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regardless of network dynamics. This is true even for the weighted betweenness measures. However, a node may very well be centrally located in terms of betweenness centrality or another centrality measure, but may not be âcentrallyâ located in the context of a network in which there is percolation. Percolation of a âcontagionâ occurs in complex networks in a number of scenarios. For example, viral or bacterial infection can spread over social networks of people, known as contact networks. The spread of disease can also be considered at a higher level of abstraction, by contemplating a network of towns or population centres, connected by road, rail or air links. Computer viruses can spread over computer networks. Rumours or news about business offers and deals can also spread via social networks of people. In all of these scenarios, a âcontagionâ spreads over the links of a complex network, altering the âstatesâ of the nodes as it spreads, either recoverable or otherwise. For example, in an epidemiological scenario, individuals go from âsusceptibleâ to âinfectedâ state as the infection spreads. The states the individual nodes can take in the above examples could be binary (such as received/not received a piece of news), discrete (susceptible/infected/recovered), or even continuous (such as the proportion of infected people in a town), as the contagion spreads. The common feature in all these scenarios is that the spread of contagion results in the change of node states in networks. Percolation centrality (PC) was proposed with this in mind, which specifically measures the importance of nodes in terms of aiding the percolation through the network. This measure was proposed by
Piraveenan et al.
769:
delivery going from the delivery site to the client's house. A second case is serial duplication, in which an item is replicated so that both the source and the target have it. An example is the propagation of information through gossip, with the information being propagated in a private way and with both the source and the target nodes being informed at the end of the process. The last case is parallel duplication, with the item being duplicated to several links at the same time, like a radio broadcast which provides the same information to many listeners at once.
52:
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flow they consider important. "Importance" can alternatively be conceived as involvement in the cohesiveness of the network. This allows centralities to be classified based on how they measure cohesiveness. Both of these approaches divide centralities in distinct categories. A further conclusion is that a centrality which is appropriate for one category will often "get it wrong" when applied to a different category.
6694:
1369:, which is defined as the number of links incident upon a node (i.e., the number of ties that a node has). The degree can be interpreted in terms of the immediate risk of a node for catching whatever is flowing through the network (such as a virus, or some information). In the case of a directed network (where ties have direction), we usually define two separate measures of degree centrality, namely
1330:
1262:. Because of the time-complexity hardness of the Shapley value calculation, most efforts in this domain are driven into implementing new algorithms and methods which rely on a peculiar topology of the network or a special character of the problem. Such an approach may lead to reducing time-complexity from exponential to polynomial.
1377:. Accordingly, indegree is a count of the number of ties directed to the node and outdegree is the number of ties that the node directs to others. When ties are associated to some positive aspects such as friendship or collaboration, indegree is often interpreted as a form of popularity, and outdegree as gregariousness.
7498:
5388:
is defined for a given node, at a given time, as the proportion of âpercolated pathsâ that go through that node. A âpercolated pathâ is a shortest path between a pair of nodes, where the source node is percolated (e.g., infected). The target node can be percolated or non-percolated, or in a partially
3279:
betweenness, which is not discussed here). Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. It was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network
1310:
Secondly, the features which (correctly) identify the most important vertices in a given network/application do not necessarily generalize to the remaining vertices. For the majority of other network nodes the rankings may be meaningless. This explains why, for example, only the first few results of
740:
The word "importance" has a wide number of meanings, leading to many different definitions of centrality. Two categorization schemes have been proposed. "Importance" can be conceived in relation to a type of flow or transfer across the network. This allows centralities to be classified by the type of
6697:
In the illustrated network, green and red nodes are the most dissimilar because they do not share neighbors between them. So, the green one contributes more to the centrality of the red one than the gray ones, because the red one can access to the blue ones only through the green, and the gray nodes
4030:
time. Normally, these algorithms assume that graphs are undirected and connected with the allowance of loops and multiple edges. When specifically dealing with network graphs, often graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between
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Transportation networks such as road networks and railway networks are studied extensively in transportation science and urban planning. A number of recent studies have focused on using centrality measures to analyze transportation networks. While many of these studies simply use generic centrality
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Where this measure permits us to quantify the topological contribution (which is why is called contribution centrality) of each node to the centrality of a given node, having more weight/relevance those nodes with greater dissimilarity, since these allow to the given node access to nodes that which
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of any network is a measure of how central its most central node is in relation to how central all the other nodes are. Centralization measures then (a) calculate the sum in differences in centrality between the most central node in a network and all other nodes; and (b) divide this quantity by the
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The attached weights to the percolation paths depend on the percolation levels assigned to the source nodes, based on the premise that the higher the percolation level of a source node is, the more important are the paths that originate from that node. Nodes which lie on shortest paths originating
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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.
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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
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The common feature of most of the aforementioned standard measures is that they assess the importance of a node by focusing only on the role that a node plays by itself. However, in many applications such an approach is inadequate because of synergies that may occur if the functioning of nodes is
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The heart of such measures is the observation that powers of the graph's adjacency matrix gives the number of walks of length given by that power. Similarly, the matrix exponential is also closely related to the number of walks of a given length. An initial transformation of the adjacency matrix
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For example, consider the problem of stopping an epidemic. Looking at above image of network, which nodes should we vaccinate? Based on previously described measures, we want to recognize nodes that are the most important in disease spreading. Approaches based only on centralities, that focus on
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A network can be considered a description of the paths along which something flows. This allows a characterization based on the type of flow and the type of path encoded by the centrality. A flow can be based on transfers, where each indivisible item goes from one node to another, like a package
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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
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The characterization by walk structure shows that almost all centralities in wide use are radial-volume measures. These encode the belief that a vertex's centrality is a function of the centrality of the vertices it is associated with. Centralities distinguish themselves on how association is
837:
Borgatti and
Everett propose that this typology provides insight into how best to compare centrality measures. Centralities placed in the same box in this 2Ă2 classification are similar enough to make plausible alternatives; one can reasonably compare which is better for a given application.
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A slew of centrality measures exist to determine the âimportanceâ of a single node in a complex network. However, these measures quantify the importance of a node in purely topological terms, and the value of the node does not depend on the âstateâ of the node in any way. It remains constant
7244:
Transportation centrality measures the summation of the proportions of paths from pairs of nodes in a network which go through the node under consideration. In this respect it is similar to
Betweenness Centrality. However, unlike Betweenness Centrality which considers only shortest paths,
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in the network. The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigenvector, e.g., such that the sum over all vertices is 1 or the total number of vertices
748:(also called walks) of some type going through a given vertex; the measures differ in how the relevant walks are defined and counted. Restricting consideration to this group allows for taxonomy which places many centralities on a spectrum from those concerned with walks of length one (
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is a generalization of degree centrality. Degree centrality measures the number of direct neighbors, and Katz centrality measures the number of all nodes that can be connected through a path, while the contributions of distant nodes are penalized. Mathematically, it is defined as
1306:
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.
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Centrality indices are answers to the question "What characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected to provide a ranking which identifies the most important nodes.
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In order to obtain better results in the ranking of the nodes of a given network, in are used dissimilarity measures (specific to the theory of classification and data mining) to enrich the centrality measures in complex networks. This is illustrated with
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6017:. A node with high cross-clique connectivity facilitates the propagation of information or disease in a graph. Cliques are subgraphs in which every node is connected to every other node in the clique. The cross-clique connectivity of a node
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Transportation
Centrality considers all possible paths between a pair of nodes. Therefore, Transportation Centrality is a generic version of Betweenness Centrality, and under certain conditions, it indeed reduces to Betweenness Centrality.
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of walks. Volume is the total number of walks of the given type. The three examples from the previous paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph.
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individual features of nodes, may not be good idea. Nodes in the red square, individually cannot stop disease spreading, but considering them as a group, we clearly see that they can stop disease if it has started in nodes
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Measures from different boxes, however, are categorically distinct. Any evaluation of relative fitness can only occur within the context of predetermining which category is more applicable, rendering the comparison moot.
1109:. Subgraph centrality replaces the adjacency matrix with its trace. A startling conclusion is that regardless of the initial transformation of the adjacency matrix, all such approaches have common limiting behavior. As
833:
centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example. Note that this classification is independent of the type of walk counted (i.e. walk, trail, path, geodesic).
812:
centralities are examples of radial centralities, counting the number of walks of length one or length infinity. Medial centralities count walks which pass through the given vertex. The canonical example is
Freeman's
5930:
from highly percolated nodes are therefore potentially more important to the percolation. The definition of PC may also be extended to include target node weights as well. Percolation centrality calculations run in
5395:
4060:. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.
2443:
1018:
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1277:, to measure the bilateral direct influence between the players. The distribution is indeed a type of eigenvector centrality. It is used to sort big data objects in Hu (2020), such as ranking U.S. colleges.
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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
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From a calculation aspect, both betweenness and closeness centralities of all vertices in a graph involve calculating the shortest paths between all pairs of vertices on a graph, which requires
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This normalisation allows comparisons between nodes of graphs of different sizes. For many graphs, there is a strong correlation between the inverse of closeness and the logarithm of degree,
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for which a non-zero eigenvector solution exists. Since the entries in the adjacency matrix are non-negative, there is a unique largest eigenvalue, which is real and positive, by the
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Sikic, Mile; Lancic, Alen; Antulov-Fantulin, Nino; Stefanic, Hrvoje (2013). "Epidemic centrality -- is there an underestimated epidemic impact of network peripheral nodes?".
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theoretically largest such sum of differences in any network of the same size. Thus, every centrality measure can have its own centralization measure. Defined formally, if
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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.
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7493:{\displaystyle TC(v)=1/((N-1)(N-2))\Sigma _{s\neq v\neq t}{\frac {\Sigma _{i\in P_{s,t}^{v}}e^{-\beta C_{s,t}^{i}}}{\Sigma _{j\in P_{s,t}^{v}}e^{-\beta C_{s,t}^{j}}}}}
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5999:
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1722:
1436:
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measures such as
Betweenness Centrality, custom centrality measures have also been defined specifically for transportation network analysis. Prominent among them is
6836:
5371:
4643:
2753:
1151:
1127:
1095:
8231:
Bauer, Frank; Lizier, Joseph (2012). "Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: A walk counting approach".
7747:
Christian F. A. Negre, Uriel N. Morzan, Heidi P. Hendrickson, Rhitankar Pal, George P. Lisi, J. Patrick Loria, Ivan
Rivalta, Junming Ho, Victor S. Batista. (2018).
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1229:
1202:
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KoschĂŒtzki, D.; Lehmann, K. A.; Peeters, L.; Richter, S.; Tenfelde-Podehl, D. and
Zlotowski, O. (2005) Centrality Indices. In Brandes, U. and Erlebach, T. (Eds.)
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time with an efficient implementation adopted from
Brandes' fast algorithm and if the calculation needs to consider target nodes weights, the worst case time is
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two people or vertices). In this case, using
Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice.
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3309:
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5926:. The values in between indicate partially percolated states ( e.g., in a network of townships, this would be the percentage of people infected in that town).
5125:
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6213:
belongs. This measure was used by Faghani in 2013 but was first proposed by Everett and Borgatti in 1998 where they called it clique-overlap centrality.
6852:
870:
allows a different definition of the type of walk counted. Under either approach, the centrality of a vertex can be expressed as an infinite sum, either
7228:
is the number of the nodes in the network. Several dissimilarity measures and networks were tested in obtaining improved results in the studied cases.
4710:
866:
allows vertices to have an external source of influence. Estrada's subgraph centrality proposes only counting closed paths (triangles, squares, etc.).
568:
5314:(or number of outbound links in a directed graph). Compared to eigenvector centrality and Katz centrality, one major difference is the scaling factor
1258:. Game-theoretic centralities try to consult described problems and opportunities, using tools from game-theory. The approach proposed in uses the
7816:
8597:
5587:{\displaystyle PC^{t}(v)={\frac {1}{N-2}}\sum _{s\neq v\neq r}{\frac {\sigma _{sr}(v)}{\sigma _{sr}}}{\frac {{x^{t}}_{s}}{{\sum {}-{x^{t}}_{v}}}}
796:
An alternative classification can be derived from how the centrality is constructed. This again splits into two classes. Centralities are either
8628:
8745:
2767:. However, when speaking of closeness centrality, people usually refer to its normalized form, given by the previous formula multiplied by
8877:
2268:
945:
8178:
da Silva, Renato; Viana, Matheus; da F. Costa, Luciano (2012). "Predicting epidemic outbreak from individual features of the spreaders".
5343:. Another difference between PageRank and eigenvector centrality is that the PageRank vector is a left hand eigenvector (note the factor
8818:
8782:
6338:
675:
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2902:
3045:
9082:"Transportation Centrality: Quantifying the Relative Importance of Nodes in Transportation Networks Based on Traffic Modeling"
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non-negative, allowing us to infer the centrality of each node in the network. Therefore, the centrality of the i-th node is
1940:
704:
within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a
6671:{\displaystyle C_{x}={\frac {\sum _{i=1}^{N}(C_{x}(p_{*})-C_{x}(p_{i}))}{\max \sum _{i=1}^{N}(C_{x}(p_{*})-C_{x}(p_{i}))}}.}
3027:(e.g. a website can have a high closeness centrality from outgoing link, but low closeness centrality from incoming links).
