}\cdot T} \end{aligned}$$

(20)

Therefore, the inter-contact time between a pair of nodes follows exponential distribution, and Theorem 1 is proven. $\square$

Table 4 Acceptance ratio of $\chi ^{2}$ tests

To validate our theoretical proof, we do a lot of $\chi ^{2}$ hypothesis tests. Each $\chi ^{2}$ test is done on tens of thousands inter-contact time samples, which are derived from CB_RWP during simulation. The results of the acceptance ratio of $\chi ^{2}$ hypothesis tests are listed in Table 4, in which $P_{s}$ is the staying probability, n is the number of nodes and $\alpha$ is the significance level.