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# KSC - K-Shell and Community Centrality

#### Definition

This model considers not only the internal properties of nodes but also the external properties of nodes, such as the community which these nodes belong to. The Susceptible-Infected-Recovered (SIR) model is used to evaluate the performance of KSC model.

The internal properties mean the classic centralities such as degree, closeness, betweenness and so forth. The external properties mean the properties based on community, such as the size and closeness of the community which the node is belong to. Given a complex network G=(V, E), the KSC value of node ν

f

f

f

α and β are set to different values according to different networks’ topology and to simplify the experiment while ensuring high performance, the authors use the following configurations:

The internal properties mean the classic centralities such as degree, closeness, betweenness and so forth. The external properties mean the properties based on community, such as the size and closeness of the community which the node is belong to. Given a complex network G=(V, E), the KSC value of node ν

_{o}is denoted by: f_{internal}(ν_{o}) represents the node’s internal influence while f_{external}(ν_{o}) represents the node’s external influence, α is the internal factor while β is the external factor, which satisfies α+β=1, α and β are determined by the actual topology and functionality of the network.f

_{internal}(ν_{o}) is denoted by: K(ν) is the internal property, which can be valued with degree, betweenness, closeness and K-shell. max_{v∈V}K(v) is the normalized factor.f

_{external}(ν_{o}) is denoted by: C is the collection composed by the communities calculated by FN algorithm, d(ν_{o}, c) is the number of ν_{o}’s neighbor in community c, |c| is the size of community c, max_{v∈V}(Σ_{c∈C}d(v,c)|c|) is the normalized factor.f

_{external}(ν_{o}) increases by the number of neighbors which lies in different communities. It indicates v_{o}’s ability to propagate messages to various communities which is related to the influence of v_{o}.α and β are set to different values according to different networks’ topology and to simplify the experiment while ensuring high performance, the authors use the following configurations:

- α = β = 0.5, K(v) = C
_{ks}(v)

#### Software

#### References

- HU, Q., GAO, Y., MA, P., YIN, Y., ZHANG, Y. & XING, C. 2013. A New Approach to Identify Influential Spreaders in Complex Networks. In: WANG, J., XIONG, H., ISHIKAWA, Y., XU, J. & ZHOU, J. (eds.) Web-Age Information Management. Springer Berlin Heidelberg.
- Xu, Jian. "KSC Centralized Index Model in Complex Network." Journal of Networks 9.5 (2014): 1245-1251.