4550:
2819:
9047:"Supplementary Information for Eigencentrality based on dissimilarity measures reveals central nodes in complex networks"
2622:
between the node and all other nodes in the graph. Thus the more central a node is, the closer it is to all other nodes.
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They are not designed to measure the influence of nodes in general. Recently, network physicists have begun developing
2601:
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287:
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Faghani, Mohamamd Reza (2013). "A Study of XSS Worm Propagation and Detection Mechanisms in Online Social Networks".
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The definition of centrality on the node level can be extended to the whole graph, in which case we are speaking of
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785:
777:
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important vertices, a measure which is optimal for such vertices is sub-optimal for the remainder of the network.
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Freeman, Linton C. "Centrality in social networks conceptual clarification." Social networks 1.3 (1979): 215â239.
8085:
4461:{\displaystyle x_{v}={\frac {1}{\lambda }}\sum _{t\in M(v)}x_{t}={\frac {1}{\lambda }}\sum _{t\in G}a_{v,t}x_{t}}
2640:
760:
focus not just on overall connectedness but occupying positions that are pivotal to the network's connectivity.
528:
4678:
that may be used to find this dominant eigenvector. Furthermore, this can be generalized so that the entries in
2448:
Also, a new extensive global measure for degree centrality named Tendency to Make Hub (TMH) defines as follows:
3880:
518:
3827:(1, if normalised) while the leaves (which are contained in no shortest paths) would have a betweenness of 0.
513:
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1290:, for which three different notions of centrality give three different choices of the most central vertex.
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317:
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3242:(2000) and then independently by Dekker (2005), using the name "valued centrality," and by Rochat (2009).
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are redundant for the red one, because it can access directly to each gray node without any intermediary.
4873:
Katz centrality can be viewed as a variant of eigenvector centrality. Another form of Katz centrality is
1803:
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all other nodes is irrelevant in undirected graphs, whereas it can produce totally different results in
3658:. The betweenness may be normalised by dividing through the number of pairs of vertices not including
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3772:, the center vertex (which is contained in every possible shortest path) would have a betweenness of
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225:
8896:"Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks"
8619:
8064:
8001:(June 1990). "Assessing the Political Landscape: Structure, Cognition, and Power in Organizations".
1562:
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5805:
3396:), determine the fraction of shortest paths that pass through the vertex in question (here, vertex
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721:
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Benzi, Michele; Klymko, Christine (2013). "A matrix analysis of different centrality measures".
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Borgatti, Stephen P.; Everett, Martin G. (2006). "A Graph-Theoretic Perspective on Centrality".
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2576:{\displaystyle {\text{TMH}}={\frac {\sum _{i=1}^{|V|}\deg(v)^{2}}{\sum _{i=1}^{|V|}\deg(v)}}}
1324:
1295:
1287:
862:
counts walks of length infinity. Alternative definitions of association are also reasonable.
701:
649:
468:
438:
327:
282:
8113:"Understanding the spreading power of all nodes in a network: a continuous-time perspective"
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4645:
component of the related eigenvector then gives the relative centrality score of the vertex
4618:
2723:
1136:
1112:
1080:
9001:
8990:"Eigencentrality based on dissimilarity measures reveals central nodes in complex networks"
8907:
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8488:
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is a constant. With a small rearrangement this can be rewritten in vector notation as the
4474:
3519:{\displaystyle C_{B}(v)=\sum _{s\neq v\neq t\in V}{\frac {\sigma _{st}(v)}{\sigma _{st}}}}
8:
9081:
7198:{\displaystyle c_{i}={\dfrac {1}{n}}\sum _{j=1}^{n}W_{ij}c_{j},\,\,\,\,\,\,i=1,\cdots ,n}
6707:, calculating the centrality of each node through the solution of the eigenvalue problem
6108:
6078:
5223:{\displaystyle x_{i}=\alpha \sum _{j}a_{ji}{\frac {x_{j}}{L(j)}}+{\frac {1-\alpha }{N}},}
4121:
3769:
3284:. In his conception, vertices that have a high probability to occur on a randomly chosen
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for any graph with the same number of nodes, then the centralization of the network is:
1757:
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8989:
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8895:
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Katz, L. 1953. A New Status Index Derived from Sociometric Index. Psychometrika, 39â43.
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of a single node in a complex graph determines the connectivity of a node to different
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2796:
2099:
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854:, then a family of centralities can be defined based on the length of walk considered.
292:
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109:
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7586:"Topological impact of negative links on the stability of resting-state brain network"
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It is shown that the principal eigenvector (associated with the largest eigenvalue of
3039:
reverses the sum and reciprocal operations in the definition of closeness centrality:
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89:
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6993:{\displaystyle D_{ij}=1-{\dfrac {|V^{+}(i)\cap V^{+}(j)|}{|V^{+}(i)\cup V^{+}(j)|}}}
9093:
9017:
9009:
8988:
Alvarez-Socorro, A. J.; Herrera-Almarza, G. C.; GonzĂĄlez-DĂaz, L. A. (2015-11-25).
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7937:
7928:
Hu, Xingwei; Shapley, Lloyd (2003). "On Authority Distributions in Organizations".
7898:
7886:
7846:
7788:
7778:
7726:
7670:
7613:
7605:
7547:
7508:
4194:
1600:
1102:
863:
348:
337:
235:
195:
179:
20:
8725:
8652:
Freeman, Linton (1977). "A set of measures of centrality based upon betweenness".
8621:
Closeness centrality extended to unconnected graphs: The harmonic centrality index
8081:
784:(vertices can be visited multiple times, no edge is traversed more than once), or
8920:
8881:
8315:
7850:
7730:
4808:{\displaystyle x_{i}=\sum _{k=1}^{\infty }\sum _{j=1}^{N}\alpha ^{k}(A^{k})_{ji}}
4695:
4671:
4069:
1354:
584:
365:
240:
149:
84:
38:
9098:
7538:
van den Heuvel MP, Sporns O (December 2013). "Network hubs in the human brain".
3260:
Hue (from red = 0 to blue = max) shows the node betweenness.
8501:
8466:
7983:
7956:
7609:
7551:
6682:
5965:
5931:
3977:
3877:
3831:
3281:
3024:
717:
705:
693:
594:
400:
375:
370:
344:
333:
210:
174:
169:
129:
67:
8966:
8434:
817:
centrality, the number of shortest paths which pass through the given vertex.
9135:
9107:
8692:
8510:
7957:"Sorting big data by revealed preference with application to college ranking"
6843:
1633:
1274:
1259:
713:
553:
458:
443:
385:
134:
124:
7783:
7749:"Eigenvector centrality for characterization of protein allosteric pathways"
4075:
720:
of disease, and brain networks. Centrality concepts were first developed in
9031:
8987:
8939:
8372:
8164:
7802:
7627:
7559:
2626:
1597:
804:
Radial centralities count walks which start/end from the given vertex. The
689:
498:
395:
250:
8443:
4615:. This greatest eigenvalue results in the desired centrality measure. The
744:
Many, though not all, centrality measures effectively count the number of
8546:
4541:
3972:
time. In the case of unweighted graphs the calculations can be done with
3239:
7661:
Bonacich, Phillip (1987). "Power and Centrality: A Family of Measures".
731:
9044:
8733:
8673:
8532:
Marchiori, Massimo; Latora, Vito (2000), "Harmony in the small-world",
8425:
8363:
8338:
8022:
7584:
Saberi M, Khosrowabadi R, Khatibi A, Misic B, Jafari G (January 2021).
4589:
2586:
where TMH increases by appearance of degree centrality in the network.
2117:
1101:
Bonacich's family of measures does not transform the adjacency matrix.
390:
380:
9013:
8848:
Bonacich, P (1991). "Simultaneous group and individual centralities".
8146:
7890:
7231:
1559:
1374:
725:
598:
154:
8665:
8014:
3256:
2438:{\displaystyle C_{D}(G)={\frac {\sum _{i=1}^{|V|}}{|V|^{2}-3|V|+2}}}
2116:
contains one central node to which all other nodes are connected (a
1013:{\displaystyle \sum _{k=0}^{\infty }{\frac {(A_{R}\beta )^{k}}{k!}}}
51:
8483:
8283:
7973:
7765:
7674:
5111:
4065:
1370:
709:
8298:
8245:
8192:
8129:
7916:
7881:
1754:-node connected graph that maximizes the following quantity (with
1558:
Calculating degree centrality for all the nodes in a graph takes
841:
724:, and many of the terms used to measure centrality reflect their
6693:
6430:{\displaystyle \max \sum _{i=1}^{N}(C_{x}(p_{*})-C_{x}(p_{i}))}
4682:
can be real numbers representing connection strengths, as in a
4061:
8467:"Linking the network centrality measures closeness and degree"
8394:
Alex Bavelas. Communication patterns in task-oriented groups.
8339:"Ranking stability and super-stable nodes in complex networks"
7583:
3288:
between two randomly chosen vertices have a high betweenness.
788:(vertices and edges can be visited/traversed multiple times).
9045:
Alvarez-Socorro, A.J.; Herrera-Almarza; GonzĂĄlez-DĂaz, L. A.
1097:
is a discount parameter which ensures convergence of the sum.
7707:
Borgatti, Stephen P. (2005). "Centrality and Network Flow".
7579:
7577:
4961:{\displaystyle x_{i}=\alpha \sum _{j=1}^{N}a_{ij}(x_{j}+1).}
2974:{\displaystyle (C(v))^{-1}\approx -\alpha \ln(k_{v})+\beta }
8893:
7911:
Michalak, Aadithya, SzczepaĆski, Ravindran, & Jennings
7071:
to ensure that the above problem has a unique solution for
5053:, the adjacency matrix) is the limit of Katz centrality as
850:
Bonacich showed that if association is defined in terms of
6688:
8045:"centrality in social networks: Conceptual clarification"
7574:
7531:
4076:
Using the adjacency matrix to find eigenvector centrality
3118:{\displaystyle H(v)=\sum _{u|u\neq v}{\frac {1}{d(u,v)}}}
8412:
Sabidussi, G (1966). "The centrality index of a graph".
8177:
1914:
Correspondingly, the degree centralization of the graph
1329:
929:{\displaystyle \sum _{k=0}^{\infty }A_{R}^{k}\beta ^{k}}
8955:
IEEE Transactions on Information Forensics and Security
6842:
matrix, defined through a dissimilarity measure, e.g.,
3185:. Harmonic centrality can be normalized by dividing by
2066:{\displaystyle C_{D}(G)={\frac {\sum _{i=1}^{|V|}}{H}}}
9080:
Piraveenan, Mahendra; Saripada, Naressa Belle (2023).
8627:. Applications of Social Network Analysis, ASNA 2009.
6440:
is the largest sum of differences in point centrality
5081:
4971:
Compared to the expression of eigenvector centrality,
3415:
More compactly the betweenness can be represented as:
2643:
1153:
approaches its maximal value, the indices converge to
8534:
Physica A: Statistical Mechanics and Its Applications
7261:
7214:
7110:
7095:
7053:
7033:
7013:
6879:
6855:
6814:
6755:
6716:
6476:
6446:
6341:
6295:
6275:
6232:
6199:
6170:
6141:
6111:
6081:
6043:
6023:
5971:
5937:
5912:
5870:
5850:
5808:
5772:
5752:
5732:
5712:
5673:
5653:
5633:
5603:
5398:
5349:
5320:
5300:
5242:
5128:
5079:
5059:
5039:
5004:
4977:
4882:
4844:
4824:
4713:
4651:
4621:
4597:
4553:
4526:
4506:
4477:
4344:
4321:
4282:
4262:
4242:
4203:
4154:
4124:
4086:
3983:
3883:
3837:
3778:
3719:
3672:
3644:
3605:
3585:
3565:
3535:
3424:
3355:
3317:
3297:
3217:
3191:
3134:
3048:
2987:
2905:
2822:
2799:
2773:
2726:
2457:
2271:
2228:
2129:
2102:
2082:
1943:
1920:
1806:
1783:
1760:
1730:
1692:
1672:
1649:
1609:
1565:
1507:
1474:
1444:
1406:
1386:
1237:
1210:
1183:
1139:
1115:
1083:
1054:
1032:
948:
879:
791:
732:
Definition and characterization of centrality indices
8894:
Piraveenan, M.; Prokopenko, M.; Hossain, L. (2013).
