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Centrality indices have two important limitations, one obvious and the other subtle. The obvious limitation is that a centrality which is optimal for one application is often sub-optimal for a different application. Indeed, if this were not so, we would not need so many different centralities. An illustration of this phenomenon is provided by the Krackhardt kite graph, for which three different notions of centrality give three different choices of the most central vertex.

Centrality indices have two important limitations, one obvious and the other subtle. The obvious limitation is that a centrality which is optimal for one application is often sub-optimal for a different application. Indeed, if this were not so, we would not need so many different centralities. An illustration of this phenomenon is provided by the Krackhardt kite graph, for which three different notions of centrality give three different choices of the most central vertex.

'''<font color="#ff8000"> 中心性Centrality</font>'''指标有两个重要的局限性，一个是显而易见的，另一个是微妙的。显而易见的局限性是，对于一个应用最优的中心性对于另一个应用常常是次优的。事实上，如果不是这样，我们就不需要这么多不同的中心性。克拉克哈特风筝图为这一现象提供了一个例证，对于这个图，三个不同的中心性概念给出了三个最中心顶点的不同选择。

'''<font color="#ff8000"> 中心性Centrality</font>'''指标有两个重要的局限性，一个显而易见，另一个则不易察觉。显而易见的局限性是，对于一个应用最优的中心性对于另一个应用常常是次优的。事实上，如果不是这样，我们就不需要这么多不同的中心性。克拉克哈特风筝图为这一现象提供了一个例证，对于这个图，三个不同的中心性概念给出了三个最中心顶点的不同选择。

The more subtle limitation is the commonly held fallacy that vertex centrality indicates the relative importance of vertices. Centrality indices are explicitly designed to produce a ranking which allows indication of the most important vertices. This they do well, under the limitation just noted. They are not designed to measure the influence of nodes in general. Recently, network physicists have begun developing  node influence metrics to address this problem.

The more subtle limitation is the commonly held fallacy that vertex centrality indicates the relative importance of vertices. Centrality indices are explicitly designed to produce a ranking which allows indication of the most important vertices. This they do well, under the limitation just noted. They are not designed to measure the influence of nodes in general. Recently, network physicists have begun developing  node influence metrics to address this problem.

The error is two-fold. Firstly, a ranking only orders vertices by importance, it does not quantify the difference in importance between different levels of the ranking. This may be mitigated by applying Freeman centralization to the centrality measure in question, which provide some insight to the importance of nodes depending on the differences of their centralization scores. Furthermore, Freeman centralization enables one to compare several networks by comparing their highest centralization scores. This approach, however, is seldom seen in practice.

The error is two-fold. Firstly, a ranking only orders vertices by importance, it does not quantify the difference in importance between different levels of the ranking. This may be mitigated by applying Freeman centralization to the centrality measure in question, which provide some insight to the importance of nodes depending on the differences of their centralization scores. Furthermore, Freeman centralization enables one to compare several networks by comparing their highest centralization scores. This approach, however, is seldom seen in practice.

Secondly, the features which (correctly) identify the most important vertices in a given network/application do not necessarily generalize to the remaining vertices.  

Secondly, the features which (correctly) identify the most important vertices in a given network/application do not necessarily generalize to the remaining vertices.  

其次，用以(正确地)识别给定网络/应用程序中最重要顶点的特征并不一定可推广到其余顶点。

其次，用以(正确地)识别给定网络/应用中最重要顶点的特征并不一定可推广到其余顶点。

For the majority of other network nodes the rankings may be meaningless.<ref name="Lawyer2015" /><ref name="daSilva2012">{{cite journal | last1=da Silva|first1=Renato |last2=Viana|first2=Matheus|last3=da F. Costa |first3=Luciano| title=Predicting epidemic outbreak from individual features of the spreaders| journal=J. Stat. Mech.: Theory Exp. | year=2012|volume=2012|pages=P07005|number=7 | doi=10.1088/1742-5468/2012/07/p07005|arxiv=1202.0024|bibcode=2012JSMTE..07..005A}}</ref><ref name="Bauer2012">{{cite journal | last1=Bauer|first1=Frank | last2=Lizier|first2=Joseph|title=Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: A walk counting approach| journal=Europhys Lett | year=2012| volume=99| pages=68007|number=6 | doi=10.1209/0295-5075/99/68007|arxiv=1203.0502|bibcode=2012EL.....9968007B}}</ref><ref name="Sikic2013">{{ cite journal| last1= Sikic| first1=Mile|last2=Lancic|first2=Alen|last3=Antulov-Fantulin|first3=Nino|last4=Stefanic|first4=Hrvoje| title = Epidemic centrality -- is there an underestimated epidemic impact of network peripheral nodes? |journal = The European Physical Journal B |volume=86 |number=10 |pages=1–13 |year=2013 | doi=10.1140/epjb/e2013-31025-5|arxiv=1110.2558 | bibcode=2013EPJB...86..440S}}</ref> This explains why, for example, only the first few results of a Google image search appear in a reasonable order. The pagerank is a highly unstable measure, showing frequent rank reversals after small adjustments of the jump parameter.<ref name="Ghoshal2011">{{cite journal | last1=Ghoshal | first1= G. | last2= Barabsi |first2= A L | title = Ranking stability and super-stable nodes in complex networks. | journal = Nat Commun | volume =2 | page = 394| year= 2011 | doi=10.1038/ncomms1396 | pmid= 21772265 | bibcode=2011NatCo...2..394G | doi-access= free }}</ref>

