Lecture 3: Network Centrality Measures

Lecture 3: Network Centrality Measures

• The lecture is about network centrality measures, focusing on understanding the importance of nodes in a network.
• Centrality is conceptually straightforward, aiming to identify the most central nodes.
• Defining 'center' in practice is complex.
• Different approaches exist, including Degree, Closeness, and Betweenness centrality.
• Graph-level measures, like centralization, help to quantify network-wide centrality patterns.
• The lecture also explores how robustness, collective behavior, information spreading, synchronization, and social dynamics are related to network functions.
• The presenter also covers centrality measures based on degree, including max degree, mean degree, and degree distribution.
• It is important to understand that degree centrality has limitations.
• The presenter highlighted that standardising degree distribution may be important.
• Centralization (skew in distribution) was presented
• Illustrative examples were given, using graphs depicting stars, modular, circle, and lines.
• This provides methods to distinguish important nodes in a network (users or actors)

Degree Based Centrality

• Degree centrality is a simple and intuitive approach focusing on the number of ties a node has.
• It measures how many connections a node has. The most connections will be important in the network.
• Other measures based on degree, such as maximum degree, mean degree, and degree distribution were discussed.
• The presenter showed examples of degrees based on centrality with different graph structures (star, circle, line structure).

Centralization

• It measures the dispersion of centrality scores among nodes to identify if the graph is centralized.
• Examples of degree centralization scores are included.
• The scores and variances of different graph structures (star, circle, line) are provided.
• Examples of high and low in-centralization in financial networks are presented.
• A high-centrality score means one node has more connections with other nodes than other nodes, whereas a low centrality means that connections are evenly distributed.
• The concept of nodes with high degrees being called "hubs" was introduced.

Degree and Degree Distribution

• Degree measures a node's centrality.
• Nodes with high degrees are called hubs.
• Degree distribution is used to measure the heterogeneity of a network. The variance of node degrees helps determine the network's heterogeneity.
• Variance provides information on the heterogeneity and dissimilarity of the nodes inside a network.
• The maximum value of entropy is obtained for a uniform degree distribution.
• The minimum value Hmin = 0 occurs when all nodes have the same degree.
• Degree-degree correlation assesses the relationship between the degrees of connected nodes.
• A positive correlation (r > 0) indicates assortative mixing (rich with rich and poor with poor connections in the network).
• A negative correlation (r < 0) indicates disassortative mixing (rich with poor connections).
• No correlation (r = 0) indicates the lack of correlations between degrees of connected nodes.

Closeness Centrality

• Closeness centrality measures how close a node is to all other nodes in a network.
• It relies on the inverse of the distances between actors in the network.
• Closeness centrality is calculated using the sum of the inverse of the shortest distances between the node and all other nodes.
• Examples in network structures are presented.
• Closeness centrality calculation in the examples were given with different graph distances.
• The closeness centrality calculations present the closeness normalized data and the distances respectively.
• The examples given illustrated the importance of centrality calculation in various situations.

Closeness Centrality in the examples

• Examples and calculations of nodes and centrality are given.

Eccentricity, Radius of Graph

• Eccentricity is the maximum distance from a node to all other nodes.
• The radius of a graph is the minimum eccentricity among all nodes.
• Illustrative examples for hospital locating problems.

Graph Theoretic Center

• The graph theoretic center (Barry or Jordan center) is the node that minimizes the maximum distance.
• The examples illustrate these key concepts.

Average Distance and Efficiency

• Describes measures related to average shortest paths and efficiency in the network, highlighting how to compute these quantities.

Reach Centrality

• It measures the portion of other nodes a node can reach in a given number of steps.
• The reach centrality considers the portion of nodes a node can reach in one step, two steps, and so on.
• The given calculation considers the approach of calculating a node's reach that can vary in numbers of steps.

Measures related to shortest paths

• Explains how closeness centrality behaves in disconnected graphs.
• Explains why closeness centrality is problematic in disconnected graphs.
• Discusses how closeness centrality does not provide useful information when a network is disconnected.
• Explains how to evaluate closeness centrality within a community.

Stress Centrality

• Stress centrality measures the number of geodesics passing through a specific node or edge.
• The related concept of geodesics and shortest linkage in the network structure.
• The presenter provided examples and calculations showing the measures of centrality in connection graph stability.

Connection Graph Stability Scores

• The methods and calculations for this concept are described and the examples for shortest paths are presented.

Betweenness Centrality

• Betweenness centrality measures the number of shortest paths between pairs of nodes that pass through a given node or edge.
• This is a model based on the communication flow.
• It is important to specify the examples regarding the geodesic, shortest paths and their count.
• The betweenness centrality calculations are provided with examples of graphs.

Network Entropy

• Entropy is an indicator of the disorder in degrees, providing an average measurement of the heterogeneity of a network.
• The entropy calculation and its calculation are provided.
• It gives the maximum of entropy occurs for uniform degree distribution and minimum entropy for all nodes with the same degree.

Vulnerability

• It measures the importance of components (nodes or edges) in a network.
• The more a component's removal impacts the efficiency, the more vulnerable/critical the component is.
• Vulnerability is defined as the reduction in network efficiency when a given component is removed.
• The ordered distribution of nodes' vulnerability reflects their importance in the network hierarchy.

Disconnecting and Cut Sets

• Explains disconnecting sets and cut sets related to removing edges or nodes to disconnect a graph.

Eigenvector Centrality

• Eigenvector centrality quantifies a node's influence based on the average centrality of its neighboring nodes.
• It is dependent on the non-negative nature of the node centrality.

Hubs vs. Authorities

• Discusses the relationships between hubs and authorities in complex networks.
• Hubs are nodes pointing to many other nodes.
• Authorities are nodes pointed to by many hubs.
• It is important to understand the difference to identify the important nodes in a network.
• Calculating the hubs and authority scores.

HITS

• Algorithm or method to distinguish hubs and authorities. - How HITS computes scores and their normalization. - Illustrative network examples highlighting the different scores.

Readings

• A list of recommended readings, including books and articles for further study about networks, crowds, and market analysis.

This lecture delves into network centrality measures, emphasizing the significance of nodes within a network. It covers various centrality concepts such as Degree, Closeness, and Betweenness centrality, as well as graph-level measures. Key relationships between centrality and network behaviors are also examined through illustrative examples.