8811:"How Google Finds Your Needle in the Web's Haystack"
7537:
6332:
is the largest such measure in the network, and if:
4578:{\displaystyle \mathbf {Ax} ={\lambda }\mathbf {x} }
2889:{\displaystyle C(v)={\frac {N-1}{\sum _{u}d(u,v)}}.}
763:
8525:
7232:
Centrality measures used in transportation networks
4315:otherwise. The relative centrality score of vertex
1286:illustration of this phenomenon is provided by the
9079:
7492:
7220:
7197:
7059:
7039:
7019:
6992:
6830:
6800:
6739:{\displaystyle W\mathbf {c} =\lambda \mathbf {c} }
6738:
6670:
6459:
6429:
6324:
6281:
6261:
6205:
6185:
6156:
6127:
6097:
6067:
6029:
5993:
5955:
5918:
5898:
5856:
5836:
5794:
5758:
5738:
5718:
5698:
5659:
5639:
5619:
5586:
5365:
5335:
5306:
5283:
5222:
5094:
5065:
5045:
5023:
4990:
4960:
4862:
4830:
4807:
4657:
4637:
4603:
4577:
4532:
4512:
4492:
4460:
4327:
4307:
4268:
4248:
4228:
4185:
4140:
4110:
4021:
3963:
3859:
3819:
3760:
3705:
3650:
3630:
3591:
3571:
3551:
3518:
3361:
3341:
3303:
3223:
3203:
3169:
3117:
3000:
2973:
2888:
2805:
2785:
2747:
2712:
2575:
2437:
2255:
2212:{\displaystyle H=(n-1)\cdot ((n-1)-1)=n^{2}-3n+2.}
2211:
2108:
2088:
2065:
1926:
1903:
1789:
1769:
1746:
1716:
1678:
1658:
1624:
1587:
1547:
1490:
1460:
1430:
1392:
1250:
1223:
1196:
1145:
1121:
1089:
1067:
1038:
1012:
928:
8777:. In Durlauf, Steven; Blume, Lawrence E. (eds.).
5906:which indicates a fully percolated state at time
1777:being the node with highest degree centrality in
1603:representation of the graph, and for edges takes
772:Likewise, the type of path can be constrained to
9133:
8586:"Conceptual Distance in Social Network Analysis"
7869:SIAM Journal on Matrix Analysis and Applications
6577:
6342:
1365:Historically first and conceptually simplest is
8769:
8697:"A faster algorithm for betweenness centrality"
8531:
7836:
7753:Proceedings of the National Academy of Sciences
5844:which indicates a non-percolated state at time
5706:is the number of those paths that pass through
3638:is the number of those paths that pass through
8887:
8336:
8038:
8036:
8034:
8032:
7832:
7830:
7828:
7826:
7702:
7700:
7698:
7696:
7694:
7692:
7656:
7654:
7652:
7650:
3403:Sum this fraction over all pairs of vertices (
3012:while α and ÎČ are constants for each network.
1666:be the node with highest degree centrality in
842:Radial-volume centralities exist on a spectrum
820:Likewise, the counting can capture either the
9128:, pp. 16â61, LNCS 3418, Springer-Verlag.
8106:
8104:
7634:
7067:are non-negative matrices, so we can use the
669:
9126:Network Analysis: Methodological Foundations
7815:: CS1 maint: multiple names: authors list (
5627:is total number of shortest paths from node
4072:are variants of the eigenvector centrality.
3559:is total number of shortest paths from node
3035:In a (not necessarily connected) graph, the
2713:{\textstyle C_{B}(v)=(\sum _{u}d(u,v))^{-1}}
1160:
8645:
8330:
8230:
8171:
8029:
7866:
7862:
7860:
7823:
7689:
7647:
1303:
1154:
859:
814:
809:
753:
8781:(2nd ed.). Springer. pp. 465ff.
8687:
8685:
8683:
8617:
8101:
7997:
7248:Transportation Centrality of a given node
6005:
676:
662:
9097:
9021:
8929:
8919:
8765:
8763:
8715:
8545:
8500:
8482:
8464:
8433:
8411:
8362:
8297:
8277:
8244:
8224:
8191:
8154:
8128:
8063:
7982:
7972:
7927:
7880:
7792:
7782:
7764:
7720:
7617:
7173:
7172:
7171:
7170:
7169:
7168:
6216:
6193:is the number of cliques to which vertex
5376:
4588:In general, there will be many different
4034:
3964:{\displaystyle O(|V||E|+|V|^{2}\log |V|)}
3245:
2618:) of a node is the average length of the
1129:approaches zero, the indices converge to
830:
8946:
8847:
8779:The New Palgrave Dictionary of Economics
7857:
7706:
7660:
6692:
3255:
1350:
1328:
1280:
1075:is the transformed adjacency matrix, and
8952:
8691:
8680:
8651:
8465:Evans, Tim S.; Chen, Bingsheng (2022).
8042:
7524:
6808:(coordinate-to-coordinate product) and
6689:Dissimilarity-based centrality measures
2589:
1105:replaces the adjacency matrix with its
780:(no vertex is visited more than once),
9134:
8808:
8804:
8802:
8760:
8583:
8110:
5105:
5095:{\displaystyle {\tfrac {1}{\lambda }}}
3030:
1130:
855:
805:
756:). Other centrality measures, such as
749:
16:Degree of connectedness within a graph
9075:
9073:
4052:) is a measure of the influence of a
3231:is the number of nodes in the graph.
7644:Oxford, UK: Oxford University Press.
7004:themselves can not access directly.
5726:. The percolation state of the node
5284:{\displaystyle L(j)=\sum _{i}a_{ji}}
3234:Harmonic centrality was proposed by
2813:is the number of nodes in the graph
1318:
8799:
6801:{\displaystyle W_{ij}=A_{ij}D_{ij}}
6269:is any centrality measure of point
5294:is the number of neighbors of node
1904:{\displaystyle H=\sum _{j=1}^{|Y|}}
1357:of the same random geometric graph.
13:
9118:
9070:
8748:from the original on March 4, 2016
8337:Ghoshal, G.; Barabsi, A L (2011).
7954:
7420:
7350:
7325:
4743:
4689:
1610:
1566:
1380:The degree centrality of a vertex
965:
896:
858:counts walks of length one, while
792:Characterization by walk structure
708:, key infrastructure nodes in the
14:
9183:
8817:. American Mathematical Society.
8704:Journal of Mathematical Sociology
5119:satisfies the following equation
3768:. For example, in an undirected
3369:vertices is computed as follows:
764:Characterization by network flows
19:For the statistical concept, see
8210:10.1088/1742-5468/2012/07/p07005
8003:Administrative Science Quarterly
6732:
6721:
4571:
4558:
4555:
1548:{\displaystyle C_{D}(v)=\deg(v)}
1265:Similarly, the solution concept
1169:
50:
9059:from the original on 2016-03-07
9038:
8981:
8868:
8841:
8832:
8821:from the original on 2018-01-11
8809:Austin, David (December 2006).
8788:from the original on 2021-01-22
8634:from the original on 2017-08-16
8611:
8600:from the original on 2020-12-04
8577:
8458:
8405:
8388:
8379:
8286:The European Physical Journal B
7991:
7948:
7921:
5802:and two special cases are when
5699:{\displaystyle \sigma _{sr}(v)}
3631:{\displaystyle \sigma _{st}(v)}
1023:for matrix exponentials, where
8875:How does Google rank webpages?
7905:
7321:
7318:
7306:
7303:
7291:
7288:
7274:
7268:
6982:
6978:
6972:
6956:
6950:
6936:
6929:
6925:
6919:
6903:
6897:
6883:
6659:
6656:
6643:
6627:
6614:
6601:
6572:
6569:
6556:
6540:
6527:
6514:
6424:
6421:
6408:
6392:
6379:
6366:
6319:
6306:
6256:
6243:
6180:
6174:
6151:
6145:
6121:
6113:
6091:
6083:
6062:
6050:
5988:
5975:
5950:
5941:
5693:
5687:
5555:
5532:
5486:
5480:
5418:
5412:
5330:
5324:
5252:
5246:
5190:
5184:
4952:
4933:
4857:
4845:
4793:
4779:
4487:
4481:
4388:
4382:
4180:
4161:
4134:
4126:
4105:
4093:
4016:
4012:
4004:
3999:
3991:
3987:
3958:
3954:
3946:
3929:
3920:
3912:
3904:
3899:
3891:
3887:
3876:may be more efficient, taking
3854:
3841:
3806:
3794:
3791:
3779:
3747:
3735:
3732:
3720:
3700:
3688:
3685:
3673:
3625:
3619:
3497:
3491:
3441:
3435:
3336:
3324:
3158:
3146:
3109:
3097:
3073:
3058:
3052:
2962:
2949:
2922:
2918:
2912:
2906:
2877:
2865:
2832:
2826:
2742:
2730:
2698:
2694:
2682:
2666:
2660:
2654:
2567:
2561:
2549:
2541:
2513:
2506:
2494:
2486:
2422:
2414:
2397:
2388:
2382:
2379:
2366:
2350:
2341:
2328:
2322:
2314:
2288:
2282:
2247:
2235:
2178:
2169:
2157:
2154:
2148:
2136:
2054:
2051:
2038:
2022:
2013:
2000:
1994:
1986:
1960:
1954:
1898:
1895:
1882:
1866:
1857:
1844:
1838:
1830:
1740:
1732:
1711:
1699:
1619:
1613:
1588:{\displaystyle \Theta (V^{2})}
1582:
1569:
1542:
1536:
1524:
1518:
1484:
1476:
1454:
1446:
1425:
1413:
990:
973:
700:assign numbers or rankings to
1:
8772:"The mathematics of networks"
8726:10.1080/0022250x.2001.9990249
8564:10.1016/s0378-4371(00)00311-3
7942:10.1016/s0899-8256(03)00130-1
7663:American Journal of Sociology
5899:{\displaystyle {x^{t}}_{i}=1}
5837:{\displaystyle {x^{t}}_{i}=0}
4838:is an attenuation factor in
4500:is a set of the neighbors of
3872:. However, on sparse graphs,
3713:and for undirected graphs is
3267:is a centrality measure of a
8921:10.1371/journal.pone.0053095
8862:10.1016/0378-8733(91)90018-o
8074:10.1016/0378-8733(78)90021-7
7851:10.1016/j.socnet.2005.11.005
7731:10.1016/j.socnet.2004.11.008
7540:Trends in Cognitive Sciences
6325:{\displaystyle C_{x}(p_{*})}
6262:{\displaystyle C_{x}(p_{i})}
5620:{\displaystyle \sigma _{sr}}
3820:{\displaystyle (n-1)(n-2)/2}
3761:{\displaystyle (n-1)(n-2)/2}
3552:{\displaystyle \sigma _{st}}
3291:The betweenness of a vertex
2096:is maximized when the graph
7:
9099:10.1109/ACCESS.2023.3339121
9052:. Nature Publishing Group.
8590:Journal of Social Structure
8043:Freeman, Linton C. (1979),
7930:Games and Economic Behavior
7502:
5795:{\displaystyle {x^{t}}_{i}}
4186:{\displaystyle A=(a_{v,t})}
3388:For each pair of vertices (
3373:For each pair of vertices (
10:
9188:
8502:10.1038/s42005-022-00949-5
8316:10.1140/epjb/e2013-31025-5
8263:10.1209/0295-5075/99/68007
8180:J. Stat. Mech.: Theory Exp
7984:10.1186/s40537-020-00300-1
7642:Networks: An Introduction.
7610:10.1038/s41598-021-81767-7
7552:10.1016/j.tics.2013.09.012
5109:
4693:
4038:
3706:{\displaystyle (n-1)(n-2)}
3249:
3170:{\displaystyle 1/d(u,v)=0}
2593:
1625:{\displaystyle \Theta (E)}
1322:
1271:Shapley-Shubik power index
18:
8967:10.1109/TIFS.2013.2280884
8884:20Q: About Networked Life
8880:January 31, 2012, at the
7239:Transportation Centrality
4308:{\displaystyle a_{v,t}=0}
4229:{\displaystyle a_{v,t}=1}
4022:{\displaystyle O(|V||E|)}
3177:if there is no path from
2625:Closeness was defined by
2256:{\displaystyle G:=(V,E),}
1298:to address this problem.
1161:Game-theoretic centrality
529:Exponential random (ERGM)
196:Informational (computing)
8770:M. E. J. Newman (2016).
8584:Dekker, Anthony (2005).
7514:Coreâperiphery structure
7069:PerronâFrobenius theorem
7027:is non-negative because
6068:{\displaystyle G:=(V,E)}
5994:{\displaystyle O(N^{3})}
5024:{\displaystyle x_{j}+1.}
4613:PerronâFrobenius theorem
4604:{\displaystyle \lambda }
4533:{\displaystyle \lambda }
4111:{\displaystyle G:=(V,E)}
3870:FloydâWarshall algorithm
3860:{\displaystyle O(V^{3})}
3342:{\displaystyle G:=(V,E)}
3008:is the degree of vertex
1717:{\displaystyle X:=(Y,Z)}
1431:{\displaystyle G:=(V,E)}
216:Scientific collaboration
7784:10.1073/pnas.1810452115
6846:dissimilarity given by
6011:Cross-clique centrality
6006:Cross-clique centrality
5373:has indices reversed).
5066:{\displaystyle \alpha }
4831:{\displaystyle \alpha }
4148:number of vertices let
722:social network analysis
645:Category:Network theory
165:Preferential attachment
9152:Algebraic graph theory
8471:Communications Physics
8111:Lawyer, Glenn (2015).
