Reconstructing complex networks without time series

In the real world there are situations where the network dynamics are transient (e.g., various spreading processes) and the final nodal states represent the available data. Can the network topology be reconstructed based on data that are not time series? Assuming that an ensemble of the final nodal states resulting from statistically independent initial triggers (signals) of the spreading dynamics is available, we develop a maximum likelihood estimation–based framework to accurately infer the interaction topology. For dynamical processes that result in a binary final state, the framework enables network reconstruction based solely on the final nodal states. Additional information, such as the first arrival time of each signal at each node, can improve the reconstruction accuracy. For processes with a uniform final state, the first arrival times can be exploited to reconstruct the network. We derive a mathematical theory for our framework and validate its performance and robustness using various combinations of spreading dynamics and real-world network topologies.

Figure 1

Schematic illustration of construction of global information matrix $S$ and local information matrix $S^j$ extracted from $S$. (a) Global information matrix $S$, where $S_{ij}=1$ if signal $i$ is received by node $j$ (marked by black); otherwise, $S_{ij}=0$ (blank). (b) Matrix $S^6$ for node 6 extracted from $S$, which contains the rows of $S$ with $S_{i6}=1$, i.e., the red boxes in (a) with element indices renumbered. Note that $S^6$ does not contain the column of node 6 in $S$. Since the state of node 5 is opposite to that of node 6 for each signal, the column corresponding to node 5 is also removed. (c) The final local information matrix $S^6$. (d) The original network with $N=8$. The eleventh signal (the sixth row of $S^6$ in (c), highlighted by dashed red box) is also received by nodes 1, 2, 3, and 8 (green nodes). However, only node 3 can directly diffuse the signal to node 6 (red node). The signals from nodes 1, 2, and 8 must reach node 3 first before arriving at node 6. Other signals must pass through node 3 or node 4 to reach node 6.

Figure 2

Illustration of a MLE-based method for network reconstruction without any dynamical trajectory or time series. (a) A network of 19 nodes, and the task is to identify the neighbors of node 11. (b) Matrix $S^{11}$ of node 11 extracted from the global information matrix $S$, where $S$ is established using 500 signals generated from 500 independent realizations of SIR epidemic process. (c) The nonzero directional connection probability values associated with node 11 to other nodes obtained from the solution of the optimization problem (3). The subset of the local information matrix $S^{11}$ which corresponds to these directional links is shown in the upper right inset of (b). (d) Inferred neighbors of node 11 from the directional connection probability values. The results are averaged over 10 independent runs.

Figure 3

Dimension reduction based on Theorems 1 and 2. For the Email network, the number of unknown parameter $n$ as a function of degree without and with dimension reduction. S0: without reduction; S1: reduction based on Theorem 1; S2: reduction based on Theorem 2. Ten independent runs are used for statistical average.

Figure 4

Effects of the number of signals on reconstruction performance for SIR dynamics. For three types of networks: ER, SW, and BA, TPR (left ordinate) and FPR (right ordinate) versus $M$. The top panels (a1–a3) are for ER networks with $\langle k\rangle =4$, 6 and 10, respectively, and the bottom panels (b1–b3) are for ER, SW, and SF networks, respectively. The network size is $N=100$ for all cases.

Figure 5

Effects of the number of signals on reconstruction performance for rumor spreading dynamics. Same as Fig. 4 except that the dynamics is of the rumor spreading type.

Figure 6

Effects of the number of signals on reconstruction performance for mixed SIR and rumor spreading dynamics. Same as Fig. 4 except that the dynamics is of the mixed type.

Figure 7

Performance of our method for empirical networks. (a–d) For four empirical networks (Karate, Dolphins, Football, and Email, respectively), TPR and FPR versus $M$. In each case, results for SIR, rumor, and mixed spreading dynamics are displayed.

Figure 8

Network reconstruction with first arrival time information. For four empirical networks, TPR (left ordinate) and FPR (right ordinate) versus $M$ when first arrival time information is incorporated into the reconstruction framework. Comparing with the case where no such information is available, we see that TPR is increased and FPR is decreased, and fewer number of signals are required for high-accuracy reconstruction. The network dynamics is assumed to be of the SIR type. (a) Karate network, (b) Dolphins network, (c) Football network, and (d) Email network.

Figure 9

Network reconstruction with SI epidemic dynamics. For SI dynamics, the information matrix $ST$ is obtained by recording the first arrival time of each signal at each node. Shown are TPR (left vertical axis) and FPR (right vertical axis) versus the number $M$ of signals for three distinct types of model complex networks: ER, SW, and BA. The top panels (a1–a3) are for $\langle k\rangle =4$, 6, and 10, respectively, while the bottom panels (b1–b3) correspond to ER, SW, and BA networks of fixed average degree, respectively.

Figure 10

Mitigation of random signal disturbances using subblocks of signals. For fixed $\rho =10\%$, F1 score versus the number of signals $M$: (a-c) ER, SW, and BA networks, respectively, where the symbol $A_{0\to 1}$ ($A_{1\to 0}$) denotes that “0” is wrongly regarded as “1” (vice versa) in the unblocked information matrix $S$, and $B_{0\to 1}$ ($B_{1\to 0}$) is for the case of blocked matrix. Each pseudomatrix is constructed with 5000 signals. The network size is $N=100$ and the average degree is $\langle k\rangle =6$. The dynamical process is of the SIR type.

Figure 11

Mitigation of random signal disturbances using subblocks of signals: F1 score versus $\rho$ for $M=25000$ for (a–c) ER, SW, and BA networks, respectively. Legends and parameters are the same as for Fig. 10.

Figure 12

Numerical test of generalization of Theorem 2. For the Email network, we implement SIR dynamics and collect the first-arrival times of signals. The left and right vertical axes represent TPR and FPR, respectively. S1 and S2 denote the reductions based on Theorem 1 and Theorem 2, respectively. The transmission rate $\beta$ is randomly chosen from the interval $[0,0.3]$, and the recovery rate is $\mu =1.0$. The results are averaged over 10 independent runs.