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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 walksto a given hub node (node with maximum degree) averaged over all startingpoints in scale-free small-world networks exhibits a sublinear or lineardependence on network order $N$ (number of nodes), which indicates that hubnodes are very efficient in receiving information if one looks upon the randomwalker as an information messenger. Thus far, the efficiency of a hub nodesending information on scale-free small-world networks has not been addressedyet. In this paper, we study random walks on the class of Koch networks withscale-free behavior and small-world effect. We derive some basic properties forrandom walks on the Koch network family, based on which we calculateanalytically the partial mean first-passage time (PMFPT) defined as the averageof MFPTs from a hub node to all other nodes, excluding the hub itself. Theobtained closed-form expression displays that in large networks the PMFPT growswith network order as $N \ln N$, which is larger than the linear scaling ofSMFPT to the hub from other nodes. On the other hand, we also address the casewith the information sender distributed uniformly among the Koch networks, andderive analytically the entire mean first-passage time (EMFPT), namely, theaverage of MFPTs between all couples of nodes, the leading scaling of which isidentical to that of PMFPT. From the obtained results, we present that althoughhub nodes are more efficient for receiving information than other nodes, theydisplay a qualitatively similar speed for sending information as non-hub nodes.Moreover, we show that the location of information sender has little effect onthe transmission efficiency. The present findings are helpful for betterunderstanding random walks performed on scale-free small-world networks.

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