7494:
7222:
7199:
7141:
7061:
7041:
7021:
6994:
6832:
6831:{\displaystyle D_{ij}}
6802:
6740:
6705:eigenvector centrality
6699:
6681:The concept is due to
6672:
6600:
6513:
6461:
6431:
6365:
6326:
6283:
6263:
6217:Freeman centralization
6207:
6187:
6158:
6129:
6099:
6069:
6031:
5995:
5957:
5920:
5900:
5858:
5838:
5796:
5760:
5740:
5720:
5700:
5661:
5641:
5621:
5588:
5386:Percolation centrality
5377:Percolation centrality
5367:
5366:{\displaystyle a_{ji}}
5337:
5308:
5285:
5224:
5096:
5067:
5047:
5025:
4992:
4962:
4919:
4864:
4832:
4809:
4768:
4747:
4659:
4639:
4638:{\displaystyle v^{th}}
4605:
4579:
4534:
4514:
4494:
4462:
4329:
4309:
4270:
4250:
4230:
4187:
4142:
4112:
4046:Eigenvector centrality
4041:Eigenvector centrality
4035:Eigenvector centrality
4023:
3965:
3861:
3821:
3762:
3707:
3652:
3632:
3593:
3573:
3553:
3520:
3363:
3343:
3305:
3261:
3252:Betweenness centrality
3246:Betweenness centrality
3225:
3205:
3171:
3119:
3002:
2975:
2890:
2807:
2787:
2749:
2748:{\displaystyle d(u,v)}
2714:
2577:
2554:
2499:
2439:
2327:
2257:
2213:
2110:
2090:
2067:
1999:
1928:
1905:
1843:
1791:
1771:
1748:
1718:
1680:
1660:
1626:
1589:
1549:
1492:
1462:
1432:
1394:
1358:
1343:Eigenvector centrality
1335:Betweenness centrality
1304:Freeman centralization
1296:node influence metrics
1267:authority distribution
1252:
1225:
1198:
1166:considered in groups.
1147:
1146:{\displaystyle \beta }
1123:
1122:{\displaystyle \beta }
1091:
1090:{\displaystyle \beta }
1069:
1040:
1014:
969:
930:
900:
758:betweenness centrality
754:eigenvector centrality
534:Random geometric (RGG)
7759:(52): E12201âE12208.
7640:Newman, M.E.J. 2010.
7495:
7223:
7200:
7121:
7062:
7042:
7022:
6995:
6833:
6803:
6741:
6696:
6673:
6580:
6493:
6462:
6460:{\displaystyle C_{x}}
6432:
6345:
6327:
6284:
6264:
6208:
6188:
6159:
6135:edges, is defined as
6130:
6100:
6070:
6032:
5996:
5958:
5956:{\displaystyle O(NM)}
5921:
5901:
5859:
5839:
5797:
5761:
5741:
5721:
5701:
5662:
5642:
5622:
5589:
5368:
5338:
5309:
5286:
5225:
5097:
5068:
5048:
5026:
4993:
4991:{\displaystyle x_{j}}
4963:
4899:
4865:
4863:{\displaystyle (0,1)}
4833:
4810:
4748:
4727:
4676:eigenvalue algorithms
4660:
4640:
4606:
4580:
4535:
4515:
4495:
4463:
4330:
4310:
4271:
4251:
4231:
4188:
4143:
4113:
4024:
3966:
3862:
3822:
3763:
3708:
3653:
3633:
3594:
3574:
3554:
3521:
3364:
3344:
3306:
3259:
3226:
3206:
3172:
3120:
3003:
3001:{\displaystyle k_{v}}
2976:
2891:
2808:
2788:
2750:
2715:
2578:
2524:
2469:
2440:
2297:
2258:
2214:
2111:
2091:
2068:
1969:
1929:
1906:
1813:
1792:
1772:
1749:
1719:
1681:
1661:
1627:
1590:
1550:
1498:edges, is defined as
1493:
1463:
1433:
1395:
1332:
1325:Degree (graph theory)
1288:Krackhardt kite graph
1281:Important limitations
1253:
1251:{\displaystyle v_{5}}
1226:
1224:{\displaystyle v_{4}}
1199:
1197:{\displaystyle v_{1}}
1155:eigenvalue centrality
1148:
1124:
1092:
1070:
1068:{\displaystyle A_{R}}
1041:
1015:
949:
939:for matrix powers or
931:
880:
860:eigenvalue centrality
752:) to infinite walks (
650:Category:Graph theory
7955:Hu, Xingwei (2020).
7525:Notes and references
7259:
7212:
7093:
7051:
7031:
7011:
6853:
6812:
6753:
6714:
6474:
6444:
6339:
6293:
6273:
6230:
6197:
6186:{\displaystyle X(v)}
6168:
6157:{\displaystyle X(v)}
6139:
6109:
6079:
6041:
6021:
5969:
5935:
5910:
5868:
5848:
5806:
5770:
5750:
5730:
5710:
5671:
5651:
5631:
5601:
5396:
5347:
5336:{\displaystyle L(j)}
5318:
5298:
5240:
5126:
5077:
5057:
5037:
5002:
4975:
4880:
4842:
4822:
4711:
4649:
4619:
4595:
4551:
4524:
4504:
4493:{\displaystyle M(v)}
4475:
4342:
4319:
4280:
4260:
4256:is linked to vertex
4240:
4201:
4152:
4122:
4084:
3981:
3881:
3835:
3776:
3717:
3670:
3642:
3603:
3583:
3563:
3533:
3422:
3353:
3315:
3295:
3215:
3189:
3132:
3046:
2985:
2903:
2820:
2797:
2771:
2724:
2641:
2612:closeness centrality
2596:Closeness centrality
2590:Closeness centrality
2455:
2269:
2226:
2127:
2120:), and in this case
2100:
2080:
1941:
1918:
1804:
1781:
1758:
1728:
1690:
1670:
1647:
1641:graph centralization
1607:
1563:
1505:
1472:
1442:
1404:
1400:, for a given graph
1384:
1339:Closeness centrality
1235:
1208:
1181:
1137:
1113:
1081:
1052:
1030:
946:
877:
9006:2015NatSR...517095A
8912:2013PLoSO...853095P
8556:2000PhyA..285..539M
8493:2022CmPhy...5..172E
8355:2011NatCo...2..394G
8308:2013EPJB...86..440S
8255:2012EL.....9968007B
8202:2012JSMTE..07..005A
8139:2015NatSR...5E8665L
7961:Journal of Big Data
7775:2018PNAS..11512201N
7602:2021NatSR..11.2176S
7484:
7450:
7414:
7380:
7007:Is noteworthy that
6128:{\displaystyle |E|}
6098:{\displaystyle |V|}
5106:PageRank centrality
4335:can be defined as:
4141:{\displaystyle |V|}
3874:Johnson's algorithm
3204:{\displaystyle N-1}
3037:harmonic centrality
3031:Harmonic centrality
2786:{\displaystyle N-1}
1747:{\displaystyle |Y|}
1491:{\displaystyle |E|}
1461:{\displaystyle |V|}
1351:Harmonic centrality
915:
454:Degree distribution
105:Community structure
8994:Scientific Reports
8815:AMS Feature Column
8435:10338.dmlcz/101401
8426:10.1007/bf02289527
8402:(6):725â730, 1950.
8396:J. Acoust. Soc. Am
8364:10.1038/ncomms1396
7590:Scientific Reports
7519:Distance in graphs
7490:
7464:
7430:
7394:
7360:
7218:
7195:
7119:
7057:
7037:
7017:
6990:
6988:
6828:
6798:
6736:
6700:
6668:
6457:
6427:
6322:
6279:
6259:
6203:
6183:
6154:
6125:
6095:
6065:
6037:for a given graph
6027:
5991:
5953:
5916:
5896:
5854:
5834:
5792:
5756:
5736:
5716:
5696:
5657:
5637:
5617:
5584:
5463:
5389:percolated state.
5363:
5333:
5304:
5281:
5267:
5220:
5154:
5092:
5090:
5063:
5043:
5021:
4988:
4958:
4860:
4828:
4805:
4655:
4635:
4601:
4575:
4530:
4510:
4490:
4458:
4431:
4392:
4325:
4305:
4266:
4246:
4226:
4183:
4138:
4108:
4080:For a given graph
4019:
3974:Brandes' algorithm
3961:
3857:
3817:
3758:
3703:
3648:
3628:
3589:
3569:
3549:
3516:
3474:
3359:
3339:
3301:
3262:
3221:
3201:
3167:
3115:
3087:
2998:
2971:
2886:
2861:
2803:
2783:
2745:
2710:
2678:
2573:
2435:
2253:
2222:So, for any graph
2209:
2106:
2086:
2063:
1924:
1901:
1787:
1770:{\displaystyle y*}
1767:
1744:
1714:
1676:
1659:{\displaystyle v*}
1656:
1622:
1585:
1545:
1488:
1458:
1428:
1390:
1359:
1273:, rather than the
1248:
1221:
1194:
1143:
1119:
1087:
1065:
1036:
1010:
926:
901:
776:(shortest paths),
638:Network scientists
564:Soft configuration
9092:: 142214â142234.
9014:10.1038/srep17095
8961:(11): 1815â1826.
8147:10.1038/srep08665
7999:Krackhardt, David
7891:10.1137/130950550
7488:
7221:{\displaystyle n}
7118:
7060:{\displaystyle D}
7040:{\displaystyle A}
7020:{\displaystyle W}
6987:
6663:
6282:{\displaystyle i}
6206:{\displaystyle v}
6030:{\displaystyle v}
5919:{\displaystyle t}
5857:{\displaystyle t}
5759:{\displaystyle t}
5739:{\displaystyle i}
5719:{\displaystyle v}
5660:{\displaystyle r}
5640:{\displaystyle s}
5582:
5503:
5442:
5440:
5307:{\displaystyle j}
5258:
5215:
5194:
5145:
5089:
5046:{\displaystyle A}
4684:stochastic matrix
4658:{\displaystyle v}
4513:{\displaystyle v}
4416:
4414:
4368:
4366:
4328:{\displaystyle v}
4269:{\displaystyle t}
4249:{\displaystyle v}
3651:{\displaystyle v}
3592:{\displaystyle t}
3572:{\displaystyle s}
3514:
3447:
3362:{\displaystyle V}
3304:{\displaystyle v}
3224:{\displaystyle N}
3113:
3064:
3015:Taking distances
2881:
2852:
2806:{\displaystyle N}
2759:between vertices
2669:
2571:
2461:
2433:
2109:{\displaystyle X}
2089:{\displaystyle H}
2061:
1927:{\displaystyle G}
1790:{\displaystyle X}
1679:{\displaystyle G}
1393:{\displaystyle v}
1367:degree centrality
1347:Degree centrality
1319:Degree centrality
1131:degree centrality
1039:{\displaystyle k}
1008:
856:Degree centrality
750:degree centrality
686:
685:
606:
605:
514:BianconiâBarabĂĄsi
408:
407:
226:Artificial neural
201:Telecommunication
9179:
9162:Network analysis
9147:Graph algorithms
9112:
9111:
9101:
9077:
9068:
9067:
9065:
9064:
9058:
9051:
9042:
9036:
9035:
9025:
8985:
8979:
8978:
8950:
8944:
8943:
8933:
8923:
8891:
8885:
8872:
8866:
8865:
8845:
8839:
8836:
8830:
8829:
8827:
8826:
8806:
8797:
8796:
8794:
8793:
8787:
8776:
8767:
8758:
8757:
8755:
8753:
8719:
8701:
8689:
8678:
8677:
8649:
8643:
8642:
8640:
8639:
8633:
8626:
8618:Yannick Rochat.
8615:
8609:
8608:
8606:
8605:
8581:
8575:
8574:
8549:
8547:cond-mat/0008357
8540:(3â4): 539â546,
8529:
8523:
8522:
8504:
8486:
8462:
8456:
8455:
8437:
8409:
8403:
8392:
8386:
8383:
8377:
8376:
8366:
8334:
8328:
8327:
8301:
8281:
8275:
8274:
8248:
8228:
8222:
8221:
8195:
8175:
8169:
8168:
8158:
8132:
8108:
8099:
8098:
8097:
8096:
8090:
8084:, archived from
8067:
8049:
8040:
8027:
8026:
7995:
7989:
7988:
7986:
7976:
7952:
7946:
7945:
7925:
7919:
7909:
7903:
7902:
7884:
7864:
7855:
7854:
7834:
7821:
7820:
7814:
7806:
7796:
7786:
7768:
7744:
7735:
7734:
7724:
7704:
7687:
7686:
7669:(5): 1170â1182.
