Maximum flow and topological structure of complex net - works

– The problem of sending the maximum amount of flow q between two arbitrary nodes s and t of complex networks along links with unit capacity is studied, which is equivalent to determining the number of link-disjoint paths between s and t. The average of q over all node pairs with smaller degree kmin is q k min ≃ c kmin for large kmin with c a constant implying that the statistics of q is related to the degree distribution of the network. The disjoint paths between hub nodes are found to be distributed among the links belonging to the same edge-biconnected component, and q can be estimated by the number of pairs of edge-biconnected links incident to the start and terminal node. The relative size of the giant edge-biconnected component of a network approximates to the coefficient c. The applicability of our results to real world networks is tested for the Internet at the autonomous system level. The analysis of network structures of complex systems has turned out to be extremely useful in exploring their large-scale organization and unveiling their evolutionary origin [1–4]. Anomalous features found by their graph-theoretical analyses [5–7] are the fingerprints of their hidden organization principles as well as the key to predicting their behaviors. Real-world networks usually serve a particular purpose which frequently can be expressed in terms of flow problems, in particular in the context of transport between the nodes along links (edges) with restricted capacities. For instance if one is interested in the maximum possible flow that can be sent from one node to another, one has to solve a maximum flow problem, for which polynomial algorithms exist. It also arises per se in a variety of situations such as assignment and scheduling problems [8]. In real-world networks one is often interested in the question of how many link-disjoint paths do exist between a particular node pair-for instance to establish as many independent transportation routes as possible between two nodes in the occasion of a sudden demand (like in the event of a natural catastrophe or in a military context). This problem is again identical to the maximum flow problem between those two nodes in the same network with unit-capacity links. Properties of complex networks related to transport are inherently connected to their topological structure, in particular to various aspects of connectedness. The existence of a single path from one node to another …