For the majority of other network nodes the rankings may be meaningless.<ref name="Lawyer2015" /><ref name="daSilva2012">{{cite journal | last1=da Silva|first1=Renato |last2=Viana|first2=Matheus|last3=da F. Costa |first3=Luciano| title=Predicting epidemic outbreak from individual features of the spreaders| journal=J. Stat. Mech.: Theory Exp. | year=2012|volume=2012|pages=P07005|number=7 | doi=10.1088/1742-5468/2012/07/p07005|arxiv=1202.0024|bibcode=2012JSMTE..07..005A}}</ref><ref name="Bauer2012">{{cite journal | last1=Bauer|first1=Frank | last2=Lizier|first2=Joseph|title=Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: A walk counting approach| journal=Europhys Lett | year=2012| volume=99| pages=68007|number=6 | doi=10.1209/0295-5075/99/68007|arxiv=1203.0502|bibcode=2012EL.....9968007B}}</ref><ref name="Sikic2013">{{ cite journal| last1= Sikic| first1=Mile|last2=Lancic|first2=Alen|last3=Antulov-Fantulin|first3=Nino|last4=Stefanic|first4=Hrvoje| title = Epidemic centrality -- is there an underestimated epidemic impact of network peripheral nodes? |journal = The European Physical Journal B |volume=86 |number=10 |pages=1–13 |year=2013 | doi=10.1140/epjb/e2013-31025-5|arxiv=1110.2558 | bibcode=2013EPJB...86..440S}}</ref> This explains why, for example, only the first few results of a Google image search appear in a reasonable order. The pagerank is a highly unstable measure, showing frequent rank reversals after small adjustments of the jump parameter.<ref name="Ghoshal2011">{{cite journal | last1=Ghoshal | first1= G. | last2= Barabsi |first2= A L | title = Ranking stability and super-stable nodes in complex networks. | journal = Nat Commun | volume =2 | page = 394| year= 2011 | doi=10.1038/ncomms1396 | pmid= 21772265 | bibcode=2011NatCo...2..394G | doi-access= free }}</ref>

For the majority of other network nodes the rankings may be meaningless. This explains why, for example, only the first few results of a Google image search appear in a reasonable order. The pagerank is a highly unstable measure, showing frequent rank reversals after small adjustments of the jump parameter.

For the majority of other network nodes the rankings may be meaningless. This explains why, for example, only the first few results of a Google image search appear in a reasonable order. The pagerank is a highly unstable measure, showing frequent rank reversals after small adjustments of the jump parameter.

对于大多数其他网络节点，排名可能是没有意义的。这就解释了为什么，例如，只有谷歌图片搜索的前几个结果以合理的顺序出现。Pagerank（网页排名） 是一个非常不稳定的度量，显示了在跳转参数小的调整之后频繁的秩逆转。

对于大多数其他网络节点，排名可能是没有意义的。这就解释了为什么，例如，只有谷歌图片搜索的前几个结果以合理的顺序出现。'''<font color="#ff8000"> 网页排名Pagerank</font>'''是一个非常不稳定的度量，显示了在跳转参数小的调整之后频繁的秩逆转。

While the failure of centrality indices to generalize to the rest of the network may at first seem counter-intuitive, it follows directly from the above definitions.

While the failure of centrality indices to generalize to the rest of the network may at first seem counter-intuitive, it follows directly from the above definitions.

Complex networks have heterogeneous topology. To the extent that the optimal measure depends on the network structure of the most important vertices, a measure which is optimal for such vertices is sub-optimal for the remainder of the network.<ref name="Lawyer2015">{{cite journal |last1= Lawyer |first1= Glenn |year= 2015 |title= Understanding the spreading power of all nodes in a network: a continuous-time perspective |journal=Sci Rep |volume=5|pages=8665|doi=10.1038/srep08665 |pmid=25727453 |pmc=4345333|arxiv=1405.6707|bibcode=2015NatSR...5E8665L}}</ref>

Complex networks have heterogeneous topology. To the extent that the optimal measure depends on the network structure of the most important vertices, a measure which is optimal for such vertices is sub-optimal for the remainder of the network.<ref name="Lawyer2015">{{cite journal |last1= Lawyer |first1= Glenn |year= 2015 |title= Understanding the spreading power of all nodes in a network: a continuous-time perspective |journal=Sci Rep |volume=5|pages=8665|doi=10.1038/srep08665 |pmid=25727453 |pmc=4345333|arxiv=1405.6707|bibcode=2015NatSR...5E8665L}}</ref>