Definition

The hierarchical degree measures the number of nodes reachable from a start node by performing walks of length ${\displaystyle h}$. For a fixed ${\displaystyle h}$ and walk type, each of these neighbors is reached with a (potentially different) probability ${\displaystyle p_{j}^{(h)}}$. Given a vector of such probabilities, the accessibility of node ${\displaystyle i}$ at scale ${\displaystyle h}$ is defined

${\displaystyle \kappa _{i}^{(h)}=\exp \left(-\sum _{j}p_{j}^{(h)}\log p_{j}^{(h)}\right)}$

The probabilities can be based on uniform-probability random walks, or additionally modulated by edge weights and/or explicit (per edge) transmission probabilities.^[7]

Applications

The accessibility has been shown to reveal community structure in urban networks,^[6] corresponds to the number of nodes which can be visited in a defined time period,^[7] and is predictive of the outcome of epidemiological SIR model spreading processes on networks with large diameter and low density.^[2]

Definition

The expected force of a node ${\displaystyle i}$ is given by

${\displaystyle \kappa _{i}=-\sum _{j=1}^{J}d_{j}\log(d_{j})}$

where the sum is taken over the set ${\displaystyle J}$ of all possible transmission clusters resulting from two transmissions starting from ${\displaystyle i}$. That is, node ${\displaystyle i}$ and two of its neighbors or ${\displaystyle i}$, one of its neighbors (called infected) and a neighbor of the infected neighbor. ${\displaystyle J}$ contains all possible orderings of the transmission events, so two clusters may contain the same nodes if they got infected in a different order. ${\displaystyle d_{j}}$ is the normalized cluster degree of cluster ${\displaystyle j\in J}$, that is, the number of edges with exactly one endpoint in cluster ${\displaystyle j}$.

The definition naturally extends to directed networks by limiting the enumeration ${\displaystyle J}$ by edge direction. Likewise, extension to weighted networks, or networks with heterogeneous transmission probabilities, is a matter of adjusting the normalization of ${\displaystyle d_{j}}$ to include the probability that that cluster forms. It is also possible to use more than two transmissions to define the set ${\displaystyle J}$.^[3]

Applications

The expected force has been shown to strongly correlate with SI, SIS, and SIR epidemic outcomes over a broad range of network topologies, both simulated and empirical.^[3][8] It has also been used to measure the pandemic potential of world airports,^[9] and mentioned in the context of digital payments,^[10] ecology,^[11] fitness,^[12] and project management.^[13]