Hodge Decomposition of Information Flow on Small-World Networks

Abstract

We investigate the influence of the small-world topology on the composition of information flow on networks. By appealing to the combinatorial Hodge theory, we decompose information flow generated by random threshold networks on the Watts-Strogatz model into three components: gradient, harmonic and curl flows. The harmonic and curl flows represent globally circular and locally circular components, respectively. The Watts-Strogatz model bridges the two extreme network topologies, a lattice network and a random network, by a single parameter that is the probability of random rewiring. The small-world topology is realized within a certain range between them. By numerical simulation we found that as networks become more random the ratio of harmonic flow to the total magnitude of information flow increases whereas the ratio of curl flow decreases. Furthermore, both quantities are significantly enhanced from the level when only network structure is considered for the network close to a random network and a lattice network, respectively. Finally, the sum of these two ratios takes its maximum value within the small-world region. These findings suggest that the dynamical information counterpart of global integration and that of local segregation are the harmonic flow and the curl flow, respectively, and that a part of the small-world region is dominated by internal circulation of information flow.

Keywords: Hodge decomposition; functional brain networks; random threshold network; small-world network; transfer entropy.

Figure 1

Time evolution of δ_t for (A) k = 3 and (B) k = 4. Each curve is the average over 100 random initial conditions for each realization of RTN, 100 realizations of RTNs on each network and 400 networks generated by the WS model with a specified value of p. p = 0.001000 (red), p = 0.003375 (green), p = 0.011391 (blue), p = 0.038443 (magenta), p = 0.129746 (cyan), p = 0.437894 (orange), and p = 0.985261 (black). These values of p were chosen so that they are arranged with an equal interval in the logarithmic scale because the small-world regime can be discriminated well in the logarithmic scale of p as we can see from Figure 5. Concretely, $p=p_0\times 1.5^{3n}$ for p₀ = 0.001000 and 0 ≤ n ≤ 6.

Figure 2

An example of edge flow on a network generated from the WS model with N = 8, k = 2 and p = 0.1 and its combinatorial Hodge decomposition into the three components. The value of each flow is rounded off to the four decimal places and multiplied by 10² for visibility.

Figure 3

The magnitude of information flow divided by the number of nodes N = 400 for k = 3 and k = 4.

Figure 4

The relative strength of gradient γ (A,D), harmonic η (B,E) and curl χ (C,F) flows together with corresponding structural ratios are shown for k = 3 (top row) and k = 4 (bottom row).

Figure 5

The loop ratio λ is compared with a small-world index ω. (A) k = 3 and (B) k = 4.