# Routing Strategies for Isochronal-Evolution Random Matching Network
## 4 Two Routing Strategies
### Planning Algorithm for the LHPMD
Although the LDPMH routing strategy leads to a shorter average posterior path length for the delivered packets in some cases, it results in fewer delivered packets, which is worthwhile. On the whole, the LHPMD routing strategy outperforms the LDPMH routing strategy. Regardless of whether an IERMN is in the free-flow state or in the congestion state, the LHPMD routing strategy is always recommended. Future research surrounding the IERMN should include its evolution direction, its spread dynamics, and its synchronization dynamics. These are our future aims and we hope that more and more researchers will study IERMNs. **Author Contributions:** Conceptualization, W.L. and Q.L. ; methodology, W.L. and Q.L. ; software, W.L. ; validation, Q.L. and Z.Z. ; formal analysis, W.L. and C.Z. ; investigation, W.L. and Z.Z. ; resources, Z.Z. and C.Z. ; writing--original draft preparation, W.L., Q.L., Z.Z. and C.Z; writing--review and editing, W.L., Q.L., Z.Z. and C.Z; supervision, Q.L. and Z.Z. ; project administration, Q.L. ; funding acquisition, Z.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China (No. 62003359). **Institutional Review Board Statement:** Not applicable. **Data Availability Statement:** Not applicable. Figure 17: Statistics on the APPTL s of Scenarios 9–12. The ordinate scale uses a logarithmic scale. ## Conflicts of Interest: The authors declare no conflict of interest. ## Abbreviations

### Abbreviations \begin{tabular}{l l} ABMS & Agent Based Modeling and Simulation \\ APPSL & Average Posterior Path Spatial Length \\ APPTL & Average Posterior Path Temporal Length \\ BDS & BeilDou Navigation Satellite System \\ LDPMH-BST & Binary Search Tree for planning LDPMH \\ DVF & Destination Vertex Field \\ FIFO & First-In-First-Out \\ FPF & Future Path Field \\ ISL & Inter-Satellite Links \\ IEN & Isochronal-Evolution Networks \\ IERMN & Isochronal-Evolution Random Matching Network \\ IERPMN & Isochronal-Evolution Random Perfect-Matching Network \\ LDP & Least Delay Path \\ LDPMH & Least Delay Path with Minimum Hop \\ LHP & Least Hop Path \\ LHPMD & Least Hop Path with Minimum Delay \\ MCN & mobile call network \\ LHPMD-OT & Ordered Tree for planning LHPMD \\ PIA & Path Iteration Avoidance \\ RMN & Random Matching Network \\ RPMN & Random Perfect-Matching Network \\ SVVF & Set of Visited Vertices Field \\ SVF & Source Vertex Field \\ SEAS & System-of-systems Effectiveness Analysis Simulation \\
**Notation** & **Meaning** \\ \(\Theta\) & number of all vertices in a network \\ \(\theta\) & number of all edges in a network \\ \(X\) & matrix that shows which edges exist at some slot \\ \(i\) & vertex number \\ \(j\) & vertex number \\ \(v_{i}\) & vertex whose number is \(i\) \\ \(k\) & slot number \\ \(x_{ki}\) & vertex that is connected by \(v_{i}\) in slot \(k\) \\ \(e_{ij}^{[k]}\) & edge between \(v_{i}\) and \(v_{j}\) in slot \(k\) \\ \(H\) & order parameter to characterize the congestion phase transition \\ \(R\) & packet generation rate \\ \(C\) & delivering capability of a vertex \\ \(P_{ijk}\) & path which is from \(v_{i}\) to \(v_{j}\) and planned at slot \(k\) \\ \(v(P_{ijk})\) & departure slot of \(P_{ijk}\) \\ \(o(P_{ijk})\) & arrival slot of \(P_{ijk}\) \\ \(\Gamma\) & planning time window \\ \(Q_{i}\) & queue of \(v_{i}\) \\ \(|Q_{i}|\) & ength of \(Q_{i}\) \\ \(Q_{i}[r]\) & \(r\)-th packet in \(Q_{i}\) \\ \(\lambda\) & level number of an LDPMH-BST or an LHPMD-OT \\ \(\xi\) & Node number in a level of an LDPMH-BST or an LHPMD-OT \\ \(V^{\prime}\) & set of vertices that have been added in a LDPMH-BST \\ \(Z\) & set of nodes of the LHPMD-OT whose value are \(j\) \\ \(g_{1}\) & scenario number \\ \end{tabular} 

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