Altruism in social networks: good guys do finish first

Abstract

Altruism in social networks was traditionally considered based on user relationships and requirements. The capability of a user to help has seldom been taken into account while studying altruism. Also, to the best of our knowledge, no quantitative analysis has been made to determine the benefits of altruism. Here we quantitatively study the amount of altruism of users based on the help extended by users to each other and the benefits they reap because of being altruistic. Results indicate that a network in which 90 % users have capabilities that are above average contains 90 % altruistic users, while a network containing 50 % of users with above-average capabilities contains only 50 % altruistic users. Results also indicate that altruistic users (the good guys) reap more benefits than selfish users and free riders (i.e., finish first).

Appendix: Optimal \(h _{i}^{*}\) and \(e_{{ij}}^{*}\)

We present the details of the analysis in obtaining the expressions for $h _{i}^{*}$ and $e_{{ij}}^{*}$ in (11) and (12), respectively. The optimal $h _{i}^{*}$ and $e_{{ij}}^{*}$ are obtained by solving for $\mathcal{H}$ in (7), i.e., by computing

$$ {\mathcal{H}}={\mathcal{A}}^{-1}{\mathcal{T}}. $$

(19)

Thus, $\mathcal{H}$ can be obtained if the inverse of $\mathcal{A}$ in (8) is computed. The following definitions from matrix theory will be used in the analysis to obtain $\mathcal{A}^{-1}.$

Definition 5

(Meyer 2010) A permutation matrix, $\mathcal{P}_{n}$ is an n × n matrix whose entries are all 0’s and 1’s, such that every row and column has exactly one 1 in it.

Definition 6

An n × n permutation matrix, $\mathbf{P}_{k},$ is called the kth circulant matrix, if $\mathbf{P}_{k}$ is obtained by k cyclic permutations of the rows of the n × n identity matrix, $\mathbf{I}_{n}.$

Lemma 1

(Meyer 2010)

• 1. $\mathbf{P}_{1}^{k}=\mathbf{P}_{k}.$

• 2. $\mathbf{P}_1^{n}=\mathbf{I}_{n}.$

• 3. $\mathbf{P}_k^{-1}=\mathbf{P}_{n-k}.$

Let $\mathbf{C}$ and $\mathbf{D}^T$ be n ² × n and n × n ² matrices, respectively, defined as follows:

$$ {\mathbf{C}}\triangleq \left[ \begin{array}{l} {\mathbf{I}}_{n}\\ {\mathbf{P}}_{1}\\ {\mathbf{P}}_{1}\\ \vdots\\ {\mathbf{P}}_{n-1} \end{array} \right ] \hbox{ {\it and} } {\mathbf{D}}^{\rm T}\triangleq \left[{\mathbf{I}}_{n} {\mathbf{P}}_{n-1} {\mathbf{P}}_{n-2} \cdots {\mathbf{P}}_1\right ]. $$

(20)

Then, $\mathcal{A}$ in (8) can be written as $\mathcal{A}=\mathbf{I}_{n^{2}}+\mathbf{C}\mathbf{D}^{\rm T}.$ Using the Sherman–Morrison–Woodbury formula (Meyer 2010), $\mathcal{A}^{-1}$ can be determined as

$$ {\mathcal{A}}^{-1}=\left ( {\mathbf{I}}_{n^{2}}+{\mathbf{C}}{\mathbf{D}}^{\rm T}\right )^{-1}= {\mathbf{I}}_{n^{2}}-{\mathbf{C}}\left ({\mathbf{I}}_{n}+{\mathbf{D}}^{\rm T}{\mathbf{C}}\right )^{-1}{\mathbf{D}}^{\rm T}. $$

(21)

From (20), and (3) in Lemma 1, $\mathbf{D}^{\rm T}\mathbf{C}=n\mathbf{I}_{n},$ i.e., $\mathbf{I}_{n}+ \mathbf{D}^{\rm T}\mathbf{C}= (n+1)\mathbf{I}_{n},$ which when applied to (21) yields

$$ {\mathcal{A}}^{-1}={\frac{1}{n+1}} \left[ \begin{array}{lllllll} n{\mathbf{I}}_{n} & -{\mathbf{P}}_{n-1} & -{\mathbf{P}}_{n-2} & -{\mathbf{P}}_{n-3} & \cdots & -{\mathbf{P}}_{2} &-{\mathbf{P}}_{1}\\ -{\mathbf{P}}_{1} & n{\mathbf{I}}_{n} & -{\mathbf{P}}_{n-1} & -{\mathbf{P}}_{n-2} & \cdots & -{\mathbf{P}}_{3} &-{\mathbf{P}}_{2}\\ -{\mathbf{P}}_{2} & -{\mathbf{P}}_{1} & n{\mathbf{I}}_{n} & -{\mathbf{P}}_{n-1} & \cdots & -{\mathbf{P}}_{4} &-{\mathbf{P}}_{3}\\ -{\mathbf{P}}_{3} & -{\mathbf{P}}_{2} & -{\mathbf{P}}_{1} & n{\mathbf{I}}_{n} & \cdots & -{\mathbf{P}}_{5} &-{\mathbf{P}}_{4}\\ \vdots & \vdots & \vdots &\vdots & \ddots & \vdots & \vdots\\ -{\mathbf{P}}_{n-2} & -{\mathbf{P}}_{n-3} & -{\mathbf{P}}_{n-4} & -{\mathbf{P}}_{n-5} & \cdots & n{\mathbf{I}}_{n} &-{\mathbf{P}}_{n-1}\\ -{\mathbf{P}}_{n-1} & -{\mathbf{P}}_{n-2} & -{\mathbf{P}}_{n-3} & -{\mathbf{P}}_{n-4} & \cdots & -{\mathbf{P}}_{1} & n{\mathbf{I}}_{n} \end{array} \right]. $$

(22)

From (19) and (22), $h_{i}^{*}(t_{i},\mathbf{c})$ and $e_{ij}^{*}(t_{j},\mathbf{c})$ can be obtained as the expressions given by (11) and (12), respectively.