We convert it to be a directed network by randomly a direction to each of the edges. The result shown in Fig. Due to the fact that node has a high value of and a large value of the , it can be considered to be a bridging node. In fact, from visual inspection of Fig. Nodes 72 and 87 have similar characteristics, while node 2 behaves more like an overlapping node.
Graph theory methods: applications in brain networks
Node diameters indicate the C index value, the color of each node is proportional to the index D. We also apply our method to another directed network: the C. The network contains nodes and edges and is divided into 3 communities, with each node representing a neuron and each edge representing a synaptic connection between neurons.
The C. The neurons with high centrality indices often have the most important functions, and all of them are inter-neurons. Applying our method to this network see Fig. We have offered an electrical-circuit-based method to ascertain and distinguish overlapping and bridging nodes that play key roles in the communications and interactions among communities in complex networks without the need to partition all communities explicitly. The two types of critical nodes can be distinguished from the other nodes within communities by the relatively high current flow passing through them, as captured by the centrality of current flow.
Further, the two types of nodes can be distinguished from each other via the imbalance of flows along their edges. In particular, the bridging edges of bridging nodes exhibit much high current flows than the other edges of the nodes. Whereas for the overlapping nodes, due to their dense connections to two communities and the absence of bridging edges, the current flows along their edges are relatively balanced. Thus the combination of the centrality of current flow passing through nodes and the imbalance of current flows along the edges of nodes offers a criterion for identifying the two types of nodes with high probability.
In contrast, we have shown that the method for community partition based on the betweenness centrality cannot be used to address this problem. We have applied our method to a number of artificial and real networks with certain community structure, finding that the two types of nodes discovered by our method are in good agreement with the inspection of small visualized networks.
Another advantage of our method is that it is available for both undirected and directed networks, accounting for its broad application scope in real situations.
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Despite the advantages of our method compared to previously established methods in the literature, there are still some open questions pertaining to explicitly inferring overlapping and bridging nodes. For example, although our method is capable of finding these nodes with high probability, we continue to lack a reasonable threshold so as to exactly distinguish the two types of nodes.
The challenge is rooted in the fact that there is only the measurement for the strength of communities rather than the exact definition of a community, accounting for the difficulty in exactly defining and recovering overlapping and bridging nodes. Nevertheless, our approach offers an alternative avenue for addressing the fundamental problem in complex networks and it is indeed effective and more efficient than existent methods in the literature based on the shortest paths and the betweenness centrality.
Taken together, our approach could motivate further effort towards detecting the key nodes pertaining to ubiquitous community structures in complex networks. Analyzed the data: FHZ.
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Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract Complex networks with community structures are ubiquitous in the real world. Introduction Many real networks typically contain components in which the nodes are of much denser connections to each other than to the rest of the network. Download: PPT. Figure 1. Schematic network composed of 32 nodes and separated into 3 parts. Methods Electrical-circuit method for undirected and directed network In an electrical-circuit network generated by placing a resistor with a specific electrical conductance on each edge of the network  , as shown in Fig.
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An electrical-circuit network with an electrical conductance on each edge. Figure 3. A simple network with four nodes and its equivalent circuit network. Method of finding and distinguishing two types of key nodes The overlapping and bridging nodes are located at conjunction positions, and the removal of these nodes will disable the interactions and communications among communities.
Results Performance on artificial networks Prior to applying our method to real-world networks, we discuss the inherent limits of the betweenness-based method for inferring the two types of nodes. Table 1. Centrality indices of the example sketched in Fig. Figure 6. The usage of our method and in the LFR benchmark network.
Real-world networks We test our method by using a number of real-world networks: the ZK network  , the SFI network  , and the C. Figure 8. The usage of our method in the directed SFI scientist collaboration network. Figure 9. The usage of our method in the C. Discussion We have offered an electrical-circuit-based method to ascertain and distinguish overlapping and bridging nodes that play key roles in the communications and interactions among communities in complex networks without the need to partition all communities explicitly.
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Nature — View Article Google Scholar New J Phys Phys Rev Lett Borgatti SP Centrality and network flow. Social Networks 55— The network is shown in Fig. The known communities are defined by conferences, each containing around 8 to 12 teams and marked with colors. Links representing intra-conference games are also marked with the same colors as the corresponding conferences. In principle, teams from one conference are more likely to play games with each other than with teams belonging to different conferences.
There also exist some independent teams that do not belong to any conference, and these teams are marked with a light-green color. Communities of college football network, using colors for conferences and spatial clusterings for identified communities. The communities identified by the proposed method are represented by spatial clusterings in Fig.
In general, the proposed method correctly clusters teams from one conference. The independent teams are clustered with conferences with which they played games most frequently, because the independent teams seldom play games between themselves. The clusters detected by the proposed method deviate from the conference segmentation in several ways. But this result is understandable given the fact that there was only one game involving teams from both these two parts. Second, one team from the Conference USA conference, marked with a dark red color, is clustered with teams from the Western Athletic conference.
This team played no games with other teams from the Conference USA conference, but played games with every team from the Western Athletic conference. Third, two teams from the Western Athletic conference are isolated from other teams from this conference, and each is grouped with part of the Sun Belt conference.