Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks

A lot of previous work showed that the sectional mean first-passage time (SMFPT), i.e., the average of mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) averaged over all starting points in scale-free small-world networks exhibits a sublinear or linear dependence on network order $N$ (number of nodes), which indicates that hub nodes are very efficient in receiving information if one looks upon the random walker as an information messenger. Thus far, the efficiency of a hub node sending information on scale-free small-world networks has not been addressed yet. In this paper, we study random walks on the class of Koch networks with scale-free behavior and small-world effect. We derive some basic properties for random walks on the Koch network family, based on which we calculate analytically the partial mean first-passage time (PMFPT) defined as the average of MFPTs from a hub node to all other nodes, excluding the hub itself. The obtained closed-form expression displays that in large networks the PMFPT grows with network order as $N \ln N$, which is larger than the linear scaling of SMFPT to the hub from other nodes. On the other hand, we also address the case with the information sender distributed uniformly among the Koch networks, and derive analytically the entire mean first-passage time (EMFPT), namely, the average of MFPTs between all couples of nodes, the leading scaling of which is identical to that of PMFPT. From the obtained results, we present that although hub nodes are more efficient for receiving information than other nodes, they display a qualitatively similar speed for sending information as non-hub nodes. Moreover, we show that the location of information sender has little effect on the transmission efficiency. The present findings are helpful for better understanding random walks performed on scale-free small-world networks.

💡 Research Summary

The paper investigates random‑walk based information transmission on the family of Koch networks, which are deterministic scale‑free graphs exhibiting the small‑world property. Starting from a triangle, each iteration replaces every existing triangle by attaching m groups of two new nodes to each of its three vertices, thereby generating at iteration t a network with Nₜ = 2(3m + 1)ᵗ + 1 nodes, Eₜ = 3(3m + 1)ᵗ edges, a power‑law degree distribution P(k) ∼ k⁻ᵞ (γ = 1 + ln(3m + 1)/ln(m + 1)), high clustering, and logarithmic average path length.

The authors focus on two mean‑first‑passage‑time (MFPT) based metrics. The first, called average sending time (AST), is the average MFPT from a hub node (one of the three highest‑degree vertices) to all other nodes. The second, the entire mean‑first‑passage time (EMFPT), is the average MFPT over all unordered node pairs. Both quantities are interpreted as measures of information‑sending efficiency, complementing the previously studied average receiving time (ART), which is the average MFPT from all nodes to a hub.

A key analytical result is the scaling relation for MFPTs when the network grows by one iteration: for any pair of nodes i, j that already exist at step t, the MFPT satisfies F_{ij}(t + 1) = (3m + 1) F_{ij}(t). This follows from a set of backward equations that account for the m new neighbors each old node acquires and the three‑step round‑trip required to reach them. Consequently, the mean return time (MRT) of a node also scales as R_i(t + 1) = (3m + 1)/(m + 1) R_i(t). For a node created at iteration t, the MRT is R_{new}(t) = 3(3m + 1)ᵗ, independent of the degree of its parent node; this matches the Kac formula R = 2E/k.

Using these relations, the authors compute the total MFPT from the hub to all nodes, T_tot(t). New nodes created at step t must first be reached via one of their older neighbors; the MFPT to a new node j′ can be written as T_j′(t) = T_j(t) + F_{jj′}(t). By expressing F_{jj′}(t) through the previously derived MRT (F_{jj′}(t) = 4⁄3 R_{j′}(t) − 2) and summing over all new nodes, a recursion for T_tot(t) is obtained and solved explicitly:

T_tot(t) = (2⁄3)(3m + 1)^{t+1}