7658:
7645:
7638:
7632:
7631:
7621:
7581:
7572:
7571:
7535:
7509:Alpha centrality
7499:
7497:
7496:
7491:
7489:
7487:
7486:
7485:
7483:
7478:
7452:
7451:
7449:
7444:
7417:
7416:
7415:
7413:
7408:
7382:
7381:
7379:
7374:
7347:
7345:
7344:
7287:
7227:
7225:
7224:
7219:
7204:
7202:
7201:
7196:
7164:
7163:
7154:
7153:
7140:
7135:
7120:
7111:
7105:
7104:
7066:
7064:
7063:
7058:
7046:
7044:
7043:
7038:
7026:
7024:
7023:
7018:
6999:
6997:
6996:
6991:
6989:
6986:
6985:
6971:
6970:
6949:
6948:
6939:
6933:
6932:
6918:
6917:
6896:
6895:
6886:
6880:
6868:
6867:
6838:is an arbitrary
6837:
6835:
6834:
6829:
6827:
6826:
6807:
6805:
6804:
6799:
6797:
6796:
6784:
6783:
6768:
6767:
6745:
6743:
6742:
6737:
6735:
6724:
6677:
6675:
6674:
6669:
6664:
6662:
6655:
6654:
6642:
6641:
6626:
6625:
6613:
6612:
6599:
6594:
6575:
6568:
6567:
6555:
6554:
6539:
6538:
6526:
6525:
6512:
6507:
6491:
6486:
6485:
6466:
6464:
6463:
6458:
6456:
6455:
6436:
6434:
6433:
6428:
6420:
6419:
6407:
6406:
6391:
6390:
6378:
6377:
6364:
6359:
6331:
6329:
6328:
6323:
6318:
6317:
6305:
6304:
6288:
6286:
6285:
6280:
6268:
6266:
6265:
6260:
6255:
6254:
6242:
6241:
6212:
6210:
6209:
6204:
6192:
6190:
6189:
6184:
6163:
6161:
6160:
6155:
6134:
6132:
6131:
6126:
6124:
6116:
6104:
6102:
6101:
6096:
6094:
6086:
6074:
6072:
6071:
6066:
6036:
6034:
6033:
6028:
6000:
5998:
5997:
5992:
5987:
5986:
5962:
5960:
5959:
5954:
5925:
5923:
5922:
5917:
5905:
5903:
5902:
5897:
5889:
5888:
5883:
5882:
5881:
5863:
5861:
5860:
5855:
5843:
5841:
5840:
5835:
5827:
5826:
5821:
5820:
5819:
5801:
5799:
5798:
5793:
5791:
5790:
5785:
5784:
5783:
5765:
5763:
5762:
5757:
5745:
5743:
5742:
5737:
5725:
5723:
5722:
5717:
5705:
5703:
5702:
5697:
5686:
5685:
5666:
5664:
5663:
5658:
5646:
5644:
5643:
5638:
5626:
5624:
5623:
5618:
5616:
5615:
5593:
5591:
5590:
5585:
5583:
5581:
5580:
5579:
5574:
5573:
5572:
5558:
5554:
5553:
5552:
5547:
5546:
5545:
5525:
5524:
5519:
5518:
5517:
5506:
5504:
5502:
5501:
5489:
5479:
5478:
5465:
5462:
5441:
5439:
5425:
5411:
5410:
5372:
5370:
5369:
5364:
5362:
5361:
5342:
5340:
5339:
5334:
5313:
5311:
5310:
5305:
5290:
5288:
5287:
5282:
5280:
5279:
5266:
5229:
5227:
5226:
5221:
5216:
5211:
5200:
5195:
5193:
5179:
5178:
5169:
5167:
5166:
5153:
5138:
5137:
5101:
5099:
5098:
5093:
5091:
5082:
5072:
5070:
5069:
5064:
5052:
5050:
5049:
5044:
5030:
5028:
5027:
5022:
5014:
5013:
4998:is replaced by
4997:
4995:
4994:
4989:
4987:
4986:
4967:
4965:
4964:
4959:
4945:
4944:
4932:
4931:
4918:
4913:
4892:
4891:
4869:
4867:
4866:
4861:
4837:
4835:
4834:
4829:
4814:
4812:
4811:
4806:
4804:
4803:
4791:
4790:
4778:
4777:
4767:
4762:
4746:
4741:
4723:
4722:
4664:
4662:
4661:
4656:
4644:
4642:
4641:
4636:
4634:
4633:
4610:
4608:
4607:
4602:
4584:
4582:
4581:
4576:
4574:
4569:
4561:
4539:
4537:
4536:
4531:
4519:
4517:
4516:
4511:
4499:
4497:
4496:
4491:
4467:
4465:
4464:
4459:
4457:
4456:
4447:
4446:
4430:
4415:
4407:
4402:
4401:
4391:
4367:
4359:
4354:
4353:
4334:
4332:
4331:
4326:
4314:
4312:
4311:
4306:
4298:
4297:
4275:
4273:
4272:
4267:
4255:
4253:
4252:
4247:
4235:
4233:
4232:
4227:
4219:
4218:
4195:adjacency matrix
4192:
4190:
4189:
4184:
4179:
4178:
4147:
4145:
4144:
4139:
4137:
4129:
4117:
4115:
4114:
4109:
4028:
4026:
4025:
4020:
4015:
4007:
4002:
3994:
3970:
3968:
3967:
3962:
3957:
3949:
3938:
3937:
3932:
3923:
3915:
3907:
3902:
3894:
3866:
3864:
3863:
3858:
3853:
3852:
3826:
3824:
3823:
3818:
3813:
3767:
3765:
3764:
3759:
3754:
3712:
3710:
3709:
3704:
3657:
3655:
3654:
3649:
3637:
3635:
3634:
3629:
3618:
3617:
3598:
3596:
3595:
3590:
3578:
3576:
3575:
3570:
3558:
3556:
3555:
3550:
3548:
3547:
3525:
3523:
3522:
3517:
3515:
3513:
3512:
3500:
3490:
3489:
3476:
3473:
3434:
3433:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3310:
3308:
3307:
3302:
3230:
3228:
3227:
3222:
3210:
3208:
3207:
3202:
3176:
3174:
3173:
3168:
3142:
3124:
3122:
3121:
3116:
3114:
3112:
3089:
3086:
3076:
3007:
3005:
3004:
2999:
2997:
2996:
2980:
2978:
2977:
2972:
2961:
2960:
2933:
2932:
2895:
2893:
2892:
2887:
2882:
2880:
2860:
2850:
2839:
2812:
2810:
2809:
2804:
2792:
2790:
2789:
2784:
2754:
2752:
2751:
2746:
2719:
2717:
2716:
2711:
2709:
2708:
2677:
2653:
2652:
2582:
2580:
2579:
2574:
2572:
2570:
2553:
2552:
2544:
2538:
2522:
2521:
2520:
2498:
2497:
2489:
2483:
2467:
2462:
2459:
2444:
2442:
2441:
2436:
2434:
2432:
2425:
2417:
2406:
2405:
2400:
2391:
2385:
2378:
2377:
2365:
2364:
2340:
2339:
2326:
2325:
2317:
2311:
2295:
2281:
2280:
2262:
2260:
2259:
2254:
2218:
2216:
2215:
2210:
2193:
2192:
2115:
2113:
2112:
2107:
2095:
2093:
2092:
2087:
2072:
2070:
2069:
2064:
2062:
2057:
2050:
2049:
2037:
2036:
2012:
2011:
1998:
1997:
1989:
1983:
1967:
1953:
1952:
1933:
1931:
1930:
1925:
1910:
1908:
1907:
1902:
1894:
1893:
1881:
1880:
1856:
1855:
1842:
1841:
1833:
1827:
1796:
1794:
1793:
1788:
1776:
1774:
1773:
1768:
1753:
1751:
1750:
1745:
1743:
1735:
1723:
1721:
1720:
1715:
1685:
1683:
1682:
1677:
1665:
1663:
1662:
1657:
1636:representation.
1631:
1629:
1628:
1623:
1601:adjacency matrix
1594:
1592:
1591:
1586:
1581:
1580:
1554:
1552:
1551:
1546:
1517:
1516:
1497:
1495:
1494:
1489:
1487:
1479:
1467:
1465:
1464:
1459:
1457:
1449:
1437:
1435:
1434:
1429:
1399:
1397:
1396:
1391:
1257:
1255:
1254:
1249:
1247:
1246:
1230:
1228:
1227:
1222:
1220:
1219:
1203:
1201:
1200:
1195:
1193:
1192:
1173:
1152:
1150:
1149:
1144:
1128:
1126:
1125:
1120:
1103:Alpha centrality
1096:
1094:
1093:
1088:
1074:
1072:
1071:
1066:
1064:
1063:
1045:
1043:
1042:
1037:
1019:
1017:
1016:
1011:
1009:
1007:
999:
998:
997:
985:
984:
971:
968:
963:
935:
933:
932:
927:
925:
924:
914:
909:
899:
894:
864:Alpha centrality
696:, indicators of
694:network analysis
678:
671:
664:
549:Stochastic block
539:Hyperbolic (HGN)
488:
487:
351:
340:
272:
271:
180:Social influence
54:
26:
25:
21:Central tendency
9187:
9186:
9182:
9181:
9180:
9178:
9177:
9176:
9132:
9131:
9121:
9119:Further reading
9116:
9115:
9078:
9071:
9062:
9060:
9056:
9049:
9043:
9039:
8986:
8982:
8951:
8947:
8892:
8888:
8882:Wayback Machine
8873:
8869:
8850:Social Networks
8846:
8842:
8837:
8833:
8824:
8822:
8807:
8800:
8791:
8789:
8785:
8774:
8768:
8761:
8751:
8749:
8699:
8690:
8681:
8666:10.2307/3033543
8650:
8646:
8637:
8635:
8631:
8624:
8616:
8612:
8603:
8601:
8582:
8578:
8530:
8526:
8463:
8459:
8410:
8406:
8393:
8389:
8384:
8380:
8335:
8331:
8282:
8278:
8229:
8225:
8176:
8172:
8109:
8102:
8094:
8092:
8088:
8065:10.1.1.227.9549
8052:Social Networks
8047:
8041:
8030:
8015:10.2307/2393394
7996:
7992:
7953:
7949:
7926:
7922:
7910:
7906:
7865:
7858:
7839:Social Networks
7835:
7824:
7808:
7807:
7745:
7738:
7709:Social Networks
7705:
7690:
7659:
7648:
7639:
7635:
7582:
7575:
7536:
7532:
7527:
7505:
7479:
7468:
7457:
7453:
7445:
7434:
7423:
7419:
7418:
7409:
7398:
7387:
7383:
7375:
7364:
7353:
7349:
7348:
7346:
7328:
7324:
7283:
7260:
7257:
7256:
7255:
7252:is defined as:
7234:
7213:
7210:
7209:
7159:
7155:
7146:
7142:
7136:
7125:
7109:
7100:
7096:
7094:
7091:
7090:
7080:
7052:
7049:
7048:
7032:
7029:
7028:
7012:
7009:
7008:
6981:
6966:
6962:
6944:
6940:
6935:
6934:
6928:
6913:
6909:
6891:
6887:
6882:
6881:
6878:
6860:
6856:
6854:
6851:
6850:
6819:
6815:
6813:
6810:
6809:
6789:
6785:
6776:
6772:
6760:
6756:
6754:
6751:
6750:
6731:
6720:
6715:
6712:
6711:
6691:
6650:
6646:
6637:
6633:
6621:
6617:
6608:
6604:
6595:
6584:
6576:
6563:
6559:
6550:
6546:
6534:
6530:
6521:
6517:
6508:
6497:
6492:
6490:
6481:
6477:
6475:
6472:
6471:
6451:
6447:
6445:
6442:
6441:
6415:
6411:
6402:
6398:
6386:
6382:
6373:
6369:
6360:
6349:
6340:
6337:
6336:
6313:
6309:
6300:
6296:
6294:
6291:
6290:
6274:
6271:
6270:
6250:
6246:
6237:
6233:
6231:
6228:
6227:
6219:
6198:
6195:
6194:
6169:
6166:
6165:
6140:
6137:
6136:
6120:
6112:
6110:
6107:
6106:
6090:
6082:
6080:
6077:
6076:
6042:
6039:
6038:
6022:
6019:
6018:
6008:
5982:
5978:
5970:
5967:
5966:
5936:
5933:
5932:
5911:
5908:
5907:
5884:
5877:
5873:
5872:
5871:
5869:
5866:
5865:
5849:
5846:
5845:
5822:
5815:
5811:
5810:
5809:
5807:
5804:
5803:
5786:
5779:
5775:
5774:
5773:
5771:
5768:
5767:
5751:
5748:
5747:
5731:
5728:
5727:
5711:
5708:
5707:
5678:
5674:
5672:
5669:
5668:
5652:
5649:
5648:
5632:
5629:
5628:
5608:
5604:
5602:
5599:
5598:
5575:
5568:
5564:
5563:
5562:
5548:
5541:
5537:
5536:
5535:
5531:
5527:
5526:
5520:
5513:
5509:
5508:
5507:
5505:
5494:
5490:
5471:
5467:
5466:
5464:
5446:
5429:
5424:
5406:
5402:
5397:
5394:
5393:
5379:
5354:
5350:
5348:
5345:
5344:
5319:
5316:
5315:
5299:
5296:
5295:
5272:
5268:
5262:
5241:
5238:
5237:
5201:
5199:
5180:
5174:
5170:
5168:
5159:
5155:
5149:
5133:
5129:
5127:
5124:
5123:
5114:
5108:
5080:
5078:
5075:
5074:
5058:
5055:
5054:
5038:
5035:
5034:
5009:
5005:
5003:
5000:
4999:
4982:
4978:
4976:
4973:
4972:
4940:
4936:
4924:
4920:
4914:
4903:
4887:
4883:
4881:
4878:
4877:
4843:
4840:
4839:
4823:
4820:
4819:
4796:
4792:
4786:
4782:
4773:
4769:
4763:
4752:
4742:
4731:
4718:
4714:
4712:
4709:
4708:
4701:Katz centrality
4698:
4696:Katz centrality
4692:
4690:Katz centrality
4674:is one of many
4672:Power iteration
4650:
4647:
4646:
4626:
4622:
4620:
4617:
4616:
4596:
4593:
4592:
4570:
4565:
4554:
4552:
4549:
4548:
4525:
4522:
4521:
4505:
4502:
4501:
4476:
4473:
4472:
4452:
4448:
4436:
4432:
4420:
4406:
4397:
4393:
4372:
4358:
4349:
4345:
4343:
4340:
4339:
4320:
4317:
4316:
4287:
4283:
4281:
4278:
4277:
4261:
4258:
4257:
4241:
4238:
4237:
4208:
4204:
4202:
4199:
4198:
4168:
4164:
4153:
4150:
4149:
4133:
4125:
4123:
4120:
4119:
4085:
4082:
4081:
4078:
4070:Katz centrality
4050:eigencentrality
4043:
4037:
4011:
4003:
3998:
3990:
3982:
3979:
3978:
3953:
3945:
3933:
3928:
3927:
3919:
3911:
3903:
3898:
3890:
3882:
3879:
3878:
3848:
3844:
3836:
3833:
3832:
3809:
3777:
3774:
3773:
3750:
3718:
3715:
3714:
3671:
3668:
3667:
3664:directed graphs
3643:
3640:
3639:
3610:
3606:
3604:
3601:
3600:
3584:
3581:
3580:
3564:
3561:
3560:
3540:
3536:
3534:
3531:
3530:
3505:
3501:
3482:
3478:
3477:
3475:
3451:
3429:
3425:
3423:
3420:
3419:
3381:), compute the
3354:
3351:
3350:
3316:
3313:
3312:
3296:
3293:
3292:
3275:(there is also
3254:
3248:
3216:
3213:
3212:
3190:
3187:
3186:
3138:
3133:
3130:
3129:
3093:
3088:
3072:
3068:
3047:
3044:
3043:
3033:
3025:directed graphs
2992:
2988:
2986:
2983:
2982:
2956:
2952:
2925:
2921:
2904:
2901:
2900:
2856:
2851:
2840:
2838:
2821:
2818:
2817:
2798:
2795:
2794:
2772:
2769:
2768:
2725:
2722:
2721:
2701:
2697:
2673:
2648:
2644:
2642:
2639:
2638:
2598:
2592:
2548:
2540:
2539:
2528:
2523:
2516:
2512:
2493:
2485:
2484:
2473:
2468:
2466:
2458:
2456:
2453:
2452:
2421:
2413:
2401:
2396:
2395:
2387:
2386:
2373:
2369:
2360:
2356:
2335:
2331:
2321:
2313:
2312:
2301:
2296:
2294:
2276:
2272:
2270:
2267:
2266:
2227:
2224:
2223:
2188:
2184:
2128:
2125:
