# Network Modeling of Murine Lymphatic System
## 5 Topological Properties of the LS Graph  
Following our previous study of the human LS [18], we characterized the topological properties of the murine LS graph model using some fundamental metrics. The quantified topological properties of the LS graph characterize the following structural and physiological features of the lymphatic network: the maximum number of lymphatics vessels entering and leaving a LN (maximum degree), the length of the shortest closed path (girth), the maximum distance between maximally separated nodes (diameter), the minimum distance between maximally separated nodes (radius), the typical distances between nodes (average path length), the characteristic of network regularity (energy and the spectral radius), characteristic of the network sparsity (edge density), the measure of how many nodes cluster together (clustering coefficient), the number of nodes critical for connectivity of the LS (number of separators), and the characteristics of lymph flow diversity (topological diversity of the vertices). Let \(G=(n,m)\) be the graph with \(n\) nodes and \(m\) edges, respectively. Consider the following characteristics:  
* The number of input nodes \(N_{inp}\), i.e., the number of nodes with degree 1 and out-degree 0;
* Maximum degree of graph \(\Delta_{G}\), i.e., the maximum degree of its vertices;
* Girth of the graph \(g\), which is the length of the shortest cycle in the simple graph;
* Diameter, i.e., the longest geodesic distance (in other terms, maximum eccentricity of any vertex) \[D=\max_{v\in V}\epsilon(v)=\max_{v\in V}\max_{u\in V}d(u,v),\] (6) where \(d(u,v)\) is the geodesic distance (shortest oriented path connecting vertices \(u\) and \(v\)), \(\epsilon(v)\) is the eccentricity of vertex \(v\);
* Radius of the graph (minimum eccentricity of any vertex), \[r=\min_{v\in V}\epsilon(v)=\min_{v\in V}\max_{u\in V}d(u,v);\] (7)
* Average path length (mean geodesic distance) \[l_{G}=\frac{1}{n(n-1)}\sum_{u,v\in V,\ u\
eq v}d(u,v);\] (8)
* The energy and the spectral radius of the graph are defined as follows, \[En(A)=\sum_{j=1}^{n}\mid\lambda_{j}\mid,\quad\rho(A)=\max\{|\lambda_{j}|\},\] (9) where \(\lambda_{j}\) stand for the eigenvalues of the adjacency matrix \(A\) of the graph;
* Edge density of the graph, i.e., the number of edges divided by the number of all possible edges, \[\rho_{d}=\frac{m}{n(n-1)};\] (10)
* The clustering coefficient \(C\) (transitivity) measures the probability that two neighbors of a vertex are connected. It can be computed as function of adjacency matrix \(A\): \[C(A)=\frac{\sum_{i=1,j=1,k=1}^{n,n,n}a_{ij}\cdot a_{jk}\cdot a_{ki}}{\sum_{i=1} ^{n}((\sum_{j=1}^{n}a_{ij})\cdot((\sum_{j=1}^{n}a_{ij})-1))};\] (11)
* Number of separators \(n_{sep}\), i.e., the vertices removal of which disconnects the graph;* Topological diversity of the vertices as a function of the Shannon entropy associated with flow rates through the incident edges, \[D_{flow}(v_{i})=\frac{H(v_{i})}{\log(k)}=\frac{-\sum_{j=1}^{k}p_{ij}\log(p_{ij})}{ \log(k)},\quad p_{ij}=\frac{|Q_{ij}|}{\sum_{j=1}^{k}|Q_{ij}|},\] (12) where \(k\) is the number of \(v_{i}\)'s incident edges and \(p_{ij}\) is the proportion of the flow between the adjacent \(v_{i}\) and \(v_{j}\) to the total flow through the edges involving \(v_{i}\). The flow diversity is defined similar to the definition of network diversity in [33]. To analyze the robustness of the mouse LS to damage, we sequentially removed individual nodes of the graph and checked how many source vertices remained connected to the sink vertex into the circulatory system. The subgraphs of the LS and whole graph were analyzed. Accordingly, the robustness of the graph was estimated as the arithmetic mean of the ratio of the number of source vertices that retained the connection to the sink to their total number in the graph/subgraphs.