Information Spreading in Random Graphs Evoving by Norros-Reittu Model

TL;DR

The paper studies how a single message spreads in a network that grows according to the Norros-Reittu PAM. It derives explicit pmfs for the number of informed nodes $\#S_{K^*}$, the total number of nodes $N_{K^*}$ at a fixed time $T^*$, and the distribution of their ratio, by linking the transmission events to a Poisson-binomial framework and a Poisson clock model with rate $\lambda$. A key result is the closed-form success probability at each step, $p_{k+1} = 1 - \prod_{w \in S_k} e^{-\frac{\Lambda_w \Lambda_{k+1}}{L_{k+1}}}$, with $L_{k+1} = \sum_{i=0}^{k+1} \Lambda_i$, which underpins the Poisson-binomial pmf for $\#S_k$ and the related distributions. Simulations across heavy-tail parameters $\tau$ and various initial conditions demonstrate how $\tau$, $N_0$, and $\#S_0$ shape diffusion speed and lead to an apparent equilibrium as the network expands. The work provides a rigorous probabilistic framework for diffusion on growing random graphs with heavy-tailed capacities and suggests avenues for time-dependent extensions and real-world applications in social, IoT, and distributed systems.

Abstract

The paper is devoted to the spreading of a message within the random graph evolving by the Norros-Reittu preferential attachment model. The latter model forms random Poissonian numbers of edges between newly added nodes and existing ones. For a pre-fixed time $T^*$, the probability mass functions of the number of nodes obtained the message and the total number of nodes in the graph, as well as the distribution function of their ratio are derived. To this end, the success probability to disseminate the message from the node with the message to the node without message is proved. The exposition is illustrated by the simulation study.

Figures (2)

• Figure 1: The evolution of a network generated using the Norros-Reittu model with regularly varying distributed node capacities (see (\ref{['0']}) with $\tau$ = 2.5), where the nodes with messages are marked in black, without message $-$ in grey. Size of circles is proportional to node weight ($\Lambda$). Multiple edges are shown as a single edge, self-loops are not shown. On the left side $-$ the graph with $N_2=3$; in the middle - Emergent Phase ($N_{19}=20$), where the preferential attachment creates hub structures; on the right $-$ Mature Network ($N_{99}=100$) with the giant connected component. The first line is responsible for $\tau = 1.5$, the second line for $\tau = 2.5$, and the third line for $\tau = 3.5$.
• Figure 2: The average value of $\#S_k / N_k$ against $N_k$ for $20$ repetitions of graph evolution by the Norros-Reittu PAM: on the left side $-$$N_0 = 10$ and $\#S_0 \in \{1, 5, 10\}$; in the middle $-$$N_0 = 50$ and $\#S_0 \in \{1, 5, 10\}$; on the right $-$$N_0 = 100$ and $\#S_0 \in \{1, 5, 10\}$, where the first line is responsible for $\tau = 1.5$, the second line for $\tau = 2.5$, and the third line for $\tau = 3.5$.