2124:
2101:
2098:
2097:
2081:
2078:
2077:
2045:
2041:
2032:
2028:
2007:
2003:
1993:
1985:
1984:
1973:
1968:
1966:
1948:
1944:
1942:
1939:
1938:
1934:is as follows:
1919:
1916:
1915:
1889:
1885:
1876:
1872:
1851:
1847:
1837:
1829:
1828:
1817:
1805:
1802:
1801:
1782:
1779:
1778:
1759:
1756:
1755:
1739:
1731:
1729:
1726:
1725:
1691:
1688:
1687:
1671:
1668:
1667:
1648:
1645:
1644:
1608:
1605:
1604:
1576:
1572:
1564:
1561:
1560:
1512:
1508:
1506:
1503:
1502:
1483:
1475:
1473:
1470:
1469:
1453:
1445:
1443:
1440:
1439:
1405:
1402:
1401:
1385:
1382:
1381:
1364:
1361:
1355:Katz centrality
1333:Examples of A)
1327:
1321:
1283:
1269:() applies the
1242:
1238:
1236:
1233:
1232:
1215:
1211:
1209:
1206:
1205:
1188:
1184:
1182:
1179:
1178:
1163:
1138:
1135:
1134:
1114:
1111:
1110:
1082:
1079:
1078:
1059:
1055:
1053:
1050:
1049:
1046:is walk length,
1031:
1028:
1027:
1000:
993:
989:
980:
976:
972:
970:
964:
953:
947:
944:
943:
920:
916:
910:
905:
895:
884:
878:
875:
874:
844:
794:
766:
734:
718:super-spreaders
682:
620:
585:Boolean network
559:Maximum entropy
509:BarabĂĄsiâAlbert
426:
343:
332:
120:Controllability
85:Complex network
72:
59:
58:
57:
56:
55:
39:Network science
24:
17:
12:
11:
5:
9185:
9175:
9174:
9172:Graph distance
9169:
9167:Network theory
9164:
9159:
9154:
9149:
9144:
9130:
9129:
9120:
9117:
9114:
9113:
9069:
9037:
8980:
8945:
8886:
8867:
8856:(2): 155â168.
8840:
8831:
8798:
8759:
8717:10.1.1.11.2024
8710:(2): 163â177.
8693:Brandes, Ulrik
8679:
8644:
8610:
8576:
8524:
8457:
8420:(4): 581â603.
8404:
8387:
8378:
8329:
8276:
8223:
8170:
8100:
8058:(3): 215â239,
8028:
8009:(2): 342â369.
7990:
7947:
7920:
7904:
7875:(2): 686â706.
7856:
7845:(4): 466â484.
7822:
7736:
7722:10.1.1.387.419
7688:
7675:10.1086/228631
7646:
7633:
7573:
7546:(12): 683â96.
7529:
7528:
7526:
7523:
7522:
7521:
7516:
7511:
7504:
7501:
7482:
7477:
7474:
7471:
7467:
7463:
7460:
7456:
7448:
7443:
7440:
7437:
7433:
7429:
7426:
7422:
7412:
7407:
7404:
7401:
7397:
7393:
7390:
7386:
7378:
7373:
7370:
7367:
7363:
7359:
7356:
7352:
7343:
7340:
7337:
7334:
7331:
7327:
7323:
7320:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7296:
7293:
7290:
7286:
7282:
7279:
7276:
7273:
7270:
7267:
7264:
7233:
7230:
7217:
7206:
7205:
7194:
7191:
7188:
7185:
7182:
7179:
7176:
7167:
7162:
7158:
7152:
7149:
7145:
7139:
7134:
7131:
7128:
7124:
7117:
7114:
7108:
7103:
7099:
7078:
7056:
7036:
7016:
7001:
7000:
6984:
6980:
6977:
6974:
6969:
6965:
6961:
6958:
6955:
6952:
6947:
6943:
6938:
6931:
6927:
6924:
6921:
6916:
6912:
6908:
6905:
6902:
6899:
6894:
6890:
6885:
6877:
6874:
6871:
6866:
6863:
6859:
6825:
6822:
6818:
6795:
6792:
6788:
6782:
6779:
6775:
6771:
6766:
6763:
6759:
6747:
6746:
6734:
6730:
6727:
6723:
6719:
6690:
6687:
6683:Linton Freeman
6679:
6678:
6667:
6661:
6658:
6653:
6649:
6645:
6640:
6636:
6632:
6629:
6624:
6620:
6616:
6611:
6607:
6603:
6598:
6593:
6590:
6587:
6583:
6579:
6574:
6571:
6566:
6562:
6558:
6553:
6549:
6545:
6542:
6537:
6533:
6529:
6524:
6520:
6516:
6511:
6506:
6503:
6500:
6496:
6489:
6484:
6480:
6454:
6450:
6438:
6437:
6426:
6423:
6418:
6414:
6410:
6405:
6401:
6397:
6394:
6389:
6385:
6381:
6376:
6372:
6368:
6363:
6358:
6355:
6352:
6348:
6344:
6321:
6316:
6312:
6308:
6303:
6299:
6278:
6258:
6253:
6249:
6245:
6240:
6236:
6223:centralization
6218:
6215:
6202:
6182:
6179:
6176:
6173:
6153:
6150:
6147:
6144:
6123:
6119:
6115:
6093:
6089:
6085:
6064:
6061:
6058:
6055:
6052:
6049:
6046:
6026:
6007:
6004:
5990:
5985:
5981:
5977:
5974:
5952:
5949:
5946:
5943:
5940:
5915:
5895:
5892:
5887:
5880:
5876:
5853:
5833:
5830:
5825:
5818:
5814:
5789:
5782:
5778:
5766:is denoted by
5755:
5735:
5715:
5695:
5692:
5689:
5684:
5681:
5677:
5656:
5636:
5614:
5611:
5607:
5595:
5594:
5578:
5571:
5567:
5561:
5557:
5551:
5544:
5540:
5534:
5530:
5523:
5516:
5512:
5500:
5497:
5493:
5488:
5485:
5482:
5477:
5474:
5470:
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5455:
5452:
5449:
5445:
5438:
5435:
5432:
5428:
5423:
5420:
5417:
5414:
5409:
5405:
5401:
5378:
5375:
5360:
5357:
5353:
5332:
5329:
5326:
5323:
5303:
5292:
5291:
5278:
5275:
5271:
5265:
5261:
5257:
5254:
5251:
5248:
5245:
5231:
5230:
5219:
5214:
5210:
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5204:
5198:
5192:
5189:
5186:
5183:
5177:
5173:
5165:
5162:
5158:
5152:
5148:
5144:
5141:
5136:
5132:
5110:Main article:
5107:
5104:
5088:
5085:
5062:
5042:
5020:
5017:
5012:
5008:
4985:
4981:
4969:
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4957:
4954:
4951:
4948:
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4935:
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4927:
4923:
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4912:
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4906:
4902:
4898:
4895:
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4853:
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4751:
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4740:
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4734:
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4694:Main article:
4691:
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4632:
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4586:
4585:
4573:
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4384:
4381:
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4375:
4371:
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4362:
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4348:
4324:
4304:
4301:
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4290:
4286:
4265:
4245:
4225:
4222:
4217:
4214:
4211:
4207:
4182:
4177:
4174:
4171:
4167:
4163:
4160:
4157:
4136:
4132:
4128:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4077:
4074:
4039:Main article:
4036:
4033:
4018:
4014:
4010:
4006:
4001:
3997:
3993:
3989:
3986:
3960:
3956:
3952:
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3936:
3931:
3926:
3922:
3918:
3914:
3910:
3906:
3901:
3897:
3893:
3889:
3886:
3868:time with the
3856:
3851:
3847:
3843:
3840:
3816:
3812:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
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3725:
3722:
3702:
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3696:
3693:
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3687:
3684:
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3678:
3675:
3647:
3627:
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3621:
3616:
3613:
3609:
3588:
3568:
3546:
3543:
3539:
3527:
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3511:
3508:
3504:
3499:
3496:
3493:
3488:
3485:
3481:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3450:
3446:
3443:
3440:
3437:
3432:
3428:
3413:
3412:
3401:
3386:
3383:shortest paths
3358:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3300:
3282:Linton Freeman
3250:Main article:
3247:
3244:
3220:
3200:
3197:
3194:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3141:
3137:
3126:
3125:
3111:
3108:
3105:
3102:
3099:
3096:
3092:
3085:
3082:
3079:
3075:
3071:
3067:
3063:
3060:
3057:
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3051:
3032:
3029:
2995:
2991:
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2964:
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2897:
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2885:
2879:
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2870:
2867:
2864:
2859:
2855:
2849:
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2843:
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2834:
2831:
2828:
2825:
2802:
2782:
2779:
2776:
2744:
2741:
2738:
2735:
2732:
2729:
2707:
2704:
2700:
2696:
2693:
2690:
2687:
2684:
2681:
2676:
2672:
2668:
2665:
2662:
2659:
2656:
2651:
2647:
2629:(1950) as the
2594:Main article:
2591:
2588:
2584:
2583:
2569:
2566:
2563:
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2551:
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2515:
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2502:
2496:
2492:
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2479:
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2310:
2307:
2304:
2300:
2293:
2290:
2287:
2284:
2279:
2275:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2220:
2219:
2208:
2205:
2202:
2199:
2196:
2191:
2187:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2105:
2085:
2074:
2073:
2060:
2056:
2053:
2048:
2044:
2040:
2035:
2031:
2027:
2024:
2021:
2018:
2015:
2010:
2006:
2002:
1996:
1992:
1988:
1982:
1979:
1976:
1972:
1965:
1962:
1959:
1956:
1951:
1947:
1923:
1912:
1911:
1900:
1897:
1892:
1888:
1884:
1879:
1875:
1871:
1868:
1865:
1862:
1859:
1854:
1850:
1846:
1840:
1836:
1832:
1826:
1823:
1820:
1816:
1812:
1809:
1786:
1766:
1763:
1742:
1738:
1734:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1675:
1655:
1652:
1621:
1618:
1615:
1612:
1584:
1579:
1575:
1571:
1568:
1556:
1555:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1515:
1511:
1486:
1482:
1478:
1456:
1452:
1448:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1389:
1323:Main article:
1320:
1317:
1282:
1279:
1245:
1241:
1218:
1214:
1191:
1187:
1162:
1159:
1142:
1118:
1099:
1098:
1086:
1076:
1062:
1058:
1047:
1035:
1021:
1020:
1006:
1003:
996:
992:
988:
983:
979:
975:
967:
962:
959:
956:
952:
937:
936:
923:
919:
913:
908:
904:
898:
893:
890:
887:
883:
843:
840:
793:
790:
765:
762:
733:
730:
714:urban networks
706:social network
684:
683:
681:
680:
673:
666:
658:
655:
654:
653:
652:
647:
641:
640:
635:
630:
622:
621:
619:
618:
615:
611:
608:
607:
604:
603:
602:
601:
592:
587:
579:
578:
574:
573:
572:
571:
566:
561:
556:
551:
546:
541:
536:
531:
526:
524:WattsâStrogatz
521:
516:
511:
506:
501:
493:
492:
484:
483:
479:
478:
477:
476:
471:
466:
461:
456:
451:
446:
441:
436:
428:
427:
425:
424:
419:
413:
410:
409:
406:
405:
404:
403:
398:
393:
388:
383:
378:
373:
368:
360:
359:
355:
354:
353:
352:
345:Incidence list
341:
334:Adjacency list
330:
325:
320:
315:
310:
305:
303:Data structure
300:
295:
290:
285:
277:
276:
268:
267:
261:
260:
259:
258:
253:
248:
243:
238:
233:
231:Interdependent
228:
223:
218:
213:
208:
203:
198:
190:
189:
185:
184:
183:
182:
177:
175:Network effect
172:
170:Balance theory
167:
162:
157:
152:
147:
142:
137:
132:
130:Social capital
127:
122:
117:
112:
107:
102:
97:
92:
87:
82:
74:
73:
71:
70:
64:
61:
60:
49:
48:
47:
46:
45:
42:
41:
35:
34:
15:
9:
6:
4:
3:
2:
9184:
9173:
9170:
9168:
9165:
9163:
9160:
9158:
9155:
9153:
9150:
9148:
9145:
9143:
9140:
9139:
9137:
9127:
9123:
9122:
9109:
9105:
9100:
9095:
9091:
9087:
9083:
9076:
9074:
9055:
9048:
9041:
9033:
9029:
9024:
9019:
9015:
9011:
9007:
9003:
8999:
8995:
8991:
8984:
8976:
8972:
8968:
8964:
8960:
8956:
8949:
8941:
8937:
8932:
8927:
8922:
8917:
8913:
8909:
8906:(1): e53095.
