# Node Conductance

#### Definition

$Node Conductance$ effectively detects spanning structural hole nodes and predict the formation of new edges. $Node Conductance$ is the sum of the probability that node $i$ is revisited at $r$-th step, where $r$ is an integer between $1$ and infinity. Moreover, with the help of node embedding techniques, Node Conductance is able to be approximately calculated on big networks effectively and efficiently.

Conductance measures how hard it is to leave a set of nodes. It named the Node Conductance as it measures how hard it is to leave a certain node. For an undirected graph $G$, and for simplicity, it assumed that $G$ is unweighted, although all of the results apply to weighted graphs equally. A random walk on $G$ defines an associated Markov chain and it defined the $Node Conductance$ of a vertex $i$, $NC_{\infty}$, as the sum of the probability that $i$ is revisited at $s$-th step, where $s$ is the integer between $1$ and $\infty$.

$$NC_{\infty}(i)={\underset{s=1}{\overset{\infty}{\sum}}} P(i|i,s).$$

Node Conductance, measuring the node influence from a global view. The intuition behind Node Conductance is the probability of revisiting the target node in a random walk. Node Conductance also show its effectiveness on mining influential node on both static and dynamic network.

#### References

• Lyu T., Sun F., Zhang Y., 2020. Node Conductance: A Scalable Node Centrality Measure on Big Networks. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 12085 LNAI, pp.529-541. DOI: 10.1007/978-3-030-47436-2_40