# Locating the Source of Diffusion in Complex Networks via Gaussian-Based Localization and Deduction
## 3 Experiments and Analysis
### Results on General Graphs
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Type** & **Name** & \(N\) & \(M\) & \(\langle k\rangle\) & \(k_{max}\) & \(H\) & \(\langle d\rangle\) & \(\langle C\rangle\) \\ \hline Random & ER(100,4) & 100 & 200 & 4.000 & 9.620 & 1.211 & 3.418 & 0.038 \\ Scale-free & BA(100,4) & 100 & 200 & 4.000 & 21.756 & 1.723 & 3.104 & 0.102 \\ Small-world & WS(100,4,0.1) & 100 & 200 & 4.000 & 6.452 & 1.047 & 4.263 & 0.286 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic topological features of three synthetic networks. \(N\) and \(M\) represent the number of nodes and edges. \(\langle k\rangle\) and \(k_{max}\) are the average degree and the maximum degree. \(H\) is the degree heterogeneity. \(\langle d\rangle\) denotes the average path length between node pairs. \(\langle C\rangle\) is the average local clustering coefficient of the network. All features are averaged over 500 times of simulations on the corresponding synthetic networks. musicians and an edge implies that two musicians have played together in a band. USAir [24] is a network of flights between US airports in 1997. Netscience [30] represents a co-authorship network of scientists working on network theory and experiment. Celegans [31] is a biological network where nodes are substrates and the edges are metabolic reactions. All networks can be downloaded from the Internet. (Dolphins, Polbooks, Football and Netscience are available on [PERSON]'s website [32]. States, Enron, Jazz, USAir and Celegans are available on Network Repository [24].) In the following experiments, we only consider the largest connected component and remove self-loops and multiple edges. A brief introduction of some basic topological features is shown in Table 3.  
Figure 7 demonstrates the average performance of source localization in real networks with signal-to-noise ratio \(\mu/\sigma=4\). Results show that GLAD-modified outperforms any other methods in all \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Type** & **Name** & \(N\) & \(M\) & \(\langle k\rangle\) & \(k_{max}\) & \(H\) & \(\langle d\rangle\) & \(\langle C\rangle\) \\ \hline Geography & States & 49 & 107 & 4.367 & 8 & 1.130 & 4.163 & 0.507 \\ Animal & Dolphins & 62 & 159 & 5.129 & 12 & 1.327 & 3.357 & 0.303 \\ Co-purchasing & Polbooks & 105 & 441 & 8.400 & 25 & 1.421 & 3.079 & 0.488 \\ Sport & Football & 115 & 613 & 10.661 & 12 & 1.007 & 2.508 & 0.403 \\ Email & Enron & 143 & 623 & 8.713 & 42 & 1.483 & 2.967 & 0.453 \\ Social & Jazz & 198 & 2742 & 27.697 & 100 & 1.395 & 2.235 & 0.633 \\ Transport & USAir & 332 & 2126 & 12.807 & 139 & 3.464 & 2.738 & 0.749 \\ Co-authorship & Netscience & 379 & 914 & 4.823 & 34 & 1.663 & 6.042 & 0.798 \\ Biological & Celegans & 453 & 2025 & 8.940 & 237 & 4.485 & 2.664 & 0.655 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Basic topological features of nine real networks. \(N\) and \(M\) are the numbers of nodes and edges. \(\langle k\rangle\) and \(k_{max}\) are the average degree and the maximum degree. \(H\) is the degree heterogeneity. \(\langle d\rangle\) denotes the average path length between nodes pair. \(\langle C\rangle\) is the average local clustering coefficient of the network. Figure 6: Source localization performance of different methods on synthetic networks with average degree \(\langle k\rangle=4\) and signal-to-noise ratio \(\mu/\sigma=3\). real networks, which is consistent with its performance in synthetic networks. One remarkable feature is that, with the same number of observers, the source in Football is easier to determine compared with that in other networks. The homogeneous structure of Football may be the reason. Considering a highly heterogeneous network, the nodes in network can be divided into center nodes and edge nodes. If the center node is the diffusion source, it is difficult to identify the source among this center node and its neighbor nodes; if the edge node is the diffusion source, it is also hard for us to locate the source among this edge node, the first-order center nodes and the second-order center nodes.