Characterizing network topology using first-passage analysis

Understanding the topological characteristics of complex networks and how they affect navigability is one of the most important goals in science today, as it plays a central role in various economic, biological, ecological, and social systems. Here we apply first-passage analysis tools to investigate the properties and characteristics of random walkers in networks with different topology. Starting with the simplest two-dimensional square lattice, we modify its topology incrementally by randomly reconnecting links between sites. We characterize these networks by first-passage time from a significant number of random walkers without interaction, varying the departure and arrival locations. We also apply the concept of first-passage simultaneity, which measures the likelihood of two walkers reaching their destination together. These measures, together with the site occupancy statistics during the processes, allowed us to differentiate the studied networks, especially the random networks from the scale-free networks, by their navigability. We also show that small-world features can also be highlighted with the proposed technique.

Figure 1

(a) Square network ($L\times L$, $L=5)$ with periodic BCs. Each site has four links and the border sites are connected to the symmetric sites of the opposite border. (b) Conservative random network. Starting from a square network two unconnected sites are randomly chosen and a new connection is established between them. A site adjacent to each of these sites is chosen randomly to disconnect from these sites and establish a new connection. These steps are repeated until all the links to be rewired and, at the end, each site keeps four links. (c) Nonconservative random network. Starting from a square network two unconnected sites are randomly chosen and a new connection is established between them. A single site next to one of these sites is chosen to lose a link. Theses steps are typically repeat to approximately $90\%$ of original links.

Figure 2

Rewiring a square network using the preferential attachment. (a) Square network ($L\times L$, $L=5)$ with periodic BCs. Each site has four links and the border sites are connected to the symmetric sites of the opposite border. In the first step $S1$ and $S2$ are connect, and $S3$ and $S4$ will be disconnect. (b) After $\mathcal{N}=20$, the site $S1$ earned more links than the others. (c) With approximately $90\%$ of links rewired, the site $S1$ is the hub of network.

Figure 3

Distribution of links during the topological change from square network to scale-free network topology. (a) One thousand links and (b) $18577$ links were rewired.

Figure 4

(a) Square lattice with mixed BCs: Black dots correspond to reflecting boundaries and white dots correspond to absorbing sites. The nodes highlighted correspond to F (far from the absorbing boundary), C (central region of the lattice), and N (near the absorbing boundary). (b) $P\left(\omega \right)$ for ${10}^4$ independent RWs for each case.

Figure 5

Results in the conservative random network with reflective BCs. $P\left(\omega \right)$ for ${10}^4$ independent RWs for each case. (a) The target site far from the start site on the network without $\mathcal{N}$. (b) The target site near from the start site on the network without $\mathcal{N}$. In both cases we increase the $\mathcal{N}$ between the links.

Figure 6

Click distribution during the topological change from square network to conservative random network topology, respectively, for $\mathcal{N}=0$, $\mathcal{N}=100$, $\mathcal{N}=1000$, and $\mathcal{N}=20000$.

Figure 7

Results in the SW random network. $P\left(\omega \right)$ for ${10}^4$ independent RWs for each case.

Figure 8

Results in the scale-free network. $P\left(\omega \right)$ for ${10}^4$ independent RWs for each case.

Figure 9

Distribution of occupation of the sites for a random walk with ${10}^6$ steps, for each case.