8905:
8901:
8897:
8890:
8883:
8879:
8876:
8871:
8863:
8859:
8855:
8851:
8844:
8835:
8820:
8816:
8812:
8805:
8803:
8784:
8780:
8773:
8766:
8764:
8747:
8743:
8739:
8735:
8731:
8727:
8723:
8718:
8713:
8709:
8705:
8698:
8694:
8688:
8686:
8684:
8675:
8671:
8667:
8663:
8659:
8655:
8648:
8630:
8623:
8622:
8614:
8599:
8595:
8591:
8587:
8580:
8573:
8569:
8565:
8561:
8557:
8553:
8548:
8543:
8539:
8535:
8528:
8520:
8516:
8512:
8508:
8503:
8498:
8494:
8490:
8485:
8480:
8476:
8472:
8468:
8461:
8453:
8449:
8445:
8441:
8436:
8431:
8427:
8423:
8419:
8415:
8414:Psychometrika
8408:
8401:
8397:
8391:
8382:
8374:
8370:
8365:
8360:
8356:
8352:
8348:
8344:
8340:
8333:
8325:
8321:
8317:
8313:
8309:
8305:
8300:
8295:
8291:
8287:
8280:
8272:
8268:
8264:
8260:
8256:
8252:
8247:
8242:
8238:
8234:
8233:Europhys Lett
8227:
8219:
8215:
8211:
8207:
8203:
8199:
8194:
8189:
8186:(7): P07005.
8185:
8181:
8174:
8166:
8162:
8157:
8152:
8148:
8144:
8140:
8136:
8131:
8126:
8122:
8118:
8114:
8107:
8105:
8091:on 2016-02-22
8087:
8083:
8079:
8075:
8071:
8066:
8061:
8057:
8053:
8046:
8039:
8037:
8035:
8033:
8024:
8020:
8016:
8012:
8008:
8004:
8000:
7994:
7985:
7980:
7975:
7970:
7966:
7962:
7958:
7951:
7943:
7939:
7935:
7931:
7924:
7918:
7914:
7908:
7900:
7896:
7892:
7888:
7883:
7878:
7874:
7870:
7863:
7861:
7852:
7848:
7844:
7840:
7833:
7831:
7829:
7827:
7818:
7812:
7804:
7800:
7795:
7790:
7785:
7780:
7776:
7772:
7767:
7762:
7758:
7754:
7750:
7743:
7741:
7732:
7728:
7723:
7718:
7714:
7710:
7703:
7701:
7699:
7697:
7695:
7693:
7684:
7680:
7676:
7672:
7668:
7664:
7657:
7655:
7653:
7651:
7643:
7637:
7629:
7625:
7620:
7615:
7611:
7607:
7603:
7599:
7595:
7591:
7587:
7580:
7578:
7569:
7565:
7561:
7557:
7553:
7549:
7545:
7541:
7534:
7530:
7520:
7517:
7515:
7512:
7510:
7507:
7506:
7500:
7480:
7475:
7472:
7469:
7465:
7461:
7458:
7454:
7446:
7441:
7438:
7435:
7431:
7427:
7424:
7410:
7405:
7402:
7399:
7395:
7391:
7388:
7384:
7376:
7371:
7368:
7365:
7361:
7357:
7354:
7341:
7338:
7335:
7332:
7329:
7315:
7312:
7309:
7300:
7297:
7294:
7284:
7280:
7277:
7271:
7265:
7262:
7253:
7251:
7246:
7242:
7240:
7229:
7215:
7192:
7189:
7186:
7183:
7180:
7177:
7174:
7165:
7160:
7156:
7150:
7147:
7143:
7137:
7132:
7129:
7126:
7122:
7115:
7112:
7106:
7101:
7097:
7089:
7088:
7087:
7085:
7081:
7074:
7070:
7054:
7034:
7014:
7005:
6975:
6967:
6963:
6959:
6953:
6945:
6941:
6922:
6914:
6910:
6906:
6900:
6892:
6888:
6875:
6872:
6869:
6864:
6861:
6857:
6849:
6848:
6847:
6845:
6841:
6840:dissimilarity
6823:
6820:
6816:
6793:
6790:
6786:
6780:
6777:
6773:
6769:
6764:
6761:
6757:
6728:
6725:
6717:
6710:
6709:
6708:
6706:
6695:
6686:
6684:
6665:
6651:
6647:
6638:
6634:
6630:
6622:
6618:
6609:
6605:
6596:
6591:
6588:
6585:
6581:
6564:
6560:
6551:
6547:
6543:
6535:
6531:
6522:
6518:
6509:
6504:
6501:
6498:
6494:
6487:
6482:
6478:
6470:
6469:
6468:
6452:
6448:
6416:
6412:
6403:
6399:
6395:
6387:
6383:
6374:
6370:
6361:
6356:
6353:
6350:
6346:
6335:
6334:
6333:
6314:
6310:
6301:
6297:
6276:
6251:
6247:
6238:
6234:
6224:
6214:
6200:
6177:
6171:
6148:
6142:
6117:
6105:vertices and
6087:
6059:
6056:
6053:
6047:
6044:
6024:
6016:
6012:
6003:
6001:
5983:
5979:
5972:
5963:
5947:
5944:
5938:
5927:
5913:
5893:
5890:
5885:
5878:
5874:
5864:whereas when
5851:
5831:
5828:
5823:
5816:
5812:
5787:
5780:
5776:
5753:
5733:
5713:
5690:
5682:
5679:
5675:
5654:
5634:
5612:
5609:
5605:
5576:
5569:
5565:
5559:
5549:
5542:
5538:
5528:
5521:
5514:
5510:
5498:
5495:
5491:
5483:
5475:
5472:
5468:
5459:
5456:
5453:
5450:
5447:
5443:
5436:
5433:
5430:
5426:
5421:
5415:
5407:
5403:
5399:
5392:
5391:
5390:
5387:
5383:
5374:
5358:
5355:
5351:
5327:
5321:
5301:
5276:
5273:
5269:
5263:
5259:
5255:
5249:
5243:
5236:
5235:
5234:
5217:
5212:
5208:
5205:
5202:
5196:
5187:
5181:
5175:
5171:
5163:
5160:
5156:
5150:
5146:
5142:
5139:
5134:
5130:
5122:
5121:
5120:
5118:
5113:
5103:
5086:
5083:
5060:
5040:
5031:
5018:
5015:
5010:
5006:
4983:
4979:
4955:
4949:
4946:
4941:
4937:
4928:
4925:
4921:
4915:
4910:
4907:
4904:
4900:
4896:
4893:
4888:
4884:
4876:
4875:
4874:
4871:
4854:
4851:
4848:
4825:
4800:
4797:
4787:
4783:
4774:
4770:
4764:
4759:
4756:
4753:
4749:
4738:
4735:
4732:
4728:
4724:
4719:
4715:
4707:
4706:
4705:
4702:
4697:
4687:
4685:
4681:
4677:
4673:
4669:
4652:
4630:
4627:
4623:
4614:
4598:
4591:
4566:
4562:
4547:
4546:
4545:
4543:
4527:
4507:
4484:
4478:
4453:
4449:
4443:
4440:
4437:
4433:
4427:
4424:
4421:
4417:
4411:
4408:
4403:
4398:
4394:
4385:
4379:
4376:
4373:
4369:
4363:
4360:
4355:
4350:
4346:
4338:
4337:
4336:
4322:
4302:
4299:
4294:
4291:
4288:
4284:
4263:
4243:
4223:
4220:
4215:
4212:
4209:
4205:
4196:
4175:
4172:
4169:
4165:
4158:
4155:
4130:
4102:
4099:
4096:
4090:
4087:
4073:
4071:
4067:
4063:
4059:
4055:
4051:
4048:(also called
4047:
4042:
4032:
4029:
4008:
3995:
3984:
3975:
3971:
3950:
3942:
3939:
3934:
3924:
3916:
3908:
3895:
3884:
3875:
3871:
3867:
3849:
3845:
3838:
3828:
3814:
3810:
3803:
3800:
3797:
3788:
3785:
3782:
3771:
3755:
3751:
3744:
3741:
3738:
3729:
3726:
3723:
3697:
3694:
3691:
3682:
3679:
3676:
3665:
3661:
3645:
3622:
3614:
3611:
3607:
3586:
3566:
3544:
3541:
3537:
3509:
3506:
3502:
3494:
3486:
3483:
3479:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3448:
3444:
3438:
3430:
3426:
3418:
3417:
3416:
3410:
3406:
3402:
3399:
3395:
3391:
3387:
3385:between them.
3384:
3380:
3376:
3372:
3371:
3370:
3356:
3333:
3330:
3327:
3321:
3318:
3298:
3289:
3287:
3286:shortest path
3283:
3278:
3274:
3270:
3266:
3258:
3253:
3243:
3241:
3237:
3232:
3218:
3198:
3195:
3192:
3184:
3180:
3164:
3161:
3155:
3152:
3149:
3143:
3139:
3135:
3106:
3103:
3100:
3094:
3090:
3083:
3080:
3077:
3069:
3065:
3061:
3055:
3049:
3042:
3041:
3040:
3038:
3028:
3026:
3022:
3018:
3013:
3011:
2993:
2989:
2968:
2965:
2957:
2953:
2946:
2943:
2940:
2937:
2934:
2929:
2926:
2915:
2909:
2883:
2874:
2871:
2868:
2862:
2857:
2853:
2847:
2844:
2841:
2835:
2829:
2823:
2816:
2815:
2814:
2800:
2780:
2777:
2774:
2766:
2762:
2758:
2739:
2736:
2733:
2727:
2705:
2702:
2691:
2688:
2685:
2679:
2674:
2670:
2663:
2657:
2649:
2645:
2636:
2632:
2628:
2623:
2621:
2620:shortest path
2617:
2613:
2610:
2606:
2603:
2597:
2587:
2564:
2558:
2555:
2545:
2535:
2532:
2529:
2525:
2517:
2509:
2503:
2500:
2490:
2480:
2477:
2474:
2470:
2463:
2451:
2450:
2449:
2429:
2426:
2418:
2410:
2407:
2402:
2392:
2374:
2370:
2361:
2357:
2353:
2347:
2344:
2336:
2332:
2318:
2308:
2305:
2302:
2298:
2291:
2285:
2277:
2273:
2265:
2264:
2263:
2250:
2244:
2241:
2238:
2232:
2229:
2206:
2203:
2200:
2197:
2194:
2189:
2185:
2181:
2175:
2172:
2166:
2163:
2160:
2151:
2145:
2142:
2139:
2133:
2130:
2123:
2122:
2121:
2119:
2103:
2083:
2076:The value of
2058:
2046:
2042:
2033:
2029:
2025:
2019:
2016:
2008:
2004:
1990:
1980:
1977:
1974:
1970:
1963:
1957:
1949:
1945:
1937:
1936:
1935:
1921:
1890:
1886:
1877:
1873:
1869:
1863:
1860:
1852:
1848:
1834:
1824:
1821:
1818:
1814:
1810:
1807:
1800:
1799:
1798:
1784:
1764:
1761:
1736:
1708:
1705:
1702:
1696:
1693:
1673:
1653:
1650:
1642:
1637:
1635:
1634:sparse matrix
1616:
1602:
1599:
1595:
1577:
1573:
1539:
1533:
1530:
1527:
1521:
1513:
1509:
1501:
1500:
1499:
1480:
1468:vertices and
1450:
1422:
1419:
1416:
1410:
1407:
1387:
1378:
1376:
1372:
1368:
1362:
1356:
1352:
1348:
1344:
1340:
1336:
1331:
1326:
1316:
1312:
1308:
1305:
1299:
1297:
1291:
1289:
1278:
1276:
1275:Shapley value
1272:
1268:
1263:
1261:
1260:Shapley value
1243:
1239:
1216:
1212:
1189:
1185:
1174:
1172:
1167:
1158:
1156:
1140:
1132:
1116:
1108:
1104:
1084:
1077:
1060:
1056:
1048:
1033:
1026:
1025:
1024:
1004:
1001:
994:
986:
981:
977:
960:
957:
954:
950:
942:
941:
940:
921:
917:
911:
906:
902:
891:
888:
885:
881:
873:
872:
871:
867:
865:
861:
857:
853:
848:
839:
835:
832:
827:
823:
818:
816:
811:
807:
803:
799:
789:
787:
783:
779:
775:
770:
761:
759:
755:
751:
747:
742:
738:
729:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
679:
674:
672:
667:
665:
660:
659:
657:
656:
651:
648:
646:
643:
642:
639:
636:
634:
631:
629:
626:
625:
624:
623:
616:
613:
612:
610:
609:
600:
596:
593:
591:
588:
586:
583:
582:
581:
580:
576:
575:
570:
569:LFR Benchmark
567:
565:
562:
560:
557:
555:
554:Blockmodeling
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
519:Fitness model
517:
515:
512:
510:
507:
505:
502:
500:
497:
496:
495:
494:
490:
489:
486:
485:
481:
480:
475:
472:
470:
467:
465:
462:
460:
459:Assortativity
457:
455:
452:
450:
447:
445:
442:
440:
437:
435:
432:
431:
430:
429:
423:
420:
418:
415:
414:
412:
411:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
372:
369:
367:
364:
363:
362:
361:
357:
356:
350:
346:
342:
339:
335:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
280:
279:
278:
274:
273:
270:
269:
266:
263:
262:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
192:
191:
188:Network types
187:
186:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
135:Link analysis
133:
131:
128:
126:
125:Graph drawing
123:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
83:
81:
78:
77:
76:
75:
69:
66:
65:
63:
62:
53:
44:
43:
40:
37:
36:
32:
28:
27:
22:
9142:Graph theory
9125:
9089:
9085:
9061:. Retrieved
9040:
8997:
8993:
8983:
8958:
8954:
8948:
8903:
8899:
8889:
8870:
8853:
8849:
8843:
8834:
8823:. Retrieved
8814:
8790:. Retrieved
8778:
8750:. Retrieved
8707:
8703:
8660:(1): 35â41.
8657:
8653:
8647:
8636:. Retrieved
8620:
8613:
8602:. Retrieved
8593:
8589:
8579:
8537:
8533:
8527:
8474:
8470:
8460:
8417:
8413:
8407:
8399:
8395:
8390:
8381:
8346:
8342:
8332:
8292:(10): 1â13.
8289:
8285:
8279:
8239:(6): 68007.
8236:
8232:
8226:
8183:
8179:
8173:
8120:
8116:
8093:, retrieved
8086:the original
8055:
8051:
8006:
8002:
7993:
7964:
7960:
7950:
7933:
7929:
7923:
7907:
7872:
7868:
7842:
7838:
7811:cite journal
7756:
7752:
7712:
7708:
7666:
7662:
7641:
7636:
7593:
7589:
7543:
7539:
7533:
7254:
7249:
7247:
7243:
7238:
7235:
7207:
7083:
7076:
7072:
7006:
7002:
6748:
6701:
6680:
6439:
6222:
6220:
6010:
6009:
5928:
5596:
5385:
5384:
5380:
5293:
5232:
5116:
5115:
5102:from below.
5032:
4970:
4872:
4817:
4700:
4699:
4679:
4667:
4587:
4470:
4079:
4049:
4045:
4044:
3976:which takes
3829:
3662:, which for
3659:
3528:
3414:
3408:
3404:
3397:
3393:
3389:
3378:
3374:
3290:
3264:
3263:
3233:
3182:
3178:
3127:
3036:
3034:
3020:
3016:
3014:
3009:
2898:
2764:
2760:
2634:
2627:Alex Bavelas
2624:
2615:
2611:
2599:
2585:
2447:
2221:
2075:
1913:
1640:
1638:
1557:
1379:
1366:
1363:
1360:
1313:
1309:
1300:
1292:
1284:
1264:
1175:
1168:
1164:
1100:
1022:
938:
868:
849:
845:
836:
825:
821:
819:
801:
797:
795:
771:
767:
743:
739:
735:
726:sociological
697:
690:graph theory
687:
544:Hierarchical
499:Random graph
433:
347: /
336: /
318:Neighborhood
160:Transitivity
140:Optimization
9086:IEEE Access
8752:October 11,
8734:10983/23603
7936:: 132â170.
7596:(1): 2176.
5073:approaches
4590:eigenvalues
4542:eigenvector
3311:in a graph
3265:Betweenness
815:betweenness
590:agent based
504:ErdĆsâRĂ©nyi
145:Reciprocity
110:Percolation
95:Small-world
9136:Categories
9063:2015-12-29
8825:2011-08-24
8792:2006-11-09
8654:Sociometry
8638:2017-02-19
8604:2017-02-18
8484:2108.01149
8477:(1): 172.
8343:Nat Commun
8095:2014-07-31
7974:2003.12198
7766:1706.02327
4236:if vertex
3770:star graph
2637:, that is
2631:reciprocal
2609:normalized
2118:star graph
810:eigenvalue
698:centrality
617:Categories
474:Efficiency
469:Modularity
449:Clustering
434:Centrality
422:Algorithms
246:Dependency
221:Biological
100:Scale-free
9108:2169-3536
9000:: 17095.
8712:CiteSeerX
8519:236881169
8511:2399-3650
8452:119981743
8299:1110.2558
8246:1203.0502
8193:1202.0024
8130:1405.6707
8060:CiteSeerX
7917:1402.0567
7882:1312.6722
7717:CiteSeerX
7715:: 55â71.
7683:145392072
7462:β
7459:−
7428:∈
7421:Σ
7392:β
7389:−
7358:∈
7351:Σ
7339:≠
7333:≠
7326:Σ
7313:−
7298:−
7187:⋯
7123:∑
6960:∪
6907:∩
6876:−
6729:λ
6631:−
6623:∗
6582:∑
6544:−
6536:∗
6495:∑
6396:−
6388:∗
6347:∑
6315:∗
5676:σ
5606:σ
5560:−
5529:∑
5492:σ
5469:σ
5457:≠
5451:≠
5444:∑
5434:−
5260:∑
5209:α
5206:−
5147:∑
5143:α
5087:λ
5061:α
4901:∑
4897:α
4826:α
4771:α
4750:∑
4744:∞
4729:∑
4599:λ
4567:λ
4544:equation
4528:λ
4425:∈
4418:∑
4412:λ
4377:∈
4370:∑
4364:λ
3943:
3801:−
3786:−
3742:−
3727:−
3695:−
3680:−
3608:σ
3538:σ
3503:σ
3480:σ
3468:∈
3462:≠
3456:≠
3449:∑
3271:within a
3236:Marchiori
3196:−
3081:≠
3066:∑
2969:β
2947:
2941:α
2938:−
2935:≈
2927:−
2854:∑
2845:−
2778:−
2703:−
2671:∑
2616:closeness
2602:connected
2559:
2526:∑
2504:
2471:∑
2408:−
2354:−
2348:∗
2299:∑
2195:−
2173:−
2164:−
2152:⋅
2143:−
2026:−
2020:∗
1971:∑
1870:−
1864:∗
1815:∑
1765:∗
1654:∗
1611:Θ
1567:Θ
1534:
1375:outdegree
1141:β
1117:β
1107:resolvent
1085:β
987:β
966:∞
951:∑
918:β
897:∞
882:∑
847:defined.
831:Closeness
774:geodesics
366:Bipartite
288:Component
206:Transport
155:Homophily
115:Evolution
90:Contagion
9157:Networks
9054:Archived
9032:26603652
8975:13587900
8940:23349699
8900:PLOS ONE
8878:Archived
8819:Archived
8783:Archived
8746:Archived
8742:13971996
8695:(2001).
8629:Archived
8598:Archived
8572:10523345
8373:21772265
8324:12052238
8165:25727453
8123:: 8665.
7803:30530700
7628:33500525
7568:18644584
7560:24231140
7503:See also
5746:at time
5647:to node
5117:PageRank
5112:PageRank
4068:and the
4066:PageRank
3579:to node
3211:, where
2793:, where
2757:distance
1371:indegree
728:origin.
710:Internet
633:Software
595:Epidemic
577:Dynamics
491:Topology
464:Distance
401:Weighted
376:Directed
371:Complete
275:Features
236:Semantic
31:a series
29:Part of
9023:4658528
9002:Bibcode
8931:3551907
8908:Bibcode
8674:3033543
8552:Bibcode
8489:Bibcode
8444:5232444
8351:Bibcode
8349:: 394.
8304:Bibcode
8271:9728486
8251:Bibcode
8218:2530998
8198:Bibcode
8156:4345333
8135:Bibcode
8117:Sci Rep
8023:2393394
7899:7088515
7794:6310864
7771:Bibcode
7619:7838299
7598:Bibcode
6844:Jaccard
6015:cliques
4197:, i.e.
4193:be the
4058:network
2755:is the
2635:farness
2633:of the
1724:be the
1353:and F)
824:or the
802:medial.
417:Metrics
386:Labeled
256:on-Chip
241:Spatial
150:Closure
9106:
9030:
9020:
8973:
8938:
8928:
8740:
8714:
8672:
8570:
8517:
8509:
8450:
8442:
8371:
8322:
8269:
8216:
8163:
8153:
8082:751590
8080:
8062:
8021:
7897:
7801:
7791:
7719:
7681:
7626:
7616:
7566:
7558:
7208:where
6749:where
6164:where
5597:where
5233:where
4818:where
4471:where
4276:, and
4062:Google
3529:where
3269:vertex
3240:Latora
3128:where
2981:where
2720:where
2607:, the
1686:. Let
1643:. Let
1231:, and
826:length
822:volume
806:degree
798:radial
782:trails
628:Topics
482:Models
439:Degree
396:Random
349:matrix
338:matrix
328:Vertex
283:Clique
265:Graphs
211:Social
68:Theory
9057:(PDF)
9050:(PDF)
8971:S2CID
8786:(PDF)
8775:(PDF)
8738:S2CID
8700:(PDF)
8670:JSTOR
8632:(PDF)
8625:(PDF)
8596:(3).
8568:S2CID
8542:arXiv
8515:S2CID
8479:arXiv
8448:S2CID
8320:S2CID
8294:arXiv
8267:S2CID
8241:arXiv
8214:S2CID
8188:arXiv
8125:arXiv
8089:(PDF)
8078:S2CID
8048:(PDF)
8019:JSTOR
7969:arXiv
7913:arXiv
7895:S2CID
7877:arXiv
7761:arXiv
7679:S2CID
7564:S2CID
7082:with
6289:, if
6075:with
4118:with
4056:in a
3349:with
3273:graph
2605:graph
2600:In a
1632:in a
1598:dense
1596:in a
1438:with
1349:, E)
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